Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

Superconductivity

K.H. Bennemann, J.B. Ketterson: Superconductivity Volume 1: Conventional and Unconventional Superconductors Volume 2: Novel Superconductors

K.H. Bennemann J.B. Ketterson Editors

Superconductivity Conventional and Unconventional Superconductors Volume 1

123

Professor Dr. K.H. Bennemann FU Berlin Department of Physics Arnimallee 14 14195 Berlin, Germany e-mail: [email protected]

Professor Dr. John B. Ketterson Northwestern University Department of Physics & Astronomy Sheridan Road 2145 60208-3112 Evanston, USA e-mail: [email protected]

ISBN 978-3-540-73252-5

e-ISBN 978-3-540-73253-2

DOI 10.1007/978-3-540-73253-2 Library of Congress Control Number: 2008926416 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: PTP-Berlin Protago-TEX-Production GmbH, Germany Production: le-tex publishing services oHG, Leipzig, Germany Cover design: WMXDesign, Heidelberg, Germany Printed on acid-free paper 987654321 springer.com

Preface

Superconductivity is a classical problem of physics. Since the discovery by Kamerlingh–Onnes in 1911 it has undergone tremendous advances in recent years. Complimentary theoretical and experimental studies have advanced our understanding of systems with strong electronic correlations and the associated superconducting phase transitions. New experimental techniques and field-theoretical methods have emerged. The symmetry of the superconducting order-parameter in certain material classes seems different from the usual singlet, s-wave form found in conventional superconductors. There is strong evidence of a Cooper-pairing mechanism differing from the conventional one arising from the electron–phonon interaction. In view of the simultaneous occurrence of an enhanced magnetic response in many novel superconductors (cuprates, ruthenates and heavy-fermion metals), pairing due to exchange of spin-excitations has been proposed and studied.With this mechanism,triplet Cooper-pairing can also occur, in addition to singlet Cooper-pairing. This may be the case in Sr2 RuO4 and heavy-fermion metals like UPt3. Most notable of course was the discovery of high temperature superconductivity in the cuprates. This discovery resulted in an enormous activity. Combining this with the discovery of superconductivity in other materials such as heavy-Fermion metals,MgB2 , various organics, fullerenes, magnetic metals under pressure, and presumably non s-symmetry triplet Cooper-pairing in ruthenates (Sr2 RuO4) we have an enrichment of the phenomenon superconductivity. Many of the new superconductors are being called unconventional, either with respect to the pairing mechanism or the symmetry of the underlying order

parameter. A central problem is the definite identification of the pairing mechanism. Yet in spite of the new superconductors continuing studies of conventional electron–phonon based superconductivity remain interesting. With the aid of historical hindsight, we expect the field to continue to advance in unexpected directions. Nonetheless it is our view that a state-of-the-art treatise on superconductivity is justified at this time, not only to summarize the present understanding, but also to introduce newcomers to the field, as was the case with the two justly-famous 1968 “bibles of superconductivity” by our former colleague R.D. Parks for an earlier generation. It is hoped that the present book will help to point the way for future work, research and discoveries. The treatise starts with a historical introduction and with the basic phenomenological and microscopic theory of conventional superconductors. This should be particularly useful for students and beginning researchers. This and the following chapters reflect the somewhat refined present state of the understanding of phonon-mediated superconductivity. However, many of the results, in particular the Cooper-pairing theory itself, have much wider validity. The phenomenological models and many-body theories are also useful for the study of other pairing mechanisms. As an example see the treatise on electron–phonon superconductivity, studies using Ginzburg–Landau theory and fluctuation phenomena. Note that appropriately generalized Ginzburg–Landau theories are useful for analyzing multi-component and competing order parameters in unconventional superconductors and for non-equilibrium behavior in general.

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Preface

Several chapters discuss recent developments of superconductors containing impurities. Of particular interest are nanostructured superconductors.The important interplay of magnetism and superconductivity is also discussed. Later chapters (see Volume II) focus on novel superconductors, in particular on cuprate superconductivity with its almost universally accepted d-wave symmetry order parameter. A wide variety of important experimental results on high superconductivity are described. In particular, experiments determining the symmetry of the superconducting order-parameter are discussed. The theory analyses the pairing mechanism, important properties and in particular vortices in high-Tc superconductors. Note that the study of vortex matter is also of interest for applications. Several chapters present a theory for cuprate superconductivity assuming Cooper-pairing due to the exchange of spinexcitations. A critical comparison with experiments is presented. Moreover, superconductivity in various novel superconductors like ruthenates, heavyFermions and organics is discussed. Many of the properties involve an interesting interplay of magnetic and superconducting excitations and interactions. The use of neutrons is very important for the analysis of the elementary excitations and of magnetic activity, in particular regarding novel superconductors. Still important open problems are discussed. This should stimulate in particular current and future research.

Berlin, February 2008

The theoretical studies illustrate our present understanding of novel superconductors and should be helpful for further theoretical development towards a general theory that plays the role of the BCS theory in the past. The connections with superfluid 3He, the best understood unconventional Fermion superfluid, and with Bose–Einstein condensation are also discussed. This treatise on superconductivity is a follow up of the previous books on the physics of superconductors published a few years ago. Several important new chapters are presented and most of the other chapters have been largely rewritten, add new results and are generally updated. Thus it is hoped that this book, like Park’s famous books on conventional superconductors, describe our present more or less definite knowledge on superconductivity one of the most interesting problem in physics. It is hoped that the book will remain useful for a long time, as was the case with the Parks treatise. A parallel goal is that the book will be helpful for both students and researchers. We wish to thank the authors for their careful work and great patience.Special thanks go to Christof Bennemann for his essential help in preparing the book. Finally we thank Dr. Ascheron from Springer for his patience and help during editing.

K. H. Bennemann and J. B. Ketterson

K. H. Bennemann is Professor of Physics at the Physics Department FUB (Berlin,Germany), Sloan Fellow, and internationally known for his research on magnetism, superconductivity, and non-equilibrium phenomena. J. B. Ketterson is Professor of Physics at Northwestern University (Evanston, Ill. USA), internationally known for his research in the area of Solid State Physics, in particular of Low Temperature Physics, on Helium. Both are Editors of the well-known books The Physics of Solid and Liquid Helium.

Contents

Volume I. Superconductivity: Conventional and Unconventional Superconductors

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV 1

History of Superconductivity: Conventional, High-Transition Temperature and Novel Superconductors K. H. Bennemann and J. B. Ketterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Novel Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Granular Superconductors, Mesoscopic Systems, Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Phenomenology and Microscopic Theory: Theoretical Foundations L. Pitaevskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Off-Diagonal Long-Range Order and Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Off-Diagonal Long-Range Order in Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The London Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Thermodynamics of Superconductors in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Intermediate State of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Ginzburg–Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Surface Energy at the Boundary Between Normal and Superconducting Phases . . . . . . . . . . . . . . . . . 2.8 Superconductors of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Quantized Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Vortex–Vortex Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Cooper Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Energy Spectrum of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Thermodynamic Properties of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Elements of the Theory of Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Green’s Function of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Temperature Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Temperature Green’s Functions for a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 9 18 23 25

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27 27 31 33 35 37 38 42 45 46 49 52 54 57 62 65 68 70 71

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Electron–Phonon Superconductivity F. Marsiglio and J. P. Carbotte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Electron–Phonon Interaction: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Critical Temperature and the Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Thermodynamics and Critical Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Anisotropy and MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coexistence of Singlet Superconductivity and Magnetic Order in Bulk Magnetic Superconductors and SF Heterostructures M. L. Kuli´c and A. I. Buzdin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ferromagnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Antiferromagnetic Superconductors (AFS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Magnetic Superconductors in the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Josephson Effect with Bulk Magnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Superconductor/Ferromagnet Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74 74 92 108 118 122 139 145 148 153

4

Theory of Superconducting Alloys L. P. Gor’kov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Averages of Green Functions Over Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Superconducting Alloys with a Small Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Paramagnetic Alloys and Gapless Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Eilenberger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164 169 180 183 188 189 196 197 199

5

Impurity Nanostructures and Quantum Interference in Superconductors D. K. Morr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Review: Single Impurity Effects in Conventional and Unconventional Superconductors . . . . . . . . . 6.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Quantum Interference and Quantum Imaging in s-wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . 6.5 Quantum Interference Phenomena in dx2 −y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Molecules on the Surface of dx2 −y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 208 216 218 223 224

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225 225 228 231 234 244 250 253 256

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Nanostructured Superconductors G. Deutscher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Nano-structured Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Granular Structure .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Normal State Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Single Grain Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Weakly Coupled Grains: Granular Insulators and Super-Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Well Coupled Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Critical Temperature of Granular Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

7

Proximity–Coupled Systems: Quasiclassical Theory of Superconductivity V. Chandrasekhar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Transport Equations in the Diffusive Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Keldysh Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Quasiclassical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Non-equilibrium Green’s Functions for Superconducting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Quasiclassical Superconducting Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 The Dirty Limit: The Usadel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Parametrization of the Quasiclassical Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Applications of the Quasiclassical Equations to Proximity-Coupled Systems . . . . . . . . . . . . . . . . . . . . . 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 260 260 262 265 269 272 275 277

8

Principles of Josephson-Junction-Based Quantum Computation S. E. Shafranjuk and J. B. Ketterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Josephson-Junction-Based Qubit Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Qubit Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Quantum Oscillations in Two Coupled Charge Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 SISIS Two-Qubit Gate with Intrinsic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 279 280 283 288 291 294 297 302 304 311 312

9

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Fluctuation Phenomena in Superconductors A. I. Larkin and A. A. Varlamov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ginzburg–Landau Formalism: Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations Below the Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ginzburg–Landau Theory of Fluctuations in Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations Near the S–I Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microscopic Derivation of the Time-Dependent Ginzburg–Landau Equation . . . . . . . . . . . . . . . . . . . . Microscopic Theory of Fluctuation Conductivity of Layered Superconductors . . . . . . . . . . . . . . . . . . . Manifestation of Fluctuations in Various Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 316 319 329 345 350 361 362 365 369 370 373 390 396 405 410 416 429

X

Contents

10.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 11

Universal Properties of Cuprate Superconductors: Evidence and Implications T. Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Critical Behavior at Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Quantum Critical Behavior and Crossover Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Concluding Remarks and Comparison with Other Layered Superconductors . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Matter G. Blatter and V. B. Geshkenbein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Ginzburg–Landau and London Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Layered Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Anisotropic Scaling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Quenched Disorder: Pinning and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Uncorrelated Disorder: Collective Pinning and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Correlated Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Surface and Geometrical Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12 Vortex Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

459 459 465 471 487 487 489

12

Unconventional Superconductivity in Novel Materials M. B. Maple, E. D. Bauer, V. S. Zapf, and J. Wosnitza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Conventional Superconductors Containing Localized Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . 13.3 f -Electron Heavy Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Layered Cuprate and Ruthenate Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Comparison of the Properties of Different Classes of Novel Superconductors . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

495 496 502 505 512 518 532 538 566 575 605 610 615 621 625

13

639 640 641 659 696 717 741 744

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XI

Volume II. Superconductivity: Novel Superconductors List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV High-Tc Superconductivity H. R. Ott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Typical Structural Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Occurrence of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Physical Properties of Copper Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Physical Properties of Non-Cuprate High-Tc Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Tunneling Spectroscopy of Conventional and Unconventional Superconductors J. Zasadzinski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Basic Tunneling Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Tunneling and Strong-Coupling Effects: Microscopic Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Tunneling Spectroscopy of Conventional Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Tunneling in High-Temperature Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Heavy Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Other Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

765 765 767 774 778 813 823 823

15

Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors C. C. Tsuei and J. R. Kirtley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Phase Sensitive Tests: Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Phase-Sensitive Tests: Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Angle-Resolved Determination of Gap Anisotropy in YBCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Universality of the d-Wave Pair State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Implications of d-Wave Pairing Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

833 833 835 841 844 847 861 862 863 864 865

16

17 17.1 17.2 17.3 17.4 17.5 17.6

Photoemission in the High-Tc Superconductors J. C. Campuzano, M. R. Norman, and M. Randeria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics of Angle-Resolved Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Valence Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal State Dispersion and the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superconducting Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

869 869 874 880 901 902 907 912 913 923 924 924 933 935 948 954

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Contents

17.7 Photoemission Lineshapes and the Electron Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963 17.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 18

Neutron Scattering and the Magnetic Response of Superconductors and Related Compounds S. M. Hayden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993 18.2 The Neutron Scattering Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994 18.3 The Static Spin Susceptibility of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 18.4 Magnetic Excitations in Metals and Weakly Coupled Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 18.5 Excitations and Superconductive Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 18.6 High Temperature Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 18.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 18.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 19

Heavy-Fermion Superconductivity P. S. Riseborough, G. M. Schmiedeshoff, and J. L. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 19.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 19.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 19.3 Properties of the Normal State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 19.4 Properties of the Superconducting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 19.5 Heavy Fermion Superconducting Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132 19.6 The Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141 20

Organic Superconductors M. Lang and J. M¨uller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 20.2 Characteristics of Organic Charge-Transfer Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 20.3 Normal-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162 20.4 Superconducting-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 20.5 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214 21

Concepts in High Temperature Superconductivity E. W. Carlson, V. J. Emery, S. A. Kivelson, and D. Orgad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227 21.2 High Temperature Superconductivity is Hard to Attain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1230 21.3 Superconductivity in the Cuprates: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234 21.4 Preview: Our View of the Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244 21.5 Quasi-1D Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245 21.6 Quasi-1D Physics in a Dynamical Stripe Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257 21.7 Electron Fractionalization in D > 1 as a Mechanism of High Temperature Superconductivity . . . 1259 21.8 Superconductors with Small Superfluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262 21.9 Lessons from Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272 21.10 Lessons from Strong Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277

Contents

XIII

21.11 Lessons from Numerical Studies of Hubbard and Related Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 21.12 Doped Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300 21.13 Stripes and High Temperature Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 A Spin Fluctuation Model for d-Wave Superconductivity A. V. Chubukov, D. Pines, and J. Schmalian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349 22.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349 22.2 Spin–Fermion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358 22.3 Summary of Strong-Coupling Theory for Electron–Phonon Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363 22.4 Strong-Coupling Approach to Spin–Fermion Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366 22.5 Fingerprints of Spin Fermion Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377 22.6 Comparison with the Experiments on Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 22.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 22.8 Note Added . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 22

23

Electronic Theory for Superconductivity in High-Tc Cuprates and Sr2 RuO4 D. Manske, I. Eremin, and K. H. Bennemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416 23.2 Electronic Theory for Hole-Doped and Electron-Doped Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428 23.3 Electronic Theory for Ruthenates (Sr2RuO4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451 23.4 Results for Hole-Doped and Electron-Doped Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1460 23.5 Results for Sr2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 23.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511 Superfluid 3 He and the Cuprate Superconductors A. J. Leggett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 24.1 Introduction: Bose Condensation and Cooper Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 24.2 The Normal State: Is the Fermi-Liquid Picture Valid? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522 24.3 Response Functions: The MIR Peak in the Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525 24.4 The Cooper-Paired States of Superfluid 3 He and the Cuprates: General Considerations . . . . . . . . . . 1526 24.5 Symmetry of the Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1531 24.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 24.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535 24

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545

List of Contributors

Bauer, E.D. Department of Physics and Institute for Pure and Applied Physical Sciences, Univ. of California, San Diego La Jolla, California, USA [email protected]

Deutscher, G. School of Physics and Astronomy, Department of Condensed Matter Physics, Tel Aviv University, Tel Aviv, Israel [email protected]

Bennemann, K.H. Institut f u¨ r Theoretische Physik, Freie Universit¨at Berlin, Berlin, Germany [email protected]

Geshkenbein, V.B. ETH Z¨urich, Theoretische Physik, H¨onggerberg CH-8093, Z¨urich, Switzerland [email protected]

Blatter, G. Theoretische Physik ETH H¨onggerberg, CH-8093 Z¨urich, Switzerland [email protected]. Buzdin, A.I. Inst. Univ. de France and Univ. Bordeaux, Talence, France [email protected] Carbotte, J.P. McMaster University, Physics Dept. Hamilton, Ontario L8S 4M1, Canada [email protected] Chandrasekhar, V. Dept. of Physics, Northwestern University, Evanston, Illinois, USA [email protected]

Gor’kov, L. Florida State University, Nat. High Magnetic Field Laboratory, Jamaica Ct. Tallahassee, FL 32308 [email protected]. Ketterson, J. Dept. of Physics, Northwestern University, Evanston, Illinois, USA [email protected] Kuli´c, M.L. Brookhaven Natl. Lab., Upton, New York, USA and Max-Planck Institute – Physics of Complex Systems, Dresden, Germany [email protected] Larkin, V.A. University of Minnesota,Laboratory of Physics, 146 Tate St. SE Minneapolis, MN 55455, USA [email protected]

XVI

List of Contributors

Maple, M.B. Dept. of Physics and Inst. for Pure and Applied Physical Sciences, Univ. of California, San Diego La Jolla, California, USA [email protected]

Schneider, T. Physik–Institut der Universit¨at Z¨urich, Switzerland [email protected] Shafranjuk, S.E. Dept. of Physics, Northwestern University, Evanston, Illinois, USA

Marsiglio, F. University of Alberta, Dept. of Physics Edmonton, AB T6G 2JI, Canada [email protected]

Varlamov, A.A. COHERENTIA-INFM, CNR Rome, Italy [email protected].

Morr, D. University of Illinois at Chicago, Dept. of Physics, USA [email protected]

Wosnitza, J. Institut f u¨ r Angewandte Physik, TU Dresden, Dresden, Germany [email protected]

Pitaevskii, L. University of Trento, Dept. of Physics I-38050 Povo (Trento), Italy [email protected].

Zapf, V.S. Dept. of Physics and Inst. for Pure and Applied Physical Sciences, Univ. of California, San Diego La Jolla, California, USA [email protected]

Volume I

Superconductivity: Conventional and Unconventional Superconductors

1

History of Superconductivity: Conventional, High-Transition Temperature and Novel Superconductors

K. H. Bennemann Dept. of Physics, Freie Universit¨at Berlin, Germany J. B. Ketterson Dept. of Physics and Astronomy, Northwestern University, Evanston, USA 1.1 1.2 1.3 1.4

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Novel Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Granular Superconductors, Mesoscopic Systems, Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Introduction

Since its discovery by H. Kamerlingh Onnes in Leiden [1] almost 100 years ago, superconductivity has remained an important area of solid state physics with continuing surprises. Its first observation in Hg, illustrated for historical reasons in Fig. 1.1, resulted from general advances in low temperature physics (liquefying He). The important continuing discovery

Fig. 1.1. Illustration of Kamerlingh Onnes’ discovery of superconductivity and vanishing of the electrical resistivity  in 1911

of new superconductors resulted from advances in material science physics. By 1980 superconductivity had been observed in many metals and alloys thereof. For an illustration see Fig. 1.2 [2]. Remarkably, the classical ferromagnets like Ni, Fe, etc. did not exhibit superconductivity. Only for the non-magnetic state and under (strong) pressure has superconductivity been reported (for example in iron, Tc = 2K) [3]. From the beginning a strong motivation was to find superconductors with a high transition temperature Tc . However, until about 1980 the A-15 compound Nb3 Ge remained the superconductor with the highest Tc at about 30 K, see Fig. 1.3. In order to achieve higher Tc values also many alloys and the effect of applying pressure were studied. Soon after 1980 exciting new superconductors belonging to rather different material classes were discovered. The situation including the high Tc cuprate superconductors discovered by Bednorz and M¨uller [4] in 1986 is illustrated in Fig. 1.4. For several reasons this tremendously stimulated the minds of the physics community. High-Tc cuprate superconductivity exhibited puzzling new behavior. Perhaps this helped the birth of new surprises which were yet to come. One may note that the history of superconductivity exhibits similar alternating periods of great ex-

4

K. H. Bennemann and J. B. Ketterson

Fig. 1.2. Overview of superconducting metals (blue) in the periodic table. Note the absence of superconductivity in the ferromagnetic transition metals and rare-earth and actinide metals. Other superconductors (under pressure) are marked in red. The superconducting transition temperature Tc is indicated

Fig. 1.3. History of the transition temperature Tc for the first 70 years following the discovery of superconductivity in 1911. The A-15 compounds were of particular interest in the search for higher Tc -superconductors

Fig. 1.4. High-Tc cuprate superconductors discovered by Bednorz and M¨uller in 1986. For La2−x Bax CuO4 a Tc  35 K, for YBa2 Cu3 O7−ı a Tc = 92 K, and for HgBa2 Ca2 Cu3 O8+ı a Tc = 133 K was observed, for example

citement and phases that were more quiet, as was the case for other important classical problems in physics. Generally, the study of superconductivity was a motor for new experimental techniques as well as for methods in theoretical physics, in many-body physics and quantum field theory, and was responsi-

ble for new concepts of quite general significance in physics. The noticeable interplay of experiment and theory was particularly fruitful as history shows, in particular regarding the electronic mechanism for phonon-mediated superconductivity and the symmetry of the superconducting state.

1

with ns ∝|

2e = ∗ | m

Cornerstones in the early history of superconductivity were: 1. Observation of vanishing resistivity (R) at a critical temperature Tc  4.2 K in Hg by Kamerlingh Onnes [1] in 1911. 2. Observation of the diamagnetic behavior of type I superconductors by Meissner and Ochsenfeld in 1933, which opened the way towards a deeper understanding of superconductivity; see Fig. 1.5 for an illustration of the Meissner effect [5]. 3. The London theory in 1935, which described the Meissner effect flux repulsion, by using for the superconducting current driven by the vector potential A the formula [6]   js = − c/4L2 A , (1.1) with L = (mc 2 /4e2ns )1/2 , and ns the density of the superfluid. Then, from rotj s and the Maxwell equations one gets the Meissner effect (see (4/c)rotj s = ∇ × ∇ × B). 4. The Isotope effect [7], Tc ∝ M −˛ , ˛ ≈ 0.5 for Hg, observed by Maxwell 1950 and which suggested that the electron–phonon coupling might be responsible for superconductivity. 5. The Ginzburg–Landau theory in 1950, which extended the London theory and introduced the order parameter [8] (r, t) =|

| ei' (r) ,

(1.2)

5

|2 and

js = 2e |

Fig. 1.5. Meissner effect for type I superconductors: If a superconductor in an external magnetic field H is cooled below its superconducting transition temperature Tc , the magnetic flux B is abruptly expelled. For particular values of B it penetrates the superconductor only within the penetration depth  at the surface (B = H + 4 M)

History and Overview

|2 v s |

2



2e ∇' − A c

 .

(1.3)

6. The breakthrough by the famous and most elegant theory of Bardeen, Cooper, Schrieffer (BCS) in 1956 which after almost 45 years gave a definite electronic explanation of superconductivity in terms of Cooper pairs (k ↑, −k ↓) forming in an energy shell !D (!D denotes the Debye frequency) around the Fermi energy "F resulting from the electron–phonon interaction [9]. The BCS theory became one of the most elegant and successful theories in physics [9]. It was further completed by the field theoretical approaches of Gor’kov [10], Abrikosov and Gor’kov [11], and Eliashberg [12]. Important in understanding (magnetic) fielddependence was Abrikosov’s analysis based on the Ginzburg–Landau theory of type I superconductors √ ( < 1/ 2, ≡ /, is the penetration depth,  the coherence length referring to the stiffness of ) and √ type II ones ( > 1/ 2), which allow magnetic flux  to penetrate the superconductor in a regular array, quantized in units of the elementary flux quantum 0 = hc/2e [13]. Important was also the observation of flux quantization in a ring, flux  = (n + 12 ) hce , (n = 0, ±1, . . .), by Doll and N¨abauer, and Deaver and Fairbank [16]. This is illustrated in Fig. 1.6. A further step was Josephson’s tunneling theory in 1962 describing also tunneling of Cooper pairs through a barrier between two superconductors [14]. The current is given by j(t) = j0 + j(' ), where ' = '2 − '1 is the phase difference between the two superconductors 1 and 2 separated by a tunnel 2 barrier (' = '20 − '10 − (2ec) 1 dxAx ). Then one may get for the Cooper pair current (A = 0 and voltage V )   2e 0 0 j(t) = j0 + j1 sin '2 − '1 − V21 t ,     V21 = '˙21 . (1.4) 2e Very important was also the study of the tunneling density of states by Schrieffer, Scalapino and Wilkens

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K. H. Bennemann and J. B. Ketterson

Fig. 1.6. Illustration of flux quantization in a superconducting Pb-cylinder. The signal I is proportional to the flux and B denotes the frozen-in field

in 1963 explaining observed detailed structure in the tunneling current and superconducting order parameter (!) as due to the electron–phonon cou pling (NT (!) = N( F )Re [!/ ! 2 − 2 (!)]) [15]. It seemed that the electronic theory for superconductivity, which had replaced largely the early phenomenological theories by Casimir, Gorter (twofluid model for ns, nn ),by London for the Meissner effect and by Ginzburg–Landau,was largely completed.

The thermodynamical and electrodynamical behavior of type I superconductors for which the Meissner effect holds up to the critical magnetic field Hc and type II superconductors with magnetic flux penetration below Bc2 and Meissner effect below Bc1 was characterized by the influence of external fields on the Cooper pairs and the phase diagrams Tc (B(T)) and Tc (j, B). Typical results are shown in Fig. 1.7. Clearly this behavior also sheds light on the interdependence of superconductivity and magnetism.Metals with parallel magnetic and superconducting activity are of central interest. Triplet Cooper pairing relates such superconductors intimately to superfluid 3 He. Superconductors with quantum-critical points are of special interest with respect to basic questions of quantum mechanics. The search for superconductors with higher transition temperatures, see Fig. 1.3, became increasingly important. In this regard of particular interest was the work by McMillan [17], which attempted to relate Tc , Tc ∝< !ph > exp{−1/( − ∗ )} ,

(1.5)

to characteristic parameters of the superconducting metals like electron–phonon coupling g or , electronic density of states at "F , etc. ( is the electron– ∞ 2 phonon coupling constant,  = 2 0 ˛ F(!) d! with ! F(!) characterizing the phonon spectrum). ∗ refers to the renormalized effective Coulomb interaction between the Cooper pairs forming electrons. For a

Fig. 1.7. Typical behavior of superconductors in an external magnetic field H: (a) type II superconductors with critical fields Hc2 , below which flux penetrates, Hc1 , below which perfect Meissner effect behavior occurs (magnetization M), (b) dependence on H and current j of type I superconductivity (B = H + 4 M)

1

Fig. 1.8. Structure of MgB2 (AlB2 , etc.). The boron planes seem to play an important role regarding superconductivity and Cooper pairing.Note that the Mg-B bonds are softer than the B-B ones

long time one observed only superconductivity due to the electron–phonon coupling and that the A-15 compounds like V3 Si, Nb3 Ge, etc. had the highest Tc . The superconducting order parameter (r, t) had swave symmetry. The prospects for finding superconductors with higher Tc was somewhat guided (and affected) by the estimates of a maximal Tc given by P.W. Anderson, Cohen, and Allen and others [18]. However, one expected that for increasing electron– phonon coupling strength , the resulting lattice instability limited essentially the occurrence of superconductivity (Tc < Tcmax ).

History and Overview

7

In view of this the recent discovery of the type II superconductor MgB2 with Tc  40K and quasi two gaps, 1  4 meV and 2  7.5 meV due to –and  -type electrons was very remarkable. Both gaps have s-symmetry and result from the highly anisotropic layer-structure of the MgB2 -lattice. Note, boron planes consisting of hexagon-B-rings characterize the structure, see Fig. 1.8. The dominating bonds of -type and  -type coupling to phonons causes superconductivity. As a consequence of the || anisotropy one gets critical magnetic fields Hc2 and || ⊥ || −2 ∼ 0.2 Hc2. Here H⊥ c2 . One estimates Hc2 ∼ ( ) || ⊥ the Hc2 and Hc2 refer to in-plane and perpendicular to B-plane upper critical field, respectively. Note that Hc2  ¥0/ 2,  ∼ −1 , ¥0 refers to the elementary flux quantum. In Fig. 1.9 we illustrate the anisotropic behaviour of Hc2 (T) indicative of two gaps. It is interesting that MgB2 can carry relatively strong superconducting currents in magnetic fields up to 3 T, which compares with that observed for high-Tc cuprates. Note the isotope effect as well as tunnel spectroscopy support phonon driven superconductivity. One gets that Mg11 −→Mg10 increases Tc by about 1 K in accordance with Tc ∝(Mg)−1/2. Interestingly AlB2 is not superconducting. Note that MgB2 is another example of the interesting role played by the lattice structure regarding superconductivity. Over the years many interesting studies of coexistence of superconductivity and magnetism (for example, the transition metals and their alloys, rare



Fig. 1.9. Critical upper magnetic fields Hc2 (T) and ⊥ Hc2 (T) referring to the B-planes and directions perpendicular to it, respectively, in the layered structure of MgB2 . The anisotropic behavior of Hc2 (T) is indicative of the approximate two-gap behavior

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K. H. Bennemann and J. B. Ketterson

Fig. 1.10. Superconductivity in heavy–fermion metals

earth (RE) compounds, etc.) and of occurrence of superconductivity in metals with strongly correlated electrons and local spins were carried out (see, for example, the heavy-fermion systems in Fig. 1.10). That singlet Cooper pairs (k ↑, −k ↓) respond sensitively to magnetism and correlation was, of course, expected on general grounds, in view of the results shown in Fig. 1.2 and of the thermodynamical behavior in an external magnetic field (see Fig. 1.7 for illustration). The studies of the effect of magnetism and magnetic fields were important for our understanding of superconductivity (for strongly renormalized quasiparticles, for polaronic type superconductivity, for Bose–Einstein vs. BCS type pair-condensation, etc.) and in particular for the search of superconductivity with the Cooper pair wave function (r, t) =| (r, t) | ei' (r,t)

CePd2 Si2 , U1−x Mx Pd2Al3, (M = Y, Th, etc.), exhibiting spin-excitations, were important [21]. Recent experimental studies of the magnetically active intermetallic compound UGe2 , of MgCNi3 (Tc = 8K), of the magnetic organometallic compound (BETS)2 FeCl4 (where superconductivity is induced by a strong external magnetic field) and similar systems, for example, clearly indicate the interdependence of magnetism and superconductivity. To illustrate the exciting new observations regarding the interplay of superconductivity and magnetism we show typical phase diagrams. In Fig. 1.11 we present results for UGe2 . Further studies should reveal the mechanism for Cooper pairing (triplet versus singlet pairing), magnetic activity and electronic

(1.6)

having non-s-wave symmetry, describing triplet pairing, and pairing resulting not from electron– phonon interaction. Here, the discovery by Osheroff et al. [19] in 1972 observing spin-triplet pairing in superfluid 3 He, another Fermi-liquid besides the superconducting metals, was very important for the further history of superconductivity. Also, the study by Berk and Schrieffer in 1966 of the interdependence of superconductivity and spin-fluctuations prepared the development of new perspectives and for new research routes [20]. In this context experimental studies of superconductivity in UPt3 , CeIn3 ,

Fig. 1.11. Phase diagram of UGe2

1

structure in the superconducting phase with relatively low Tc . Note, the behavior of Tc (p). First Tc increases with pressure p, but then decreases again, and Tc → 0 for p ≈ 17 kbar. Another compound with interesting magnetic and superconducting properties is ZrZn2 , which exhibits itinerant ferromagnetism. It seems that superconductivity exists up to 22 kbar, but not in the paramagnetic phase. The formation and character of the Cooper pairs (triplet pairing?) must still be studied. One can expect that there may be many compounds with an interesting interdependence of superconductivity and magnetism. In Fig. 1.12 we show the interesting results for iron (Fe) under pressure. It is remarkable that Fe becomes superconducting at low temperatures and for pressure between 15 and 30 GPa in the non-magnetic hexagonal-close-packed -phase. The Cooper pairing needs to be studied as well as important properties revealing the electronic basis of the phase diagram.

History and Overview

9

tively weakly, while paramagnetic impurities have a strong destructive effect and may destroy superconductivity. External magnetic fields and electrical currents destroy the superconducting state, break up Cooper pairs, and may even cause gapless superconductivity.

1.2

Novel Superconductors

(a) Cuprates (High-Transition Temperature Superconductors) The superconductivity research changed dramatically when the high Tc -cuprate superconductors with CuO2 -layers like La2−x Srx CuO4 , YBa2 Cu3O7−8 , Tl2 Ba2 CaCu2 O8 , etc. were discovered by Bednorz and M¨uller in 1986 with transition temperatures ranging from Tc  35 K to Tc  160 K (under pressure) in HgBa2 CaCu2 O6+ı [4]. The carriers in the cuprates, with typical layered structure shown in Fig. 1.13, are strongly correlated. As a result one observes many unusual properties, non-Fermi-liquid behavior, a rich phase diagram and antiferromagnetism. Superconductivity depends sensitively on hole doping in the CuO2 -planes, see Fig. 1.13 for illustra-

Fig. 1.12. Superconducting hexagonal -phase Fe under strong pressure. ˛, ˇ, refers to the various Fe phases

In all conventional superconductors phase coherent Cooper pairing occurs at the transitiontemperature Tc . The structure of the order parameter k (!), seen in the spectral-density, for example, for conventional superconductors reflects characteristic phonon frequencies involved in Cooper pairing. The Meissner effect occurs at Tc for phase coherent Cooper pairs. Also, in conventional superconductors typically non-magnetic impurities reduce Tc rela-

Fig. 1.13. Structure of the cuprate YBa2 Cu3 O7−ı with Tc up to 92 K depending on hole doping ı. Singlet Cooper pairing occurs essentially in the CuO2 -planes like in the other members of the cuprate family

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K. H. Bennemann and J. B. Ketterson

tion [22]. It was important that experiments indicated singlet Cooper pairing and d-wave symmetry, see phase-sensitive measurements by Tsuei and Kirtley [23]. Type II superconductivity and non-ssymmetry of the order parameter, dx2−y 2 -symmetry, is observed. Tunnel spectroscopy, ARPES, and many other experiments support this. Parallel activity (antiferromagnetic and superconducting) occurs. Typically an interdependence of these activities is observed. Singlet Cooper pairing is present. Due to strong correlations unusual properties are exhibited, see, for example, the temperature dependence of the electrical resistivity ( ∝ T) and other transport properties, a pseudo gap of dx2 −y 2 -symmetry in the quasi-particle dispersions for lower doping, nonFermi-liquid behavior (self-energy £(!) ∝ !, etc.), and so on. As a consequence, the elementary excitations in the cuprates seem to behave anomalously. The doping dependence of Tc indicates that phase fluctuations of Cooper pairs play a role, in particular in underdoped cuprates with stronger correlations among the quasi-particles. This seems reflected by T c ∝ ns ,

(1.7)

where ns denotes the doping dependent superfluid density (ns = ns (x, T)). Whether Tc ∝ ns also occurs for electron doping needs to be verified.It is expected for low Cooper pair density. Figure 1.14 illustrates the doping-dependent phase diagram of hole doped (La2−x Srx CuO4 )- and electron (Nd2−x Cex CuO4 )-doped cuprates. In hole doped cuprates for increasing doping x one gets that Tc increases first due to increase of hole concentration and itinerancy and then Tc decreases again due to the disappearance of the antiferromagnetic spin-excitations. Note that electron doping consists largely of occupying the hybridized d-orbitals (upper Hubbard-band) of Cu,of quenching the Cu-spins, while hole doping of the oxygen p-states, destroying long-range antiferromagnetism due to frustration, consists mainly of emptying the oxygen p-band. Thus with increasing hole doping antiferromagnetic excitations are weakened and itinerancy of the correlating hole-carriers is improved. One might expect somewhat different behaviour of hole and electron doped cuprates. Note too that due to correlations

Fig. 1.14. Doping dependence of the superconducting transition temperature Tc (x) of (a) hole (La2−x Srx CuO4 , . . .) and (b) electron (Nd2−x Cex CuO4 , . . .) doped cuprates. Tc∗ neglects C.P. phase fluctuations, T ∗ refers to the onset of pseudo-gap.A.F.refers to the anti-ferromagnetic phase and ns to superfluid density.The inset in (a) is a calculated spectral density. Note the asymmetry in the spectral density. In (b) the inset refers to calculations of Tc . Here, Tc ∝ ns , see the dashed curve, for electron doping is an open question at present

one gets asymmetric peaks in the spectral density. This asymmetry increases for decreasing hole doping. Hence, it is more difficult to add an electron than to extract one in accordance with photoemission experiments. For increasing pd-hybridization this electron-hole asymmetry is expected to decrease. Typical behavior of cuprate superconductors in a magnetic field reflecting the stiffness of the superconducting wave-function (r, t) is shown in

1

History and Overview

11

Fig. 1.16. Cooper pairing in the cuprates due to coupling of carriers (holes or electrons) in Cu-O planes to a.f. spin fluctuations characterized by the spin susceptibility (q, !) (G refers to matrix Green’s function of quasi-particles, Ueff is the effective coupling)

Fig. 1.15. The critical magnetic fields in high Tc superconductors. Note the steep slope of Hc2 near Tc . Various phases with vortices occur. Above Hm the melting of the vortex system occurs. Hi refers to the irreversibility field, below which vortices are pinned

Fig. 1.15. New phases of the vortex state were observed. The behaviour in underdoped cuprates is particularly interesting, since the vortex formation is related to the Cooper pair phase ' and its stiffness, (r, t) =| (r, t) | ei' (r,t) . For decreasing doping x the density of Cooper pairs, ns , decreases and thus phase disorder of the Cooper pairs is expected to play an important role. The debate on the Cooper pairing mechanism is still under way [24]. Theories mainly by Pines et al. [25], Emery et al. [26], Tewordt et al. [27], Bennemann et al. [28] and others attempt to explain many properties and in particular d-wave superconductivity as resulting from the coupling of the holes in the CuO2 -planes to antiferromagnetic spin fluctuations. In Fig. 1.16 an Eliashberg-type pairing theory is indicated. In agreement with many experiments one gets for the order parameter (k, !) the results shown in Fig. 1.17. Furthermore, as expected on general grounds one gets also for the electrondoped cuprates like Nd2−x Cex CuO4 a d-wave symmetry [29]. For results see Fig. 1.17, and experiments in particular by Kirtley and Tsuei [23]. Note, after the discovery of high Tc superconductors the theoretical developments based on hole-spin-fluctuation coupling were initiated by Scalapino, Bickers [29, 30], Schmalian et al. [31],

Fig. 1.17. d-symmetry order-parameter (k, !) for hole doped cuprates. In accordance with the CuO2 -planes we refer to the 2D Brillouin zone with kx and ky

and others extending the study by Berk, Schrieffer [20]. Important results were first achieved when careful numerical results were obtained from a self-consistent solution of the spin-fluctuation-type Eliashberg-theory (see Fig. 1.16) [31]. The interesting d-wave symmetry of the superconducting order parameter (r, t) or the renormalized one (k, !) can be most simply understood from the general linearized gap function with pairing potential Vk,k : 

k 2Ek k   Ek = "k2 + 2k ,

k  −

Vk,k

(1.8)

Clearly, for Vk,k < 0 (attractive) one may get for constant potential V a k-independent s-symmetry superconductivity. However, for Vk,k > 0 (repulsive, as expected for the cuprates due to the strong correlations and occurrence of antiferromagnetism) one gets superconductivity, i.e. solutions of Eq. (1.8), only for k-dependent pairing Vk,k and (k). In the case of

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K. H. Bennemann and J. B. Ketterson

Fig. 1.18. Illustration of d-wave symmetry Cooper pairing due to exchange of antiferromagnetic spin fluctuation with wave vector Qpair between two quasi-particles at opposite parts of the Fermi surface. A quarter of the first Brillouin zone is shown and the signs + and − refer to areas where k is positive or negative and a node k = 0 is indicated by dashed lines. If pairing occurs also due to exchange of phonons, for example with wave vector qpair , then deviations of d-symmetry are expected

Vk,k  Vk−k > 0 and k −k  ≈ Q, Q being the antiferromagnetic wave vector at which the spin susceptibility (q, !) is maximal,one then gets from the combination of the Fermi-surface topology in the cuprates and (q, !) an order parameter of mainly d-wave  symmetry (k = 20 cos(kx ) − cos(ky ) ). Note that

(q, !) controls the transitions across the Fermisurface. Note that V = V { }. The behaviour of (k) is illustrated in Fig. 1.17 [32]. If pairing mainly occurs due to exchange of antiferromagnetic spin fluctuations with wave vector Q, then one expects this to be the strongest for directions in the Brillouin Zone (BZ) where Q can bridge corresponding portions of the Fermi surface. This is more the case for (0, ) than (, ) directions. Then, the dx2 −y 2 -wave symmetry is expected with nodes for the superconducting order parameter along the diagonals of the BZ,see Fig.1.18 for an illustration. Obviously, Fermi-surface topology (nesting) plays an important role. In view of the Fermi-surface topology and scattering largely by Q one expects for underdoped cuprates that the spectral density of the quasi-particles is broades for Q along (0, ) than nodal direction (, ). However, for increasing doping and overdoped cuprates Q does not bridge any more the Fermi-surface along (0, ) and thus the spectral-density gets narrows and broades for Qdirections (, ). One expects in agreement with experiment significant feedback effects of superconductivity on the dynamical spin susceptibility, (q, !), see inelastic neutron scattering experiments, and on the elementary excitations, see optical conductivity measurements. This interesting interdependence of spin excitations and superconductivity in the cuprates is clearly seen by the change of the dynamical spin Fig. 1.19. Spin excitation spectrum (q = Q, !) versus frequency as observed by inelastic neutron scattering (INS) experiments in cuprates [33]. Note the feedback of superconductivity for the spin susceptibility and formation of a so-called “resonance” peak

1

susceptibility as observed in inelastic neutron scattering experiments (INS), see Fig. 1.19 for results. Of course, this feedback of superconductivity on the spin-susceptibility depends on the topology of the Fermi-surface, distortions, number of CuO2 -layers per unit cell, doping, etc. Details may serve as a fingerprint of the pairing mechanism. [33] In Fig. 1.20 results are shown for  in electron doped superconductors like Nd2−x CeCuO4 . Due to different dispersions k and Fermi-surface nesting one gets a lower Tc (x) than in the hole-doped cuprates. Again, in electron-doped cuprates one also has a d-wave symmetry order parameter, as has been found by Tsuei and Kirtley [34]. Assuming pairing due to an exchange of antiferromagnetic spin excitations this is, of course, expected as shown in Fig. 1.20. Further experiments are necessary to learn about possible different behavior of the electron-doped and hole-doped cuprates. Since in general electron– phonon coupling may also cause superconductivity, this should possibly be taken into account in particular when Tc is smaller. Note that phonon-driven superconductivity yields an s-wave symmetry order parameter. Hence, it might be possible that superconducting transitions due to spin excitations and phonons act together and compete with each other.

History and Overview

13

(b) Ruthenates One may say that the study of the high Tc superconductors (exhibiting many interesting features like stripe-structures) during the last 15 years largely occupied the research activity and amusingly enough this might have helped the preparation of new surprises like the triplet Cooper pairing occurring likely in Sr2RuO4 , UPt3 and possibly other systems, i.e. superconductivity in organic materials, and also of the new singlet high Tc superconductor MgB2 with Tc  39 K. Due to advances in crystal growth the ruthenates have become a very interesting material class in condensed matter physics [35–38]. There are structural similarities with the cuprates [39, 40]. In Sr2 RuO4 layers of RuO2 are separated by Sr and O atoms, in SrRuO3 one has RuO2 -layers with Sr in between, and in Sr3 Ru2O7 a similar, but a more 3Dtype structure is present. Sr2RuO4 is a novel superconductor with Tc =1.5 K observed in 1994 by Maeno [36,37].Several measurements indicate triplet Cooper pairing, see the experimental results (NMR– Knight–Shift) shown in Fig. 1.21. Spin triplet Cooper pairing with orbital angular momentum l = 1 yields for the order parameter (k) = a |↑↑ + b |↓↓ + c |↑↓ + ↓↑. It seems that in Sr2 RuO4 due to spin-orbit coupling only the last spin state with zero spin (along the zdirection (perpendicular to the RuO2-planes) is in-

Fig. 1.20. Superconductivity in electron-doped cuprates (such asNd2−x CexCuO4 ): Symmetry of order–parameter in the Brillouin zone (doping x = 0.15, T/Tc = 0.8) [32]. The phase diagram for Tc (x) is shown in Fig. 1.14

Fig. 1.21. Results for the uniform spin susceptibility in the superconducting state of ruthenates as measured by NMR Knight shift [35]. This seems to demontsrate that Sr2 RuO4 is a triplet superconductor. Note that the dotted line would occur in case of singlet Cooper pairing (K-Knight shift)

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K. H. Bennemann and J. B. Ketterson

volved. In the presence of a magnetic field also the states |↑↑ and |↓↓ may play a role. In Sr2 RuO4 one observes a likely p-symmetry for :   k ∝ z kx + iky . (1.9) In Fig. 1.22 illustrates results showing the magnetic anisotropy (due to spin–orbit coupling and lattice structure). As a consequence, the anisotropy of Hc2 is even larger than in the cuprates. SrRuO3 is a ferromagnetic metal. Upon applying a magnetic field one observes a magnetic quantum critical point in Sr3 Ru2O7 .The Fermi surface of Sr2 RuO4 is illustrated in Fig. 1.23. These facts already suggest that the ruthenates are a most important class of new materials in condensed matter physics. In particular triplet Cooper pairs (k ↑, −k ↑) in Sr2 RuO4 may stimulate further studies of triplet pairing in superconducting heavyfermion metals, in organic systems and ferromagnetic metals like Fe under pressure and a unifying view on superconductivity and 3 He-superfluidity. Note that superconductivity in Sr2RuO4 seems related to both antiferromagnetic and ferromagnetic activity. The superconducting order parameter seems to be of p-symmetry [38,40]: ky a kx a kz c k = 0 z cos sin cos 2 2 2 ky a kx a cos , (1.10) + i sin 2 2 with no nodes in the RuO2 -planes, but with nodes along the z-direction perpendicular to the planes. The possible form of k is illustrated in Fig. 1.24. Note that the spin of the Cooper pairs is parallel to the RuO2-planes, but with no preferable direction in these planes. The orbital angular momentum l of the Cooper pairs points in the z-direction perpendicular to the planes [38,40–42]. Of course, triplet Cooper pairing (breaking time reversal symmetry) is reflected by corresponding thermodynamical and optical behavior. Due to the usual coupling between the orbital angular momentum l and the external magnetic field h one expects a rich response and different critical magnetic fields (hc2(T)) depending on whether h is parallel to the RuO2-planes or h||z perpendicular to the RuO2-

Fig. 1.22. Magnetic anisotropy +− (T) and zz (T) referring to in plane and perpendicular to RuO2 -plane response, respectively, in Sr2 RuO4 . The experimental results refer to spin relaxation (T1 , Knight-shift)

Fig. 1.23. 2D-Fermi surface topology of Sr2 RuO4 and RuO2 planes. ˛, ˇ and denote the FS of the corresponding hybridized bands

planes. Furthermore, the breaking of time reversal symmetry will be seen by corresponding optical response, for example, its dependence on light polar-

1

History and Overview

15

c) Heavy-Fermion Metals The heavy-fermion metals like CeIn3 , UPt3, etc., are characterized by an unusually large electronic density of states (DOS) near the Fermi energy, N(0). Thus, the effective mass m of the carriers is much larger then the bare electronic mass m0 . Note, m may range up to 100 m0 or more. Typical heavy-fermion metals are listed in Table 1.1 [42–45]. Table 1.1. List of heavy-fermion metals. m and m0 are the effective and bare electron mass and Tc the superconducting transition temperature. Note the interesting dependence of Tc on pressure

Fig. 1.24. Illustration of a possible form of the p-type superconducting order parameter in Sr2 RuO4 . (a) Spin triplet states in superconducting Sr2 RuO4 corresponding to  ∼ z(kx + iky ), with angular momentum parallel to z-axis and spins perpendicular, and  ∼ xkx + yky , with vanishing angular momentum. Clearly, an external magnetic field and spin–orbit coupling will control the most stable triplet state. (b) Illustration of the amplitude of the order parameter. In principle incommensurability along the z-axis may occur

ization. The magnetic anisotropy is expected to lift the degeneracy of the three triplet states (|↑↑, |↓↓, |↑↓+ |↓↑) [40]. Note that it is of utmost significance to identify definitely triplet pairing and the pairing field (spinexcitations).

m/m0 Tc (K)

UPt3 360 0.55

UBe13 1100 0.85

UPd2Al3 210 2

UGe2 100 0.7

Due to the large electronic density of states (DOS) the heavy-fermion metals exhibit simultaneously interesting magnetic and superconducting behaviour. Multiple magnetic and superconducting phases and generally complex thermodynamical behaviour exist [45,46]. For example, this is demonstrated by the phase diagrams of Tc (h) and Tc (P) of UPt3 shown in Fig. 1.25. Note, while in conventional type II superconductors there are two superconducting phases in the h–T plane (a low field Meissner phase and a high field vortex phase above hc1 (T)); in UPt3 there are five phases: vortex phase C, and phases A and B each exhibiting a Meissner and vortex phase. The phase diagram Tc (P) also exhibits the superconducting phase A,appearing for pressure P=0 at Tc ≥ 5 K, and phase B appearing

Fig. 1.25. The phase diagram (a) Tc (P) and (b) Tc (h) of the heavy-fermion metal UPt3 is shown (h denotes the external magnetic field, p the pressure). A, B, C refers to different superconducting states

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K. H. Bennemann and J. B. Ketterson

for P=0 at Tc  0.44 K. The symmetry of the order parameter is certainly not simply s-wave like, but different for Tc (P,h). This is still being analyzed as well as the mechanism for Cooper pairing. However, spin-excitations are probably involved. Several studies have been performed for analyzing the superconducting phases by using a Ginzburg– Landau theory and allowing a more complex order parameter including triplet Cooper pairing. Note that the free-energy change may be described by F = ˛1 (T − Tc1)|1 |2 + ˛2 (T − Tc2)|2 |2 + ˇ1 |1 |4 + ˇ2|2 |4 + ˇ12 |1 |2 |2 |2 + . . . (1.11) Here, 1 , 2 refer to the order parameter of the different phase transitions.The coefficients are pressure dependent. Then in the usual way one attempts to derive the observed phase diagrams from Eq. 1.11. In Fig. 1.26 we show more data on heavy-fermion metals indicating the role played by magnetic excitations. The superconducting transition-temperature Tc (p) of CeIn3 may be of particular interest due to quantum criticality at p28 kbar where the N´eel temperature decreases (TN → 0) and superconductivity

Fig. 1.26. Phase diagram of the heavy-fermion CeIn3 . Note the appearance of superconductivity at larger pressure. A quantum critical point occurs at about 28 kbar

appears [46]. This, however, needs further studies. Note the interplay of a.f. and superconductivity and the changes due to pressure (itineracy of carriers is expected to increase with pressure). As in CePd2 Sr2 and other heavy-fermions a quantum critical point seems present. An interesting phase diagram is also observed for UGe2 with an interplay of ferromagnetism and superconductivity. Here a ferromagnetic quantum critical point possibly plays a role. The behaviour of Fe under pressure effects in particular the important interplay of structure,ferromagnetism and superconductivity. The origin of superconductivity needs to be studied. It is of utmost interest to determine for UPt3 and the other heavy-fermion, for Fe and UGe2, see Fig. 1.27, and related metals the interplay of mag-

Fig. 1.27. (a) Phase diagram of UGe2 . (b) Superconducting hexagonal -phase Fe under strong pressure. ˛, , refers to the various Fe phases (˛: bcc, ": hcp)

1

History and Overview

k ∼ cos (k · r)

17

(1.12)

also indicates that magnetism and superconductivity may seek a compromising arrangement. This is also known from Ginzburg–Landau theory with two order parameters, which may coexist or exclude each other, depending on the parameters controlling the energetics. Further interesting phase diagrams are shown in Fig.1.28.Not much is known about the pairing mechanism and symmetry of the superconducting state appearing upon alloying and applying pressure.Such typical phase diagrams are again indicative of the interplay of structure, antiferromagnetism or ferromagnetism and Cooper pairing. Note that further experiments may somewhat modify these interesting phase diagrams and offer new insight. Organic Superconductors

Fig. 1.28. Superconductivity in heavy-fermion systems: (a) Typical complex phase diagram, (b) pressure induced superconductivity. (Here, TN refers to the Ne´el temperature, TSG to the spin-glass temperature, NFL to nearly Fermi-liquid behavior, and  to the electrical resistivity)

netism and superconductivity, the symmetry of the superconducting order parameter, the thermodynamical properties and the role played by quantum critical points (Q.C.P.). Besides Ginzburg–Landau type analysis electronic theories are needed to understand superconductivity in these systems. To summarize,the interdependence of magnetism and superconductivity has been a classical problem and nowadays seems to have regained interest due to technical advances. Note that the Larkin– Ovchinnikov state of a superconductor in the presence of an exchange field [47]

Recently, many studies have shown that organic metals are most interesting. A large class of crystalline organic systems are quasi-two-dimensional chargetransfer salts. Many become superconducting [48]. Various properties of these organic metals are shared with the cuprates, such as layered structures, strong correlations causing also antiferromagnetism and non-s-symmetry Cooper pairing.The organic metals are very clean, nearly impurity free systems. Moreover, astonishingly, superconductivity may be induced by strong magnetic fields. In Fig. 1.29 the interesting phase diagram of (BEDT-TTF)2 -Cu[N(CN)2 ]Br, a typical composition of a charge-transfer salt, is shown. Such molecules are stacked into layers. Planes of Cu[N(CN)2 ]Br anions separate these layers. Within these layers the molecules form dimers and electrons or holes can then hop easily from one molecule to the next, but not perpendicular to the layers (hence: quasi 2Dbehavior). Thus, salts consist of conducting layers separated by a non-conducting environment (anions). Clearly,bond-length changes,also due to pressure, will sensitively change the electrical properties. Also applying a magnetic field perpendicular to the layers (along the c-axis) affects the system due to the formation of Landau-levels that will pass the Fermi

18

K. H. Bennemann and J. B. Ketterson

Fig. 1.29. Phase diagram of the organic metal (BEDT-TTF)2 -Cu[N(CN)2 ]Br (Bisethylenedithio–terathiofulene–X type salt) from N.M.R. and A.C. susceptibility

energy as the external magnetic field varies (see de Haas–van Alphen oscillations). In stronger magnetic fields quantum mechanical interband tunneling also occurs, see Singleton et al. [48]. Superconductivity occurs at relatively low temperatures usually upon applying pressure, see Fig. 1.29. Various experiments indicate unconventional Cooper pairing (20 ∼7kB Tc , non-s-symmetry of k ). Note superconductivity occurs close to the antiferromagnetic-state like in the cuprates and heavy-fermion metals. Experiments indicate that the long range magnetic order is due to localized spins on the dimer and not due to the hopping carriers (holes or electrons). Antiferromagnetic spin-fluctuations occur as a precursor of superconductivity.How much these and phonons are involved in Cooper pairing must be clarified by further analysis. Theoretical studies are presented by Schmalian and others. In an external magnetic field interesting behaviour may result due to the layered structure, Landau-level formation and spin-split bands. Possibly Cooper pairs (k ↑, −k+q ↓) may form.A LarkinOvchinnikov state [47, 48] with k ∼ cos(kr) may occur. In general, the response to an external magnetic field is highly anisotropic. Furthermore, it is interesting that superconductivity may be induced in -(BETS)2 FeCl4 by a magnetic field h, which destroys long range magnetic order of the Fe3+ . Field induced superconductivity may also occur in ˛(BEDT-TTF)2 KHg(SCN)4 by affecting with the external magnetic field h the charge-density-wave (CDW) present in this system.

Clearly further experiments are needed to clarify the situation. However, rich behaviour may be expected in general, since CDW, SDW excitations, Landau levels and 2D properties are present. The different excitations and phases may cause inhomogeneities (as a compromise to gain maximal energy).

1.3

Granular Superconductors, Mesoscopic Systems, Josephson Junctions

As early studies by Buckel and Hielsch and others have shown strongly disordered lattices and amorphous metals may affect superconductivity due to changes in the electron–phonon coupling, the phonon-spectrum, the electronic parameters (DOS:N(0)) and surface effects.As a result superconductivity changes and might be strengthened, there is an increase of Tc , and generally thermodynamical properties might change favorably, see Garland, Bennemann [49], and Deutscher and references therein. In the amorphous state some non-metallic systems become metallic and then also superconducting. In Table 1.2 we illustrate the situation. Remarkable is also the occurrence of superconductivity in PdH-systems where dramatic changes in Tc (that are dependent on H-doping and disorder) are observed,see experiments by Buckel et al.[49,50]. This is a good example of the potential of material science regarding superconductivity studies. Similarly this is the case for superconductivity in fullerides.

1

History and Overview

19

Table 1.2. Superconductivity in disordered and amorphous metals.Estimates of the superconducting transitiontemperature Tc by Bennemann et al. [49] (Tc0 refers to the crystalline structure) material

(Tc /Tc0 )expt

(Tc /Tc0 )calc.

Al Pb Ga Sn In

∼5 ∼0 ∼8 ∼ 1.3 ∼ 1.3

4.9 0 8 2 1.2

Extending these studies of granular materials, small particles, nanostructures and metals consisting of an ensemble of small metallic grains and clusters have been investigated [51], see Fig. 1.30. Thus, quantum size effects occur (discretization of electronic energy spectrum: k → n, granular size l ∼ , ;  is the coherence length and  the penetration depth). In such granular superconductors surface effects and proximity effects become important. As a function of grain size metal insulator transitions and strong quantum-mechanical behaviour occurs. Note that on general grounds one expects, for example, that in reduced dimensions fluctuations play a more important role.(Also,different Cooper pairing mechanisms and singlet and triplet pairing are expected to depend differently on particle size.) The discretization of the electronic energy spectrum, k → n, is illustrated in Fig. 1.31. Of course, the level spacing ı(T) affects various properties such as the coherence-length (n) and penetration depth (n) and thus the thermodynamical and optical properties. Note that /R∼N2/3 and hence  R due to N2/3  1. One expects that superconductivity disappears below a critical particle size (R< ). The behaviour of small particles due to changes in the number n of electrons, for example n → n± 1, and Cooper pairs by 1, for example, is very interesting and may play a role in achieving a two-level superconducting state (two charged states). This may possibly be used for information technology (information storage, optical imprinting technology). The charging of small particles is controlled by the electrostatic energy. The change of the charge

Fig. 1.30. Illustration of an ensemble of small particles, grains which are in the normal (n) or superconducting (s) state. For small particle size (radius R) and volume V the electronic energy spectrum gets discrete with level spacing ı(ı = (N(0)V )−1 ≈ F /n = (VF /R)(RkF )−2 ; N(0) = DOS at F , n = no of electrons). The grains may be in the normal (n) or superconducting (s) state. In such a grain structure some sites may be empty (0). Note, such a lattice-like structure may resemble an alloy of s and n state grains and also a nanostructure if small particles are removed irregularly from the lattice (empty sites)

Fig. 1.31. Illustration of size effects in small particles having a diameter of the order of R and n–electrons. The level spacing is ı ≈ F /n ≈ (VF /R)(kF R)−2 and the coherence length  ∼ (Rı/kTc )(RkF )2 =ı0 N 2/3 R. Here, we introduced the dimensionless size parameter ı0 = ı/kTc

Q = en → Q ± 1 causes an energy change e2 /2C; C is the capacitance of the small particle and this may lead to a Coulomb blockade (in tunneling, for example).Approximately at temperatures T  0 due to the electrostatic energy (Q = en, Q = C  V being the charge at the electrodes of the applied voltage) it is En = 

Q2 Q  − CV 2C C

(1.13)

(C, C capacitances, V =external potential). The Coulomb blockade is periodically lifted as a function of n and V when En+1 = En . In the superconducting

20

K. H. Bennemann and J. B. Ketterson

Fig. 1.32. Charging up behavior of a small superconductor with capacitance C and charge Q = en, n-even, in a potential ( > e2 /2C). This behavior (2e-jumps and 2eperiodicity in V) follows from En = En+2 and n-even and reflects Cooper pair formation. One has also En = En+1 + , where En+1 has one unpaired electron. Then Q changes by 2e at corresponding external voltage. Again one has a 2eperiodicity in the energy level structure

state for T 0 and  > e2 /2C one has the situation illustrated in Fig. 1.32. From En+2 = En , n-even and a large Cooper pair sea, one gets a period-doubling with respect to the normal state of the charge periodically of energy levels, of the jumps in Q at voltages V which lift the Coulomb blockade (V∝ (2n + 2)). Note that En+2 = En might not hold for smaller density of Cooper pairs. For En+1 with one unpaired electron at energy  above the ground-state one has En = En+1 +  to determine the 2e charge periodicity. This gives the even–odd number parity, see Tinkham et al.and Nozarov [52].Charge transfer occurs for the small particle when energy levels cross. In view of the strong quantum-mechanical behavior of small particles structures consisting of a larger number of small particles seem very interesting (mesoscopic systems). Note that the charging up behavior will reflect spin and thus may be different for singlet and triplet Cooper pairing. Also the behaviour of grains is related to the one of Fermions in optical lattices. An ensemble of grains may exhibit transitions to superconductivity at Tc1, where the single grains are superconducting, and at Tc2 where globally and phase coherently the whole ensemble becomes superconducting (Tc2 ≤ Tc1 and zEJ ≥ Ec ); for an illustration see Fig. 1.33. Generally an ensemble of grains may resemble an alloy like array of S/S and

Fig. 1.33. Transitions in an ensemble of superconducting quantum dots controlled by the Josephson energy EJ and the Coulomb energy Ec = e2 /2C. Tc refers to the global superconducting transition-temperature of the ensemble. Phase O below Tc refers to a globally ordered superconducting state of the ensemble and phase d above Tc to superconducting grains with no global Cooper pair phase coherence. z is the coordination number

S/N junctions. (S = superconducting, N = normal state grains.) Of course, the behavior of an ensemble of grains depends on the coupling,i.e. on electron hopping between the grains. The structural order of the grains also plays a role. A regular lattice-like arrangement of the grains (1D, 2D-topology) is of particular interest. For small distances between the cluster particles electron tunneling, inelastic Cooper pair tunneling (Josephson coupling) occurs. Hence, charge fluctuations are present in such granular superconductors. The resultant interesting Josephson effect can be described by [52]   H = HT − £i,j EJ i, j cos ¥ij  2 2e2 £i,j Ni − Nj + . . . + (1.14) C In Eq. (1.14) the first term refers to normal electron hopping between the grains, the second term to Cooper pairs tunneling between neighboring superconducting grains (Josephson effect), and the last term is electrostatic energy due to different charges of neighboring grains i, j. For simplicity, the same capacitance C and EJ -Josephson energy is taken. The phase difference ¥ij = ¥i − ¥j refers to the phases ¥i of the superconducting order parameter i = |i |ei¥i of grain i. It is important to note that

1

[¥ , N] = i ,

(1.15)

where N is the Cooper pair number operator [51,53]. Hence, ¥ and N are canonically conjugate variables. Treating these as classical variables from the Hamilton–Jacobi equations one immediately finds the Josephson equations jJ = (EJ /e) sin ¥ and

•

¥ = 2eV .

(1.16) (1.17)

Note that the commutator Eq. (1.15) implies the uncertainty relation ¥ N ∼ 1 .

(1.18)

Consequently, large phase fluctuations, i.e. the phase incoherence of the small particles imply a N nd thus N is a good quantum number. Hence, in view of Eq. (1.18) in an ensemble of superconducting grains one may expect an ensemble transition to a Mottinsulator if the capacitance C becomes smaller and thus the Josephson energy (∼ EJ ) becomes smaller than the electrostatic energy ∼ 2e2/C) [51, 52]. The charge transport between the grains is suppressed while each grain is still superconducting. This behaviour is illustrated in Fig. 1.33 [52]. Note that if the capacitance C becomes larger, N increases and then all grains become phase coherent (¥ → 0) [51,53]. These thoughts apply also to fermions in optical lattices and corresponding phase transitions. For small particles (of size <10 nm, for example) the proximity effect and the Andreev-reflection [51,54] play an increasingly important role and may cause anomalous behaviour of the diamagnetic susceptibility and of the maximal Josephson currents as a function of particle size, temperature, etc. For example, the Josephson currents (jJ = j0 sin ¥ , j0 = /2eRT , RT resistance of tunnel barrier) may reverse (jJ → −jJ ,-junctions) and the maximal current j0 reflects quantum size effects. In a Josephson lattice of quantum dots one has j0 = z e , where z is  the lattice coordination number [51]. The ensemble topology of the small particles and their separation plays a role, since mean free path effects and phase

History and Overview

21

coherency matter.A ring-like arrangement of Josephson coupled quantum-dots is expected to be an interesting physics toy regarding behaviour of two-state superconductors and flux-pinning [52]. Tunnel Junctions It is obvious that tunnel junctions have become an interesting new area of solid state physics [55, 56]. The tunnel medium (metals, insulators, molecules) can be manipulated, in particular optically, and new non-equilibrium physics may result. In Fig. 1.34 we illustrate a tunnel junction describing in particular S/N/S, N/S/N junctions. Here, S may refer to singlet or triplet superconductors and N to normal state systems, metals and ferromagnets [54]. In the case of an S1 /N2 /S3 sandwich, where S1 and S3 are singlet BCStype superconductors,the spin tunnel current is zero. If S1 is a singlet and S3 a triplet superconductor, one expects for Josephson tunneling [54]  jj = (j1n sin n + j2n cos n) . n

Here, n = 1, 2, . . . and  refers to the phase difference of the order parameter of S1 and S3 . We assume that the thickness 2d of N2 does not destroy phase coherence of the tunneling electrons.Equation (1.19) corresponds to a general Fourier-expansion of jJ . If time reversal symmetry is present (both S1 and S3

Fig. 1.34. Illustration of a tunnel junction N1 /N2 /N3 , where Ni may refer to singlet and triplet superconductors and to normal state metals, including ferromagnets, respectively. The tunnel current jT may refer to single electrons or Cooper pairs and transport of charge and spin. Depending on the thickness 2d of the tunnel medium proximity effect and Andreev-reflection at S/N interfaces plays a role. Note that applying a voltage to the tunnel junction may yield a two-level system with no or one extra Cooper pair on superconductor 3 (qubit)

22

K. H. Bennemann and J. B. Ketterson

are singlet B.C.S. superconductors) the second term in Eq. (1.19) disappears, since  → − implies jJ → −jJ . Then one has approximately jJ ∼ sin . If S3 , for example, refers to a triplet superconductor, then characteristically jJ (n = 1) = 0 ,

(1.19)

since the superconducting wave functions i = spin i 'iorbital of i = 1 and i = 3 are orthogonal [54]. Thus to lowest order for a (singlet/N2 / triplet) sandwich jJ ∼ sin(2) + . . . (1.20) Clearly spin-active interfaces of the sandwich resulting from spin-orbit coupling or spin-excitations in N2 may change the symmetry arguments and then jJ (n).Note,in the case of spin-flip scattering selection rules change and spin and orbital angular momentum need not to be conserved separately, but only the total angular momentum. As a consequence, one might get contributions from j11 and j21 and from higher jin . In summary, one expects characteristic properties and differences for the (singlet/N2 /singlet), (singlet/N2 /triplet), and (triplet/N2 /triplet) junctions. Similarly, if N2 is a ferromagnetic metal or insulator it will affect differently and characteristically junctions involving only singlet superconductors and those involving a triplet superconductor. Clearly, Josephson tunneling of singlet Cooper pairs is more characteristically affected by a ferromagnet N2 tunneling medium than jJ for triplet superconductors. In particular, recent studies have shown the physical richness of behaviour expected for tunnel currents, see Morr et al. for an illustration [56]. Sandwiches (F1 /S/F3) involving a superconductor S between two ferromagnets F1 and F3 with parallel or antiparallel orientation of the magnetization will also exhibit interesting behaviour. (We assume for simplicity that the thickness 2d of S is smaller than the spin-mean free path in S.) Then charge (jc )-tunnel and spin (jsp )-tunnel currents are expected. Due to spin-polarized tunnel currents provided by the exchange splitting of majority and minority bands in the ferromagnets one gets, in general, the functional behaviour

Fig. 1.35. Phase-diagram of an (F1 /S/F3 ) sandwich. S refers to a singlet superconductor and F1 ,F3 to ferromagnets with Tc at P for parallel (P) orientation of their magnetization and Tc at AP for antiparallel (AP) one. L.O. is the Larkin– Ovchinnikov phase, h the exchange field and Tc and Tco the superconducting transition-temperature in the presence and absence, respectively, of the exchange field. dF and dS refer to the thickness of the ferromagnetic and superconducting film, respectively

jT = jT {, . . .} ,

 = {jT , . . .} .

(1.21)

The spin-polarized tunnel current jT may cause the presence of unpaired single electrons at the Fermi energy F besides Cooper pairs, and thus superconductivity is weakened. The effect should be different for singlet and triplet-Cooper pairs. A rich thermodynamical behaviour is expected. Due to the proximity effect and Andreev-reflection [51,54] the F/S interface coupling may cause the induction of a Larkin–Ovchinikov state [47] into the singlet superconductor (k → 0 cos kx + . . .), for example if  ∼ 2d. Then the electrons in the superconductor may feel the exchange field present in the ferromagnets. Also the exchange field accompanying the spin-polarized tunnel current might cause a similar change of the Cooper pairing. For sandwiches (F1 | S | F3 ) with a singlet superconductor one gets the results shown in Fig. 1.35. Corresponding results are expected for such a sandwich with a triplet superconductor. Then the phase diagram is determined by the coupling of the Cooper pair spin (angular momentum) to the exchange field h. Applying Usadel-type and Eilenberger-type equations [54] for Cooper pairing in external fields h one has

1



 1   !n + ih(r) + vF · ∇ F =  (r)G , 2  d§ 1  G(r) , !n = !n + 2 4

and  (r) = (r) +

1 2



23

scaling follows immediately from the general formula s = (m/e2)! Im (!). As a special result one gets the Uemura scaling s ∼ Tc for underdoped high Tc -superconductors. Note,  s = (!  /DC Tc )!=0 · DC Tc = const.(DC Tc ),

d§ F(r) . 4

(1.22)

Here, G and F are the usual Green’s functions and  is the elastic scattering term. Thus, one obtains the interesting phase diagram for (F1 /S/F3 ) sandwiches shown in Fig. 1.35. [54,55] Recently, Nogueira and Bennemann discussed a spin Josephson-like spin current jJ for (F1/N2 /F3 )- sandwiches [53]. If the thickness 2d is smaller than the spin-mean free path the results could also be applied if N2 is a superconductor.

1.4

History and Overview

Outlook

In summary, this overview shows that the history of superconductivity has been full of surprises and that superconductivity is a stimulating and continuing problem of physics. A theory like the BCS-theory for phonon-mediated superconductivity would be of utmost significance for the novel superconductors. The pairing mechanism, symmetry of the order parameter and the superfluid density are central issues. More detailed knowledge on the magnetic activity in the cuprates, etc., on local versus itinerant magnetism (see (q, !), etc.), and on short-range spin ordering is needed. Correlations (local repulsion) cause a particle-hole asymmetry reflecting that it is more difficult to add an electron than to remove one and this affects many properties. Future studies may change some parts of our present physical picture, will extend our knowledge, and will certainly bring new discoveries.As was recently demonstrated by Homes [57], the superconducting transition temperature Tc may be scaled for many materials in a unifying way according to s ∼ DC Tc .

(1.23)

Here, s and DC refer to the superfluid density at T=0 and the electrical conductivity,respectively.This

if DC T is independent of temperature. We have attempted to summarize recent developments in the area of superconductivity.Superconductivity in MgB2, cuprates, ruthenates, heavy-fermion metals (material) and organic systems shows that new metals continue to keep the field alive. This view is further supported by the results on small particles, nanostructures and sandwich-structures. The interdependence of structure and superconductivity and further of magnetism and superconductivity remains a key physical issue. In particular tunneling probing sensitively the superconducting state is of interest [56]. In tunnel junctions (SC/FM/SC) the magnetic medium FM may affect (tune) the phase between the superconductors (SC) and cause spin-polarization of the tunnel current and the current may be manipulated by an external magnetic field. Extreme quantum-mechanical behaviour due to T →0, reduced dimensionality, correlations, quantum-fluctuations and in particular quantum critical points (QCP) might present new surprises in the future. New non-equilibrium behaviour, optical studies probing the superconducting state and (fast) switching phenomena involving superconductivity will be interesting. Of course, the search for new superconductors still remains interesting.Superconductivity in intercalated systems like Nax CoO2 (H2O)y , (for x0.35 and y1.3 Tc ≤5K [58] may serve as an example. In liquid metallic hydrogen (H), obtained upon applying pressure, besides other phases superconductivity may occur [59]. Under pressure hydrogen becomes liquid and exhibits an insulator to metal transition. One gets a groundstate quantum-liquid metal. In the presence of a magnetic field liquid metallic hydrogen (free energy:  2 F = i=e,p [| (∇ ± ieA) i |2/2mi + V (| i |2 )] + B2 , B = ∇ × A, i refers to electrons and protons) exhibits several phase transitions with strong quantum mechanical features, superconductivity and superfluity as was shown by Babaev, Sudbo, Ashcroft et al. [59].

24

K. H. Bennemann and J. B. Ketterson

A liquid metal close to the ground state (T → 0) and ordered quantum mechanical fluids (a superconductor or a superfluid of protons and electrons, two-fluid system) are obtained. In the presence of an external magnetic field several phase transitions to ordered states, between superconductor and superfluid, occur. This may be of general significance, in particular for astrophysics. Note, hydrogen amounts to about 90% of all elements and constitutes due to the light mass and zero–point energy strongly quantum–mechanical systems. Metallic H may become a type II superconductor and under higher pressures electrons and protons form a 2 component Fermi–liquid, Cooper pairs of protons and of electrons form and may coexist. In a magnetic field interesting novel structures of vorties may occur. The pairing potentials needs to be studied by a microscopic theory. A unifying field theoretical analysis of non-Fermi liquid behaviour and of superfluidity and of Bose– Einstein condensation (BEC) remains a challenging goal.Ultracold Fermi gas may become superfluid and then tuning the interactions (in an optical lattice, for example) one observes a BCS-state, BEC and a crossover transition.As the size of the singlet Cooper pairs decreases these are expected to behave more and more like bosonic molecules and consequently a BEC should occur (see such transitions in optical lattices). The studies of this by Greiner et al. and on the crossover transition BCSBEC are of utmost interest [60]. For describing phase transitions (in optical lattices with sites i) one uses the Bose–Hubbard hamiltonian   1 H =− ta+1 aj + ("i − )ni + Uni (ni − 1). 2 i,j

i

tion technology, see the review by Sch¨on et al. [61]. The charge changes in Fig. 1.32 are due to Cooper pairs and describe the way to get two-level superconducting states [52,62]. Such quantum-state engineering using low capacitance Josephson tunneling junctions may yield quantum bits (qubits) for quantum information processing. As discussed by Sch¨on et al. single-qubit and two-qubit quantum states can be controlled by gate voltage (or in the case of phase flux degrees of freedom by magnetic fields), see [62]. Applying Eq. (1.14) to the tunnel junction shown in Fig. 1.34 one has H = 4Ec (n − ng )2 − EJ cos 13 + . . . ,

(1.24)

if for low capacitance and large superconducting gap energy  only tunneling of Cooper pairs occurs. Ec is the charging energy, n the number of extra Cooper pairs on system 3 (nop = −i@/@) and ng = Cg Vg /2e acts as a control parameter, see Fig 1.34. Here, we have applied a gate voltage Vg connecting an electrode to the superconductor 1 and another one that acts on the superconductor 3 via a gate capacitance Cg . This is a qubit yielding approximately a two-level system as illustrated in Fig. 1.36.A qubit with Cooper–pair states zero or one is similar to the 2 spin states (|↑ = (10), |↓ = (01)) and may be described by the model Hamiltonian 1 1 HQ = − Bz z − Bx x , 2 2

i

Here, t denotes the hopping integral between (atomic) sites i, the chemical potential acts as a Lagrange multiplier and the interaction U between particles (atoms) tends to localize these. Note that triplet Cooper pairs do not show a BEC transition. Phase coherence of the Cooper pairs might affect the BCS→BEC transition. Coulomb interactions might suppress the pairing and cause a Mott transition. Systems consisting of superconducting quantum dots may become very useful for quantum informa-

Fig. 1.36. Charging energy of superconducting electron box (see (3) in Fig. 1.34) as a function of a gate voltage Vg (˛ng ). The degenerate parabola (dashed curves) split due to Josephson coupling mixing and one gets a two-level quantum system with states a, b and so on. The superconducting box (3) becomes a qubit with states n = 0, n = 1 similar to two spin-states

1

with Bz ≡ 4Ec (1 − 2ng ) and Bx ≡ EJ . Combining several such systems (several tunnel junctions) one may achieve a behaviour that is useful for quantummechanical information technology [61, 62]. Interesting is also the case of a spin-polarized Josephson current (triplet superconductivity, a ferromagnetic medium 2, see Fig 1.34, etc.). A magnetic field could manipulate the tunnel current. If strong correlations occur in the tunnel medium (described by a Hubbard like Hamiltonian) then in particular irradiation may affect the electronic occupation of levels in the tun-

History and Overview

25

nel medium and thus manipulate the tunnel current (ultrafast switching).

Acknowledgements We thank D. Manske, I. Eremin, F. Nogueira for helpful discussions and last but not least E. v. Sulzbach for encouragement. In particular we are also grateful to L. Tewordt, W. Buckel, R.D. Parks, M. Peter and B. M¨uhlschlegel for many discussions over the years.

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31. J. Schmalian, M. Langer, S. Grabowski, and K.H. Bennemann, Comp. Phys. Comm. 93, 141 (1996) 32. D. Manske, I. Eremin, and K.H. Bennemann, Phys. Rev. B 63, 054517 (2001); D. Manske and K.H. Bennemann, Mol. Phys. Rep. 24, 52 (1999); D. Manske, I. Eremin, and K.H. Bennemann, Phys. Rev. B 62, 13922 (2000); I. Eremin, D. Manske, C. Joas, and K.H. Bennemann, cond-mat 0102074; D. Manske and K.H. Bennemann, In: Fluctuating Paths and Fields, ed. W. Janke et al. (World Scientific, Singapore, 2001) 33. P. Bourges, In: Gap Symmetry and Fluctuatios in High-Tc Superconductors, eds. J. Bok, G. Deutscher, D. Pavuna and S. Wolf (Plenum Press, New York, 1998) 34. C.C. Tsuei, and J.R. Kirtley, Phys. Rev. Lett. 85, 182 (2000) 35. K. Ishida, H. Mukuda, Y. Kitaoka, Z.Q. Mao, H. Fukuzawa, and Y. Maeno, Phys. Rev . B 63, 060507 (2001) 36. Y. Maeno et al., Nature 372, 532 (1994) 37. Y. Maeno, T.M. Rice, M. Sigrist, Physics Today 42-47, (2001) 38. Y. Maeno et al, Nature 372, 532 (1994); T.M. Rice, Nature 396, 627 (1998); Y. Maeno, T.M. Rice, M. Sigrist, Physics Today 54, 42, Physica C 341-348 695 (2000) 39. J. Nagamatsu, N. Nakagawa, T. Musanaka, Y. Zenitana, J. Akimitsu, Nature 410, 63 (2001) 40. D. Manske, I. Eremin, and K.H. Bennemann, In: The Physics of Superconductors, Vol 2 (Springer, 2003) 41. A. Mackenzie, Y. Maeno and S. Julian, Phys. World 15 (No. 4), 33 (2002) 42. M.B. Maple, Physica C 341-348, 47 (2000) 43. H.v. L¨ohneysen, Physica B 197, 551 (1994); N.D. Mathur et al., Nature 394, 39 (1998) 44. H.R. Ott, In: The Physics of Superconductors Vol. 1, Eds. K.H. Bennemann, J.B. Ketterson (Springer, 2002) 45. L. Degiorgi, Rev. Mod. Phys. 71 687 (1999); F. Steglich et al. J. Phys. Chem. Sol. 59, 2190 (1998); M.B. Maple, Physica C 341-348, 47 (2000); C.R. Stewart, Rev. of Mod. Physics 73, 797 (2001); R. Joynt and L. Taillefer, Rev. Mod. Phys. 74, 235 (2002) 46. M. Vojta, Phys. Bl. 55, March 2002; S. Sachdev: Quantum Phase Transitions (Cambridge Univ. Press, 1999); J. Flouquet and A. Buzdin, Phys. World 15, 41(2002) 47. A.I. Larkin, Yu.N. Ovchinnikov, JETP 20, 762 (1965) 48. J. Singleton and C. Mielke, Phys. World 15 (No1) 35 (2002); T. Ishiguo et al., Organic Superconductors (Springer 1998); M. Lang, In: The Physics of Superconductors Vol. 2, Eds. K.H. Bennemann, J.B. Ketterson (Springer, 2003) 49. J.W. Garland, K.H. Bennemann, F.M. Mueller, Phys. Rev. Lett. 21, 1315 (1968); W. Buckel, and R. Hilsch, Z. Phys. 138, 109 (1954) 50. K.H. Bennemann, Z. Physik 260, 367 (1973) 51. W. Belzig and C. Bruder, Phys. Bl. 56 (No.5) 35 (2000); C.W.J. Beenakker, In: Mesocopic Quantum Physics (North Holland, Amsterdam 1995); B. M¨uhschlegel, Surface Rev. and Lett. 3 (No.1) 115 (1996), W.P. Halperin, Rev. Mod. Phys. 58, 533 (1986); B. Janko, L. Smith, V. Ambegaokar, Phys. Rev. B 50, 1152; M.T. Tuominen, J.M. Hergenrother, T.S. Tighe, and M. Tinkham, Phys Rev. Lett. 69 (1992) 1997; Mesoscopic Superconductivity Physica B 203, 201 (1994); Y. Imry, Introduction to Mesoscopic Physics (Oxford Univ. Press, 1997); B.L. Altshuler et al., Mesoscopic Phenomena in Solids (North-Holland, Amsterdam 1991) 52. J.B. Ketterson and S.N. Long, Superconductivity (Cambridge Univ. Press, 1999) 53. F.S. Nogueira and K.H. Bennemann, Europhys. Lett. 67, 620 (2004) 54. I. Baladi´c, A. Buzdin, N. Ryzhanova and A.Vedyayev, Phys. Rev. B 63, 54518 (2001); Y. Asano, Y. Tanaka, M. Sigrist, and S. Kashiwaya, cond-mat /0212353v1 55. L.B. Ioffe and M.V. Feigel’man, cond-mat /0203011; B. Doucot, M.V. Feigel’man and L.B. Ioffe, Phys. Rev. Lett. 90, 107003 (2003) 56. B. Kastening, D. Morr, D. Manske, and K. Bennemann, Phys. Rev. Lett. 96, 47009 (2006) 57. C.C. Homes, Nature 430, 539–541 (2004); T. Timusk, Physics World 18, p. 31, (2005) 58. T. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R.A. Dilanian, and T. Sasaki, Nature 422, 53 (2003) 59. E. Babaev, A. Sudbo, N.W. Ashcroft, Nature 431, 666 (2004) 60. M. Greiner, I. Bloch, O. Mandel, T. H¨ansch, T. Esslinger, Phys. Rev. Lett. 87, 160405 (2001), M. Greiner et al., Nature 415, 39 (2002); C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. 92, 40403 (2004) 61. Y. Makhlim, G. Sch¨on, A. Shnirman, Rev. Mod. Phys. 73, 357 (2001) 62. M. Trinkham, Introduction to Superconductivity (Dover, Mineola, NY, 1996)

2

Phenomenology and Microscopic Theory: Theoretical Foundations

L. Pitaevskii University of Trento Dept. of Physics, Trento, Italy

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13

2.14

2.15 2.16 2.17

2.1

Off-Diagonal Long-Range Order and Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Off-Diagonal Long-Range Order in Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The London Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Thermodynamics of Superconductors in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 The Intermediate State of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 The Ginzburg–Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Surface Energy at the Boundary Between Normal and Superconducting Phases . . . . . . . . . . . . . . . . . . 42 Superconductors of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Quantized Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Vortex–Vortex Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Cooper Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Energy Spectrum of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Thermodynamic Properties of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.13.1 Temperature Dependence of the Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.13.2 Thermodynamic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.13.3 Microscopic Meaning of the GL Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Elements of the Theory of Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 2.14.1 General Properties of Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.14.2 Green’s Function of an Ideal Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.14.3 Two-Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Green’s Function of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 Temperature Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 Temperature Green’s Functions of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Off-Diagonal Long-Range Order and Superfluidity

Many important properties of superconductors follow from an understanding of the difference between the symmetry of the superconducting and normal states, i.e. an understanding of the nature of the or1

der parameter arising from the phase transition between these states. Essential hints can be obtained from the deep physical analogy between two physical phenomena – superfluidity and superconductivity. For this reason it is instructive to examine some properties of a superfluid liquid.1

In this and in the next chapter we use units such as the Boltzmann constant kB = 1, so that temperature has dimensions of energy. We also use the CGS system of units. Operators are denoted by a circumflex. Hermitian conjugation is denoted by a † symbol.

28

L. Pitaevskii

Superfluidity was first discovered in liquid 4 He at temperatures below 2.18 K. A superfluid has the property that it can flow through narrow capillaries or slits without exhibiting viscosity. The viscosity  of a superfluid liquid is exactly zero. At T = 0 the liquid consists entirely of superfluid.At non-zero temperature part of the liquid behaves as a superfluid without viscosity; the remaining part behaves as a normal viscous liquid. However, these two parts fully interpenetrate each other. Thus the density of a superfluid can be written as a sum  = s + n of superfluid and normal parts with associated hydrodynamic velocities vs and vn . There is no friction between these two parts of the liquid. We will not discuss here details of the theory of superfluidity.What is important for us is the relation of this phenomenon with the Bose–Einstein condensation of atoms in the superfluid liquid. It is wellknown that Bose–Einstein condensation is associated with a thermodynamic state in which a finite number of atoms occupy a state with zero momentum, i.e. atoms “condense" in a p = 0 state. Thus, the momentum distribution of atoms contains a deltafunction term: N (p) = (2)3 n0ı (p) + n (p) .

(2.1)

The distribution function N (p) defines the number of atoms in the element of momentum space: dN = N (p)

Vd 3p . (2)3

(2.2)

Thus, the number of atoms in the condensate is N0 = n0V where n0 is the density of the condensate atoms. N0 is proportional to the total number of atoms N and depends on temperature. For an ideal Bose gas n0 (T) can easily be calculated explicitly. For a general system it can be related to properties of the one-body density matrix of the system  (1) (r1 , r2). This matrix can be expressed in terms of the particle, creation and annihilation operators according to  (1) (r1, r2 , t) = ¦ˆ † (r2, t)¦ˆ (r1 , t) .

(2.3)

(Here and below we will consider these operators in the time-dependent Heisenberg representation.) For a uniform system in equilibrium this function de-

pends on the difference of coordinates r = r1 − r2 only and does not depend on time t. The momentum distribution can then be calculated as the Fourier transform of this function:  N (p) =  (1) (r) e−ip·r/ d 3 x . (2.4) The presence of the delta function in equation (2.1) requires that the one-body density matrix does not tend to zero as |r1 − r2 | → ∞, but to a constant value n0 :  (1) (r1 − r2 ) → n0, |r1 − r2 | → ∞ .

(2.5)

This equation expresses the existence of so-called off-diagonal long-range order. This is the symmetry property that distinguishes the superfluid and nonsuperfluid phases. (L.D. Landau 1951 [1], O. Penrose 1951 [2].) Now, at large distances there can be no correlations between ¦ˆ † and ¦ˆ and therefore, the density (1)   have the  property for  (r1 , r2, t) → matrix must ¦ˆ † (r2, t) ¦ˆ (r1 , t) for |r1 − r2 | → ∞. In the equilibrium uniform system  that we  have considered √ † ˆ ˆ n0 so far ¦ (r1, t) = ¦ (r2, t) = const = apart from an arbitrary phase. However, if the liquid moves, or if it is subjected to non-uniform and non-stationary external conditions, the asymptotic expression for the density matrix will now be a function of coordinates and time:  (1) (r1 , r2, t) → ¦0 (r1 , t)¦0∗ (r2 , t) , |r1 − r2 | → ∞ ,

(2.6)

where we ¦0(r, t) ≡   introduced  the notations:  ¦ˆ (r, t) , ¦0∗ (r, t) ≡ ¦ˆ † (r, t) . The function ¦0(r, t) is interpreted as a wave function of the atoms in the condensate.It plays the role of the order parameter associated with the phase transition from the normal to the superfluid state. Its definition implies that the operator ¦ˆ (r, t) can be written in the form ¦ˆ (r, t) = ¦0 (r, t) + ¦ˆ  (r, t) , (2.7)   with ¦ˆ  (r1, t) = 0. One must not, however, take these definitions too literally. The operator ¦ˆ has

2

no diagonal matrix elements with respect to the number of atoms N since it generates a state with N − 1 atoms which, by definition, is orthogonal to the N atom state. Thus, strictly one  speaking,  ˆ  must define ¦0 (r, t) = N − 1 ¦ (r, t) N where the state N − 1 can be obtained from N by adiabatically removing one atom. However, in the thermodynamic limit, where N → ∞, the difference between these two states disappears and one can consider ¦0 as an average value. However, there is a “remnant" of this “off-diagonal" coupling. The Heisenberg matrix element between two stationary states depends on time according

 to N − 1 |. . .| N ∼ exp − it (E (N) − E (N − 1)) . Taking into account that (E (N) − E (N − 1)) ≈ ,where is the chemical potential of the system, one finally gets:       N − 1 ¦ˆ (r, t) N ∝ e −i t/ . (2.8) ¦0 (r, t) is a complex function. Its modulus defines the density of the condensate atoms according to |¦0 (r, t)|2 = n0 (r, t). Thus, we may write  ¦0(r, t) = n0 (r, t)ei (r,t) . (2.9) The phase of the order parameter also has a deep physical meaning. Note first of all that according to (2.8) @

=− . (2.10) @t  It is natural to expect that this equation is valid not only at equilibrium where the chemical potential

is independent of coordinates and time, but also if

is a slowly varying function of r and t. The gradient of is also fundamental; to establish its significance let us calculate the particle current density for the condensate atoms.According to the general equation of quantum mechanics j0 =

 i   ¦0 ∇¦0∗ − ¦0∗ ∇¦0 = n0 ∇ , 2m m

(2.11)

where m is the mass of an atom. Thus, the condensate moves with a velocity vs =

 ∇ . m

(2.12)

Theoretical Foundations

29

Since this motion can exist in a state of thermodynamic equilibrium characterized by the quantity ¦0 , it is non-dissipative; this is the physical origin of the vanishing viscosity associated with the motion of the superfluid. Equation (2.12) therefore defines the velocity of the superfluid motion. We have already taken this into account in our definition of vs . However,to avoid misunderstanding,we must emphasize that, although the velocity of the superfluid part of the liquid is the same as the condensate velocity, the density of the superfluid component s and the density of the condensate mn0 are not the same. This is clearly evident from the fact that at T = 0 all of the liquid is superfluid, but not all its atoms are in the condensate. One may say that the condensate atoms in their motion carry other atoms creating an “effective”superfluid density. Note that the appearance in the theory of a complex order parameter ¦0 with some phase implies a spontaneous breaking of the gauge symmetry in the superfluid state. One must remember, however, that the absolute value of the phase has no physical meaning. One can always add to an arbitrary constant and this constant, which cancels out in (2.6), can be dropped in the definition of ¦0 . Observable physical quantities such as and vs are defined by derivatives of the phase. The difference in the behavior of normal and superfluid liquids is manifested in a very spectacular way in their properties under rotation. An ordinary liquid, placed in a cylindrical vessel rotating about its axes, rotates as whole with the vessel. The velocity field in the liquid is v = § × r, where § is the angular velocity; then curl v =2§. This type of rotation is impossible for a superfluid. According to (2.12), for the superfluid flow one has curl vs = 0 corresponding to potential flow. This equation is one of the most important results of the Landau theory of superfluidity. It says that a superfluid remains at rest even in a rotating vessel. Such a state, however, becomes thermodynamically unfavorable for sufficiently large rates of rotation. The reason is that in a rotating coordinate frame thermodynamic equilibrium corresponds to a minimum of the thermodynamic potential Erot = E − M · §, with E and M being the energy and an-

30

L. Pitaevskii

gular momentum in a laboratory coordinate frame. As a result the state with M · §0 turns out to be energetically more favorable than that with M = 0 for sufficiently large §. The apparent contradiction between this statement and (2.12) is removed by assuming that the potential flow is lost only at certain lines of singularity in the liquid, so-called vortex lines (L. Onsager 1949 [3], R.P. Feynman 1955 [4]). Let us consider the simplest case of a straight vortex line along the axis of a cylindrical vessel. In this case the streamlines are circles in planes perpendicular to the vortex line, with centers lying on the  line. Integrating (2.12) around such a circle gives vs · dl = (/m) ı , where ı is the change in the circulation phase on going around the contour and returning to the original point. Since the wave function ¦0 is single valued, it must be an integer multiple of 2. Thus the circulation velocity is governed by  2 s, (2.13) vs · dl = m where s is an integer. The integral on the l.h.s. of this equation gives 2rvs , where r is the distance from the axis so that  . (2.14) vs = mr Here we set s = 1 since it is not difficult to show that vortices with higher values are thermodynamically unstable. Note that (2.13) is valid for integration on any contour enclosing the line. Since this must hold for arbitrarily small r, it follows that the vorticity, curl vs is concentrated on the axis according to curl vs =

2 (2) sı (r) . m

(2.15)

Here,r is a two-dimensional vector in a plane perpendicular to the line and ı (2) (r) is the two-dimensional delta-function. We emphasize that the existence of such a vortex line as a stable object clearly demonstrates that the viscosity of the superfluid is zero. In the presence of even a small viscosity the vorticity will diffuse from the axis and will eventually spread over the whole volume. The vortex line possesses an energy "v and an angular momentum Mv . The appearance of the line is

favorable if it decreases the energy Erot = E0 + "v − §Mv , i.e. if §§c = "v /Mv . To calculate this critical angular velocity we have to know "v and Mv . The energy is simply the kinetic energy of rotation of the liquid:   1 2 dr "v = s vs2 dV = Ls 2 , (2.16) 2 m r where L is the length of the cylinder. The integration with respect to r has to be taken between the value r ∼ a of the order of an atomic distance,below which this macroscopic theory becomes invalid, and the radius R of the cylinder. Because of the logarithmic divergence of the integral, the result is not sensitive to the precise value of a and so   2 R "v = Ls 2 log . (2.17) m a The angular momentum can be calculated in a similar way:  Mv = s vs rdV   s dV = LR2 s . (2.18) = m m Thus §c =

   R . log 2 mR a

(2.19)

As the angular velocity of the vessel increases further new vortex lines appear and eventually a “rigid body” type of rotation characteristic of an ordinary liquid with curl v =2 § will result. According to (2.15), the mean vorticity is curlvs =  2/m, where  is the number of the vortex lines per unit area. Equating these values we obtain  =

m§ . 

(2.20)

The thermal properties of a uniform body at low temperatures result from elementary excitations or quasiparticles. Every excitation has a definite mo  mentum p and energy ". The function " p characterizes the system. It is important to note that in a superfluid liquid the excitations are viewed as a sort of gas that can move through the rest of the liquid

2

with some velocity vn . Thus at every point in the superfluid there exist two independent motions with velocities vs and vn. The gas of excitations possesses quite normal properties. Excitations can collide with the walls of a capillary and exchange momentum. The flow of the excitations is thus a viscous one and the excitations constitute the “normal part” of the liquid. The mass current, i.e. the momentum density, can be separated into superfluid and normal parts: j =s vs + nvn ,

(2.21)

where s and n correspond to the densities of the normal and superfluid parts. Their sum is, of course, the total density of the liquid  = s + n . One can calculate the normal density using the Galilean invariance. According to this principle the momentum P and energy E of a body in a coordinate frame K  which moves with a velocity V, are related to P and E in a stationary frame K by 1 P = P − MV, E = E − V · P+ MV 2 . 2

(2.22)

By definition, the function "(p) is the dispersion law of the excitations in the frame where the superfluid part is at rest. Let us suppose that there is a single elementary excitation   in the liquid with momentum p and energy " p . In this case the momentum of  the liquid is P = p and the energy E = E0 + " p , where E0 is the ground state energy. Let us now make a transformation to the frame moving with the velocity V = −v s , where the superfluid velocity is vs . According to (2.22) we have P = Mvs + p ,   1 E = E + MV 2 + " p + vs · p . 2

Theoretical Foundations

31

In this frame the superfluid velocity is (vs − vn ) if vs is the velocity in the laboratory frame. Thus, the equilibrium distribution function of the elementary excitations is

   n (p) = n0 " p + (vs − vn ) ·p , (2.25)   −1 where n0 ["] = exp T" − 1 is the Bose distribution function. Note that (2.25) has a meaning only if the argument of the function n0 is positive for all values of p. This results in a condition   " p |vs − vn | min . (2.26) p The l.h.s. of this equation defines the so-called Landau critical velocity. Equation (2.25) is valid for an arbitrary coordinate frame. We now go to the frame where the superfluid part vs = 0. In this frame    is at rest, n (p) = n0 " p − vn ·p and according to (2.21) the momentum density is j = nvn . Thus, the density of the normal part can be calculated as 

   d3p n vn = pn0 " p − vn ·p . (2.27) (2)3 If the velocity vn is small, we can expand the integrand. Integrating over the solid angle we obtain 1 n = − 2 6

∞

dn0 4 p dp . d"

(2.28)

0

This equation is one of the most important results of the Landau’s theory of superfluidity [5].

2.2

Off-Diagonal Long-Range Order in Superconductors

(2.23)

From these equations we see that the momentum of the excitation in the new coordinate frame is still p but the energy is now     (2.24) "˜ p = " p + vs · p . By definition the velocity of the normal part vn is the velocity of the coordinate frame where the gas of excitations is in the thermodynamic equilibrium.

We now turn our attention from Bose superfluids to superconductors. The early theories of superconductivity were phenomenological and we will follow this development. The full microscopic theory of superconductivity was constructed by J. Bardeen, L.N. Cooper and J.R. Schrieffer (BCS) in 1957 [6]. We will present this theory later. General ideas of this theory, however, provide a deep insight into the microscopic basis underlying the phenomenological theories.

32

L. Pitaevskii

According to the BCS theory the essence of the transition into the superconducting state is the formation of bound states, or pairs, involving two electrons with opposite values of their momenta. These pairs behave as Bose particles and exhibit a phenomenon analogous to the Bose–Einstein condensation. This analogy is sufficient to define the nature of the off-diagonal long-range order for superconductors. Since the condensation takes place for the pairs, one must consider the two-particle density matrix for electrons (2)   ˛ˇ ı (r1 , r1 , r2 , r2 , t) =

¦ˆ † (r2 , t)¦ˆ ı† (r2 , t)¦ˆ ˛ (r1, t)¦ˆ ˇ (r1 , t) . (2.29) Here ¦ˆ † (r, t) and ¦ˆ ˛ (r, t) are the electron creation and annihilation operators.These operators are spinors with indices ˛ , ˇ, etc. Let us consider this matrix for the particular values r1 = r1 and r2 = r2 : (2) ˛ˇ ı (r1 , r2 , t) =

¦ˆ † (r2 , t)¦ˆ ı† (r2 , t)¦ˆ ˛ (r1, t)¦ˆ ˇ (r1 , t) . (2.30) The operator ¦ˆ † (r2, t)¦ˆı† (r2 , t) creates an electron pair. The condensate of the pairs involves the behavior of (2.30) for large separations, which we write in the form (2) ∗ ˛ˇ ı (r1 , r2 , t) →   ı (r2 , t)  ˛ˇ (r1 , t) ,

|r1 − r2| → ∞ ,

(2.31)

where we introduced the function  ˛ˇ (r, t) according to      ˛ˇ (r, t) = N − 2 ¦ˆ ˛ (r, t)¦ˆ ˇ (r, t) N ,      ∗ ı (r, t) = N ¦ˆ † (r, t)¦ˆ ı† (r, t) N − 2 . (2.32) The state |N − 2 in (2.32) is the state obtained by adiabatically removing two electrons from the initial state |N of N electrons. In the thermodynamic limit N → ∞ the difference between these two states is negligible and one can consider the matrix element 2

  as an average value. However, for stationary conditions this function has the time dependence corresponding to the transition N → N − 2  ˛ˇ (r, t) ∝ e−2i t/ ,

(2.33)

where is the chemical potential of the electrons. In this chapter we consider only this stationary case. The function  ˛ˇ has the physical meaning of a wave function of the superconducting pairs. The spinor structure of  ˛ˇ is defined by the value of the spin of the pairs. For the majority of superconductors the bound states form with spin zero. The wave function of a particle with zero spin is proportional to an antisymmetric unit spinor tensor   01 I˛ˇ = . (2.34) −1 0 In this case  ˛ˇ (r, t) = I˛ˇ   (r, t) .

(2.35)

The complex function   (r, t) is an order parameter of which the appearance distinguishes the superconducting state of the metal from the normal state.2 Note that as in the case of superfluidity, the existence of the complex order parameter can be interpreted as spontaneous breaking of the gauge invariance. The function   can be represented in the form   (r, t) = |  (r)| ei (r,t). The gradient of the phase (r, t) defines the velocity of the superconductive pairs  vs = ∇ , (2.36) 2m where m is the mass of an electron. (Compare (2.12). Here we have taken into account the fact that the mass of a pair is 2m . This equation, however, cannot be correct in the presence of a magnetic field, because it is not invariant with respect to a gauge transformation. We write the magnetic induction as B = curl A ,

(2.37)

where A is the vector potential. We will restrict ourselves to the case of a time-independent magnetic field. In this case gauge invariance states that all observable physical quantities must remain unchanged

This definition of the order parameter is due to L.P. Gor’kov [7].

2

by a transformation of the vector potential of the form A → A + ∇ , (2.38) where (r) is an arbitrary function of the coordinates. The ¦ -operators of electrons must be transformed accordingly as   ¦ˆ ˛ → ¦ˆ ˛ exp ie /c ,   (2.39) ¦ˆ ˛† → ¦ˆ ˛† exp −ie /c . Then, according to (2.32), the order parameter transforms as     →   exp i2e /c . (2.40) Thus,the order parameter is transformed in the same way as the wave function of a particle of charge 2e. One can rewrite this equation in the form of the phase transformation: → + 2e /c .

(2.41)

Equation (2.36) is not invariant under such a transformation. To restore the required invariance, one must include a further term containing the vector potential   2e  vs = ∇ − A . (2.42) 2m c It is easy to check that this equation is invariant under the transformations (2.38) and (2.41). The superconducting current density, which is a directly observable physical quantity, is proportional to the velocity vs . In this chapter we will consider the simplest case of superconductors with cubic crystal symmetry. In this case the direction of the current coincides with that for the velocity and thus js = ens vs . The quantity ns defined in this way is called the number density of superconducting electrons. This density, of course, depends on temperature and disappears at the transition point. Using a microscopic theory we will later show that near this point ns is proportional to | |2 . Finally one gets for the superconducting current density:   2e e js = ns ∇ − A . (2.43) 2m c This is a fundamental equation for the phenomenological description of superconductors.

Theoretical Foundations

33

It is worth noting that js represents only part of the total current density. In a superconductor a normal current jn can also flow, which results in the production of Joule heat and dissipation of energy. The total current density is j = js + jn . However, in the absence of the external electromagnetic forces the normal current will eventually decay. In a state of thermodynamic equilibrium only a superconducting current can exist. The time derivative of the phase defines, according to (2.33), the chemical potential of the electrons @ 2

=− @t  (B.D. Josephson 1962 [8]).

2.3

(2.44)

The London Equation

The fundamental problem of how the magnetic field is distributed inside a superconductor can be solved using (2.43). Let us consider a uniform superconductor in the state of thermodynamic equilibrium. In this case there is no normal current and js = j. Applying the curl operator to both sides of (2.43) and using (2.37), we obtain the London equation e 2 ns B (2.45) mc (F. London and H. London 1935 [9]). Using this equation we can calculate the current distribution resulting from a known distribution of the magnetic induction. In this sense for superconductors it plays an analogous role to that of Ohm’s law j =  E, for normal metals, where the latter defines the current density due to an electric field E.To obtain the full system of equations we must incorporate the Maxwell equations 4 curl B = j (2.46) c and div B = 0 . (2.47) curl j = −

Let us substitute the current density j from (2.46) into (2.45). Taking into account that curl curl B = grad div B − B = −B, we obtain B =

1 B, ı2

(2.48)

34

L. Pitaevskii

where

mc 2 ı = . (2.49) 4e2ns Equations (2.47) and (2.48) determine the magnetic field in a superconductor. These equations are sufficient to explain a fundamental property of a superconductor: the Meissner effect. For simplicity, let us consider a superconductor occupying the half-space x0. Then the problem is a one-dimensional one and the induction B depends only on the coordinate x. From (2.47) one gets dBx /dx = 0 and (2.48) gives Bx = 0; i.e.the vector B is parallel to the surface. Equation (2.48) now becomes 2

d2 B B = 2. 2 dx ı

(2.50)

B = 0 inside the bulk of the body. One then says that a superconductor is an ideal diamagnetic, i.e. a body with magnetic permeability = 0. Then, the usual equation B = H gives B = 0 for any H. Such a formal description is convenient, because it permits us to consider a superconductor as the particular case of ordinary magnetism. In an external magnetic field a superconductor acquires a magnetic moment. Let us calculate this moment for the important example of a long cylinder. Let H0 be the external field parallel to its axis. From the condition that the tangential component of H is continuous on the surface it follows that H = H0 both outside and inside the cylinder. It then follows that the magnetization (magnetic moment per unit volume) defined by

From the two solutions of this equation one must choose the one which decreases at x → ∞:

m = (B − H) /4 ,

B (x) = H0 e−x/ı ,

is −H0 /4 inside the body . Finally, we have for the total magnetic moment of the cylinder

(2.51)

where the vector H0 is parallel to the surface. Outside the superconductor in the half-space x < 0 one has B = H = H0 . We see that a magnetic field decreases exponentially into the superconductor, penetrating a distance of the order of ı. This distance is called the London penetration depth. A simple estimate shows that this distance is a macroscopical one that is much larger than the mean distance r between electrons in the metal. Indeed, we always have ns ≤ n ∼ r −3 . Then it follows from (2.49) that ı 2 ≥ (r/4r0 ) r 2 , where r0 = e2/mc 2 is the “classical radius” of an electron. Thus, ı  r, since r  r0. Still the distance ı is small compared to the typical dimensions of samples (ı ∼ 10−6 − 10−5 cm). The decrease of the magnetic induction described by (2.51) is a result of a “screening” of the external field by the superconducting currents. From (2.45) it follows that these currents flow within a surface layer of thickness ı. Note that for a bulk superconductor with characteristic dimensions Rı (2.51) defines the field distribution near the surface independent of the shape of the body. In this case, the axis x must be directed along the internal normal to the surface. Thus,the magnetic field does not penetrate deeply into a superconductor. The magnetic induction is

M = −V

H0 , 4

(2.52)

(2.53)

where V is its volume. This magnetic moment in the field H0 can be calculated also by differentiation of the energy of the system M = −@E/@H0 .Hence,a long cylinder in a field H0 has an additional magnetic energy H2 Emag = V 0 . (2.54) 8 This energy is positive. The presence of a superconductor in a magnetic field increases the energy. In contrast the energy of a normal metal is practically independent of the field, because its permeability

≈ 1. It is clear that in a sufficiently strong magnetic field the superconducting state becomes thermodynamically less favorable than the normal state and at some critical field Hc superconductivity will be destroyed.We will discuss this in detail in the next section. The above derivation of the London equation is subject to some restrictions. In particular, (2.43) for the superconducting current is valid only under some conditions. First of all, the magnetic field must be weak compared to the critical field Hc . Further,

2

the equation is valid only if all quantities, the current density for example, vary sufficiently slowly in space. Microscopic theory shows that the characteristic distances over which such quantities vary must be large in comparison to the so-called correlation or coherence length 0 ∼ vF /Tc ,

(2.55)

where vF is a characteristic Fermi velocity of the electrons in the metal (see Sects. 2.6 and 3.3.3 below).According to (2.51) the magnetic induction,and consequently the current, change on the scale of the penetration depth. Thus, we obtain the condition ı  0 .

(2.56)

This condition is always fulfilled near the transition point. The density of superconductive electrons ns → 0 as T → Tc and hence ı increases indefinitely. Whether the condition (2.56) is fulfilled far from Tc depends on specific properties of the given superconductor. Superconductors in which the inequality (2.56) is fulfilled are called London superconductors. A phenomenological description of superconductors in the opposite limit ı  0 was suggested by A.B. Pippard in 1953 [10]. In this case one speaks of Pippard superconductors. Note that the existence of the inequality (2.56) does not affect the proof that B = 0 inside a superconductor. The assumption that the field is not expelled would result in a contradiction. In conclusion we consider the remarkable magnetic-flux quantization effect in a multiplyconnected superconductor (F. London, 1950 [11]). This effect is also a consequence of (2.43) for the superconducting current. Let us consider a superconducting torus with dimensions that are large compared with both ı and 0. A very important property of such a torus is that the magnetic flux through it cannot be changed without the destruction of superconductivity. Indeed, any change of ¥ would induce an electromotive force E according to the equation of the electromagnetic induction E = − 1c d¥ . In the dt absence of resistivity this, however, would result in an infinite current. This is impossible and therefore d¥ /dt = 0. The flux remains constant when the external field changes or when the shape of the torus

Theoretical Foundations

35

is altered. It is an adiabatic invariant of the system. Such invariants are subject to quantization in quantum theory. Consider a closed contour inside the torus encircling its aperture and passing within the torus far from the surface. Then the magnetic induction B = curl A and the current density j = (c/4) curl B are equal to zero on the contour. The vector potential, however, is not zero. Equation (2.43) takes the form: A=

c ∇ . 2e

(2.57)

Integrating along the contour we have  c A · dl = ı , 2e

(2.58)

where ı is the change in the phase of the wave function of the pairs on passing around the contour.Since this function must be single-valued, this change can only be an integer multiple of 2. On the other hand, the circulation of the vector A is equal to the magnetic flux  through a surface spanning the contour, i.e. the flux through the torus:    A · dl = curl A·df = B·df ≡  . (2.59) Finally we have  = s0

,

0 =

s = 0, ±1, ±2 .

c , |e| (2.60)

The quantity 0 is the quantum of magnetic flux. It numerical value is 2 × 10−7 G · cm2 . It is important to note that the factor 2 in (2.57) is due to the fact that the charge of a pair is 2e.Thus an experimental check of the quantization rule (2.60) gives a direct prove of the existence of the electron pairing.

2.4

Thermodynamics of Superconductors in a Magnetic Field

In this section we will discuss effects of an external magnetic field on the phase transition between the normal and superconducting states. We will consider a long cylinder in a longitudinal external field

36

L. Pitaevskii

H0 . More complex behavior in a magnetic field of superconductors having an arbitrary shape will be considered in the next section. We have already mentioned that in a sufficiently high magnetic field the superconducting state becomes unfavorable and so the magnetic field penetrates into the cylinder and superconductivity is destroyed. However, there are two kinds of behavior with respect to this field permeation, depending on the properties of the superconductor. Superconductors of the first kind undergo a sharp transition when the magnetic field reaches the critical field Hc and the field penetrates throughout the cylinder destroying the superconductivity. In the superconductors of the second kind the magnetic field penetrates only at the surface. We will consider peculiar properties of such superconductors in Sect. 2.8. In the absence of a magnetic field the phase transition between the normal and superconducting state is a second order phase transition; i.e., the normal phase becomes completely unstable below the critical temperature Tc , which, of course, depends on the pressure P, and the superconducting phase is unstable above Tc . Abrupt penetration of the magnetic field in a first kind superconductor implies that the phase transition in the field is a first order phase transition. In this case both the superconducting and normal phases can be characterized for some interval of the field around Tc by their thermodynamic potentials ¥s and ¥n . The phase with lower value of ¥ is stable at a given H0 .The second one is metastable. Superconductivity is a relatively weak phenomenon in the sense that its characteristic energy Tc is small compared to the Fermi energy of the electrons "F . Therefore, the magnetic energy (2.54) is a small correction to the total energy of the superconductor. In this case one can easily write an equation for ¥s . Taking into account that a magnetic field does not penetrate into the superconductor and that small corrections to different thermodynamics potentials are the same if expressed in terms of the proper thermodynamic variables, we have H2 ¥s (T, P, H0) = ¥s0 (T, P) + Vs0 (T, P) 0 , (2.61) 8 where ¥s0 (T, P) and Vs0 (T, P) are the thermodynamic potential and the volume of the superconduc-

tive cylinder in the absence of the magnetic field. On the other hand, the magnetic field penetrates into the cylinder in the normal state. However, the thermodynamic potential in this state does not depend on the field if one neglects the small paramagnetism or diamagnetism of a normal metal. Hence, ¥n = ¥n (T, P). At the critical field; i.e. on the transition line, the thermodynamic potentials of the normal and superconducting phases are equal: ¥n (T, P) = ¥s0 (T, P) + Vs0 (T, P)

Hc2 . 8

(2.62)

Here, Hc (T, P) is the critical field expressed in terms of temperature and pressure. Tc (P) is the transition temperature in the absence of a magnetic field; i.e. Hc (Tc , P) = 0. At T → Tc the critical field tends to zero. The corresponding law will be established below (see (2.73)). Since small corrections to all thermodynamic potentials are equal if expressed in corresponding variables, one can rewrite (2.62) also for the free energy Fn (T, V ) = Fs0 (T, V ) + V

Hc2 (T, V ) . 8

(2.63)

The function Hc (T, P) defines the fundamental properties of the first order transition – the latent heat and the change of the volume at the transition point. To calculate the latent heat note that by defini n tion the entropy in the normal state is Sn = − @¥ @T P and in the superconducting state   @¥s Ss = − @T P,H0     H02 @¥s0 @Vs0 . (2.64) =− − @T P @T P 8 On the other hand, differentiation of (2.62) with respect to temperature gives       @¥n Hc2 @¥s0 @Vs0 , = + @T P @T P @T P 8   @ Hc2 + Vs0 . (2.65) @T 8 P Using the expression for Sn and (2.64) (taken at H0 = Hc ) we get the change in the entropy at the phase transition

2

Ss − Sn =

V s Hc 4



@Hc @T

 .

(2.66)

P

Correspondingly the latent heat of the transition Q = T (Sn − Ss ) is equal to (W.H. Keesom, 1924)   Vs Hc T @Hc Q=− . (2.67) 4 @T P The  @Hc  critical field decreases with temperature, 0 so Q0. Thus, heat is absorbed at the transi@T P tion from the superconducting to the normal phase. Note that in the absence of a magnetic field (at T = Tc ), the r.h.s. of (2.66) and (2.67) vanish with Hc as they must, because the transition becomes second order. On the other hand, according to Nernst’s theorem both Ss and Snare zero at T = 0. Then it c follows from (2.66) that @H @T P → 0 as T → 0. Similarly one can calculate the change of volume of the cylinder at the phase transition. We recall the following thermodynamic identities   @¥n Vn = , @P T   @¥s Vs = @P T     H02 @¥s0 @Vs0 . (2.68) = + @P T @P T 8 Differentiating (2.62) with respect to the pressure gives       @¥n Hc2 @¥s0 @Vs0 = + @P T @P T @P T 8  2 @ Hc + Vs0 . (2.69) @P 8 T Putting H0 = Hc in (2.68) and combining it with (2.69) we have   Vs Hc @Hc . (2.70) Vs − Vn = − 4 @P P To find the change of the specific heat at the transition line, one must differentiate (2.66) with respect the temperature. Taking into account that CP = T (@S/@T)P and neglecting the term with (@Vs /@T)P ,

Theoretical Foundations

37

which is usually very small, we get the change in the specific heat at constant pressure as   TVs @Hc 2 Cs − Cn = 4 @T P   TVs Hc @2Hc + . (2.71) 4 @T 2 P In the absence of the magnetic field (T = Tc , Hc = 0) we find the change in specific heat at the secondorder transition   Tc Vs @Hc 2 Cs − Cn = (2.72) 4 @T P (A.J. Rutgers, 1933). It may be noted that in the Landau mean field theory for the second-order phase transition the specific heat has a finite discontinuity at the transition line. Equation (2.72) then implies that the derivative (@Hc /@T)P is finite, i.e. that the critical field tends to zero as Hc = constant× (Tc − T)

(2.73)

(we will discuss this problem in Sect. 2.6).

2.5

The Intermediate State of Superconductors

In the previous section we assumed that the superconducting body placed in the magnetic field is a cylinder with the external magnetic field H0 directed along its axis. In this section we discuss the general case of a superconductor of any shape. However, the qualitatively important features of this general case can be understood by studying a superconducting ellipsoid. In this case the magnetic field H inside the superconductor is uniform. Let the external field be parallel to one of the ellipsoid axes.The vectors B and H then have the same direction and one can write nB + (1 − n)H = H0 ,

(2.74)

where n is the so-called demagnetizing factor. This factor lies in the range 0 ≤ n ≤ 1. For a sphere n = 1/3. Since B = 0 inside the body, one has H=

H0 . 1−n

(2.75)

38

L. Pitaevskii

The field outside the ellipsoid is not uniform.We will not present here the corresponding equations. It is enough to note that in the equatorial plane of the ellipsoid the field H has the same value as in (2.75). This follows immediately from the fact that the field both at internal and external surfaces is tangential and that the tangential components of H are continuous. Obviously the magnetic field is greatest on the equator and this value is larger than H0 . When the external field reaches the value H0 = (1 − n) Hc , the field at the surface reaches Hc . When H0 increases further, the sample cannot remain in a pure (homogeneous) superconducting state. However, it cannot pass as a whole into the normal state, because then the field H0 would be less than Hc everywhere. At first sight one might assume that the superconductivity will be initially lost in a small “belt” near the equator, the size of which would gradually increase with increasing the field. A simple analysis shows, however, that such a picture is not self-consistent. Creation of the normal belt would decrease the field near the equator and such a belt cannot exist. This paradoxical situation has the following solution. When the field reaches the value Hc on the surface, the body is divided into thin parallel alternating normal and superconducting layers (L.D.Landau, 1937 [12]). For the case of an ellipsoid all of the body is in this intermediate state for external fields in the range (1 − n)Hc H0 Hc . (2.76) In view of this one can build a simple phenomenological theory for describing the intermediate state (R.E. Peierls, 1936 [13]; F.London, 1936 [14]). The crucial idea of this theory is to introduce a magnetic induction B averaged over distances that are large compared to the layer thickness and a corresponding “field” H. Our goal is to establish relations between these quantities. It is obvious from the symmetry considerations that these vectors are parallel to the direction of the external field and that the layers are also parallel to this direction. Note first of all that the equation H = Hc must be fulfilled on the boundaries of the normal layers. Indeed, for such a condition the thermodynamic potentials of the normal and superconducting phases are equal and any displacement of the

boundary does not change the thermodynamic potential; i.e. the surface is in neutral equilibrium with respect to this displacement. If H = Hc one of the phases is energetically more favorable than another. Then the boundary would move in the direction of the less favorable phase. Since we assumed that the layers are thin, one gets for the magnetic field B = H = Hc everywhere in the normal layers. On the contrary, in the superconducting layers one has B = 0. This means that the average induction is B = xHc , (2.77) where x is the fraction of the normal phase, i.e. the fraction of the volume that is in the normal state. In addition, H = Hc in the superconducting layers due to the boundary conditions for the tangential components. Hence, H = Hc . Combining these equations with (2.74), we have for the magnetic induction and the normal phase concentration: H0 1 − n − Hc . B = xHc = (2.78) n n It follows from this equation that the averaged magnetic induction depends linearly on the external magnetic field in the interval (2.76). Thus B = 0 at H0 = (1 − n)Hc and B = Hc at H0 = Hc . Analogous phenomena take place for a body of non-ellipsoidal shape. However, in this case the body contains regions in both the pure superconducting and pure normal states separated by regions in the intermediate state.

2.6

The Ginzburg–Landau Theory

In this section we present the Ginzburg–Landau (GL) theory of superconductivity (1950) [15]. This theory gives a quantitative description of superconductors near the transition point. This was the first theory to properly take into account the quantum nature of superconductivity and it has been used for the solution of numerous problems. It was constructed before the microscopic theory of superconductivity. When the microscopic theory was created it was shown that the Ginzburg–Landau equations can be derived from this theory. This derivation yielded a

2

physical interpretation of the basic quantities entering the Ginzburg–Landau theory. In particular, the microscopic derivation allows a calculation of the coefficients that enter in the GL theory. The Ginzburg–Landau theory is based on general ideas associated with the Landau theory of secondorder phase transitions. The crucial point of this theory is the expansion of the free energy of the system in powers of an “order parameter” that describes the difference in symmetry of the two phases. This order parameter differs from zero below the transition point and vanishes above. One assumes that the parameter is small near Tc and changes slowly in space. For the case of a superconductor the order parameter is given by the complex wave function of the superconducting pairs  . The free energy of the system cannot depend on the phase of the function  , i.e. it must be invariant under the transformation   →  eiˇ where ˇ is a constant. The expansion of the free energy in terms of   and its gradients can be written in the form B f = fn + d |∇ |2 + A | |2 + | |4 , (2.79) 2 where the phenomenological coefficients d, A and B are functions on the temperature and the density of the body. The coefficients d and B must be positive. However, the theory has a more natural form if one introduces a new quantity that is proportional to   in a such way that the gradient term in (2.79) corresponds to the quantum mechanical kinetic energy  2 2  of a particle of mass 2m, i.e. 4m ∇  , where m is 1/2   . the mass of electron. Therefore, = 4m/2d  2 n s We also introduce the notation   = 2 . We will see below that for this definition of the quantity ns is just the density of superconducting electrons as  2 it was introduced in (2.43) in Sect. 2.2.   itself is then the density of superconducting pairs. Finally, in the absence of a magnetic field, one gets for the free energy of a superconductor: F = Fn    2  2 b  4   2 ∇ + + a   +   dV . (2.80) 4m 2 Here, Fn is the free energy at energy of the normal state.

= 0, i.e. Fn is the free

Theoretical Foundations

39

Let us consider a uniform superconductor of volume V . Then  2 bV  4   . (2.81) F = Fn + aV   + 2 The equilibrium value of the order parameter can be found by minimization of F. The function can be chosen as real. The equation @F/@ = 0 has two solutions: = 0 and 2 = −a/b. The first corresponds to the normal state, the second to the superconducting state. At a given temperature the stable phase is one where the free energy has a minimum. One can easily check that @2 F/@ 2 = 2aV for the normal and @2 F/@ 2 = −6aV for the superconducting phases. Respectively,the solution corresponds to a minimum if this derivative is positive. Therefore the coefficient a must be positive for TTc and negative for TTc . In the Landau theory of phase transitions it is assumed that the coefficient a can be expanded near the point Tc in integral powers of (T − Tc ). We therefore write a = ˛ (T − Tc ) ,

(2.82)

where ˛ is a positive function and |T − Tc |  Tc . Thus for TTc one gets  2 2˛ ns = 2   = (2.83) (Tc − T) . b Substituting this into (2.81), we find a2 ˛2 = Fn − V (Tc − T)2 . (2.84) 2b 2b Near the transition the second term on the r.h.s. of (2.84) is small. Thus the thermodynamic potential can be written as ˛2 (2.85) ¥s = ¥n − V (Tc − T)2 . 2b A comparison   of this expression with (2.62) shows that a2 /2b = Hc2 /8 and 1/2  4˛ 2 Hc = (2.86) (Tc − T) . b F s = Fn − V

Of course this equation is in accordance with (2.73). Differentiating both sides of (2.84) twice, we find the discontinuity of the specific heat: Cs − Cn = VTc

˛2 . b

(2.87)

40

L. Pitaevskii

In accordance with the general theory of phase transitions, the specific heat of the less symmetric, i.e. superconducting, phase is higher. The following remark should be made in connection with (2.80). The presence of the gradient term in the equation means that the Ginzburg– Landau theory contains a characteristic parameter, (t), with the dimensions of length. Let us consider a non-uniform distribution where the function (r) changes with distance. Such non-uniformity will significantly change the energy if the gradient term  2  2 (1/4m) ∇  is of the order of a   . Thus, it is natural to define the length  as 1/2  Tc  (T) ∼ 0  0 . (2.88) Tc − T This quantity is called the correlation or healing length. It defines a typical scale for the change of the wave function.We will see that the ratio / (m˛Tc )1/2 is of the order of 0 ∼ vF /Tc , see microscopic expressions for the coefficients of the Ginzburg– Landau equations. Thus, in the range of applicability of the Ginzburg–Landau theory 1/2  Tc  (T) ∼ 0  0 . (2.89) Tc − T Equation (2.89) is valid near the transition point. However, it must give the correct order of magnitude for T → 0. Thus, the quantity 0 has the physical meaning of the correlation length at zero temperature. Let us now consider the behaviour in the presence of a magnetic field. The density of magnetic energy B2 /8 must be added in the integrand (2.80). But this is insufficient in that the gradient term in (2.80) is not invariant with respect to the gauge transformation (2.38)-(2.41) . To the invari restore 2   ance, one must substitute for ∇ the combina 2 tion [∇ − i (2e/c) A]  ,which is obviously gaugeinvariant. The final expression for the free energy then takes the form:   2   2e 2   ∇ − i F = Fn + A   4m c   2 b  4 B2 dV . (2.90) + a  +   + 2 8

Here, the magnetic induction must be expressed as B = curl A. One can obtain the basic equations of the Landau–Ginzburg theory by varying this functional with respect to A and ∗ . Carrying out first the variation with respect to A, we find after a simple calculation:    ie  ∗ ıF = c ∇ − ∇ ∗ 2m  curl B 2e2  2 ıAdV A+ + m 4  dV + div (ıA × B) = 0. (2.91) 4 The second integral can be transformed into an integral over a remote surface and disappears. To minimize the free energy, the expression in the brackets must be equal to zero. This results in the Maxwell equation c curl B = j, (2.92) 4 provided that the current density is given by  2e2  2 ie    A . (2.93) ∇ ∗ − ∗∇ + 2m mc According to the of ns we can substitute  definition  √ = ns /2 exp i . Then (2.93) becomes   2e e ns ∇ − A . (2.94) j= 2m c j=

Equation (2.94) coincides with (2.43). This justifies  2 our identification of 2   with ns .Variation of (2.90) with respect ∗ gives, after a simple integration by parts,    2   2e 2 ıF = − ∇ −i A 4m c   2   +a +b ı ∗ dV +

2 4m

  ∇ −i

2e A c

 ı

∗

· dS = 0 . (2.95)

The second integration is over the surface of the sample. The volume integral vanishes when    2 2 2e 2 − ∇ −i A +a +b  = 0 . (2.96) 4m c

2

Equations (2.92) and (2.96) form the complete system of equations of the Ginzburg–Landau (GL) theory. It is interesting to note that in the original paper these equations were written with the electric charge e instead of 2e.The reason is that at that time the phenomenon of the pairing of electrons was not known. The correct version of the equations was established by L.P. Gor’kov (1959). The surface integral in (2.95) yields the boundary condition   2e ∇ −i A ·n=0 (2.97) c on the surface of the superconductor, where n is the vector normal to the surface. This condition guarantees the absence of a supercurrent through the surface. It is worth noting that (2.97) does not imply that = 0 on the surface as it would be natural to assume for a wave function. Actually decreases to zero within atomic distances at the surface. Such distances, however, cannot be considered on the basis of the GL theory. The induction B does not need any special boundary conditions. Equations (2.92), (2.93) are valid in the entire space. (Outside the body, of course, j = 0.) Thus the vector B is continuous at the surface. The GL equations are non-linear with the result that the distribution of the field in the superconductor depends on its strength. Because of this the density of superconducting electrons in (2.94) depends on position even in a uniform superconductor. However, this effect is small if the field is weak; i.e. B  Hc . In this case one can assume that ns is constant and is given by (2.83). Then (2.94) is equivalent to the London equation (2.45) and the penetration of the magnetic field is described by (2.51), where the penetration depth is given by:  ı=

mc 2 b 8e2 ˛ (Tc − T)

1/2 .

(2.98)

Note that this quantity has the same temperature dependence as the coherence length . However the physical meaning of these parameters ı and  is different. The parameter  defines the scale of the spatial change of the wave function , while ı defines

Theoretical Foundations

41

the scale over which the magnetic field changes. The ratio of these two lengths, , does not depend on temperature: =

ı mcb1/2 = .  (2)1/2  |e|

(2.99)

Using (2.86) one can also express  in terms of the directly observable quantities: √ |e|  = 2 2Hc ı 2 . (2.100) c The Ginzburg–Landau parameter  depends on the properties of the material and characterizes the superconductor. We will see in the next section that the behavior of superconductors in a magnetic field with small and large values of  is completely different. Actually in the framework of the GL theory  is the only parameter necessary to describe a superconductor. This can be seen after transforming the equations to dimensionless variables. It is convenient to introduce these variables as  r A @ b , r= , A= ,∇ = , = |a| ı Hc ı @r B B = rot A = , (2.101) Hc with rot A = ∇ × A. Substituting these variables into (2.96) and (2.92) yields   2  2  + 2 − +   = 0 (2.102) −i∇ − A and rot rot A =

i ∇ ∗− 2  2 −    A.

∗

∇ (2.103)

As said, the values of the parameter  differ widely among superconductors. The majority of the “good”, or rather pure metals have a relatively small  ∼ 0.01 to 0.2. For superconducting alloys large values of  are typical. The new HTC superconductors are very anisotropic substances. Correspondingly the values for  depend on direction and can range from 50 to 500. In conclusion, let us consider briefly the conditions for the applicability of the GL theory.As we have

42

L. Pitaevskii

already mentioned these equations are valid only near the transition, i.e. for (Tc − T)  Tc . However, this is insufficient for superconductors with small . The point is that the penetration depth ı must be large compared to the correlation length 0 (see (2.55)–(2.56)). It is not difficult to show that this condition can be written as (Tc − T)   2 Tc .

(2.104)

The GL equations are also not valid too close to Tc , because the validity of the Landau mean-field theory of phase transitions breaks down in this region due to fluctuations of the order parameter. The associated theory gives the criterion (2.80) (Tc − T) 

b2 Tc2 m3 . 6 ˛

(2.105)

We will estimate the coefficients ˛ and b using the microscopical theory in Sect. 3.3.3. Substitution of the corresponding expressions in (2.105) shows that the r.h.s. of this inequality is very small for usual superconductors. However, this condition may be important for the HTC materials.

2.7

Surface Energy at the Boundary Between Normal and Superconducting Phases

As we showed in Sect. 2.4, a superconductor in the intermediate state is divided into alternating layers of normal and superconducting phases. The boundaries between these regions involve a surface energy. In this section we will calculate this energy with the help of the Ginzburg–Landau theory. This energy is an important characteristic of a superconductor and a knowledge of it is necessary for developing a full theory for the intermediate state. A positive value of this surface energy is a necessary condition of the stability of this state. However, we will see that this is not the case for sufficiently large values of the GL parameter . This result plays an important role in understanding type II superconductors. To calculate the surface energy one must consider the structure of the transition layer between the normal and superconducting regions. In Sect. 2.4 this layer was considered as infinitesimally thin. Because

the thickness of this layer is much less than the distances between layers, it is sufficient to consider a single boundary separating normal and superconducting half-spaces. Let us take the axis x perpendicular to the layers and directed into the superconducting phase.All quantities under consideration will depend on the x coordinate only; i.e. the problem is one-dimensional. We can assume that the induction vector B is directed along the z-axis. Then the vectorpotential A can by chosen along the y-axis and B=

dA . dx

(2.106)

We must solve the GL equations (2.92) and (2.96) for this one-dimensional problem subject to the boundary conditions for the s-n boundary established in Sect. 2.4: x → −∞, x → ∞,

→ 0, B → Hc , → (|a| /b)1/2, B → 0 . (2.107)

Note that (2.106) defines A only up to an arbitrary constant. We will fix this constant by the condition A → 0 at x → ∞. Before proceeding with the calculations one must formulate the exact thermodynamic definition of the surface energy. This is a delicate problem. The main point is to use the correct thermodynamic potential. As a first step let us define the magnetic field strength H at every point of the body. To do this let us introduce the magnetization m by curl m = j/c. (This is possible because in the stationary case div j = 0.) Then H = B − 4m and (2.92) takes the form curl H = 0 .

(2.108)

In our one-dimensional problem this means that dH/dx = 0 and H = const. The boundary conditions yield H = Hc (2.109) everywhere. Equation (2.109) is our condition for thermodynamic equilibrium together with the condition of constant temperature. Our problem is analogous to the condition of constant chemical potential

at the liquid–gas phase transition. The thermodynamic potential must be written with respect to the variables (T, H). The condition (2.109) can then

2

be imposed explicitly. (Note that in analogy to the definition of the surface tension at the liquid–gas boundary one must use the thermodynamic potential § with respect to the variables T, ; see, for example [16]). The free energy density f is the thermodynamic potential with respect to the variables T, B: df = −SdT + H · dB/4 ,

(2.110)

(see [17]). We suppressed the gradient term here. We introduce a new thermodynamic potential H.B  f =f − 4

d f = −SdT − B.dH/4 .

(2.112)

Thus,  f (T, H) is the desired thermodynamic potential with respect to the variables T, H. Adding the gradient term we can write an expression for  f . Prior to doing this we introduce some simplifications. The point is that for the particular symmetry of our problem A.∇ = ∇.A = 0. As a result, the imaginary terms drop out of the GL equations and can be chosen as real. Taking this into account we get 

2 4m



4e2 2 A 2 c 2  b 4 B2 Hc B − + + 2 8 4

 f (T, Hc ) = fn +

2

+



  f (x) −  f∞ dx or ∞  2   sn = 4m −∞

b + 2

 2

+a

2

(2.113)

(the prime denoting differentiation with respect to x). Using (2.86) and the boundary conditions (2.107) it is not difficult to show that the function  f tends to the same constant value when x → ∞ and x → −∞: a2 H2  f → f∞ = f n − = fn − c , 2b 8 x → ±∞ .

(2.114)

This constant term gives a contribution proportional to the volume. For calculating the surface energy one has to subtract this contribution. Finally, the surface energy (or the surface tension) of the boundary between the normal and superconducting phases is

4

2

−∞

+

2

4e2 + 2 2 A2 c

43

 2

+a

2

 B2 Hc B Hc2 − + dx . (2.115) + 8 4 8

This can be written as ∞  Hc2 ı 2 sn = 8 2

(2.111)

from which we obtain

Theoretical Foundations

+

4

2

  2 + A −2

 2   dx . + A −1

2

(2.116)

In the rest of this section we will omit the overbars, because only dimensionless quantities will be used. The GL equation (2.102) and (2.103) for our onedimensional problem take the form:    1 2  2 3 A −1 =0 (2.117) − + 2 and

A − A 3 = 0 . The boundary conditions (2.107) are = 0, B = A = 1, for x → −∞ , = 1, A = 0, for x → ∞ .

(2.118) and

Note that the boundary condition A = 0 at x → ∞, which we imposed previously, follows from (2.118) and (2.119). This means that this condition and our restriction that is real are actually not independent. Equations (2.117) and (2.118) give   2 2 + A2 − A2 − 2 2 − 4 = constant . (2.119) 2  This can be verified directly. One can also use a mechanical analogy. Equations (2.117) and (2.118) can  x2 be derived by minimization of 2the2 integral Ldx with the “Lagrangian” L = (2/ ) + A2 +  x12  A − 2 2 + 4 . Expression (2.119) is then the “energy”, which is obtained from L by changing the sign of the “potential energy”. It follows from the boundary conditions that this energy must equal unity. Thus, we have   2 2 + A2 − A2 − 2 2 − 4 = 1 . (2.120) 2 

44

L. Pitaevskii

The expression (2.116) for the surface tension can be presented in different forms as H 2ı sn = c 4 H 2ı = c 8

∞  −∞ ∞

2 2





2







+A A −1 2

A −1 −

4



 

dx .

dx

(2.121)

−∞

The first form is obtained with the help of (2.120), while the second follows from integrating the term 2 by parts and substituting  from (2.117). Calculating sn requires numerical integration of the equations. An analytical result can be obtained in the limiting case 1 when ı (T)  (T). In this case the magnetic field penetrates only slightly into the superconducting phase. (It is not difficult to show that the penetration depth √ in this non-uniform problem is of the order 1/ .3) The wave function is small in this region and gives only a small contribution to sn . The main contribution arises from the region where changes rapidly, which is of the order of 1/. The situation is illustrated in Fig. 2.1.

 = √ (1 − 2 ) . 2 This equation has a simple solution  √  = tanh x/ 2 . 

(2.122)

(2.123)

We can now calculate the surface tension. It is convenient to use the second expression (2.121). However the integration must be taken from 0, because (2.123) was derived for distances x ∼ 1/1, which give the main contribution,and has no meaning for negative x. Then, √ 2 ∞  2Hc ı  ns = 1 − tanh 4 x dx 8 0 √ 4 2 Hc2 . (2.124) = 3 8 It already follows from (2.124) that ns decreases as  increases. For sufficiently large  the surface tension becomes negative. This follows from the first expression (2.121). At large  one can neglect the first term in the integrand. (It is important that the transition region, where changes significantly, cannot be thinner than the penetration depth for the magnetic field, since any change in B causes a change in .) Then the integral is negative because A = B is positive and less than 1. In terms of unscaled quantities this means that BHc . Actually sn becomes zero for 1 = √ . (2.125) 2 This result was established By V.Ginzburg and L.Landau by means of a numerical calculation.Here we will present an analytical derivation. Let us introduce the notations r = A +

Fig. 2.1. Distribution of the magnetic field and order parameter in the transition layer between the normal and superconducting phases

There is no magnetic field in this region and one can put A = 0 in (2.120). Solving this equation for  , we have 3

2

− 1, s = 2



+A

2

.

(2.126)

Then the second expression (2.121) can be rewritten as ∞ 

   Hc2ı sn = A − 1 − 2 rdx . (2.127) 8 −∞

2 2 Let us consider the distance x  1/ and side of (2.117) and its √  A  1. Then one can neglect the right-hand solution matched to (2.123) is = x/ 2. Substituting this into (2.118), we find A =  2 x 2 A/2. It follows from this equation that the characteristic length of variation of A is of the order of 1/ 1/2 .

2

Correspondingly we can also rewrite (2.117) and (2.118) as   2 2 − 4 + s 2 − 2As + r 2 2   + 2 1 − 2 r = 0 (2.128) and r − s = 0 .

(2.129)

It is now obvious that for  = 1/2 both (2.128) and (2.129) are fulfilled if 2

r = s = 0.

(2.130)

Furthermore, these equations are compatible with the boundary conditions (2.119), because r and s are, according to these conditions, equal to zero both in the normal and superconducting phases. Finally, we have sn = 0 by virtue of (2.127).

2.8

Superconductors of the Second Kind

Results of the previous section explain the existence of two kinds of superconductors: superconductors of the first and second kinds or type I and type II. Superconductors of the first kind have a positive surface energy sn . Superconductors with sn < 0 are superconductors of the second√kind. Therefore, near Tc superconductors √ with 1/ 2 are of the first kind and those with 1/ 2 are of the second kind. The main difference between these two types of superconductors is in the character of the phase transition. A sharp boundary between two phases is possible only if the surface tension of the interface is positive. Thus, our discussion of the phase transition in previous sections relates only to type I superconductors. We now consider the phase transition in superconductors of the second kind. As in Sect. 2.4 we consider a cylindrical superconductor in a longitudinal magnetic field.For a superconductor of the first kind superconductivity is destroyed when the external magnetic field H0 reaches the critical value Hc , which is defined by the thermodynamic properties of the material. For H0Hc the sample will be entirely

Theoretical Foundations

45

in the normal state. For a sample with negative surface tension, however, it is favorable to have a “mixture”of normal and superconductive phases for magnetic fields around Hc . The loss in the volume energy can be compensated by a gain in surface energy. The magnetic field penetrates the superconductor over a range of fields Hc1H0 Hc2. The thermodynamic field Hc must lie inside this interval. However, it has no special significance for the phase transition of superconductors of the second kind. A cylindrical superconductor in this range of fields is said to be in the mixed state. The nature of this state was established by A.A. Abrikosov [18]. We will discuss thus in the next sections. However, the value of the upper critical field Hc2 can be calculated even without the detailed theory of the mixed state (V.L. Ginzburg, L.D. Landau, 1950 [15]). It is enough to note that just below Hc2 the value of in the superconducting regions must be small. Therefore this region can be described by a linearized Ginzburg– Landau equation. For the same reason we can also neglect perturbations of the external field. Omitting  2 the non-linear (  ) term in (2.96) and assuming that the vector potential corresponds to the uniform external field H0 , we have  2 2e 1 −i − A0 = |a| . (2.131) 4m c We assume that TTc , so that a0. We seek a solution where → 0 at infinity. Equation (2.131) coincides with the Schr¨odinger equation for a particle of mass 2m and charge 2e in a magnetic field H0. The boundary conditions are also the same.The quantity |a| plays the role of the energy. The minimum energy of a such particle in a uniform magnetic field is 1 1  |e| H0 !B = . (2.132) 2 2 mc Hence, equation (2.131) has a solution only if |a| |e| B0 /2mc or equivalently if the magnetic field is less than an upper critical field "0 =

Hc2 =

0 2mc |a| = .  |e| 2 2

(2.133)

Using (2.86) and (2.99) one can rewrite this equation in the form

46

L. Pitaevskii

Fig. 2.2. Phase diagrams of superconductors of the first (a) and second (b) kinds

√ Hc2 =  2Hc

(2.134)

(A.A. Abrikosov, 1952 [21]). Equation (2.134) confirms the above conclusion that √ for superconductors of the second kind, when  2, a superconducting order parameter can appear in a normal sample at fields larger than Hc . Equation (2.134) defines the upper√critical field for type II superconductors where 1/ 2. However, this equation also has a meaning for type I superconductors when Hc2Hc . Here Hc2 defines the “boundary for supercooling” of the normal state. For H0Hc2 the thermodynamically unfavorable normal state becomes absolutely unstable because of the possibility of creating a superconducting nucleus.In the interval Hc2 H0 Hc the normal phase is metastable. In conclusion, we note that the condition for the appearance of superconducting regions is different near the surface of a superconductor. Here one has to use the boundary condition d /dx = 0, where the x-axis is a normal to the surface. In this case (2.131) gives H0 Hc3 = 1.7Hc2 = 2.4Hc (2.135) (D. Saint–James and P.G. de Gennes, 1963 [19]). For fields Hc2H0 Hc3 one has a state with surface superconductivity. The density of superconducting electrons is different from zero for a surface layer with a thickness of the order of  (T). The case where Hc2 Hc Hc3 corresponds to a special kind of superconductor. The mixed phase does not exist in this

case, but there is surface superconductivity for the interval Hc B0 Hc3. The phase diagrams for the superconductors of the first and the second kinds are shown schematically in Fig. 2.2(a) and (b), respectively.

2.9

Quantized Vortex Lines

In this section we will consider the structure of the mixed state. (For more details see [20].) The main problem is to understand how the magnetic field can penetrate into the bulk of the superconductor. Let us again consider a superconducting cylinder in a magnetic field H0 . It is natural to expect that the normal regions, with their accompanying magnetic field, are cylindrical tubes parallel to the field. To obtain the maximum gain in the (negative) surface energy, the number of these tubes must be as large as possible. However, there is a restriction. The magnetic flux inside such a tube must be an integral multiple n of the flux quantum 0 = c/ |e| introduced in Sect. 2.3. As we will see later the total gain in surface energy is largest for n = 1, which results in the largest number of tubes. The proof follows immediately by applying the arguments of this section to each tube. This is obvious when there is no overlapping of the magnetic tubes, i.e. when their number is small enough, a condition which in any case will apply near the lower critical field Hc1.

2

Thus the magnetic tube must possess a minimal flux 0 . The magnetic field is concentrated inside the tube.At large distances from the tube it is shielded by the annular superconducting current flowing around the tube. This current is the analog of the superfluid velocity field surrounding the vortex lines in a superfluid liquid which we discussed in Sect. 2.1. We can then picture the mixed state as an array of quantized vortex lines. Such vortex lines were predicted by A.A. Abrikosov in 1957. Their existence is crucial for explaining the properties of type II superconductors. Our first task is to calculate the lower critical field Hc1 for our superconducting cylinder. This field can be found from the condition that the penetration of a single vortex line becomes thermodynamically favorable. Again the problem is to choose the correct thermodynamical potential. The situation here is analogous to the one in Sect. 2.7. We must again define the magnetic strength H = B − 4M at every point of the cylinder. Because of the symmetry of the problem, H is directed along the axis of the cylinder and on the boundary one has continuity of the tangential components H = H0 . The Maxwell equation curl H = 0 will be satisfied if H = H0

(2.136)

everywhere. As in Sect. 2.7 this means that we must consider the thermodynamic equilibrium at fixed H (as well as temperature and volume), i.e. use the thermodynamic potential  1 F˜ = F − H · BdV , (2.137) 4 where F is the free energy. Taking into account equation (2.136), we can rewrite (2.137) as  H0 BdV . (2.138) F˜ = F − 4 There are two contributions in (2.138) in the presence of a vortex line. The magnetic induction inside the tube gives the magnetic energy in the external  field −H0 BdV /4 = −H0 0 L/4. On the other hand, the presence of a vortex line increases the free energy of the superconducting media, F = Fs + "L, where " is the energy per unit of length. Thus, F˜ = Fs + "L − 0 H0 L/4 .

(2.139)

Theoretical Foundations

47

The presence of the vortex line is thermodynamically favorable if the contribution is negative; i.e. if "L − 0 H0L/40, or H0 Hc1 =

4" . 0

(2.140)

This is a general equation which defines the lower critical field Hc1 at arbitrary temperatures. Near the transition temperature Tc one can calculate the energy " using the Ginzburg–Landau theory. In this approximation one must numerically integrate the set of equations (2.92)–(2.96). However, the problem can be solved analytically in the important case where ı

(2.141)

(near Tc this means   1).In this case one has a natural separation of the scales over which the quantities entering the problem vary. The coherence length  determines the distance over which the order parameter varies from zero on the axis of the vortex to its constant bulk value deep in the superconducting phase. Thus, at distances r   the density of superconducting electrons is equal to its bulk value ns . On the contrary, the magnetic induction B (r) varies over the larger distances ı  . Thus most of the magnetic flux passes through the region where ns ∼ = const. This is important for the calculation of the energy ". In this region we can use the London equation. We begin the calculation with (2.43). The behavior of the magnetic field and the order parameter near the core of the vortex is shown schematically in the Fig. 2.3. Substituting js from the Maxwell equation, we can rewrite (2.43) as: A + ı 2curl B = 0 ∇ /2 .

(2.142)

The phase in the presence of a vortex line is not a single-valued function of the coordinates. For a vortex line with the minimum flux 0 , the phase increases by 2 on traversing a closed contour that encloses the line. (Actually for a single line the order parameter is proportional to ei' , where ' is the angle in cylindrical coordinates.) Thus the integral along such a contour is  ∇ · dl =2 . (2.143)

48

L. Pitaevskii

We can now calculate the energy " of a vortex line. The magnetic part of the free energy corresponding to the London equations is given by the integral 

2  1 FB = B + ı 2 (curl B)2 dV . (2.148) 8

Fig. 2.3. Distribution of the magnetic field and order parameter near the core of the vortex line in a superconductor of the second kind

Integrating (2.142) we find    A + ı 2 curl B · dl =0 .

(2.144)

We will see below that for the calculation of the energy, relatively small distances from the axis are important. It is not difficult to check that in the range ır

(2.145)

the second term on the l.h.s. of (2.144) gives the main contribution.We take as the contour of integration in (2.144) a circle of radius r. For this geometry the vector curl B has only one component (curl B)' along the contour. The integration is then simple and we have 0 dB = . (2.146) (curl B)' = − dr 2rı 2 Equation (2.146) has a very simple physical meaning. According to the Maxwell equation we have (curl B)' = 4c ens vs' . Equation (2.146) then gives vs' = /2mr for the superfluid velocity as it must be for a vortex line in a superfluid of particles with mass 2m. Integrating of (2.146) for the induction gives 0 ı B (r) = (2.147) log . 2ı 2 r This equation is valid in the interval (2.145) with logarithmic accuracy. A more exact expression will be obtained in the next section.

Indeed, by varying the expression with respect to B we immediately obtain the London equation (2.45). The coefficient follows from the condition that for ı = 0 (2.148) coincides with the energy of the magnetic field in vacuum. The main contribution to the integral is due to the second term, which contains a logarithmic divergence. Substituting (2.146) into (2.148) and integrating in the range (2.145), we obtain for the energy per unit length of the vortex line     0 2 ı "= . (2.149) log 4ı  This expression also has only logarithmic accuracy, i.e. the constant under the logarithm is not defined. Equation (2.149) explains why only vortex lines with the minimum flux 0 are most favorable. The energy of a line is proportional to the square of its magnetic flux. Thus the fragmentation of one line with the flux n0 into n lines with flux 0 results in an n-fold gain in energy. Substituting (2.149) in (2.140), we find the lower critical field   0 ı Hc1 = . (2.150) log 4ı 2  It is worth noting that accordingto (2.147) near the axis, for r ∼ , the induction is B  ≈ 2Hc1. Near Tc in the Ginzburg–Landau region this equation can be expressed in terms of the parameter  log  Hc1 = Hc √ . (2.151)  2 A comparison of (2.151) and (2.134) shows that for 1 the following inequalities are satisfied: Hc1  Hc  Hc2 .

(2.152)

Numerical solution of the Ginzburg–Landau equations allows us to calculate the coefficient under the logarithm. The result is: Hc1 = Hc

log (1.08) √ .  2

(2.153)

2

Note that the more exact equation (2.153) is still only valid for large . For an arbitrary value of  the lower critical field near Tc can be expressed as Hc1 =√ Hc h (). The function is h () = 1 for  = 1/ 2. For this critical value of  one has Hc1 = Hc2 = Hc . As the external field increases, the number of the vortex lines also increases. These lines form a twodimensional lattice in the plane perpendicular to the axis of the cylinder. The density of the vortex lines and the symmetry of the lattice depend on the interaction between lines, which will be considered in the next section. Here we will present some qualitative considerations. If distances between the vortex lines are large compared to the radius of the cylinder, one can describe the superconductor in the mixed state in an averaged manner introducing the mean value of the induction B¯ over the cross-section of the cylinder. Correspondingly one can define the magnetization M = (B¯ − H)/4 = (B¯ − H0 )/4. The magnetization is related to H0 by the equation @f˜ /@H0 = −M. Let us integrate this equation over H0 from 0 to Hc2.We find Hc2   MdH0 = − fn − fs0 ,

(2.154)

0

where fn is the free energy density of the normal metal and fs0 is the free energy density of the superconductor in absence of the field. Both quantities do not depend on the field and one need not distinguish between f and f˜ . In the field range 0H0 Hc1 , the field does not penetrate into the cylinder and the magnetization is simply M = −H0 /4. Then,   Hc2  2 + Hc1 /8. According to Hc1 MdH0 = − fn − fs0 equation (2.63), the quantity in the parenthesis is equal to Hc2 /8. Thus, we finally obtain the important relation Hc2  1  2 Hc1 − Hc2 . MdH0 = (2.155) 8 Hc1

Theoretical Foundations

49

where  is the number of lines per unit area. This result is invalid near Hc2 where the cores of the vortex lines begin to overlap.

2.10

Vortex–Vortex Interactions

In this section we will examine the relation between the mean induction in a type II superconductor and the external field. According to (2.156) the induction is defined by the number of vortex lines per unit area. To calculate this number we have to take into account the interaction between the vortex lines. As a first step we have to find the magnetic induction through a loop of arbitrary radius surrounding the line without the restriction (2.145). Let us calculate the curl of both sides of (2.142). Note that curl ∇ = 2nz ı (r) ,

(2.157)

where r is the two-dimensional radius-vector in the x, y plane and nz is a unit vector along the axis z. (We assume that the axis of the vortex line coincides with the z-axis.) Indeed, integrating ∇ along the contour encircling the line and transforming the integral by Stokes’ theorem into an integral over a surface spanning the contour, we have according to (2.143)   ∇ · dl = curl ∇ · dS = 2 . (2.158) Since this equation must be satisfied for any such contour of integration, we have (2.157). Finally we obtain B + ı 2 curl curl B = nz 0 ı (r) .

(2.159)

In the presence of several vortex lines the r.h.s. of this equation contains a sum of ı-functions corresponding to the positions of the lines. Using the vector identity curl curl B = ∇div B − B = −B, we obtain B − ı 2 B = 0 ı (r) .

(2.160)

This equation is valid at all distances

Notice also that every vortex carries the flux 0 , and hence the mean value of the induction over the crosssection of the cylinder is B¯ =  0 ,

(2.156)

r .

(2.161)

Throughout all the space except the line r = 0 equation (2.160) coincides with the London equation (2.45). The ı-function on the r.h.s. defines

50

L. Pitaevskii

the character of the singularity of the solution at r → 0. Actually this singularity has already been defined in the equation (2.147), which is valid at small r. The solution of this equation at r → ∞ is B (r) = const K0 (r/ı), where K0 is the Hankel function of an imaginary argument. The coefficient must be defined by matching with the solution (2.147). Note that B(r) is finite.  Using the asymptotic formula K0 (x) ≈ log 2/ x for x  1, where = eC ≈ 1.78 (C is Euler’s constant), we finally have B (r) =

02 2ı 2

K0 (r/ı) .

(2.162)

Using equation (2.162) we can rewrite (2.147) as B (r) =

2ı 0 , log 2ı 2

r

r ı.

(2.163)

In the opposite limit of large distances one can use the asymptotic expression K0 (x) ≈ (/2x)1/2 e−x for x  1. Thus, at large distances from the axis of the vortex line the induction decreases according to B (r) =

0 (8rı 3 )1/2

e−r/ı ,

r ı.

(2.164)

Accordingly the superconductive current density decreases as j' = −

c0 c dB = e−r/ı . 4 dr 8 (2 3 rı 5)1/2

(2.165)

There is an important difference with respect to the vortex line in superfluid helium. In the latter case the superfluid velocity follows the 1/r law for arbitrarily large distances. This difference is due to the charged nature of the electron liquid. The motion of electrons creates a magnetic field, which in turn screens the current. Let us apply the results obtained to the calculation of the energy of interaction of vortex lines. It is important that equation (2.160), which defines the distributionof the induction, is a linear one. It means that under the condition (2.161) the fields produced by different vortex lines are additive. Let us consider two vortex lines placed at r1 and r2 separated by a distance d from each other. Then, B = B1 + B2 . The energy of the lines is given by (2.148). Let us transform

the first term in the integrand by means of (2.159) which gives B2 + ı 2 (curl B)2

 = ı 2 −B·curl curl B + (curl B)2 + 0 Bz (r) [ı (r − r1 ) + ı (r − r2)] . (2.166) The first term on the r.h.s. can be transformed into the form −B·curl curl B + (curl B)2 = div (B×curl B) .

(2.167)

The volume integration of this term in (2.148) can be reduced to an integration over a remote surface. This integral disappears, because of the fast decrease of the field. We need not worry here about singularities of the field; they are taken into account properly by the ı-functional terms in (2.166). Because we are interested here in the energy of interaction of the lines, we must take into account only the “mixed terms” of the type B2z (r) ı (r − r1 ). (Terms like B1z (r) ı (r − r1 ) contribute to the self-energy of the vortex lines (2.149).) Now the integration in (2.148) is trivial. We have for the interaction energy L"int =

L0 (B2 (r1) + B1 (r2 )) . 8

(2.168)

Both terms on the right contribute equally and using (2.162) we have   02 0 d B (d) = . (2.169) K "int (d) = 0 4 8 2 ı 2 ı Now we can define the number  of the vortex line per unit area in the thermodynamic equilibrium. When the interaction between the vortex lines is taken into account, this equilibrium corresponds to an ordered configuration of the lines, forming a twodimensional lattice in the plane perpendicular to the cylinder axis.It is impossible to predict the symmetry of this lattice from general considerations.One has to compare the thermodynamic potentials for different configurations and choose the most favorable. For isotropic superconductors the most favorable lattice is formed from equilateral triangles with the vortex lines at their vertices. We will consider only this case.

2

The expression for the thermodynamic potential per unit volume f˜ can be written in the form   1 0 H 0 f˜ = fs +  " − 4 1 + "int (|ri − rk |) , (2.170) 2 i,k

where the summation of the interaction energies is over all filaments passing through a unit area. One can express the energy " in this equation according to (2.140): " = 0 Hc1/4.For distances d  ı,as will be the case well below the upper critical field Hc2, it is sufficient to consider only pairs of neighboring lines. One can also use the asymptotic expression for "int :  1/2 2 ı "int = 7/2 03/2 2 e−r/ı , r  ı (2.171) 2  ı d (see (2.164)). For the triangular lattice each vortex has six nearest neighbors. Then, 1 "int (|ri − rk |) = 3 "int (d) , (2.172) 2 i,k

where d is the length  a side. The area of an equilat√ of 3/4 d 2 and the number of lines is eral triangle is a half of the number of triangles in the lattice. Hence, 2 √ . (2.173) 3 With the help of (2.172) and (2.173) we can write (2.170) in the form   H0 − Hc1 30 e−l , + f˜ (l) = fs + A − l2 23/2 1/2ı ı 2 l5/2 (2.174) where we introduced a dimensionless parameter  √  l = d/ı  1 and A = 0 / 2 3ı 2 . Minimizing f˜ (l) with respect to l yields H0:  =

d2

30 l1/2 e−l , (2.175) 25/2 1/2ı ı 2 where we omitted terms of higher order in 1/l. By means of the equation B¯ =  0 , one can express l in terms of B¯  1/2 20 . (2.176) l= √ 3ı 2B¯ H0 − Hc1 =

Theoretical Foundations

51

Equations (2.175)–(2.176) define the “magnetization curve”, i.e. the function B¯ (H0 ) for a superconductor of the second kind. Notice that this function has an infinite derivative near Hc1. It is not difficult to check that 1 dB¯ 1 ∝ log−3 , dH0 H0 − Hc1 H0 − Hc1

(2.177)

as H0 → Hc1 . Results (2.175) and (2.176) are valid for a small density of the vortex lines, i.e. d  ı. The problem can also be solved if d ∼ ı  . The only difference is that we must use the general equation (2.169) for the interaction energy and not restrict the summation in (2.170) to nearest neighbors. The situation becomes completely different near the upper critical field Hc2.The disappearance of the superconductivity at H0 → Hc2 takes place as a phase transition of the second order.Accordingly the square of the modulus  2 of the order parameter   ∝ (Hc2 − H0). Similarly, the superconducting current and the magnetization M are also proportional to (Hc2 − H0 ): −M =

H0 − B¯ ∝ Hc2 − H0 . 4

(2.178)

Let us estimate the mean distance between the vortex lines. For H0 ∼ Hc2 the mean induction B¯ ∼ Hc2. It then follows from (2.78) and (2.133) that  ∼ 1/ 2 and d ∼ , which means that the cores of vortex lines overlap.The linear equation (2.159) is no longer valid. However, (2.156) is still valid. Let us consider a closed contour near the surface of the cylinder. The change of the wave function on passing round the contour is 2 S, where S is the cross–section area of the cylinder. One obtains from (2.94) that the magnetic flux is  = 0  S −

2m e



j · dl . ns

(2.179)

It is possible to show using the symmetries of the vortex lattices that one can choose a contour sufficiently near the surface so that the integral in (2.179) is zero. Thus, we obtain (2.156) as an approximate equation.

52

L. Pitaevskii

2.11

Cooper Pairing

(2) Vnm =

We now present the general ideas and methods of the microscopic theory of superconductivity developed by J. Bardeen, L.N. Cooper and J.R. Schrieffer.4 As we have already mentioned,the main point of this theory is the formation of bound states or pairs of electrons due to their interaction. Obviously, to create a bound state the interaction must be attractive, which raises the problem of the origin of such an attraction. Let us consider electron–electron scattering in a metal. We will consider two electrons with opposite values of the momenta, because just such electrons create the pairs. Let p and −p be the initial values of the momenta of the electrons and p and −p the final values . For simplicity, we will consider an isotropic model of the metal. The electron–electron interaction in metals has two different contributions. One is the Coulomb repulsion between electrons screened on the scale of the Debye radius rD , which is of the order of an interatomic distance in good metals. The corresponding potential is e2e−r/rD / r and its Fourier component, describing electron–electron scattering, is Uq =

4e2 , q2 + (rD )−2 

where q = p− p is the change of the momentum of an electron and is the dielectric constant of the lattice. The repulsive nature of this interaction is expressed by the positive sign of this Fourier component. The second contribution to the interaction is the electron–phonon interaction. The Hamiltonian of this interaction describes emission and absorption of a phonon by an electron. Electron–electron interaction arises in second-order perturbation theory, which describes processes of emission and absorption of virtual phonons. We will show that these processes result in an effective attraction between electrons (J. Bardeen, 1950 [24], H. Fr¨ohlich (1950) [25].) The general equation for the scattering amplitude in the second Born approximation is 4

(2.181)

where m and n are, correspondingly, the initial and final states and the sum is taken over all intermediate states i. In our case the initial state contains two electrons and has energy 2" (p).The final state has energy 2" (p ). There are two types of intermediate states allowed by momentum conservation. In the first of these,the electron with momentum p emits a phonon of momentum q and acquires the momentum p− q = p; the energy of this state is " (p) + " (p) + !q , where !q is the frequency of the phonon with wave vector q. In the second state, the electron with momentum −p emits a phonon with momentum −q and acquires a momentum −p−q = −p . Because " (−p ) = " (p), this state has the same energy. Then we can use the fact that matrix elements for emission and absorption of a phonon are complex-conjugates   and depend only on q. As a result, the contribution of the virtual phonons to the scattering amplitude is Vp(2)  = 1p 1

(2.180)

1 Vni Vim 2 i   1 1 , × + En − Ei Em − Ei

1   2 M q 2   1 1 , (2.182) × + ı" − !q −ı" − !q

where ı" = " (p) − " (p) is the energy transfer. The total scattering amplitude is   2 Vp1 p1 = Uq + M q 

!q  2 . (ı")2 − !q

(2.183)

The phonon contribution is negative when |ı"| !q . This corresponds to attraction. If the Coulomb part Uq is not too large,the total interaction will be attractive. Note that !q is in any case less than the Debye frequency !D . In fact, the phonons with frequencies !q ∼ !D are most important. Qualitatively, attraction takes place when |ı"| !D . These considerations explain the origin of the attractive interaction. However, there is a problem. The

Here we give a concise introduction to this very rich subject. A more comprehensive presentation can be found in the excellent books [22] and [23].

2

smallness of the transition temperatures shows that this attraction is weak. According to a well-known theorem of quantum mechanics there is no bound state of two particles if the interaction is small. This problem was solved in the paper by Cooper in 1956 [26]. He showed that although this theorem is valid for free particles, it does not apply to the particles of a degenerated Fermi gas. The difference is due to the existence of the Fermi sphere. The formation of a bound state of two particles in vacuum for a weak interaction is prevented by the smallness of the density of states of these particles at small energy ". This density of states ∼ " 1/2 . The situation in the Fermi gas at T = 0 is different. Let us consider two particles excited from the Fermi sphere with opposite momenta p1 = −p2 . Their total energy E must be larger than 2"F because of the Pauli principle. If as a result of the interaction the energy E becomes less than 2"F the particles will be in a bound state. The difference (2"F − E) is the binding energy. Thus in the case of the Fermi gas, the boundary that separates bound states from states of the continuous spectrum is 2"F . However, the density of states at 2"F is not necessarily small and the above quantum mechanical theorem for particles in vacuum does not apply. Let us consider an ideal Fermi gas. It is convenient to introduce excitation energies. An excitation in such a gas can be produced by transfer of a particle from the Fermi surface to a state of momentum ppF . Thus, energy will be  the excitation     p2 p2F " p = 2m − 2m ≈ vF p − pF for ppF . Here we assumed that this energy is small. An excitation with ppF can be created by transferring a particle from a state with momentum p inside the Fermi sphere to the surface of the Fermi sphere. Then,     p2F   p2 " p = 2m − 2m ≈ vF pF − p for ppF . From this point of view the gas has the elementary excitation spectrum     " (p) = vF p − pF  ≡ p  , (2.184) where we introduced    a useful notation p = vF p − pF = pF p − pF /m.Let us suppose now that the particles interact by means of a weak attractive potential U (|r1 − r2|). Then the binding energy will

Theoretical Foundations

53

be small and the wave function will spread well beyond the range of the potential. One may take for the potential U (|r1 − r2 |) = −gı (r1 − r2 ) , g0 .

(2.185)

The energy of the bound state of two excitations can be found from the Schr¨odinger equation

     " pˆ 1 + " pˆ 2 − gı (r1 − r2 ) ¦ (r1 , r2) = (E − 2"F ) ¦ (r1, r2 ) .

(2.186)

Here we took into account that the excitations energies are defined with respect to the Fermi energy. We seek a bound state with zero total momentum. Then the wave function must be an invariant with respect to translations in space and can be written in the form ¦ (r1, r2 ) = ¦ (r1 − r2 ) . (2.187) Let us transform (2.186) to the momentum representation by multiplying both sides with eip·r/ and integrating over r = r1 − r2. We find 

2 [" (p) + ] ¦p = g ,

(2.188)

where ¦p = eip·r/ ¦ (r) d3 x . Here, we introduced 2 for the binding energy: E − 2"F = −2. One can rewrite (2.188) in the form ¦p = g¦0/2 [" (p) + ] .

(2.189)

We integrate both sides of this equation with respect to the momentum. Taking into account that  d3p ¦0 = ¦p (2.190) (2)3 and integrating with respect to angles, we find the following equation for   g p2 dp = 1. (2.191) 4 2 3 " (p) +  The integral (2.191) diverges. However, this divergence is non-physical. It is related to the substitution of the ı-function for the real potential. The divergence can be eliminated by a renormalization of the coupling constant g (which can be expressed in terms of the scattering amplitude of particles). We will not discuss this procedure. Instead we note that in our

54

L. Pitaevskii

problem the attractive interaction is due to the exchange of phonons that is possible only for particles sufficiently near the Fermi surface, " (p) ≤ !D  "F ; here !D is the maximum (Debye) frequency of phonons. Thus, the main contribution to the integral (2.191) involves the interval     vF p − pF   !D . (2.192) One can substitute p2F for the factor p2 in the integrand anduse (2.184) for " (p). Cutting off the inte gration at p − pF  ∼ !D /vF , in place of (2.191) we have gmpF log (!D /) = 1 . (2.193) 2 2 3 Finally we obtain    = !D exp −2/gF , (2.194) where we introduced the notation F = mpF / 2 3 .

(2.195)

The quantity F is the number of states per unit energy (the density of states) of a particle on the Fermi surface which is given by 2

d 3p (2)

3

=

p2 dp mp = d" = F d" .  2 3  2 3

(2.196)

The factor 2 is due to the twofold spin degeneracy. Equation (2.194) clearly shows that the Cooper pairing phenomenon depends on the finite value of the density of states near the Fermi surface. To simplify the notation, we did not write the spin indices of the wave function in the previous equations. A solution of (2.188) automatically corresponds to a pair with spin equal to zero, i.e. a singlet state. Indeed, the isotropic wave function (2.189) has the orbital angular momentum equal to zero and all states of two identical fermions with even values of the angular momentum are singlets.Note also that it is important that we considered particles with opposite momenta. One can easily check that the binding energy tends to zero quickly when the total momentum increases. Note that for a particle in two-dimensional space the density of states is constant at small energy and a bound state can also be formed for an arbitrary weak

attractive potential. The energy of the state is exponentially small with respect to the potential strength similar to (2.194). Thus, we have shown that an attraction between particles must lead to the formation of bound pairs, regardless of how weak the attraction is. When the formation of pairs is energetically favorable, the state of non-paired particles is no longer a ground-state of the system. It must be rebuilt using pairs. To excite this new ground-state one has to break a pair. This requires the energy 2, i.e. a gap in the excitation spectrum appears.Finally,the pairs in the rebuilt ground state behave like Bose particles and can accumulate on the lowest energy level in an analogy with the phenomenon of Bose–Einstein condensation.We have already used this analogy. In the next sections we will present a quantitative theory of superconductivity where these properties will be demonstrated explicitly.

2.12

Energy Spectrum of a Superconductor

We now undertake a study of the theory of superconductivity based on the Cooper pairing phenomenon. This theory was developed by J. Bardeen, L. Cooper and J.R. Schrieffer and is referred to as the “BCS theory” [6]. Here we will present a method for deriving the pairing which is due to N.N. Bogoliubov (1958) [27]. A similar method was also developed by J.G. Valatin (1958) [28]. The first step is to choose a proper model. The model must be sufficiently simple to allow an analytical solution and must take into account essential features of the Cooper pairing. We will consider electrons in a metal as a Fermi gas with a weak ıfunctional attraction (2.185) between particles. In the second quantization representation the Hamiltonian of the system can be written as   2  † ˆ H= − (2.197) ¦ˆ (r) ¦ˆ ˛ (r) 2m ˛  g − ¦ˆ ˛† (r) ¦ˆ ˇ† (r) ¦ˆ ˇ (r) ¦ˆ ˛ (r) d3 x , 2 where g0 and the ¦ˆ -operator can be represented as an expansion

2

1  aˆ p u˛ ( ) eip·r . ¦ˆ ˛ (r) = √ V p,

(2.198)

Also, the operators obey the anticommutation rules aˆ p aˆ p   + aˆ p   aˆ p = 0 , (2.200)

Substituting (2.198) into (2.198) transforms the Hamiltonian into the form Hˆ =

 p2 aˆ †p aˆ p 2m p, g  † † − aˆ p + aˆ p − aˆ p2 − aˆ p1 + , 1 2 V 



55

p aˆ †p aˆ p

p,

Here, aˆ p is the annihilation operator for a Fermi particle with momentum p and spin projection  = ±1/2, and u˛ ( ) is the corresponding spinor amplitude. These amplitudes satisfy the completeness condition   u˛ ( ) u∗˛   = ı   . (2.199)

aˆ p aˆ †p   + aˆ †p   aˆ p = ı   ıpp .

Hˆ  =

Theoretical Foundations

(2.201)

p1 p2 p1

where p2 = p1 + p2 − p1 and the suffixes + and − replace the spin variables +1/2 and −1/2. Note that the products aˆ †p 1 aˆ †p 2 with 1 = 2 are canceled in 1 2 the second sum in (2.201) due to the anticommutation relations (2.200). Physically, the Bose condensation analogy is related to the fact that the ı-function potential acts only between pairs of particles in an l = 0 state. Such pairs have total spin equal zero, thus 1 = −2 . The model of a superconductor based on this Hamiltonian is known as the BCS model. The next simplifying step is based on the crucial role of the Cooper pairing. Taking into account that this pairing exists only for particles with opposite values of the momentum, we will retain in the second sum in (2.201) only the terms with p1 = −p2 ≡ p and p1 = −p2 ≡ p .We will also use the“grand canonical” ˆ where is the chemiHamiltonian Hˆ  = Hˆ − N,  cal potential of the gas, and Nˆ = ˆ †p aˆ p the p, a particle number operator. This permits us to relax the condition requiring the conservation of the number of particles. We can substitute = "F = p2F /2m with sufficient accuracy. Then we obtain the reduced Hamiltonian

−

g  † † aˆ  aˆ  aˆ −p− aˆ p+ , V  p + −p −

(2.202)

pp

where, as in (2.184), p =

  p2 − ≈ vF p − pF . 2m

(2.203)

As we explained in the previous section, the Cooper instability changes the spectrum of elementary excitations. An initially surprising result is that the annihilation and creation operators aˆ , aˆ † are no longer represent elementary excitations (even approximately). We now denote the operators that destroy and create excitations by bˆ and bˆ † and assume they are related to the destruction and creation operators for particles aˆ and aˆ † by the following (Bogoliubov) transformation: aˆ p+ = up bˆ p+ + vp bˆ †−p− , aˆ p− = up bˆ p− + vp bˆ †−p+ ,

(2.204)

where up and vp can be chosen as real functions.(N.N. Bogoliubov used an analogous transformation in his ˆ bˆ † theory of dilute Bose gas [29].) The operators b, must obey the same anticommutation rules (2.200). As a result, the parameters u and v must be normalized according to u2p + vp2 = 1 .

(2.205)

The ground state of the system |0 is defined as the state where there are no elementary excitations: bˆ p |0 = 0|bˆ †p = 0 .

(2.206)

For an excited state the average of the operator bˆ †p bˆ p defines the average number of excitations: bˆ †p bˆ p  = np ,

(2.207)

where here and below . . . means averaging with respect to the grand canonical ensemble with the Hamiltonian H  . The commutation relations then give bˆ p bˆ †p  = 1 − np . (2.208)

56

L. Pitaevskii

Let us express the Hamiltonian H  in terms of the ˆ Substituting (2.204) into quasiparticle operators b. (2.201) and using the anticommutation relations, we obtain  H = 2 p vp2 p

+

 p

+2

   p u2p − vp2 bˆ †p+ bˆ p+ + bˆ †p− bˆ p−



  p up vp bˆ †p+ bˆ −p− + bˆ †p− bˆ p+

p

g ˆ, − Dˆ † D V

(2.209)

where we introduced the notation  ˆ = u2 bˆ −p− bˆ p+ − v2 bˆ † bˆ † D p

p

(2.210)   + up vp 1 − bˆ †p+ bˆ p+ − bˆ †−p1 bˆ −p− . p p+ −p−

Let us calculate the average energy of the system. There is no problem to average terms that are quadratic in the b-operators. The diagonal matrix element are non-zero only for the products (2.207) and (2.208). In calculating the fourth-order term we will ˆ i.e.we will subneglect fluctuations of the quantity D;    2 ˆ †D ˆ for Dˆ . Such a procedure corresponds stitute D to the well-known mean field approximation. Thus, we obtain     E = H   = (2.211) p 2vp2 + u2p − vp2

S=−

 p

np log np     + 1 − np log 1 − np . (2.213)

It follows from (2.213) that minimizing the energy for a given entropy is equivalent to minimizing for given occupation numbers of the excitations np . Varying expression (2.212) and taking into account that according to (2.205) up ıup = −vp ıvp , we find   ıE = −2 1 − np+ − np−   ıu  p × 2p up vp −  u2p − vp2 , (2.214) vp where we introduced an important quantity g   = Dˆ V   g  up vp 1 − np+ − np− . (2.215) = V p The condition ıE = 0 yields   2p up vp =  u2p − vp2 .

(2.216)

Equations (2.205) and (2.216) allow us to express the parameters up , and vp in terms of p and : ⎞ ⎛  1 p ⎠ u2p = ⎝1 +   , 2 2 2  + p ⎞ ⎛  1 p ⎠ (2.217) vp2 = ⎝1 −   . 2 2  + 2 p

p

  g  2  Dˆ , × np+ + np− − V with

     ˆ = D up vp 1 − np+ − np− .

Using these expressions in (2.215) we find g  1 − np+ − np−   = 1. 2V p 2 + p2 (2.212)

p

The coefficients up and vp will be determined by minimizing the energy E. This minimization according to the general rule of thermodynamics must be performed at constant entropy. In terms of the distribution function of the elementary excitations obeying the Fermi statistics, the entropy can be written as

(2.218)

The quantity  plays a crucial role in the theory. It defines the energy spectrum of the elementary excitation. The characteristic feature of theory is the dependence of the spectrum on the distribution of the excitations. It is worth noting that this does not change equation (2.213) for the entropy. This equation is combinatorial in character and requires only that an excitation can be characterized by definite momentum and spin projection.

2

Theoretical Foundations

57

We will define the energy of an elementary excitations using Landau’s theory of a Fermi liquid. Ac  cording to this theory one can find the energy " p of an excitation with momentum p and spin projection  by varying the total energy with respect to the distribution function np :    " p ınp . (2.219) ıE = p

The expression for the energy is given by (2.212) and the variation with respect to np can be carried out at constant up and vp due to our minimization with respect to these variables. A simple calculation gives   " p = 2 + p2 . (2.220) The most important property of (2.220) is that the excitation energy cannot be less than .This means that the excited states of the system are separated from its ground state by an energy gap. The existence of this gap is closely related to the Cooper pairing considered in the previous section. The energy 2 needed to create a pair of excitations can be interpreted as the “binding energy” of the Cooper pair. This energy is needed in order to break the pair. The excitation energy (2.220) does not depend on the projection of the spin  will omit the index  . If  = 0 we  and  we have " p = p , which corresponds to (2.184) for the excitation energy of the normal Fermi gas. The gap  is an experimentally observable quantity.It can be measured in experiments on microwave  and sound absorption. The function " p is presented schematically in Fig. 2.4. The dashed line shows for comparison the spectrum (2.184) of the normal Fermi gas. In a state of thermodynamic equilibrium the distribution function np does not depend on the spin projection and is given by the Fermi distribution np+ = np−

&−1   % " p + 1 . (2.221) ≡ n (") = exp T $

Note that this expression is nothing more than the distribution of Fermi particles with zero chemical potential. The reason is that we used the “grand

Fig. 2.4. Energy spectrum of a normal metal (dashed line) and a superconductor (solid line).  is the energy gap

ˆ − Nˆ and the value canonical” Hamiltonian Hˆ  = H

is subtracted from the quasi-particle energy. Substituting (2.220) and (2.221) in (2.218) and changing from a summation to an integration over p-space, we can finally write the equation for the gap in the form  g 1 − 2n (") d 3 p   = 1. (2.222) 2 " p (2)3 Note that

 "  . (2.223) 2T It is obvious that (2.222) has a solution only if g0, since np ≤ 1/2. If the interaction is repulsive the Cooper pairs are not created and electrons remain in the normal state. The magnitude of the gap  depends on the temperature through the distribution function np . We will construct an approximate solution of (2.222) in the next section. 1 − 2n (") = tanh

2.13

Thermodynamic Properties of Superconductors

2.13.1

Temperature Dependence of the Energy Gap

In this section we solve (2.222) for the gap .At T = 0 there are no quasi-particles and np = 0. Then (2.222) takes the form  1 g d3p   = 1, (2.224) 2 " p (2)3

58

L. Pitaevskii

or g 4 2 3

∞ 0

2

p dp = 1. 20 + p2

(2.225)

Here,we introduced 0 for the value of  (T) at T = 0. When g is small, equation (2.225) has a solution only if the integral is larger. When 0 → 0, the integral diverges logarithmically near the point p = 0. This means that at small 0 only values p ≈ pF are important. Within logarithmic accuracy we can substitute p2 → p2F in the numerator and cut the integration off at  ∼ ". ˜ The value of the cut off parameter "˜ depends on the nature of the physical system. In the superconducting metal, where attraction is due to the exchange of phonons, "˜ is of the order of the maximum frequency of phonons, "˜ ∼ !D , where !D is the Debye frequency of the metal.5 Equation (2.225) then becomes gp2F 2 2 3 vF

!D 0

dp 20 + p2

= 1.

(2.226)

gp2F 2!D log 2 23 vF 0 Finally

 = 1.

!D /2Tc

tanh x

dx = 1. x

(2.229)

0

Integrating by parts and taking into account that tanh (!D /2Tc ) ≈ 1, we obtain  gF

log (!D /Tc ) − I/2 = 1 , (2.230) 2 where ∞ I= 0

  log x dx = 2 log /2 , 2 cosh (x/2)

= eC = 1.78 .

(2.231)

(To obtain rapid convergency of the integral,the limit was extended to ∞.) Hence the transition temperature is  

Tc = 0 = 0.570 = 1.14!D exp −2/gF , (2.232) 

  0 = 2!D exp −2/gF ,

(2.227) gF 2

!D     d + tanh 2T 

(2.233)

0

(2.228)

where F = mpF / 23 is the density of states near the Fermi surface. According to (2.228) the gap 0 is exponentially   small with regard to !D . The function 0 g has an essential singularity as a function of the coupling constant at g = 0 and cannot be expanded in powers of this parameter. Actually, the BCS theory corresponds to the sum of an infinite number of terms in the perturbation theory series. We will begin the investigation of the temperature dependence of the gap by a calculation of the transition temperature Tc at which  is zero. Since  = 0 5

gF 2

with 0 given by (2.228). The behavior of  near Tc can be found in an analogous way. One gets

Integration gives 

for T = Tc we can easily transform (2.222) to the form

gF 2

! D 

0

  1  " 1 − tanh d = 1 . tanh 2T " 2T 

Note that the first integral on the l.h.s. differs from the integral (2.229) only in that Tc is replaced by T. This permits us to rewrite (2.234) as Tc = T ∞   "  1   1 − tanh d .(2.234) tanh 2T  2T " log

0

Recent progress in experimental techniques has permitted one to trap and cool rarefied Fermi gases of neutral atoms. In such a system the coupling constant g is proportional to the s-wave atom–atom scattering length. If this length is negative, the gas at low temperature undergoes a phase transition to a superfluid state. In this case "˜ ∼ . A consistent theory of this phenomenon was given in [30].

2

The limit of integration is again extended to ∞. If  is small, one can expand the r.h.s. with respect to 2 obtaining 2 Tc = − 2 I1 , (2.235) log T 8T ∞ d  tanh x  dx 7(3) with I1 = dx x x = −  2 . Then, near the 0

transition point we obtain to first order in (Tc − T) 7 (3) 2 Tc − T −  = 0, Tc 8 2Tc2

(2.236)

or finally  1/2 T 8 2 1−  (T) = Tc 7 (3) Tc  1/2 T = 3.06Tc 1 − . Tc 

(2.237)

2.13.2

Theoretical Foundations

59

Thermodynamic Functions

In this section we discuss the thermodynamics properties of the electron gas in a superconductor.We begin with the derivation of several useful relations for the grand canonical thermodynamic potential       ˆ (2.242) § T, V , = −T log Tr e−H /T . Let us calculate the derivative of § with respect to the coupling constant g. According to the general theorem concerning the differentiation of thermodynamic potentials, one has '  (    ˆ @§/@g T,V, = @H /@g . (2.243)

  @Hˆ  /@g

Let us now consider the limit of low temperature. Rewriting (2.222) in the form   1 n (") d 3p g d3p   −1 = g , (2.238) 2 " (2)3 " p (2)3

From (2.209) and (2.212) we have

we note that the first term on the left differs from that at T = 0 only in that 0 is replaced by  (T). According to (2.227)  this means that the l.h.s. of the equation is gF /2 log (0 /) . The integral on the r.h.s. converges rapidly due to the exponential decrease of np far from the Fermi surface. Because of this we can safely substitute d 3p → 4mpF dp and extend the limits of integration p to ±∞. After the change of variable p = x the equation takes the form   √ ∞ n  1 + x 2 2 log (0 /)−2 dx = . (2.239) √ 2 gF 1+x

We recall that thermodynamic derivatives of different thermodynamic potentials are equal when expressed in terms of the appropriate variables. For example,the thermodynamic identity for the free energy   @F dF = −SdT − PdV + dN + dg (2.245) @g T,V,N

0

At low temperatures (T  0 ) the distribution function is exponentially small and small values of x give the main contribution to the integral. Using the approximate expressions n (") ≈ e−"/T ts, " ≈ 0 + 2 /20 and integrating, we find     0 .  ≈ 0 1 − (2T/0 ) exp − T

(2.240)

(2.241)

ˆ † D/V ˆ −D and   @§/@g T,V, = −V 2 /g 2 .

can be rewritten as   d§ = d F − N = −SdT − PdV   @F − Nd + dg . @g T,V,N



=

(2.244)

(2.246)

    Thus @F/@g T,V,N = @§/@g T,V, . Hence   @F/@g T,V,N = −V 2 /g 2 .

(2.247)

The energy gap  is defined by equation (2.222) and depends on pF as a parameter. In (2.247) pF must be expressed as a function of N (not !) according to the ideal-gas equation

60

L. Pitaevskii

p3F N = . V 3 3 3

(2.248)

Let us integrate (2.247) over g from 0 to g. Taking into account that at g = 0 the gap  = 0 and the free energy refers to the normal metal at the same temperature, we find: g F s = Fn − V

2 dg . g2

(2.249)

0

At absolute zero the free energy coincides with the energy and  = 0 . From (2.228) we have d log 0 = d0 /0 = (2/F ) dg/g 2 .

(2.250)

Substituting into (2.249) and integrating we find the difference between the ground state energies of the superconducting and normal states: Es = En − V F 20 .

(2.251)

We note, a simple physical meaning of this equation: En − Es = 0 ıN, where ıN = V F 0 is of the order of the number of electrons in the energy interval 0 . This equation shows that the superconducting state is more stable than the normal state as it must be under the given conditions. A comparison with (2.63) gives the expression for the critical field: Hc2 = F 20 . 8

(2.252)

Let us now consider the opposite case T → Tc . Differentiating (2.235) with respect to g and taking into account (2.235) we find 7 (3) 2 dg d = . 4 2 T 2 F g 2

(2.253)

Substituting this equation in (2.249) and calculating the integral, we get Fs = F n − V

7 (3) F 4 . 32 2 T 2

(2.254)

Substituting here the limiting expression (2.237), we finally obtain   T 2 2mpF Tc Fs = Fn − V 1− . (2.255) 7 (3) 3 Tc

Calculation of   the specific heat according to CV = −T @2 F/@T 2 gives the discontinuity in the specific heat at the transition point: Cs − Cn = V

4mpF Tc . 7 (3) 3

(2.256)

The specific heat of the normal state in the model under consideration is simply the specific heat of the ideal Fermi gas of the same density, i.e. Cn = VmpF T/33. Thus, the ratio of the specific heats at the transition point is   12 Cs + 1 = 2.43 . (2.257) = Cn T=Tc 7 (3) To calculate the specific heat in the low temperatures region, it is more convenient to use (2.219) for the variation of the energy with respect to the distribution function. Changing from a summation to an integration we can write  @E @n Vd 3 p C= =2 " @T @T (2)3 ∞ 2VpF @n (2.258) ≈ 2 3 " d .   @T 0

Then, we can use the approximate form (2.240) and an integration gives √   5/2 0 2mpF 0 C = V 3/2 3 3/2 exp − . (2.259)  T T In concluding this section we calculate the number of superconductive electrons ns .According to the definition, ns = n − nn = n − n /m, where n is the total density and the normal mass density n is given by (2.28). With the usual simplifications and using equation (2.248) for the total density we transform this equation to the form nn = −2 n

∞

dn d . d"

(2.260)

0

There is a very useful expression which relates the normal density and the gap  (T). To derive it, we

2

differentiate (2.239) with respect to temperature. After changing back to an integration over , we find by comparison with (2.260): n  =1− , nn T

and 2 ns = (Tc − T) , Tc − T  Tc . n Tc 2.13.3

(2.263)

Microscopic Meaning of the GL Equations

One of the important results of the BCS theory is that it allows us to understand the microscopic meaning of the coefficients in the phenomenological Ginzburg–Landau equations (L.P. Gor’kov, 1959 [31]). First of all, a comparison of (2.237) for the temperature dependence of the gap near Tc with (2.263) √ allows us to find the relation between  and = ns /2:   √ 7 (3) 1/2  = n . (2.264) 8 2 Tc Furthermore, comparison of (2.86) and (2.256) for the discontinuity in the specific heat gives 4mpF ˛2 = . b 7 (3) 3

(2.265)

Comparison of (2.82) and (2.263) gives a second equation ˛ n p3F = = . (2.266) b Tc 3 2 3 Tc Solving these equations with respect to ˛ and b, we find ˛=

˛Tc 12 2mTc , = 7.04Tc / , b = 2 7 (3) pF n

(2.267)

61

where is the chemical potential of the ideal Fermi gas, = p2F /2m. Substitution this value of b in (2.98) for the GL parameter  gives  = 1.09

(2.261)

where  = d/dT. The result (2.261) allows us to write down the equations for the temperature dependence of nn for different limiting cases without additional calculations. We obtain directly from (2.241) and (2.237):   nn 20 1/2 −0/T = e ,T → 0 (2.262) n T

Theoretical Foundations

Tc mc 1/2

n1/2  |e|

.

(2.268)

Note also that the quantity 0 which gives the correlation length at T = 0 is 0 =

  1/2 vF = ∼ . 1/2 1/2 (m˛Tc ) Tc (7m) Tc

(2.269)

Expression (2.85) for the critical field can now be written in the form   mpF 1/2 (2.270) Hc = 2.44 (Tc − T) . 3 Let us now express the condition (2.105) for the validity of the GL theory in terms of the microscopic theory. Substitution from (2.267) gives:  4 Tc − T Tc  . (2.271) Tc

With the possible exception of high temperature superconductors very close to Tc , this inequality is generally satisfied in practice ((2.271)). The theory of superconductivity which we have developed is of course quite crude. Real metals are anisotropic bodies and the energy spectrum of their electrons depends on the interaction with the crystal lattice. To some extent this fact can be taken into account by replacing the free electron mass in the above equations by some effective mass m∗ . Further, the BCS Hamiltonian is only a crude approximation of the real electron–phonon Hamiltonian.Finally,the that the coupling constant g is small theory assumes  gF  1 . According to (2.232) this condition requires log (!D /Tc )  1 . (2.272) In practice this condition is satisfied only with limited accuracy. A more sophisticated theory of superconductivity, which is free of these defects, will be presented in other chapters of this book.It is remarkable, however, that even the simple theory based on the BCS model in many respects gives a good description of the properties of superconductors, both qualitatively and quantitatively.

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L. Pitaevskii

2.14

Elements of the Theory of Green’s Functions

2.14.1

General Properties of Green’s Functions

The Green’s function method is an important part of the modern theory of superconductivity. This method permits a formulation of the theory in a very transparent and convenient form and provides a powerful tool to solve more complicated problems in superconductivity. In this section we will introduce basic notions of the Green’s function theory. Of course, our review cannot replace a systematic textbook (see, for example, [32] and [33]). However, we hope that our presentation can serve as a useful introduction to the subject. The Green’s function G˛ˇ of a system of fermions at T = 0 is defined as   G˛ˇ (X1 , X2 ) = −i Tˆ ¦ˆ ˛ (X1 ) ¦ˆ ˇ† (X2 ) , (2.273) where . . . denotes averaging with respect the ground state of the system. Here and below ¦ˆ ˛ (X) and ¦ˆ ˛† (X) are the annihilation and creation and creation operators for electrons in the time-dependent Heisenberg representations and X denotes the time t and coordinates r, ¦ˆ ˛ (X) ≡ ¦ˆ ˛ (r, t).As in the previous section we will use the grand canonical Hamiltonian Hˆ  = Hˆ − Nˆ . Then6   ¦ˆ ˛ (r, t) = exp iHˆ  t ¦ˆ ˛ (r)   × exp −iHˆ  t , (2.274) where ¦ˆ ˛ (r) is the operator in the time-independent Schr¨odinger representation. The symbol Tˆ is the time ordering operator: the operators to the right of this symbol are to be arranged from right to left in the order of increasing times of their arguments. The products are also to be multiplied by the factor (−1)P , where P is the number of permutations of the fermionic operators needed to obtain the chronological product from their original order. In our case of two operators we have explicitly 6

G˛ˇ (X1 , X2 ) = ⎫ ⎧   ⎨ −i ¦ˆ ˛ (X1 ) ¦ˆ † (X2 ) , for t1 t2 ⎬ ˇ   . (2.275) ⎩ i ¦ˆ † (X2 ) ¦ˆ ˛ (X1 ) , for t1 t2 ⎭ ˇ In the absence of a magnetic field, the spin dependence of the Green’s function reduces to a unit spinmatrix: G˛ˇ (X1 , X2 ) = ı˛ˇ G (X1 , X2 ) .

(2.276)

If, as we assume, the system is in stationary external fields, then Green’s function depends only on the difference t = t1 − t2 of its time arguments. If in addition the system is microscopically homogeneous in space, Green’s function depends only on the difference r = r1 − r2 and G (X1 , X2 ) = G (X1 − X2 ) ≡ G (t, r) .

(2.277)

This condition is not valid for the electrons in the lattice of a real metal. However, it is valid for a model of a uniform superconducting gas, and then it is convenient to go to the Fourier representation  d!d 3 p G (t, r) = G (!, p) ei(p·r−!t) , (2.278) (2)3 with G (!, p) =

 G (t, r) e−i(p·r−!t) dtd 3 r .

(2.279)

From a knowledge of the Green’s function one can calculate the one-body density matrix of the system   (1) ˛ˇ (2.280) (r1 , r2, t) = ¦ˆ ˇ† (r2 , t) ¦ˆ ˛ (r1, t) . For a uniform system in the ground state we have according to (2.275) (1) ˛ˇ (r) = ı˛ˇ  (1) (r) = −iı˛ˇ G (t → −0, r) ,

r = r1 − r2 .

(2.281)

The Fourier expansion of the density matrix defines the momentum distribution of the particles  N (p) =  (1) (r) eip·r d 3 r . (2.282) Taking into account (2.281) and (2.279) we find:

In order to simplify the equation, we will use below units such that  = 1. However, we will include  in final results.

2

∞ N (p) = −i

G (!, p) e−i!t

−∞

(2.283)

Important properties of Green’s function can be established by inserting a complete set of states and rewriting it in the form (see (2.275)) G (X1 , X2 ) =      i       − 0 ¦ˆ ˛ (X1 ) m m ¦ˆ ˛† (X2 ) 0 , 2 m for t1 t2 . (2.284) Since the operator ¦ˆ ˛† creates an electron, |m is a state of the system of N + 1 electrons. The state |0 is the ground state of the system of N electrons. For t1 t2 we have an analogous equation. The crucial point is that the time and coordinate dependence of the matrix elements can be found from general considerations.    First of all, the time dependence ˆ  of 0 ¦˛ (X1 ) m has to correspond to the timedependence of matrix elements in the Heisenberg representation, i.e. 0 |. . .| m ∼ ei!0m t . Since we use the grand canonical Hamiltonian H  , the frequencies are  !0m = E0 (N) − Em (N + 1)

= E0 (N) − Em (N + 1) + .

(2.285)

Further, for a uniform system any transformation of the form r → r+a can only change the wave function of the system by a constant phase factor. This means    that 0 ¦ˆ ˛ (X1 ) m ∼ eipm ·r , where pm is a constant vector interpreted as the momentum of the state |m. Thus       0 ¦ˆ ˛ (X1 ) m =  ∗     m ¦ˆ ˇ† (X2 ) 0 =       0 ¦ˆ ˛ (0) m ei(!0m t+pm ·r) . (2.286) Substituting into (2.275) gives

63

G (t, r)

d! , 2

t → −0 .

Theoretical Foundations

(2.287) ⎫ ⎧  i(!0m t+pm ·r) A e , for t1 t2 ⎬ i⎨ m m  − = , −i(! t−p ·r)   2 ⎩ − Bm e 0m m , for t1 t2 ⎭ m

where the summation with respect to m is over states with N + 1 particles and the summation with respect to m is over states with N − 1 particles and we introduced the notation    2     Am =  0 ¦ˆ ˛ (0) m  ,    2     Bm =  m ¦ˆ ˛ (0) 0  , !0m = Em (N − 1) − E0 (N) + .

(2.288)

Note that the coefficients Am and Bm are real and positive. It is convenient to transform the energy differences !0m and !m0 taking into account that our system contains a large number of particles. In the first case we can write E0 (N) + ≈ E0 (N + 1) , !0m ≈ E0 (N + 1) − Em (N + 1) ≡ −"m , (2.289) and analogously !0m = Em (N − 1) − E0 (N − 1) ≡ "m .

(2.290)

It is obvious in view of the definition of the ground state (Em E0 ) that the quantities "m and "m are positive. Let us now calculate G (!, p) according to (2.279). The spatial integration gives delta functions. In performing the integration over t, one must add to ! an infinitesimal imaginary part to insure convergence. This imaginary part must be positive for integration from 0 to ∞ and negative for integration from −∞ to 0. Then G (!, p) =

(2)3 2   × m

  Am ı p − pm ! − "m + i0 +

(2.291)

   Bm ı p + pm . ! + "m − i0

We do not distinguish now the summation over m and m , since it cannot result in a misunderstanding.

64

L. Pitaevskii

Equation (2.292) has an important physical meaning. It shows that the poles of the Green’s function give the energy levels of the system. However, in the thermodynamic limit the energy levels of a macroscopical body become continuous. This corresponds to the change of the summation over m into an integration. Still if the system has well-defined elementary excitations with the dispersion relation " (p) between states m (2.292) there is a state with one excitation of the momentum p and energy " = " (p). This state gives an “individual” pole contribution to the sum. Thus, the excitation dispersion relation " (p) is determined by the equation G−1 (" (p) , p) = 0 .









¦ˆ ˛ (r, t) ¦ˆ ˇ r , t + ¦ˆ ˇ r, t ¦ˆ ˛ (r, t) =     ¦ˆ ˛† (r, t) ¦ˆ ˇ† r, t + ¦ˆ ˇ† r , t ¦ˆ ˛† (r, t) = 0 . (2.294) The grand canonical Hamiltonian of an ideal gas is   2  ∇  † ˆ ˆ − ¦ˆ ˛ (r, t) d3 x . (2.295) H0 = ¦˛ (r, t) − 2m The equation of motion for the operator ¦ˆ ˛ (r, t) can be obtained according to the general rule as   @ (2.296) i ¦ˆ ˛ = − Hˆ 0 ¦ˆ ˛ − ¦ˆ ˛ Hˆ 0 . @t

(2.297)

Let us differentiate Green’s function (2.273) with respect to t1 . Here, we have to take into account that Green’s function has a discontinuity at t1 = t2 . According to (2.275) one has

 G˛ˇ = G˛ˇ |t1 =t2 +0 − G˛ˇ |t1 =t2 −0  = − i ¦ˆ ˛ (r1 , t1) ¦ˆ ˇ† (r2 , t1 )

 + ¦ˆ ˇ† (r2 , t1 ) ¦ˆ ˛ (r1, t1 ) . (2.298)

 Equation (2.293) then gives G˛ˇ = −iı˛ˇ ı (r1 − r2). Differentiation of this discontinuity

 gives a contribution G˛ˇ ı (t1 − t2 ) and 0 / @¦ˆ ˛ (X1 ) † @ (0) ¦ˆ ˇ (X2 ) i G˛ˇ = −i T @t @t1

Green’s Function of an Ideal Fermi Gas

We begin our discussion of Green’s function for a non-superconducting Fermi system with the calculation of Green’s function G(0) of an ideal Fermi gas. Recall first that the operators ¦ˆ ˛ (r, t) and ¦ˆ ˛† (r, t) at coinciding times obey the commutation rules     ¦ˆ ˛ (r, t) ¦ˆ ˇ† r, t + ¦ˆ ˇ† r , t ¦ˆ ˛ (r, t) =   (2.293) ı˛ˇ ı r − r , and

  ∇2 @ ˆ − ¦ˆ ˛ (r, t) . i ¦˛ (r, t) = − @t 2m

(2.292)

On the contrary, for the states which contain several excitations with total momentum p, the energy of the system is not uniquely determined by the value of p and the total energy of these excitations covers a continuous range of values in the thermodynamic limit. In this case the pole is removed by the integration over m. 2.14.2

The commutator on the r.h.s. of this equation is calculated by means of (2.293) and (2.294). As a result we obtain the “Schr¨odinger-like equation”:

− iı˛ˇ ı (r1 − r2) ı (t1 − t2 ) . (2.299) Substituting (2.297) we finally obtain   ∇2 @ + G(0) (t, r) = ı (r) ı (t) . (2.300) i + @t 2m The Fourier transform of this equation gives G

(0)

−1  p2 + + i0sgn ! , (2.301) (!, p) = ! − 2m

where the infinitesimal imaginary part, which is important only near the pole, is chosen according to (2.292). Comparing with (2.292) we also find that       p2  −  , " p = 2m

(2.302)

as it must be in the absence of an interaction. Near the  Fermi surface this equation can be rewritten as " p = pF p − pF  /m.

2

Green’s function of a normal Fermi system in the presence of an interaction cannot be calculated in a general form. However, the energy spectrum of such a system is qualitatively similar to the spectrum of the ideal Fermi gas. Near the Fermi surface it can be presented as     " p = pF p − pF  /m∗ , (2.303) where m∗ is the effective mass of an elementary excitation which depends on the interaction. The corresponding Green’s function is G (!, p) =

(2.304)

Z   + g (!, p) , ! − pF p − pF /m∗ + i0sgn !

where   g (!, p) is a function that is regular at ! = " p . One can show also that the “constant of renormalization” Z is restricted according to 0Z ≤ 1. The transition from the normal state to the superconducting state, of course, changes the situation completely. 2.14.3

Two-Particle Green’s Function

It is impossible to obtain a closed equation similar to (2.300) for Green’s function of a system in the presence of an interaction. However, one can derive an equation that connects the Green’s function (2.273) with a more complicated two-body Green’s function. To derive this equation, we consider the simplest case where the electrons interact by means of the twobody potential U (|r1 − r2 |). One must then add to the Hamiltonian (2.295) the two-body term  1 ¦ˆ ˇ† (r2 , t) ¦ˆ ˛† (r1 , t) U (|r1 − r2 |) Vˆ 2 = 2 × ¦ˆ ˛ (r1, t) ¦ˆ ˇ (r2 , t) d3 x1 d3 x2 . (2.305) Keeping in mind, however, future applications to the BCS model of superconductivity, we can simplify V2 by assuming a ı-functional interaction (2.185). We then have  g ˆ ¦ˆ ˇ† (r, t) ¦ˆ ˛† (r, t) ¦ˆ ˛ (r, t) ¦ˆ ˇ (r, t) d3 x . V2 = − 2 (2.306) 7

Theoretical Foundations

65

The time dependence of the ¦ -operator is now deˆ 0 has to be fined by the equation (2.296), where H ˆ 0 + Vˆ 2. Calculating the commutachanged to Hˆ  = H tor gives instead of (2.297) the operator equation @ i ¦ˆ ˛ (r, t) = @t   ∇2 † ˆ ˆ − − g ¦ˇ (r, t) ¦ˇ (r, t) ¦ˆ ˛ (r, t) (2.307) − 2m with which we can calculate @G˛ˇ /@t. Using (2.307) and noting that the discontinuity of Green’s function does not depend on the interaction,instead of (2.300) we easily obtain   @ ∇12 i + G˛ˇ (X1 − X2 ) + @t1 2m   −ig Tˆ ¦ˆ † (X1 ) ¦ˆ (X1 ) ¦ˆ ˛ (X1 ) ¦ˆ ˇ† (X2 ) = ı˛ˇ ı (X1 − X2 ) , (2.308) where ı (X1 − X2 ) = ı (r1 − r2 ) ı (t1 − t2 ). The Tproduct on the l.h.s. of this equation is a particular case of the two-particle Green’s function K ı;˛ˇ (X3 , X4 ; X1 , X2 ) =   Tˆ ¦ˆ (X3 ) ¦ˆ ı (X4 ) ¦ˆ ˛† (X1 ) ¦ˆ ˇ† (X2 ) . (2.309) There is no general prescription to express the twoparticle Green’s function K in the terms of the oneparticle Green’s function G. Still, for many problems this is possible by using valid approximations.

2.15

Green’s Function of a Superconductor

The power of the Green’s functions method in the theory of superconductivity is associated first of all with the possibility of providing a rigorous formulation of the phenomenon of Bose condensation of Cooper pairs, which we used earlier.7 Consider the following operator that creates a pair of electrons ¦ˆ ˛† (X1 ) ¦ˆ ˇ† (X2 ) .

(2.310)

The technique described below is due to L.P. Gor’kov (1958) [7]. A compact matrix formulation of Green’s theory of superconductivity was developed by Y. Nambu [34].

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L. Pitaevskii

When this operator acts on the ground state wave function of a system of N electrons, ¦ˆ ˛† (X1 ) ¦ˆ ˇ† (X2 ) |N it creates, generally speaking, a superposition of excited states of the system. However, in the presence of the condensate of the Cooper pairs, a pair can be added to the condensate. Because of the large number of pairs, such an addition does not excite the system; i.e. in the presence of the Bose condensate the operator (2.310) has a nonvanishing matrix element between ground states of system, which differ only by the numbers of electrons: N + 2|¦ˆ ˛† (X1 ) ¦ˆ ˇ† (X2 ) |N .

(2.311)

In the thermodynamic limit the difference between these two states is negligible and this quantity can be considered as an average of the operator (2.310). The appearance of such“anomalous averages” is a special feature of a superconducting system. Let us consider the two-particle Green’s function K defined according to (2.309). The average of the products of the ¦ -operators can be written according to the matrix multiplication rule. However, for the reconstruction of the ground state of the system in lowest approximation only the “anomalous” matrix elements (2.311) are important. The BCS theory corresponds to the following approximation of the function K: K ı;˛ˇ (X3 , X4 ; X1 , X2 )   ≈ N|Tˆ ¦ˆ (X3 ) ¦ˆ ı (X4 ) |N + 2   × N + 2|Tˆ ¦ˆ ˛† (X1 ) ¦ˆ ˇ† (X2 ) |N . (2.312) Thus, in a natural way, so-called anomalous Green’s functions appear:   F˛ˇ (X1 , X2 )= N|T ¦ˆ ˛ (X1 ) ¦ˆ ˇ (X2 ) |N +2 , (2.313) and

  + F˛ˇ (X1 , X2 ) = N + 2|T ¦ˆ ˛† (X1 ) ¦ˆ ˇ† (X2 ) |N . (2.314) We must make an important remark here. In addition to the anomalous matrix elements (2.311), the

function K also contains “normal” terms, for example matrix elements of the operator ¦ˆ ˛ (X1 ) ¦ˆ ˇ† (X2 ). They are not at all small compared with the anomalous terms. (On the contrary, the anomalous functions F and F + are proportional to the gap  and consequently are exponentially small with respect to the coupling constant g.) However, we will nonetheless ignore these non-anomalous matrix elements in lowest approximation, because they do not cause the qualitative change in the nature of the ground state of the system (which is not the case for the anomalous terms). They only give “regular" corrections to the chemical potential and the effective mass. Note that the wave function of the superconducting pair ˛ˇ (r, t), which we introduced in Sect. 2.2, is expressed in terms of F˛ˇ as  ˛ˇ (r, t) = F˛ˇ (X, X) , X = (r, t) .

(2.315)

Like  ˛ˇ , the function F˛ˇ is proportional to the antisymmetric unit spinor of rank two I˛ˇ : F˛ˇ (X1 , X2 ) = I˛ˇ F(X1 , X2 ) , + F˛ˇ (X1 , X2 ) = I˛ˇ F +(X1 , X2 ) .

(2.316)

The difference in the spinor structures of G˛ˇ and F˛ˇ is related to the fact that G˛ˇ is a mixed contracovariant spinor while F˛ˇ is a contravariant one. Note also that due to the commutation relations (2.294) the functions (2.316) are symmetrical in X1 and X2 : F (X1 , X2 ) = F (X2 , X1 ) , F + (X1 , X2 ) = F + (X2 , X1 ) .

(2.317)

In the absence of an external field the functions G, F and F + depend only on the coordinate difference X = X1 − X2 . Then the function F (0) =  

(2.318)

is a constant. Note also that F + (0) =  ∗ . Our notation here is different from the one that we used previously. Here, we use the grand canonical Hamiltonian H  . Because of this, (2.315) for  ˛ˇ does not contain the factor e−2i t , which we had in (2.33). Let us now derive equations for the BCS model of a superconductor. Using the decomposition (2.312)

2

and taking into account that I 2 = −1, we can rewrite (2.308) in the form   ∇2 @ + G (X)+ig F + (X) = ı (X) . (2.319) i + @t 2m This equation contains the second unknown function F + . To derive the equation of motion for this function, we require the equation of motion for ¦ˆ ˛† , which is the Hermitian conjugation of (2.307): @ † ¦ˆ (r, t) @t ˛   2 ∇ + + g ¦ˆ ˛† (r, t) ¦ˆ ˇ† (r, t) ¦ˆ ˇ (r, t) . (2.320) = 2m i

Differentiating (2.314) with respect to t1 gives an equation with the anomalous two-particle Green’s function   Tˆ ¦ˆ a† (X1 ) ¦ˆ † (X1 ) ¦ˆ (X1 ) ¦ˆ ˇ† (X2 ) = iI˛ˇ  ∗ G (X1 − X2 ) .

(2.321)

As a result, we obtain the following equation for F + (X):   ∇2 @ − F + (X) − ig ∗ G (X) = 0 . (2.322) i − @t 2m A delta-function term similar to the right-hand side of (2.319) does not arise here, since the function F + (X1 , X2 ),unlike G (X1 , X2 ),is continuous at t1 = t2 . Equations (2.319) and (2.322) can be easily solved in the momentum representation. Taking the Fourier components of these equations, we obtain   ! − p G (P) + ig F + (P) = 1 ,   ! + p F + (P) − ig ∗ G (P) = 0 , (2.323) where P = (p, !) and p = p2 /2m − . Eliminating F + (P) from (2.323) gives the formal solution ! + p   − "2 p up vp  +  , = !−" p ! +" p

G (P) =

where

!2

(2.324)

Theoretical Foundations

  " p = 2 + p2 .

67

(2.325)

Here, we used the notation  = g | | ,

(2.326)

and up and vp are given by (2.217). The solution (2.324) is only a formal one, since it does not define the way to pass around the poles. In this section we consider only the case of zero temperature. The case of finite temperatures will be considered in the next section by a different method. At T = 0 the way to pass around the poles is given by the general equation (2.292). Adding infinitesimal imaginary parts according to this equation we obtain up vp     G (P) = + ! − " p + i0 ! + " p − i0 ! + p       . (2.327) =  ! − " p + i0 ! + " p − i0   Comparing this with (2.292) also shows that " p is the energy of elementary excitations. Thus we have recovered equation (2.220) for the spectrum. Equation (2.326) establishes a relation between the gap in the spectrum and the anomalous Green’s function of a superconductor. Substituting (2.327) gives F + (P) = 

ig        . (2.328) ! − " p + i0 ! + " p − i0

Then, rewriting (2.318) in momentum space, we find  d!d 3p  ∗ = F + (P) . (2.329) (2)4 Substituting (2.328), we can integrate with respect to ! by closing the contour with an infinite semicircle in the upper or lower half plane. The integral is expressed in terms of the residue at the pole:  1 d3p g    ∗ =  ∗ . (2.330) 2 " p (2)3 Thus we recapture (2.224) for the gap 0 at T = 0. The function  d!d 3 p + F (r) = F + (P) e−ip·r (2)4  −ip·r 3 g e d p   =  ∗ (2.331) 2 " p (2)3

68

L. Pitaevskii

F + (X1 , X2 ) → F + (X1 , X2 )    ie  × exp −

(r1 ) + (r2) . c

describes Cooper pairs and has the physical meaning of a pair wave function. It follows from (2.331) that thisfunction  decreases for large r as sin pF r/ e−r/0 , with 0 =

pF . m0

Accordingly, one has

and

(2.333)

  1 @ 2 i + (∇1 + ieA (r1 ) /c) + F + @t 2m × (X1 , X2 ) − ig ∗ (r1 ) G (X1 , X2 ) = 0 .

(2.334)

It is not difficult to show that these equations are invariant with respect to the gauge transformation A (r) → A (r) + ∇ (r) , G (X1 , X2 ) → G (X1 , X2 )    ie 

(r1) − (r2 ) , × exp c 8

  2ie ∗ (r) → ∗ (r) exp − (r) . c

(2.332)

Thus, the coherence length 0 introduced in (2.55) has the physical meaning of the size of a Cooper pair. Since 0 is exponentially small, the coherence length is very large in comparison to the interelectronic spacings /pF . So far we have considered a superconductor in the absence of a magnetic field. The introduction of the magnetic field is achieved by replacing the operator ∇ in (2.307) by (∇ − ieA/c) and correspondingly by (∇ + ieA/c) in (2.320).In the presence of a stationary magnetic field Green’s functions depend on r1 and r2 separately, but continue to depend on t1 and t1 only through the difference t = t1 −t2 .We can keep the old spin structure of the Green’s functions, since direct interaction of the magnetic field with spin magnetic momenta of electrons is small and can be neglected. Finally, the equations for the Green’s functions in the field take the form:   @ 1 2 i + (∇1 − ieA (r1 ) /c) +

@t 2m × G (X1 , X2 ) + ig  (r1) F + (X1 , X2 ) = ı (X1 − X2 )

(2.337)

(2.335)

(2.336)

(2.338)

However, the momentum cut-off actually breaks the gauge invariance. A consistent gauge-invariant theory can be formulated only by using a realistic electron–phonon interaction.

2.16

Temperature Green’s Functions

There are several approaches to using Green’s function methods to solve finite-temperature problems. We will present here the general and elegant method of so-called imaginary-time temperature Green’s functions.This method is based on introducing a new imaginary-time representation for the ¦ -operators of the electrons. So far we used these operators in the ordinary time-dependent Heisenberg representation ( 2.274). The imaginary-time operators ˆ (r, ) and ˆ + (r, ) can be obtained from the Heisenberg operators ¦ˆ ˛ (r, t) and ¦ˆ ˛† (r, t) by replacing the variable t by the imaginary variable (−i):     ˆ˛ (r, ) = exp Hˆ   ¦ˆ ˛ (r) exp −Hˆ   ,     ˆ˛+ (r, ) = exp Hˆ   ¦ˆ ˛† (r) exp −Hˆ   . (2.339) We will denote the imaginary-time operators by oblique (“Italic”) letters.We emphasize that the operator ˆ˛+ (r, ) is not the Hermitian-conjugate of the  † operator ˆ˛ (r, ); actually ˆ˛ (r, ) = ˆ˛+ (r, −). The temperature Green’s function G is defined in terms of these operators.8 The definition includes averaging with respect to the statistical distribution: G˛ˇ (r1 , 1; r2 , 2) =   − Tˆ  ˆ˛ (r1 , 1) ˆˇ+ (r2 , 2 ) ,

(2.340)

The mathematical technique based on using these Green’s functions was introduced by T. Matsubara in 1955 [35], see also Abrikosov et al. [36].

2

where Tˆ is the symbol for “-chronological ordering”. It arranged the operators from right to left in order of increasing . The sign is changed for every interchange of Fermi operators. The brackets . . . mean averaging over the grand canonical distribution: 1  % 2 § − Hˆ  . . . = tr exp ... , (2.341) T where tr denotes the sum of the diagonal matrix elements, and § is the grand canonical thermodynamic potential, which we already used in Sect. 3.3. Like G˛ˇ the function G˛ˇ in the absence of a magnetic field is proportional to a unit spinor: G˛ˇ = ı˛ˇ G. Thus, G (r1, 1 ; r2, 2 ) =   eˇ§ −ˇ Hˆ  ˆ ˆ + ˆ tr e − T ¦˛ (r1 , 1)¦˛ (r2, 2 ) . (2.342) 2 Here and below ˇ means the inverse temperature: ˇ = 1/T .

(2.343)

Next,we will prove an important property of the temperature Green’s functions. The function G depends only on the “time” difference  = 1 − 2 . Consider the case 1 2 . Then, according to (2.339) and (2.342): eˇ§ tr (2.344) G (r1, 1 ; r2, 2 ) = − 2   ˆ ˆ ˆ ˆ × e−ˇ H e2 H ¦ˆ ˛† (r2 ) e−(2 −1 )H ¦ˆ ˛ (r1 )e−2 H . A cyclic permutation of operators in the trace reduces this equation to: eˇ§ G (r1 , 1; r2 , 2) = tr (2.345) 2   ˆ ˆ × e−(ˇ+ )H ¦ˆ ˛† (r2 ) e H ¦ˆ ˛ (r1 ) ts, 0 . Analogous transformations give for 0: eˇ§ tr (2.346) G (r1, 1 ; r2, 2 ) = − 2   ˆ ˆ e− H ¦ˆ ˛† (r2 ) e−(ˇ− )H ¦ˆ ˛ (r1 ) , 0 .

Theoretical Foundations

69

These equation show that indeed G is a function only of . Moreover, comparison of (2.346) and (2.346) gives an important symmetry relation:   G (0) = −G  + ˇ . (2.347) Hence, it is sufficient to consider the variable  in the interval −ˇ ≤  ≤ ˇ . (2.348) Then (2.347) expresses G for negative  through its value for 0. The function G defines the thermodynamic properties of the system. For example, one can calculate the density of the system from the equation   N 2G (r; r;  → −0) = − ¦ˆ ˛† (r) ¦ˆ ˛ (r) = − . (2.349) V The density N/V can be expressed in terms of the chemical potential , volume V and temperature T. One can calculate the potential § by integration of the equation N = − (@§/@N)T,V . The periodicity condition (2.348) allows us to represent the temperature Green’s function as a Fourier series with respect to  (as opposed to a Fourier integral) of the form ∞ 

G (r1 , r2, ) = T

  e−in  G r1 , r2, n ; (2.350)

n=−∞

this greatly simplifies the application of the formalism. The symmetry condition (2.347) will be automatically fulfilled if n = (2n + 1)T , n = 0, ±1, ±2, ±3, . . . (2.351) The inverse transformation is 



ˇ

G r 1 , r 2 , n =

ein  G (r1 , r2, ) d .

(2.352)

0

  In a spatially-uniform system G r1 , r2, n depends only on the coordinate difference r = r1 − r2 . Then     d 3p  . (2.353) G r, n = eip·r G p, n (2)3 In concluding this section note that differentiation of (2.339) gives the equation of motion for ¦

70

L. Pitaevskii

  @ (2.354) ¦ˆ ˛ = − Hˆ 0 ¦ˆ ˛ − ¦ˆ ˛ Hˆ 0 , @ which can be obtained from (2.296) by t → −i. An equation of the same form is valid also for ¦ + . For an ideal gas we find instead of (2.300):   @ ∇2 − + + G (0) (r, ) = ı (r) ı () . (2.355) @ 2m −

The Fourier transform of this equation gives   1 . (2.356) G (0) p, n = in − p

The equation for F + is derived analogously to (2.322). Thus, we obtain   ∇2 @ + + F + (X) − ∗ G (X) = 0 , (2.361) @ 2m where

Temperature Green’s Functions for a Superconductor

Equations for the temperature Green’s functions of a superconductor can be derived in the same way as was done for T = 0. An equation for the imaginarytime ¦ -operator can be obtained from (2.307) and (2.320) by replacing t → −i. Differentiating (2.340) for the Green’s function leads to the two-particle temperature Green’s function K ı;˛ˇ (X3 , X4 ; X1 , X2 ) =   Tˆ  ¦ˆ (X3 ) ¦ˆ ı (X4 ) ¦ˆ ˛+ (X1 ) ¦ˆ ˇ+ (X2 ) ,

(2.357)

where X = (r, ) and . . . denotes the statistical average. In the superconducting state, the system is characterized by the existence of the anomalous averages   + F˛ˇ (X1 , X2 ) = Tˆ  ¦ˆ ˛+ (X1 ) ¦ˆ ˇ+ (X2 ) = I˛ˇ F + (X1 , X2 ) ,

(2.358)

and a similar form for F˛ˇ . Instead of the decomposition (2.312) we then have K ı;˛ˇ (X3 , X4 ; X1 , X2 ) = + F ı (X3 , X4 ) F˛ˇ (X1 , X2 ) .

(2.359)

As a result we obtain for G the equation   ∇2 @ + G (X)+F + (X) = ı (X) . (2.360) − + @ 2m

(2.362)

One can easily solve this equation by Fourier transformation. Then, ∞   + F (X) = T ei(p·r−n  )

Since this function has no poles on the real axes of n , it is uniquely defined by (2.355) without any additional conditions.

2.17

∗ = gF + (0) .

n=−∞

  d3p , (2.363) × F + p, n (2)3 with n defined by (2.351). The system of equations (2.360) to (2.362) takes the form       in − p G p, n + F + p, n = 1 ,       in + p F + p, n + ∗ G p, n = 0 , (2.364) with ∗ = gT

∞   n=−∞

  d3p F + p, n . (2)3

(2.365)

The solutions of (2.364) are     in + p ∗ + p,  = , F , G p, n = − 2 n n + " 2 n2 + " 2 (2.366) 2 2 where again " =  + p . Equation (2.365) for ∗ is reduced to ∞  d 3p gT  (2.367)   = 1. (2)3 n=−∞ n2 + " p 2 The sum with respect to n is given by ∞ 

−1 x 1 tanh . (2.368) = (2n + 1)2  2 + x 2 2x 2 n=−∞

Finally we obtain the gap equation  " d3p g 1 tanh = 1, 2 " 2T (2)3 which coincides with (2.222).

(2.369)

2

Theoretical Foundations

71

References 1. L.D. Landau and E.M. Lifshitz, Statisticheskaya Fizika (Fizmatgiz, Moscow, 1951); In: Statistical Physics (Pergamon Press, Oxford 1958) Sect. 133 2. O. Penrose, Phylos. Mag. 42, 1373 (1951) 3. L. Onsager, Nuove Cimento 6, Suppl. 2, 249 and 281 (1949) 4. R.P. Feynman, In: Progress in Low Temperature Physics, ed. by C.J. Gorter (North-Holland, Amsterdam 1955) 5. L.D. Landau, J. Phys. (USSR) 5, 71 (1941) 6. J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 106, 162; 108, 1175 (1957) 7. L.P. Gor’kov, Sov. Phys. JETP 7, 505 (1958) 8. B.D. Josephson, Phys. Lett. 1, 251 (1962); Rev. Mod. Phys. 36, 216 (1964) 9. F. London and H London, Proc. Roy. Soc. 149, 71 (1935); Physica 2, 341 (1935) 10. A.B. Pippard, Proc. Roy. Soc. A 216, 547 (1953) 11. F. London, In: Superfluids (Wiley, New York 1950) Vol. 1 12. L.D. Landau, Phys. Z. (USSR) 11, 129 (1937) 13. R.E. Peierls, Proc. Roy. Soc. A 155, 613 (1936) 14. F. London, Physica 3, 450 (1936) 15. V.L Ginzburg and L.D. Landau, Zh. Exp. Teor. Fiz. 20 1064 (1950) 16. L.D. Landau and E.M. Lifshitz, In: Statistical Physics, Part 1, 3rd edn. (Pergamon Press, Oxford 1986) 17. L.D. Landau and E.M. Lifshitz, In: Electrodynamics of Continuous Media, 2nd edn. (Pergamon Press, Oxford 1984) 18. A.A. Abrikosov, Zh. Exp. Teor. Fiz. 32, 1442 (1957) 19. D. Sain-James and P.G. De Gennes, Phys. Lett. 7, 306 (1963) 20. A.A. Abrikosov, In: Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1988) 21. A.A. Abrikosov, Dokl. AN SSSR 86, 489 (1952) 22. J.R. Schrieffer, In: Theory of Superconductivity (W.A. Benjamin, New York 1964) 23. P.G. de Gennes, In: Supercpnductivity of Metals and Alloys (W.A. Benjamin, New York 1966) 24. J. Bardeen, Phys. Rev. 79, 167; 80, 567 (1950) 25. H. Fr¨ohlich, Phys. Rev. 79, 1189 (1956) 26. L.N. Cooper, Phys. Rev. 104, 845 (1950) 27. N.N. Bogoliubov, Sov. Phys. JETP 7, 41 (1958); Nuovo Cimento 7, 794 (1958) 28. J.G. Valatin, Nuovo Cimento 7, 843 (1958) 29. N.N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947) 30. L.P. Gor’kov, T.K. Melik-Barkhudarov, Sov. Phys. JETP 13, 1018 (1961) 31. L.P. Gor’kov, Sov. Phys. JETP 9, 1364 (1959) 32. E.M. Lifshitz, L.P. Pitaevskii, In: Statistical Physics, Part 2 (Pergamon Press, Oxford 1986) 33. A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinski, In: Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, NJ, 1963) 34. Y. Nambu, Phys. Rev. 117 648 (1960) 35. T. Matsubara, Prog. Theor. Phys. 14, 351 (1955) 36. A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinski, Sov. Phys. JETP 9, 636 (1959)

3

Electron–Phonon Superconductivity?

F. Marsiglio University of Alberta, Dept. of Physics, Edmonton, Canada J. P. Carbotte McMaster University, Physics Dept. Hamilton, Canada

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 The Electron–Phonon Interaction: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 3.2.1 Historical Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.2 Electron–Ion Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.3 Migdal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.4 Eliashberg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 The Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.1 Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.2 The Eliashberg Function, ˛ 2F( ): Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 3.3.3 Extraction from Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4 The Critical Temperature and the Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.1 Approximate Solution: The BCS Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.4.2 Maximum Tc , Asymptotic Limits, and Optimal Phonon Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109 3.4.3 Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.4.4 The Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.4.5 The Energy Gap: Dependence on Coupling Strength Tc /!n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.4.6 Optimal Phonon Spectra and Asymptotic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.5 Thermodynamics and Critical Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5.1 The Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5.2 Critical Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.6 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.6.1 Formalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.6.2 BCS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6.3 Eliashberg Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.6.4 Phonon Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.7 Anisotropy and MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Appendix. Microscopic Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.1 Migdal–Eliashberg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.2 The Polaron Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.3 Many Electrons on a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153



A Brief Note on This Revision. The original article has been revised in several areas, mainly concerning the older literature. No attempt has been made to account for all the updates since the first version was written. We have added material related to structure in the bare electronic density of states (EDOS), and concerning anisotropy. In particular the anisotropic aspects of MgB2 have been emphasized.

74

3.1

F. Marsiglio and J. P. Carbotte

Introduction

A fairly sophisticated description of electron–phonon superconductivity has existed since the early 1960s, following the work of Eliashberg [1], Nambu [2], Morel and Anderson [3], and Schrieffer et al. [4]. All of this work extended the original ideas of Bardeen, Cooper, and Schrieffer [5] on superconductivity, to include dynamical phonon exchange as the root cause of the effective attractive interaction between electrons in a metal. For certain superconducting materials, Eliashberg theory (as this description is generally called) provides a very accurate description of the superconducting state. Nonetheless, as B.T. Matthias was fond of iterating [6], this description was never considered (by him and others) particularly helpful for discovering new, high temperature superconductors. Note, however, that Bednorz and M¨uller [7, 8] were motivated by BCS theory and various ideas that would lead to a stronger electron–phonon coupling in metallic oxides. Part of the problem remains that a truly accurate description of the normal state has not been forthcoming. Part of that problem is the “curse” of Fermi Liquid Theory. To the extent that the electron–phonon coupling causes relatively innocuous corrections to most normal state properties, its underlying characteristics remain undetectable (indeed, as will be reviewed here, the characteristics of the electron–phonon interaction are made more apparent in the superconducting state). An exception may be the A15 compounds, whose anomalous normal state properties might help us achieve further understanding of the electron–phonon interaction in these materials [9]. This review will barely touch upon normal state properties influenced by the electron–phonon interaction.A considerable literature continues to develop on this topic, including a more microscopic treatment of model systems with simple electron–ion interactions. There have been many theoretical developments in the last two decades, many of which have been directed towards understanding the high temperature oxides. Some references will be provided in the Appendix, but, for the bulk of the chapter, we will focus primarily on the superconducting state in “conventional” superconductors. In the past, many

reviews have been written on the role of the electron– phonon interaction in superconductors.The reader is directed in particular to the reviews by Carbotte [10], Rainer [11], Allen and Mitrovi´c [12], and Scalapino [13] (they are listed here in inverse chronological order). While we have repeated much of what already exists in these reviews, we felt it was important for completeness in the present volume, and because the material is presented with a slightly different outlook than has been done in the past. The first section provides an overview of the subject as we see it,with some details relegated to the Appendix.This is followed by a discussion of our knowledge of the electron–phonon interaction in metals, including an update on old ideas to use the optical conductivity to extract this information. The next two sections provide a very brief review of the impact of the electron phonon interaction on the superconducting critical temperature, the energy gap, the specific heat, and critical magnetic fields. The next section examines dynamical response functions. Again, largely because of the discovery of the high temperature superconductors, workers were prompted to re-examine in more detail the effect of stronger electron–phonon coupling on various response functions. For example, as will be discussed in the pertinent subsection, the lack of a coherence peak in the NMR relaxation time was observed. Does this (on its own) indicate an exotic mechanism, or can it be explained by damping effects due to substantial electron–phonon coupling? Answers to such questions are reviewed in this section. Finally, we end with a summary, including some remarks on various non-cuprate but non-conventional superconductors. The Appendix will sketch some derivations and provide references to more recent literature.

3.2

The Electron–Phonon Interaction: Overview

3.2.1

Historical Developments

The history of superconductivity is an immense and fascinating subject [14]. While the discovery of superconductivity occurred in 1911 [15], from a theoretical point of view, a first breakthrough occurred

3

with the discovery of the Meissner–Ochsenfeld effect [16], and the understanding that this implied that the superconducting state was a thermodynamic phase [17]. During this time a few attempts were made at proposing a mechanism for superconductivity (see, for example, [18]), but, by 1950, when London’s book [19] appeared, nothing concerning mechanism was really known [20]. In 1950 several important developments took place [21]; first, two independent isotope effect measurements were performed on Hg [22, 23], which indicated that the superconducting transition was intimately related to the lattice, probably through the electron–phonon interaction. These experiments were all the more remarkable because in 1922 Kamerling Onnes and Tuyn had looked for an isotope effect in superconducting Pb, and, within the experimental accuracy of the time, had found no effect [24]. Secondly, Fr¨ohlich [25] adopted, for the first time, a field-theoretical approach to problems in condensed matter. In particular, he studied the electron–phonon interaction in metals, and demonstrated, through second order perturbation theory, that electrons exhibit an effective attractive interaction through the phonons. Although the theory as formulated was incomplete,it did lay the foundations for subsequent work. In fact one of the essential features of this mechanism was summarized in his introduction [25]: “Nor is it accidental that very good conductors do not become superconductors, for the required relatively strong interaction between electrons and lattice vibrations gives rise to large normal resistivity”. His theory correctly produced an isotope effect (recognized in a Note Added in Proof), and, moreover, foreshadowed the discovery of the perovskite superconductors, by suggesting that the number of free electrons per atom should be reduced. After hearing about the isotope effect measurements, Bardeen also formulated a theory of superconductivity based on the electron–phonon interaction, wherein he determined the ground state energy variationally [26]. Both of these theories failed to properly explain superconductivity, essentially because they focussed on the single-electron self-energies, rather than the two-electron instability [21]. Another breakthrough occurred a little later

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when Fr¨ohlich [27] used a self-consistent method to determine an energy lowering proportional to exp (−1/), where  is the dimensionless electronphonon coupling constant. This showed how essential singularities could enter the problem, and why no perturbation expansion in  would succeed in this problem (although in fact the energy lowering is due to a Peierls instability, not superconductivity). A parallel development meanwhile had been taking place in the problem of electron propagation in polar crystals, i.e. the study of polarons. In fact, this problem dates back to at least 1933 [28],when Landau first introduced the idea of a“polarization”cloud due to the ions surrounding an electron, which, among other things, renormalized its properties. Fr¨ohlich also addressed this problem, first in 1937 [29], and then again in 1950 [30]. Lee, Low and Pines [31] subsequently took up the problem, also using fieldtheoretic techniques, to provide a solution to the intermediate coupling polaron problem. This problem was taken on later by Feynman [32], then by Holstein and others [33], along with many others to the present day. In fact, as described in the Appendix, a small group of physicists continues to emphasize polaron physics as being critical to high temperature superconductivity in the perovskites. Pines, having worked with Bohm on electronelectron interactions, and having just used fieldtheoretic techniques in the polaron problem, now combined with Bardeen to derive an effective electron–electron interaction, taking into account both electron–electron interactions and lattice degrees of freedom [34]. The result was the effective interaction Hamiltonian between two electrons with wave vectors k and k  and energies k and k [35]: eff Vk,k  =

4e2 (k − k )2 + k◦2

 2 ! 2 (k − k ) × 1+ , ( k − k )2 − 2 ! 2 (k − k ) 

(3.1)

where k◦ is the Thomas-Fermi wave vector, and !(q) is the dressed phonon frequency. Equation (3.1) is an effective interaction; a more formal and general approach, utilizing Green functions, will be given later. Nonetheless, it is clear that this effective interaction

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F. Marsiglio and J. P. Carbotte

Fig. 3.1. In (a) one electron polarizes the lattice (indicated by dashed circles displaced towards uppermost electron); in (b) that electron has moved away. In the meantime a second electron (seen below in (a)) is attracted to the polarized region, which has remained polarized long after the first electron has left the region. The figure is schematic only, and does not, for example, properly convey the opposite momenta such a pair should possess

captures the essence of “overscreening”, i.e. for electronic energy differences less than the phonon energy, the phonon contribution to the screened interaction has the opposite sign from the electronically screened interaction, and exceeds it in magnitude. Physically [36], one electron makes a transition, which excites a phonon, accompanied by an ionic charge density fluctuation. A second electron undergoes a transition caused by this induced charge

density fluctuation. If the differences in the electron energies is small compared to the phonon excitation energy, the second electron is actually attracted to the first. This is shown in Fig. 3.1. Equation (3.1) represents the starting point for the two-electron interaction in metals. It was further simplified for both the Cooper pair calculation [37] and the Bardeen–Cooper–Schrieffer (BCS) [5] calculation. The progression of events that ultimately led to a successful theory for BCS has been well documented [21].Most of this part of the story had little to do with the details of the attractive mechanism, but rather with the pairing theory itself. Thus, one can divide the theory of superconductivity into two separate conquests: first the establishment of a pairing formalism, which leads to a superconducting condensate, given some attractive particle–particle interaction, and secondly, a mechanism by which two electrons might attract one another. BCS, by simplifying the interaction, succeeded in establishing the pairing formalism. They were able to explain quite a number of experiments, previously performed, in progress at the time of the formulation of the theory, and many that were to follow. However, one might well ask to what extent the experiments support the electron–phonon mechanism as being responsible for superconductivity. The electron–phonon mechanism has been questioned many times over the years since BCS, not just on new superconductors, but on conventional ones as well. Indeed, one of the elegant outcomes of the BCS pairing formalism is the universality of various properties; at the same time this universality means that the theory really doesn’t distinguish one superconductor from another, and, more seriously, one mechanism from another. Fortunately, while many superconductors do display universality, some do not, and these, as it turns out, provided very strong support for the electron–phonon mechanism, as initially motivated by Fr¨ohlich [25] and by Bardeen and Pines [34]. Much of this chapter will be concerned with these deviations from universality. After the BCS paper appeared, several workers rederived their results using alternative formalisms. For example,Anderson used an RPA treatment of the reduced BCS Hamiltonian in terms of pseudospin operators [38], and Bogoliubov and others [39, 40]

3

developed more general methods, later to be adapted to inhomogeneous superconductivity by de Gennes [41]. Finally, Gor’kov [42] developed a Green function method, from which both the BCS results, and the Ginzburg–Landau phenomenology [43] could be derived, near the transition temperature, Tc . The Gor’kov formalism proved to be the most useful,for the purposes of generalizing BCS theory (with its model effective interaction) to the case where the electron–phonon interaction is properly taken into account in the superconducting state. This was done by Eliashberg [1], as well as Nambu [2], and later partially by Morel and Anderson [3] and more completely by Schrieffer and coworkers [4,44,45].Around the same time tunneling became a very useful spectroscopic probe of the superconducting state [46]; besides providing an excellent measure of the gap in a superconductor, it also revealed the fine detail of the electron–phonon interaction [47], to such an extent that tunneling data could be “inverted”to tell us about the underlying electron–phonon interactions [48]. These developments have been well documented in the Parks treatise [49]. In particular retardation effects are covered in the articles by Scalapino [13] and McMillan and Rowell [50]. An interesting historical perspective is provided in the article by Anderson [51]. In the meantime, developments in our understanding of the polaron were occurring in parallel. The problem of phonon-mediated superconductivity and the problem of the impact of electron–phonon interactions on a single electron are obviously related, but, after the initial work by Fr¨ohlich and Pines and coworkers, the two fields seem to have parted ways. Indeed, an excellent summary of the status of polarons at that time is [52], where, however, there is essentially no “cross-talk” with the theory of superconductivity. Similarly, in the treatise by Parks [49] there is essentially no discussion of polarons (only one sentence in the article by Gladstone et al.[53] (see p.684),where,however,the polaron concept is simply dismissed from a discussion of metals),in spite of the fact that the “polaron” really is the essential building block of the BCS theory of superconductivity. So, for example, a perusal of the index of the classic texts on superconductivity,by Schrieffer [44],Blatt [54],Rick-

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77

ayzen [55], de Gennes [41], and Tinkham [56] reveals not a single entry [57]. The reason for this is that the electron–phonon coupling strength in all known superconductors was deemed to be sufficiently weak that the only effect on normal state properties was a slightly increased electron effective mass. Thus, the electronic state is presumed to be well described by Fermi Liquid Theory, upon which the BCS theory (and its modifications) is based. It is important to keep this in mind; for this reason we will refrain from referring to Eliashberg theory as a strong coupling theory (we ourselves have used this term in the past). Eliashberg theory goes beyond BCS theory because it includes retardation effects; however, it is still a weak coupling theory, in the sense that the Fermi energy is the dominant energy, and the quasiparticle picture remains intact. We make this distinction because in recent years polaron theory has experienced a renaissance, and some attempts to explain high temperature superconductivity have utilized polaron and bipolaron concepts. The bipolaron is simply a bound state of two polarons, analogous to the Cooper pair, except that the latter requires a Fermi sea to exist (at least in three dimensions) whereas the former exists as a tightly bound pair in the absence of a Fermi sea. In this respect bipolaron theories resemble the quasichemical theory advocated by Schafroth and coworkers [54,58] in the 1950s. Tightly bound electron pairs are now recognized as the strong coupling limit of the BCS ground state; the transition to the normal state is, however, governed by very different (and as yet undetermined) excitations compared to BCS theory. We will refer to some of this work in the course of this chapter. To complete this brief historical tour, we should add that in 1964, with the suggestion of a theorist [59], what has emerged as a new class of superconductors was discovered [60]. The actual superconducting compound was doped Strontium Titanate (SrTiO3),a perovskite with low carrier density. This compound, along with BaPb0.75 Bi0.25O3 , another doped perovskite discovered in 1975 [61] with a transition temperature of 12 K, were the precursors to the modern high temperature superconductors discovered by Bednorz and M¨uller [7]. In fact, with for-

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tuitous foresight, Schooley et al. [62] remarked, “If SrTiO3 had magnetic properties, a complete study of this material would require a thorough knowledge of all of solid state physics”. Little did they know that in 1986 perovskites would be discovered, that not only had high superconducting transition temperatures, but also exhibited a plethora of magnetic phenomena. We should also note that the so-called cuprates, which presently exhibit superconducting transition temperatures up to 160 K (under pressure), all contain CuO2 layers, whereas the cubic oxides (such as SrTiO3, BaPb0.75 Bi0.25O3 , and Ba1−x Kx BiO3 [63] (with Tc ≈ 30 K)) do not. For this reason many workers have come to regard the layered cuprates and the cubic oxides as belonging to two completely separate (and unconventional) classes, even though they are both essentially low carrier density perovskites. 3.2.2

Electron–Ion Interaction

Overview A useful ab initio theory has to begin from some fundamental starting point. In condensed matter systems the starting point is usually taken to be electrons and ions (with their charges, and masses, etc.) along with the chemical composition of the material [11].Given these ingredients,the prescription for calculation is, in principle, straightforward. One has to solve the many-body Schrodinger equation, with a Hamiltonian consisting of one-body kinetic energy terms and the two-body Coulomb interaction. The form of these terms, along with all the constants involved, are known, so all that is required to solve the problem is perhaps some ingenuity along with unlimited computer resources. This has been referred to by Laughlin as the Condensed Matter version of “The Theory of Everything” (see, for example, [64]). Of course the difficulty is that, even if one could solve this problem, one would not recognize what the solution represented. The notion of ionic collective modes (i.e. phonons), for example, would not be very transparent in such an approach. More obscure still would be the distinction between a superconducting state versus a metallic state.

Instead, an approach which separates the complex many-body problem into smaller, more tractable pieces, has traditionally been adopted in condensed matter, and in particular in the problem of superconductivity [5, 11, 13]. The most systematic approach has been discussed by Rainer [11]. The premise in this approach is the observation that many metals (amongst which many undergo a transition to a superconducting state) are well described by Landau Fermi Liquid Theory. This allows for an asymptotic expansion in small parameters like kB Tc /EF , !phon /EF and 1/kF , where EF (kF ) is the Fermi energy (wavevector), !phon is a typical phonon frequency, and  is the electron mean free path. He separates the problem into the “high energy problem” (effect of Coulomb interactions amongst the electrons themselves as well as between the electrons and the fixed nuclear potentials), and the“low energy problem” (the dressing of conduction electrons with phonons), and the eventual formation of the superconducting state.Most of this review will concern the low energy problem. In our opinion the high energy problem is not at all solved at present, from a truly “ab initio” approach. For example, strictly speaking, one cannot rely on any of the expansion parameters mentioned above, because one does not know, in principle, whether one has a metal with a welldefined Fermi surface, to begin with. Nonetheless, by appealing to experimental observation, one can use for many cases the fact that nature has already solved the high energy problem, and proceed from there to solve the low energy part. This has been the dominant philosophy throughout most of the last four decades towards understanding superconductivity. The difficulty with this approach was exemplified by the discovery of superconductivity in the layered perovskites; band structure calculations for the parent compound (La2 CuO4 ) demonstrated that it was a metal, when in fact the real material was an antiferromagnetic insulator. This problem was later repaired [65], but it remains the case that band structure calculations fail to properly take into account strong Coulomb correlations, and remain somewhat powerless to reliably predict a breakdown of the Fermi Liquid picture.

3

With these caveats, the“ab initio”approach of [11] has experienced excellent success in cases where a metallic state is known to exist, and experimental input has been used in the theory. We will comment in particular on the “low energy” part of the theory later in this chapter. A thorough discussion is available in [11]. Models The net result of a proper handling of the “high energy”problem in the case of a well-behaved metal is a set of input parameters for the low energy problem that are simple enough to make the remaining part of the problem appear to have arisen from a noninteracting model. The distinction is that the input parameters (band structure, phonon spectrum, etc.) come not directly from specified model parameters, but rather from previous calculation and/or experiment. For this reason, we now discuss possible models for the electron–phonon interaction, which, for the moment, we view as fundamental models in their own right,and not as models which somehow parameterize (and disguise) the “high energy problem”. The reason for this is that we hope to accomplish several tasks simultaneously. First, we will in effect work through the “low energy problem” discussed in the previous subsection. Secondly, we will touch upon some of the more recent work on electron– phonon Hamiltonians, which are characterized not so much by comparison with experiment as comparison with some “exact” solution, as attained, for example, by Quantum Monte Carlo methods [66,67]. Thirdly, we will also be able to make contact with recent ongoing work on the polaron (and bipolaron). These latter two topics are presented here more by way of a digression. Some further detail is presented in an Appendix, but for a more thorough discussion the cited literature will have to be consulted. It is always tempting to immediately compare the results of a calculation with experiment; agreement justifies the starting model (in this context this would mean the Hamiltonian, with associated parameters), whereas disagreement would tend to rule out the starting model as a candidate. In the many-body problem, however, life is not so simple. For one thing,

Electron–Phonon Superconductivity

79

we know the starting Hamiltonian, as emphasized in the previous subsection. We will get agreement with experiment if we were only able to routinely calculate any observable. However, in our endeavour to understand many-body systems, we have grown to utilize effective Hamiltonians, which would capture the essence of the phenomenon under investigation. The purpose of this strategy is twofold; we make sense of the many-body system in terms we can understand, and we make the calculation itself more tractable in practice. There are many Hamiltonians in condensed matter physics, which were derived as effective Hamiltonians for some particular problem, but, which have since taken on a life of their own. This is true because (a) they have withstood solution in spite of their simplicity, and (b) they epitomize some qualitative aspect of the more general problem. Famous examples are the Heisenberg/Ising model for spins, and the Hubbard model for fermions with spin degrees of freedom. In the electron–phonon problem several effective models have arisen over the years, the three most prominent of which have been the Fr¨ohlich Hamiltonian [25], the Holstein model [33], and the BLF (Bariˇsi´c–Labb´e–Friedel) model [68] (also known as the SSH (Su–Schrieffer–Heeger) model [69]). The Fr¨ohlich Hamiltonian was derived in a continuum approximation (see [70] or [71] for a derivation), and results in a coupling between the electron density and the ionic momentum (a canonical transformation changes this to the ionic displacement) which diverges as the momentum transfer between electron and ions goes to zero. This Hamiltonian has been the subject of many investigations of the polaron. Holstein proposed his model as a simplification in which the interaction between electron and ion is more local; in fact in some ways the simplification Hubbard [72] invoked to replace the longrange Coulomb interaction is analogous to the simplification that the Holstein model represents compared to the Fr¨ohlich Hamiltonian. Both the Fr¨ohlich and Holstein models represent couplings of the electron to an optical phonon mode. We will focus on the Holstein model since it is particularly amenable to numerical simulations. In contrast, the BLF (SSH) model couples the electron to the relative displace-

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Fig. 3.2. Schematic of ionic displacements in (a) the Holstein model, and (b) the BLF model. In (a) neighbouring chains are distorted in the vicinity of the electron,and in (b) neighbouring ions, when displaced while undergoing oscillations, lead to an increased (or decreased) overlap region (shaded in black), which leads to an altered hopping amplitude for the electron

ment of nearby ions, i.e. an acoustic phonon mode. The physics is simple; in the Holstein model ionic distortions affect the electron energy level at a particular site, while in the BLF model ionic displacements affect the electron hopping amplitude. These are represented pictorially in Fig. 3.2, although of course the coupling is dynamic. The BLF model gained prominence in the 1980s [73] when it was used to describe solitons in conducting polymers; otherwise comparatively little effort has been expended towards an understanding of its properties, particularly in two or three dimensions. The BLF Hamiltonian is H=

 p2 1  2 i + K ui − uj (3.2) 2M 2 i   †  cj + h.c. , − tij − ˛ · (ui − uj ) ci 

where the first line refers to the ions, with mass M and spring constant K. The ionic degrees of freedom are described by the ion momentum, pi , and displacement, ui , at site i. The electrons are described by † creation (annihilation) operators ci (ci ) for an electron with spin  at site i.The electron hopping amplitude is given by tij ; this in turn is modulated by ionic vibrations, and therefore results in the electron–ion coupling with strength |˛|.The coupling constant |˛| is proportional to the gradient of the hopping overlap integral between electron orbitals on two neighbouring sites.

Equation 3.2 gives rise to the standard electron– phonon Hamiltonian, as written in momentum space:  †  1  H= k ck ck + !q a†q aq + √ g(k, k ) N q kk  k 

  × ak−k + a†−(k−k ) ck†  ck .

(3.3)

We have used the conventional oscillator operators,  M!  aq = 2 q uq + ipq and the standard Fourier ex † † pansions, ci = √1N k eik·Ri ck , etc. The phonon dispersion is given by !q , where, in principle, q includes branch indices as well as momenta within the first Brillouin zone, and g(k, k ) is the coupling function.For the BLF Hamiltonian,this coupling function has a very specific form (involving sine functions).A more general consideration of the electron–ion interaction yields a Hamiltonian of essentially the same form [12,13], but where the parameters involved are understood to already contain the “high energy” effects alluded to earlier.State-of-the-art computations of the electron–ion coupling strength, are given, for example, in [74] (for La2−x Srx CuO4 ) and in [75] (and references therein, for A3 C60). The Holstein Hamiltonian is  † H = −t (ci cj + H.c.) 

+

    p2 1 i + Kxi2 − ˛ xi ni , (3.4) 2M 2 i

i

3

where the parameters are as before except that the displacement variable xi represents the (onedimensional) displacement of some optical mode (say a breathing mode) associated with the ith site, and the electron–ion coupling ˛ represents the change in site energy (per unit displacement) associated with this mode.In momentum space this Hamiltonian is particularly simple:  †  H= k ck ck + !E a†q aq  † g  +√ ck , aq + a†−(q) ck+q N kq

(3.5)

£(k, i!m) = −

3.2.3

Migdal Theory

The primary language of many-body systems is the Green function, or propagator. Many books have been written (see for example Refs. [76–81]) about the Green function formalism, so we will bypass a thorough discussion here. A sketch of the derivation of the Migdal [82] equation for the electron selfenergy is given in the Appendix. Migdal argued that all vertex corrections are O(m/M)1/2 compared to the bare vertex, and therefore can be ignored. Here m (M) is the electron (ion) mass. This represents a tremendous simplification, and allows one to solve a theory which should work for arbitrary coupling strength (this is, in fact, not the case, for reasons that will become apparent in the next section). An “exact” formulation of the electron–phonon problem can be summarized [82–84] in terms of the Dyson equations (written in momentum and imaginary frequency space):

−1 G(k, i!m ) = G◦ (k, i!m )−1 − £(k, i!m ) (3.6) for the electron, and

−1 D(q, in ) = D◦ (q, in )−1 − ¢(q, in )

(3.7)

1  Nˇ   k ,m

× gk,k D(k − k , i!m − i!m )G(k , i!m ) (3.8) ×  (k , i!m ; k, i!m ; k − k , i!m − i!m ), and



where !E is the Einstein mode frequency and g ≡ ˛ 2 !E . This model has been studied extensively in 2K the last twenty years, at least partly due to its simplicity. Some of this work is reviewed in the Appendix.

81

for the phonon, where G(k, i!m ) is the one-electron Green function, D(q, in ) is the phonon propagator, and £(k, i!m ) is the electron and ¢(q, in ) the phonon self-energy. Then,

q

k

Electron–Phonon Superconductivity

¢(q, in ) =

2  Nˇ k,m

× gk,k+q G(k + q, i!m + in )G(k, i!m) (3.9) ×  (k + q, i!m + in ; k, i!m; q, in ), where the vertex function  can only be defined in terms of an infinite set of diagrams (i.e. not in closed form). The non-interacting propagators are

−1 G◦ (k, i!m ) = i!m − ( k − )

(3.10)

for the electron and

−1 D◦ (q, in ) = −M(! 2 (q) + n2 )

(3.11)

for the phonon, where k is the single electron dispersion (band indices are implicit here and in the following), is the chemical potential, and !(q) is the phonon dispersion. In writing these relations we have adopted the finite temperature Matsubara formalism, with Fermion (i!m ≡ iT(2m − 1)) and Boson (in ≡ i2Tn) Matsubara frequencies, where m and n are integers and T is the temperature (kB ≡ 1). The Matsubara sums in Eqs. (3.8) and (3.9) extend over all integers, and the momentum sums extend over the first Brillouin zone. This convention will be maintained unless noted otherwise. Migdal’s approximation was to set the vertex function  equal to the bare vertex, g. Then, the electron self-energy can be written:

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F. Marsiglio and J. P. Carbotte

1  2   |gk,k  | Nˇ k ,m × D(k − k  , i!m − i!m )

£(k, i!m ) = −

(3.12)

× G(k , i!m ). Migdal [82] also included renormalization effects in the phonon propagator. With an application to real materials in mind, however, the electron dispersion relations will have been obtained from a band structure calculation, and the phonon properties will generally have been taken from experiment. In this case the phonon self-energy is omitted entirely (to avoid double counting). In addition electron–electron effects have been omitted, as they have been presumed to be included already in the band structure and phonon calculations (to the best extent possible). Alternatively, (3.12) can be viewed as having been derived from some microscopic electron–ion Hamiltonian. For example, in the case of the Holstein Hamiltonian, Eq. (3.4), gk,k → g, the constant appearing in (3.5), and the electron band structure is given by k = −2t cos (kx ) (in one dimension, and for nearest-neighbour hopping only). In addition, the phonon frequency becomes dispersionless (!(q) → !E ) and the phonon self-energy is given by some appropriate approximation. Such an identification is useful for comparison to exact results (usually done numerically – see the Appendix for references). In the classical literature [12, 82, 83, 85, 86], Eq. (3.12) is simplified in the following way. First, very often the phonon propagator is provided separately, usually by inelastic neutron scattering measurements [87, 88]. To see how, one first writes the phonon propagator in terms of its spectral representation [12]: ∞ d B(q,  )

D(q, in ) = 0

2 (in )2 −  2

,

(3.13)

where B(q,  ) is the phonon spectral function B(q,  ) ≡ −

1 ImD(q,  + iı). 

(3.14)

The spectral function is positive definite, and obeys a sum rule; it is the quantity that is constructed with

fits to high-symmetry phonon dispersion curves measured by inelastic neutron scattering [87]. Following this tact a calculation of the phonon selfenergy is no longer required. Another simplification was recognized in [83]; this is the use of the non-interacting electron Green function G◦ (k, i!m ) in the right hand side of (3.12) instead of the full self-consistent choice, G(k, i!m ). This approximation is valid when particle-hole symmetry is present and the infinite bandwidth approximation is invoked. This latter approximation is used extensively in the early literature on metals and superconductors; a systematic explanation of the logic is provided in [12], and requires the usual hierarchy of energy scales, !phon  EF ( ≡ 1). The result is £(k, i!m) =  1  d |gk,k |2 B(k − k ,  ) Nˇ   ∞

(3.15)

k ,m 0

×

2 G◦ (k , i!m ) . (!m − !m )2 +  2

The form of (3.15) allows one to introduce the electron–phonon spectral function, 2 2  ˛k,k  F( ) ≡ N( )|gk,k  | B(k − k ,  ) ,

(3.16)

where N( ) is the electron density of states at the bare chemical potential. At this point one can introduce “Fermi surface Harmonics” [12,89], and define an electron self-energy with Fermi momentum which depends on Matsubara frequency, and on the angle around the Fermi surface. Elastic impurities would act to homogenize the self-energy (as well as other properties), so a more useful function for dirty superconductors is the Fermi-surface-averaged spectral function 1  2 ˛ 2 F( ) ≡ ˛  F( ) N( )2  k,k (3.17) k,k ×ı( k − )ı( k − ) . For many purposes it is the isotropic electronphonon function of Eq. (3.17) which is needed. This function requires a double Fermi surface average over the initial momentum k and final momentum k

3

Electron–Phonon Superconductivity

83

In Fig. 3.3 we show results for ˛k2 F( ) calculated by Leung et al. [90] in a multiple plane wave calculation for Al.Two directions on the Fermi surface defined by the solid angles (, ) are shown. The top frame is for  = 1◦ ,  = 1◦ and the bottom is for  = 15◦ ,  = 23◦ . The solid curves are ˛k2 F( ) and the dashed curves are for comparison. They give the frequency distribution itself, i.e. |gk,k |2 is replaced by unity in the definition [90]. Note the considerable differences between the two frames. This difference leads to the anisotropy in the electron-phonon mass renormalization and in the gap function as one moves along the Fermi surface of Al. To gain an understanding of electron–phonon effects, Englesberg and Schrieffer [83] solved this model for two simple phonon models, the Einstein and Debye models. Here we summarize their results for the Einstein model, with unmodified phonon spectrum, a simpler case since both the phonon spectrum and the bare vertex function are independent of momentum. In this case gk,k ≡ g and B(q,  ) ≡ ı( − !E ). Using, in addition, the prescription  1  → d N( ) (3.19) N k

Fig. 3.3. The calculated spectral function for different momentum directions, ˛k2 F(§) (solid curves) and the directional phonon frequency distribution, Fk (§) (dashed curves) vs. frequency. Reproduced from [90]

states. The reason that the averaging is usually confined to the Fermi surface is that the phonon energy is generally much smaller than the Fermi energy. It should be noted, however, that in metals the di2 rectional electron-phonon spectral density,˛k,k  F( ) of Eq. (3.16) is highly anisotropic in k and k . This anisotropy has two sources. One is the multiple plane wave nature of the electronic states represented by k, and the other is the anisotropy in the phonons themselves,particularly in the phonon Umklapp processes.We can define a spectral density specific to the electron state k by 1  2 ˛k2 F( ) ≡ ˛  F( )ı( k − ). (3.18) N( )  k,k

along with a constant density of states approximation, extended over an infinite bandwidth, one obtains for the electron self-energy ∞ £(i!m ) = !E2 ×

 m

d −∞

1 ˇ

(3.20)

1

!E2 + (!m − !m )2 i!m

1 , − ( − )

where we have used the standard definition for the electron–phonon mass enhancement parameter, : ∞ ≡2

d 0

˛ 2 F( ) , 

(3.21)

which for the Einstein spectrum used here reduces to  = 2N( F )g 2 /!E . (3.22)

k

Performing the Matsubara sum yields

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F. Marsiglio and J. P. Carbotte

!E £(i!m ) = 2

∞ d

(3.23)

−∞

×



n(!E ) + f ( − ) n(!E ) + 1 − f ( − ) + i!m − !E − ( − ) i!m + !E − ( − )



,

where f ( − ) is the Fermi function and n(!E ) is the Bose distribution function. The remaining integral can also be performed [12] £(z) =  ×

−2i(n(!E ) + 1/2) 

+

!E 2

!E − z 1 +i 2 2T



(3.24) 

−

!E + z 1 −i 2 2T

 ,

where (x) is the digamma function [12, 91] and the entire expression has been analytically continued to a general complex frequency z. Because we performed the Matsubara sum first, before replacing i!m with z, this is the physically correct analytic continuation [92]. At zero temperature one can use well-documented properties of the digamma function, or, more simply, refer to the analytic continuation of Eq. (3.23), since the Bose and Fermi functions may be more familiar. Since n(!E ) → 0 and f ( − ) → ( − ) as T → 0 ((x) is the Heaviside step function), the self-energy at T = 0 is   !E !E − z £(z) = ln . (3.25) 2 !E + z Spectroscopic measurements yield properties as a function of real frequency; because of the analytic properties of the Green function, this corresponds to a frequency either slightly above or below the real axis. We will use frequencies slightly above, and designate the infinitesimal positive imaginary part by “iı”. Thus, £(! + iı) =

 





 !E − !  !E  − i(| ! | −!E ) . ln  !E + !  2 (3.26)

The real and imaginary parts of this self-energy are shown in Fig. 3.4, along with the non-interacting inverse Green function (! − ( k − )) to determine the poles of the electron Green function (see Eq. (3.6)) graphically.A quantity often measured in single particle spectroscopies is the spectral function, A(k, !) defined by 1 A(k, !) ≡ − ImG(k, ! + iı). 

(3.27)

With this definition, we obtain, through Eqs. (3.6) and (3.26),    !E  !E − !  A(k, !) = ı ! − ( k − ) − ln  , 2 !E + !  if | ! |< !E , !E /2   E −! ! − ( k − ) − !2 E ln  ! !E +!

=

2 , 2   !E +  2

if | ! |> !E . (3.28) Plots are shown in Fig. 3.5. Each spectral function displays a quasiparticle peak, whose strength ak and frequency !k is implicitly dependent on wavevector −1   , (3.29) ak = 1 + 1 − (!k /!E )2 where !k is the solution (between −!E and !E ) to the zero of the delta-function argument in Eq. (3.28). For all momenta (or equivalently all k − ) there is a solution, whose frequency approaches !E asymptotically as k − → ∞.The weight of this peak starts at the Fermi surface ( k = ) as 1/(1 + ) and quickly goes to zero according to Eq. (3.29) as !k → !E , which occurs for k > 2!E . For larger k a quasipar∼ ticle peak forms once again, albeit with non-zero width, at approximately the non-interacting electron energy, k = . At intermediate k ≈ !E , the quasiparticle picture has broken down, and a description as described here is required for a complete picture. How well the Migdal approximation works in specific circumstances is the subject of ongoing research (see, for example, Refs. [93–97], and the Appendix. For example, Alexandrov et al. [98] found an apparent breakdown (for coupling strengths greater

3

Fig. 3.4. Real and imaginary parts of the electron self-energy in the normal state, for an Einstein spectrum ( = 1). The dotted lines are the inverse non-interacting electron Green functions, ! − ( k − ), for ( k − )/!E = −4, 0, and 4, from top to bottom, respectively

than 1, within the Holstein model) to the approximation when a finite electronic bandwidth was taken into account. Very recent work following this line of thought can be found in Refs. [99–104], for example. We have focussed on the modifications to the electron spectral function due to the electron–phonon interaction. For excitations at the Fermi level ( k =

),the quasiparticle pole remains there (!kF = 0),remains infinitely long-lived (it is a delta-function),but has a reduced weight, by a factor of 1 + . This same factor enhances the effective mass, and alters various normal state properties in a similar way [86, 105]. For example, the low temperature electronic specific heat is linear in temperature with coefficient usually denoted by , which is proportional to the electron density of states. The electron–phonon interaction enhances this coefficient by the same factor, 1 + . Other renormalizations are reviewed in [86].

Electron–Phonon Superconductivity

85

Fig. 3.5. The spectral function for an electron interacting with phonons (Einstein spectrum with  = 1) for various momenta as labelled. Note that for each momentum there is a delta function contribution (artificially broadened in this figure) whose weight diminishes as one moves away from the chemical potential and whose frequency approaches the Einstein phonon frequency. The incoherent component grows with increasing k − , and approaches a reasonably well-defined peak centered around k − for large values (e.g. dot-dashed curve)

3.2.4

Eliashberg Theory

Eliashberg theory is the natural development of BCS theory to include retardation effects due to the “sluggishness” of the phonon response. In fact, insofar as BCS introduced an energy cutoff, !D (the Debye frequency), they included, in the most minimal way, retardation effects. However, Eliashberg theory goes well beyond this approximation, and handles momentum cutoffs and frequency cutoffs separately. We begin this section with a very brief review of BCS theory, followed by a more detailed discussion of Eliashberg theory.

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BCS Theory Before one establishes a theory of superconductivity, one requires a satisfactory theory of the normal state. In conventional superconductors, Fermi Liquid Theory appears to work very well, so that, while we cannot solve the problem of electrons interacting through the Coulomb interaction,experiment tells us that Coulomb interactions give rise to well-defined quasiparticles, i.e. a set of excitations which are in one-to-one correspondence with those of the freeelectron gas. The net result is that one begins the problem with a “reduced” Hamiltonian,  † Hred = k k ck ck (3.30)  † † + kk Vk,k ck↑ c−k↓ c−k ↓ ck ↑ , where, for example, the electron energy dispersion k already contains much of the effect due to Coulomb interactions.The important point is that well-defined quasiparticles with a well-defined energy dispersion near the Fermi surface are assumed to exist, and are summarized by the dispersion k . The pairing interaction V (k, k ) is assumed to be “left-over” from the main part of the Coulomb interaction, and this is the part that BCS simply modelled,based on earlier work by Fr¨ohlich [25] and Bardeen and Pines [34]. Complete derivations of BCS theory have been provided elsewhere in this volume; here we state the final result [44]: 1  ˇEk k k = − , (3.31) Vk,k tanh  N  2Ek 2 k

where Ek =

( k − )2 + 2k

(3.32)

is the quasiparticle energy in the superconducting state, and k is the variational parameter used by BCS. An additional equation which must be considered alongside the gap equation (3.31) is the number equation, 1  k −

ˇEk n=1− . (3.33) tanh N Ek 2 k

Given a pair potential and an electron density, one has to “invert” these equations to determine the variational parameter k and the chemical potential. For

conventional superconductors the chemical potential hardly changes on going from the normal to the superconducting state,and the variational parameter is much smaller than the chemical potential, with the result that the second equation was usually ignored. BCS then modelled the pairing interaction as a negative (and therefore attractive) constant with a sharp cutoff in momentum space: Vk,k ≈ −V (!D − | ( k − ) |)(!D − | ( k − ) |). (3.34) Using this potential in Eq. (3.31), along with a constant density of states assumption over the entire range of integration, we obtain 1 = 

!D

ˇE d tanh , E 2

(3.35)

0

where  ≡ N( )V .At T = 0, the integral can be done analytically to give  = 2!D

exp (−1/) . 1 − exp (−1/)

(3.36)

In weak coupling this becomes the more familiar  = 2!D exp (−1/),

(3.37)

while in strong coupling we obtain  = 2!D .

(3.38)

Both of these results are within the realm of BCS theory (at zero temperature) [106,107], although the latter generally requires a self-consistent solution with the number equation (3.33). Close to the critical temperature, Tc ,the BCS equation becomes 1 = 

ˇ!  D /2

dx

tanh x , x

(3.39)

0

which can’t be solved in terms of elementary functions for arbitrary coupling strength. Nonetheless, in weak coupling, one obtains Tc = 1.13!D exp (−1/), and in strong coupling

(3.40)

3

Tc = !D /2.

(3.41)

It is clear that Tc or the zero temperature variational parameter  depend on material properties such as the phonon spectrum (!D ), the electronic structure (N( )) and the electron–ion coupling strength (V ). However, it is possible to form various thermodynamic ratios, which turn out to be independent of material parameters. The obvious example from the preceding equations is the ratio k2 . In weak couB Tc pling (most relevant for conventional superconductors), for example, we obtain 2 = 3.53, kB Tc

(3.42)

a universal result, independent of the material involved. Many other such ratios can be determined within BCS theory, and the observed deviations from these universal values contributed to the need for an improved formulation of BCS theory. For example, the observed value of this ratio in superconducting Pb was closer to 4.5,a result that is readily understood with Eliashberg theory. It is worth noting that simply extending BCS theory to the strong coupling limit (see Eqs. (3.38) and (3.41) above) results again in a universal constant, k2 = 4, which is the maximum B Tc value attainable within BCS theory with a constant interaction [108], and is still clearly too low. Other aspects of BCS theory, particularly those which prove to inadequately account for the superconducting properties of some materials (notably Pb and Hg) will not be reviewed here. Instead, we will make reference to the BCS limit as we encounter various properties within the experimental or Eliashberg context. Eliashberg Equations In most reviews and texts that derive the Eliashberg equations, the starting point is the Nambu formalism [2]. While this formalism simplifies the actual derivation, it also provides a roadblock to further understanding for the uninitiated. For this reason we have followed the conceptually much more straight forward approach (provided by Rickayzen [55], for example) in the derivation outlined in the Appendix.

Electron–Phonon Superconductivity

87

The result can be summarized by the following set of equations: £(k, i!m) ≡

(3.43)

1  kk (i!m − i!m ) G(k , i!m ) , Nˇ   N( ) k ,m

(k, i!m) ≡ 1 Nˇ

 k  ,m

(3.44) 

kk (i!m − i!m ) − Vkk F(k , i!m ) , N( )

G(k, i!m) =

(3.45)

Gn−1 (−k, −i!m ) , ¯ i!m ) Gn−1 (k, i!m)Gn−1 (−k, −i!m ) + (k, i!m)(k,

with F(k, i!m) =

(3.46) (k, i!m) ¯ + (−k, −i!m )(−k, −i!m )

Gn−1 (k, i!m )Gn−1 (−k, −i!m )

Gn−1 (k, i!m ) = G◦−1 (k, i!m ) − £(k, i!m ).

(3.47)

Another couple of equations identical to Eqs. (3.45) and (3.46), except with ¯ and F¯ instead of  and F, have been omitted; they indicate that some choice of phase is possible,which will be important for Josephson effects [109] but not for what will be considered in the remainder of this chapter. Therefore, we use ¯ = . Note, that G◦−1 (k, i!m) is the inverse of the non-interacting Green function, in which Hartree– Fock contributions from both the electron–ion and electron–electron interactions are assumed to be contained. Following the standard practice we have used a kernel given by ∞ kk (z) ≡ 0 2 ˛kk  F( )

2 2 ˛kk  F( ) d , 2  − z2

(3.48)

where is given by Eq. (3.16). Equations (3.43)–(3.48) have been written in a fairly general way; in this way they can be viewed as having arisen from a microscopic Hamiltonian as in

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F. Marsiglio and J. P. Carbotte

Eqs. (3.2)–(3.4) (although electron–electron interactions have been included in the pairing channel only, and not in the single electron self-energy), or, alternatively, from a treatment of real metals, where, as mentioned earlier, the electron and phonon structure come from previous calculations and/or experiments. These equations emphasize the electron– ion interaction; attempts to explain superconductivity through the electron–electron interactions have been proposed in the past, mainly through collective modes [110–117]; some of these attempts will be treated elsewhere in this volume in the context of high temperature superconductivity. Assuming the electron and phonon structure is given, Eqs. (3.43)–(3.48) must be solved for the two functions, £(k, i!m) and (k, i!m). The procedure is as follows: it is standard practice to separate the self-energy, £(k, i!m ), into its even and odd components [12]:

 1

i!m 1 − Z(k, i!m ) ≡ £(k, i!m ) (3.49) 2  − £(k, −i!m ) , 1

£(k, i!m ) 2  + £(k, −i!m ) ,

(k, i!m ) ≡

where Z and are both even functions of i!m (and, as we’ve assumed all along, k). Then, Eq. (3.43) becomes two equations,

(k, i!m) =

  1  kk (i!m − i!m ) − Vkk Nˇ   N( )

(3.52)

k ,m

×

2 2  !m  Z (k , i!m )

(k , i!m ) .  2 + k − + (k , i!m ) +  2 (k , i!m )

These are supplemented with the electron number equation, which determines the chemical potential, : 2  + G(k , i!m )ei!m 0 , (3.53) n= Nˇ   k ,m

or n=1− ×

2 Nˇ

(3.54)

 2 2  k  ,m !m Z (k , i!m )

k − + (k , i!m ) .  2 + k − + ((k , i!m ) +  2 (k , i!m )

These constitute general Eliashberg equations for the electron–phonon interaction, in which electron– electron interactions enter explicitly only in the pairing equation. Very complete calculations of these functions (linearized, for the calculation of Tc ) were carried out for Nb by Peter et al. [118], and for Pb by Daams [119]. The more standard practice is to essentially confine all electronic properties to the Fermi surface; then only the anisotropy of the various functions need be considered. Often these are simply averaged over (due to impurities, for example), or the anisotropy may be very weak and therefore neglected. In this case Eqs. (3.50)–(3.54) can be written  Zm = 1 + T (i!m − i!m ) (3.55) m

1  kk (i!m − i!m ) Z(k, i!m ) = 1 + Nˇ   N( )

(!m /!m )Zm A0 (m ) , × 2 2 2 !m Zm + m

(3.50)

k ,m



×

 !m /!m Z(k , i!m ) ,  2 2 2   !m +  2(k  , i!m )  Z (k , i!m ) + k  − + (k , i!m )

1  kk (i!m − i!m ) (k, i!m) = − Nˇ   N( )

(3.51)

k ,m

k − + (k , i!m ) ×  2 2 2   !m +  2 (k , i!m )  Z (k , i!m ) + k  − + (k , i!m )

along with the gap equation (3.44):

m = −T



(i!m − i!m )A1 (m )

,(3.56)

m

m = T



 (i!m − i!m ) − N( )Vcoul (3.57)

m

m A0 (m ) , × 2 2 2 !m Zm + m  A1 (m ) , n = 1 − 2TN( ) m

(3.58)

3

where we have adopted the shorthand Z(i!m ) = Zm , etc, (z) and Vcoul represent appropriate Fermi surface averages of the quantities involved, and the functions A0 (m ) and A1 (m ) are given by integrals over appropriate density of states, using the prescription (3.19) to convert from Eqs. (3.50)–(3.54) to Eqs. (3.55)–(3.58). If the electron density of states is assumed to be constant, then, with the additional approximation of infinite bandwidth, A0 (m ) ≡ 1 (actually a cutoff, (!c − | !m |), is required in Eq. (3.58)), and A1 (m ) ≡ 0. This last result effectively removes m (and Eqs. (3.56) and (3.58) ) from further consideration. An earlier review by one of us [10] covered the consequences of the remaining two coupled equations in great detail. All of the equations discussed so far have been developed on the imaginary frequency axis. Because practitioners in the field at the time were interested in tunneling spectroscopy measurements [47], the theory was first developed on the real frequency axis [4,45]. The resulting equations are complicated, even for numerical solution. It wasn’t until quite a number of years later that numerical work returned to the imaginary axis [120], where, for thermodynamic properties, the numerical solution was very efficient [121–124]. The difficulty, however, was that imaginary axis solutions are not suitable for dynamical properties. We will return to the interplay between imaginary and real frequency axis solutions as we encounter them throughout the chapter. The other complication we have mentioned is an energy variation in the EDOS, as seems to exist in some A15 compounds. If this energy dependence occurs on a scale comparable to !D , then N( ) cannot be assumed to be constant, and cannot be taken outside of the integrals in Eqs. (3.50)–(3.54). Such EDOS energy dependence is thought to be responsible for some of the anomalous properties seen in A15 compounds — their magnetic susceptibility and Knight shift [125], and the structural transformation from cubic to tetragonal [126–128]. Several electronic band structure calculations [129–132] also find sharp structure in N( ) at the Fermi level.An accurate description of the superconducting state thus requires a proper treatment of this structure. This was first undertaken to understand Tc by Horsch

Electron–Phonon Superconductivity

89

and Rietschel [133] and independently by Nettel and Thomas [134]. A more general approach to understanding the effect of energy dependence in N( ) on Tc was given by Lie and Carbotte [135], who formulated the functional derivative ıTc /ıN( ); they found that only values of N( ) within 5 to 10 times Tc around the chemical potential have an appreciable effect on the value of Tc . More specifically they found that ıTc /ıN( ) is approximately a Lorentzian with center at the chemical potential; the function becomes negative only at energies | − | > 50Tc . ∼ Irradiation damage experiments illustrate some of this dependency. For example, irradiation of Mo3Ge causes an increase in Tc [136]. Washing out gap anisotropy with the irradiation cannot possibly account for an increase in Tc ; instead, this result finds a natural explanation in the fact that the chemical potential for Mo3Ge falls in a valley [137] of the EDOS, and irradiation smears the EDOS, thus increasing N( ), and hence Tc . In what follows we will sketch some of the theory behind this work; More details on the formulation of Eliashberg theory with an energy dependent N( ) can be found in the work of Pickett [138] and Mitrovi´c and Carbotte [139], and references therein. To treat the possible energy dependence in the band structure EDOS one can return to Eqs. (3.43)– (3.47). Beginning with the normal state, already an important point is apparent: namely the Green function and self-energy must be solved self-consistently when there is a significant energy dependence in the bare EDOS.For example,in Eq.(3.43),if anisotropy in the interaction is neglected,then the resulting k -sum can be converted into a single integral over energy  with the integrand weighted by the EDOS N(  ). In the zero temperature limit, one obtains, on the real axis, +∞ ∞ d!  N˜ (! ) d§ ˛ 2F(§)

£(! + iı) =  ×

−∞

(3.59)

0

 (−!  ) (!  ) + , ! − !  − § + iı ! − !  + § + iı

where the renormalized EDOS is defined by   +∞ 1 N( ) ˜ − Im G( , ! + iı) . (3.60) d N(!) = N( )  −∞

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F. Marsiglio and J. P. Carbotte

In principle the interaction, though isotropic, could in fact depend on energy; this leads naturally to the electron-phonon spectral function ˛ 2F( , , §) =

2    ˛k,k  F(§)ı( − k )ı( − k ) k,k

N( )N( )

. (3.61)

Usually the dependence on and  is not significant so that often these energies are simply evaluated at the Fermi energy, and the Eq. (3.59) results. The reader is referred to the work of Mitrovi´c and Carbotte [140] for details concerning the effects of the energy dependence in Eq. (3.61) on superconducting properties. Equations (3.59) and (3.60) must be solved selfconsistently. Solutions for a model bare density of states N( ) (used to mimic the calculated bare EDOS for Nb3 Sn [129]) are given in [139, 141]. The model (dotted curve) along with the calculated renormalized EDOS (solid curve) is shown in Fig. 3.6 in an energy range near the Fermi energy.The model consists of a triangular peak superposed on a constant background. The calculation uses an electron-phonon spectral function extracted from tunneling measurements [142]. Note that the calculated renormalized EDOS is sharper than the bare EDOS. Also, there are

Fig. 3.6. The renormalized EDOS (solid curve) and the model bare EDOS (normalized to its value at the bare chemical potential) (dotted curve) vs. energy near the chemical potential (! = 0). The solid circle indicates the (bare) chemical potential corresponding to the bare EDOS at zero temperature. The actual chemical potential for the interacting system has been defined to be zero.Reproduced from [139]

slight modulations in the ±20−30 meV region which correspond to phonon structure. Finally, note that the bare EDOS does not have particle-hole symmetry; this results in a shift in the chemical potential (from the filled dot, which corresponds to the chemical potential for a given electron density in the bare EDOS, to zero, which is the chemical potential corresponding to the renormalized EDOS). The shift is given by the value of the real part of £(! + iı) at zero frequency. More detailed structure is visible in the selfenergy. The real and imaginary parts of the selfenergy are shown in the top and bottom frames of Fig. 3.7, respectively. The solid curves result from the full calculation, consistent with Fig. 3.6. The dotted curves are obtained with a constant EDOS with value equal to that of the bare EDOS at the chemical potential. These curves have electron-hole symmetry,

Fig. 3.7. The real (top frame) and imaginary (bottom frame) parts of the electron–phonon self-energy computed for the model EDOS in Fig. 3.6 (solid curve) and for a constant EDOS (dotted line) with value equal to that at the chemical potential. Reproduced from [139]

3

Electron–Phonon Superconductivity

91

while the solid curves obviously do not. For example, in the real part of the self-energy the maximum value at negative frequency is clearly much higher in magnitude than the minimum value at positive frequency. Also note that Re £(! + iı) for ! = 0 is no longer zero. For this case it has a significant value of 6.43 meV. Another noteworthy result is that +∞ d

1 ˜ N(! = 0) = − Im 

N( )/N( ) − − Re£(0 + iı) + iı

−∞

= N( = −Re£(0 + iı))/N( ).

(3.62)

The value of the renormalized EDOS at the chemical potential remains equal to unity. This means that the chemical potential is shifted by an amount equal to Re £(0 + iı) with respect to the “bare” chemical potential ( ). As in the case with a constant bare EDOS, here we can define a frequency dependent mass renormalization parameter (!) and a damping factor  (!) through the relations (!) ≡ −

@ Re£(! + iı) @!

(3.63)

and  (!) ≡ |Im£(! + iı)|.

(3.64)

In Fig. 3.8 we show (!) vs. ! for the parameters discussed above. Note the lack of particle-hole symmetry. According to the prescription (3.63) (0) = 1.33, which is 22% smaller than that given by the standard definition, Eq. (3.21) (which gives 1.7 for the Nb3 Sn spectrum). For a constant bare EDOS these two definitions would agree. One can also derive an expression for  (!) (at T = 0):

Fig. 3.8. The electron–phonon renormalization parameter as a function of frequency near the chemical potential, at zero temperature. Reproduced from [139]

imaginary part of the self-energy gives ˛ 2F(!) directly. This is no longer true for a non-constant bare d EDOS, as shown in Fig. 3.9 where we plot d! | (!)| vs.!.Once again the result is asymmetric,and differs considerably from the actual ˛ 2 F(!), particularly for |!| > ∼ 20 meV, where the derivative is negative. More details are provided in [139]. Results concerning far-infrared spectroscopy, obtained by Mitrovi´c and Perkowitz [143] for V3Si thin films in the normal state, will be discussed in a later section following the “standard” description of far-infrared spectroscopy.

|!|  (!) =  d! 

0  × (!)N˜ (!  ) + (−!)N˜ (−!  )

(3.65)

× ˛ 2 F(|!| − !  ), which reduces to the “standard” formula when ˜ N(!) = 1. Then the frequency derivative of the

Fig. 3.9. The derivative of the electron scattering rate with respect to frequency as a function of frequency at zero temperature. Reproduced from [139]

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F. Marsiglio and J. P. Carbotte

3.3

The Phonons

3.3.1

Neutron Scattering

When dealing with model Hamiltonians, the phonon dispersion relations (before interaction with the electrons) are generally given, and simple: they are Einstein modes, or Debye-like modes, for example. A notable exception is the case where the model contains anharmonic forces,in which case even the“noninteracting” phonon spectrum is unknown. In the case of real solids, and in particular metals, the situation is much worse. In this case the electrons cannot be ignored, though they can be treated in the Born–Oppenheimer approximation. Nonetheless the results require parametrization (with input from other experiments) and are generally not reliable. Pseudopotential methods [144,145] can be applied to this problem, again, with limited success. In contrast, the spectacular success of inelastic neutron scattering techniques [87,88] to simply measure the phonon dispersion curves in real metals effectively eliminates the need to calculate them quantitatively. Various qualitative effects, like the impact of electronic screening on the long wavelength ionic plasma mode [146], as well as the existence of Kohn anomalies [147], all due to the presence of electrons, are understood theoretically. For detailed results, however, Born–von Karman fits to high symmetry phonon dispersions suffice for an excellent description of the low temperature phonon properties. At temperatures of order 10 K, the phonons in most conventional superconductors are completely determined, and no longer changing with temperature. Hence, as far as understanding (low temperature) superconductivity is concerned, these higher temperature measurements are sufficient. The measured dispersion curves, !q (again, branch indices are suppressed), are summarized in the frequency distribution 1  F( ) = ı( − !q ), (3.66) N q where N is the number of ions in the system, and q is a wavevector which ranges over the entire First Brillouin Zone (FBZ),(and implicitly contains the branch index). It should be stressed that this procedure is

Fig. 3.10. A set of “constant q” scans in Pb taken at various points along the diagonal in the Brillouin zone.Reproduced from [87]

an idealization; in actual fact a set of “constant q” scans are performed (usually along high symmetry directions). A typical result [87] is shown in Fig. 3.10 for Pb, for a set of wavevectors along the diagonal in reciprocal space. Note that the neutron counts tend to form a peak as a function of energy transfer (to the neutron),  . In general these peaks have a finite width, i.e. broader than the spectrometer resolution; these are due to a variety of effects, for example, anharmonic effects. Nonetheless, because the peaks are relatively sharp compared to the centroid energy, (i.e. the phonon inverse lifetimes are small compared to their energies), these data are usually presented in the form of Fig. 3.11, as a set of disper-

3

Electron–Phonon Superconductivity

93

Fig. 3.11. The dispersion curves for Pb at 100 K, as a function of momentum along various high symmetry directions. Reproduced from [87]

sion curves. Figure 3.11 does obscure, however, the lifetimes of the various phonons,and hence the validity of Eq. (3.66), where infinitely long-lived phonons are assumed throughout the Brillouin zone, is called into question. Nonetheless, for most of the Brillouin zone the approximation of infinitely long-lived excitations is a good one (hence, the name, phonon), and so the spectrum of excitations can be constructed according to Eq. (3.66). Such a procedure relies on coherent neutron scattering.An alternative is to use incoherent neutron scattering, whereby one measures the spectrum more or less directly. This latter procedure has advantages over the former, but also includes multiphonon scattering processes, and for non-elemental materials, weighs the contribution from each element differently, according to their varying scattering lengths. The result is often denoted the “general-

ized density of states” (GDOS). A comparison for a Thallium-Lead alloy is shown in Fig. 3.12 [148,149]. Also shown is the result from tunneling, to be discussed in the next subsection. There is clearly good agreement between the various methods. Amongst the two neutron scattering techniques, inelastic coherent neutron scattering produces the sharpest features, but requires a model (i.e. a Born–von Karman fit) to extract the spectrum F( ) from the dispersion curves measured along high symmetry directions. 3.3.2

The Eliashberg Function, ˛2 F(): Calculations

First-principle calculations of the electron–phonon spectral function, ˛ 2F( ) require a knowledge of the electronic wave functions,the phonon spectrum,and the electron–phonon matrix elements between two single–electron Bloch states. A fairly comprehensive

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F. Marsiglio and J. P. Carbotte

Fig. 3.12. The electron–phonon spectral function ˛ 2 F(!) (solid curve) for Pb.40 Tl.60 determined from tunneling experiments and convoluted by instrument resolution of the neutron spectrometer compared with the neutron results for the phonon frequency distribution F(!) (dashed curve) measured by incoherent inelastic neutron scattering [149] (upper frame). The lower frame shows the tunneling results (solid curve) compared with the phonon frequency distribution (dashed curve) determined from a Born von Karman analysis of the phonon dispersion curves in Pb.4 Tl.6 [148]

review is given in [86]. For our purposes, we note that, since the phonon spectrum will come from experiment, Eq. (3.16) requires calculation of gk,k . It is [10,86] gk,k j = < k | j (k − k ) · ∇V | k >  1/2  (3.67) × ,  2M!j (k − k ) where,for this equation we have included the phonon branch index j explicitly. The Bloch state is denoted | k >, and j (k) is the polarization vector for the (jk)th phonon mode.The crystal potential is denoted V , and as one might expect, the electron–phonon coupling depends on its gradient. Tomlinson and Carbotte [150] used pseudopotential methods [151, 152] to compute gk,k j and, from Eq. (3.16), ˛ 2F( ), for Pb. The phonons were taken from experiment [87,88,153,154] through Born–von K´arm´an fits. The result is plotted in Fig. 3.13, along

with results from tunneling experiments (to be described below). The agreement is qualitatively very good; this provides very strong confirmation of the electron–phonon mechanism of superconductivity. Further details of more modern calculations of electron–phonon coupling constants can be found in, for example, Refs. [74] and [75] and references therein. Their reliability appears to remain an issue, both with the high temperature cuprates, and perhaps less so with the fulleride and more conventional superconductors. The spirit of these calculations is somewhat different than the older ones, in that coupling constants are extracted from the phonon linewidths, where it is assumed that the phonon broadening is entirely due to the electron– ion interaction (and not, say, anharmonic effects). Allen [155, 156] derived a formula (Fermi’s Golden Rule) for the inverse lifetime, q ( ), of a phonon with momentum (and branch index) q:

3

Fig. 3.13. The electron–phonon spectral function ˛ 2 F(!) measured in tunneling experiments (dotted curve) compared with that which is calculated from first principles (solid curve) [150]

q = 2!q 



|gk,k+q |

k

 (3.68)

where again we have suppressed both phonon branch indices and electron band labels. Using this equation, in the approximation that the expres

 sion f ( k+q − ) − f ( k − ) /(!q ) is replaced by ı( k − ) makes it resemble Eq. (3.17), one can write 1 1  1 q ˛ 2 F( ) = ı( − !q ) N( ) N q 2 !q =

1 1 !q q ı( − !q ) , 3N q 2

(3.69)

where the second line serves to define a q-dependent coupling parameter:

95

Fig. 3.14. A histogram of phonon linewidths in Nb. The linewidths represent the full width at half-maximum for both longitudinal modes (solid) and transverse modes (dashed and dotted). The circles and triangles are from experiment. Reproduced from [157]

q ≡

2

f ( k+q − ) − f ( k − ) × !q ×ı( k+q + !q − k ) ,

Electron–Phonon Superconductivity

q 3 . N( ) !q2

(3.70)

It is through these relations that coupling parameters are often determined. Butler et al. [157, 158] have made extensive calculations of electron-phonon linewidths in Nb. Results of their calculated phonon lifetimes, based on extended band structure calculations, are presented in Fig. 3.14 for two high symmetry directions, [, 0, 0] and [, , 0] for transverse (dashed and dotted curves) and longitudinal (solid curve) branches. A comparison with experimental results is also given. The reader is referred to their papers for details. The electron-phonon spectral density ˛ 2F( ) obtained in this way through Eq. (3.69) is shown in Fig. 3.15 (histogram), where it is compared with an early experimental spectral function extracted from tunneling inversion by Robinson and Rowell [159] (solid curve).

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F. Marsiglio and J. P. Carbotte

Fig. 3.15. Calculated (histogram) and experimental (solid curve) electron–phonon spectral function, ˛ 2 F(§) for Nb. Reproduced from [157]

It is worth noting at this point that several moments of the function ˛ 2F( ) have played an important role in characterizing retardation (and strong coupling) effects in superconductivity. Foremost amongst these is the mass enhancement parameter, , already defined in Eq. (3.21); in addition, the characteristic phonon frequency, !ln is given by ⎤ ⎡ ∞  2 ˛ F( ) ⎦ 2 d ln ( ) !ln ≡ exp ⎣ . (3.71)   0

Further discussion of these calculations can be found in Refs. [10,86]. 3.3.3

Extraction from Experiment

Experiments which probe dynamical properties do so as a function of frequency, which is a real quantity.However,the Eliashberg equations as formulated in the previous section are written on the imaginary frequency axis. To extract information from these equations relevant to spectroscopic experiments,one must analytically continue these equations to the real frequency axis. Mathematically speaking, this is not a unique procedure; one can often imagine several functions whose values on the imaginary axis are

equal, and yet differ elsewhere in the complex plane (and in particular on the real axis). For example, replacing unity by − exp (ˇi!m ), in any number of places in the equations does not affect the imaginary axis equations, or their solutions, and yet on the real axis the corresponding number of factors − exp (ˇ!) will appear. Physically speaking, however, the Green functions involved have to satisfy certain conditions; complying with these conditions determines the function uniquely [92]. This allows a unique determination of the analytic continuation of the Eliashberg equations on the real axis. This procedure will be discussed in the following subsection, followed by subsections on experimental spectroscopies, and how they can be used to extract the Eliashberg function, ˛ 2F( ). The Real-Axis Eliashberg Equations We begin with Eqs. (3.43)–(3.47). To analytically continue Eqs. (3.46)–(3.47) is trivial; one simply replaces the imaginary frequency i!m wherever it appears with ! + iı. The iı remains to remind us that we are analytically continuing the function to just above the real axis; it is important to specify this since there is a discontinuity in the Green function as one crosses the real axis.A simple replacement of i!m with ! + iı in Eqs. (3.43) and (3.45) (leaving the summations over m ) would in general be incorrect. The correct procedure is to first perform the Matsubara sum, and then make the replacement. To perform the Matsubara sum, however, one has to introduce the spectral representation for the Green functions, G and F. These are given by ∞ G(k, i!m) =

d!

A(k, !) , i!m − !

(3.72)

d!

C(k, !) , i!m − !

(3.73)

−∞ ∞

F(k, i!m) = −∞

where A(k, !) is given by Eq. (3.27) and C(k, !) is given by a similar relation: 1 C(k, !) ≡ − ImF(k, ! + iı). 

(3.74)

3

The spectral representation for the phonons is already present in Eqs. (3.43) and (3.45). Therefore the Matsubara sum can be performed straightforwardly (see, for example, [12, 81]), and the analytical continuation can be done. Upon integrating over momentum (using, as in Eqs. (3.55)–(3.58) electronhole symmetry and a constant (and infinite in extent) density of electron states),one arrives at the standard real-axis Eliashberg equations [4, 12]. These equations are much more difficult to solve than the imaginary axis counterparts.They require numerical integration of principal value integrals and square-root singularities, and the various Green function components are complex. In contrast the imaginary axis equations are amenable to computers (the sums are discrete) and the quantities involved are real. Moreover a considerable number of thermodynamic and magnetic properties can be obtained directly from the imaginary axis solutions. The discrepancy in computational ease between the two formulations led to an alternative path to dynamical information, namely the direct analytic continuation of the solutions of the imaginary axis equations to the real axis by a fitting procedure with Pad´e approximants [160]. This method is in general very sensitive to the input data, and has (surmountable [161,162]) difficulties at high temperatures and frequencies. More recently yet another procedure was formulated [163], which first requires a numerical solution of the imaginary axis equations, followed by a numerical solution of analytic continuation equations. This latter set is formally exact (i.e. no fitting required) and yet avoids the complications of the realaxis equations. These equations are £(k, z) =

1 Nˇ

∞  k  m =−∞ ∞

kk (z − i!m ) G(k , i!m ) N( )

˛ 2  F( ) 1  (3.75) d kk + N  N( ) k 0   × f ( − z) + N( ) G(k , z −  ) 







+ f ( + z) + N( ) G(k , z +  ) ,

Electron–Phonon Superconductivity

1 (k, z) = Nˇ

97

  ∞  kk (z − i!m )  − Vkk N( )  

k m =−∞

× F(k , i!m ) ∞ ˛ 2  F( ) 1  (3.76) d kk + N  N( ) k 0   × f ( − z) + N( ) F(k , z −  )    + f ( + z) + N( ) F(k , z +  ) , where z can actually be anywhere in the upper halfplane. Thus, for example, Eqs. (3.43) and (3.45) can be recovered by substituting z = i!m . On the other hand, once these equations have been solved, one can substitute z = ! + iı, and iterate the resulting equations to convergence. When the “standard” approximations for the momentum dependence are made (i.e. Fermi surface averaging, constant density of states, particle-hole symmetry, etc.) the result is Z(! + iı) = 1 + × i + ! ×

∞ iT  (! − i!m ) ! m=−∞

!m Z(i!m ) 2 Z 2 (i! ) !m m

∞

+  2 (i!m )

(3.77)



 d ˛ F( ) N( ) + f ( − !) 2

0

(! −  )Z(! −  + iı)

−  + iı) −  2(! −  + iı)

 + N( ) + f ( + !) ×

×

(! −

 )2 Z 2 (!



(! +  )Z(! +  + iı) (! +  )2 Z 2 (! +  + iı) −  2(! +  + iı)

,

(! + iı) = ∞  

T (! − i!m ) − ∗ (!c )(!c − |!m |) m=−∞

×

(i!m ) 2 Z 2 (i! ) +  2 (i! ) !m m m

(3.78)

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F. Marsiglio and J. P. Carbotte



∞ d ˛ 2F( )

+ i

 N( ) + f ( − !)

0

(! −  + iı) × 2 2 (! −  ) Z (! −  + iı) −  2 (! −  + iı)

 + N( ) + f ( + !) (! +  + iı)

× (! +  )2 Z 2 (! +  + iı) −  2 (! +  + iı)

 .

Note that in cases where the square-root is complex, the branch with positive imaginary part is to be chosen. One important point has been glossed over in these derivations. Because of the infinite bandwidth approximation, an unphysical divergence occurs in the term involving the direct Coulomb repulsion, Vk,k , both in the imaginary axis formulation, (3.58), and in the real-axis formulation, Eq. (3.78). The solution to this difficulty is to introduce a cutoff in frequency space (even though the original premise was that the Coulomb repulsion was frequency independent), as is apparent in the two equations. In fact, this cutoff should be of order the Fermi energy, or bandwidth. However, this requires a summation (or integration) out to huge frequency scales. In fact one can use a scaling argument [3,164,165] to replace this summation (or integration) by one which spans a small multiple (≈ 6) of the phonon frequency range. Hence the magnitude of the Coulomb repulsion is scaled down, and becomes [164]

tion is identically zero and the Fermi function becomes a step function: f ( − !) → (! −  ). Once the imaginary axis equations have been solved, solution of Eqs. (3.77) and (3.78) no longer requires iteration. One can simply build up the solution by construction from ! = 0 (assuming ˛ 2F( ) has no weight at  = 0); in fact, if the phonon spectrum has no weight below a frequency, min , then only the first lines in Eqs. (3.77) and (3.78) need be evaluated. In particular, if the gap (still to be defined) happens to occur below this minimum frequency (often a good approximation for a conventional superconductor) then the gap can be obtained in this manner.1 In the following two sections we explore the possibility of using Eqs. (3.77) and (3.78) to obtain information about the microscopic parameters of Eliashberg theory. Tunneling Perhaps the simplest, most direct probe of the excitations of a solid is through single particle tunneling. In this experiment electrons are injected into (or extracted from) a sample, as a function of bias voltage, V . The resulting current is proportional to the superconducting density of states [46,168–170]:    IS (V ) ∝ d!Re √ 2|!| 2 ! − (!) (3.80)

 f (!) − f (! + V ) ,

(3.79)

where we have used the gap function, (!), defined as (!) ≡ (! + iı)/Z(! + iı). (3.81)

where U is a double Fermi surface average of the direct Coulomb repulsion. This reduction is correct physically, in that the retardation due to the phonons should reduce the effectiveness of the direct Coulomb repulsion towards breaking up a Cooper pair. It does appear to overestimate this reduction, however [166]. The analytic continuation of this part of the equations has been treated in detail in [167]. In the zero temperature limit, Eqs. (3.77) and (3.78) are particularly simple. Then the Bose func-

The proportionality constant contains information about the density of states in the electron supplier (or acceptor), and the tunneling matrix element. These are usually assumed to be constant. If one takes the zero temperature limit,then the derivative of the current with respect to the voltage is simply proportional to the superconducting density of states, %      dI dI |V | , (3.82) / = Re  dV S dV N V 2 − 2 (V )

∗ (!c ) ≈

1

N( )U , 1 + N( )U ln ! Fc

The reader may notice that this amounts to simply changing the imaginary frequency i!m where it occurs in Eqs. (3.55) and (3.56) to ! before doing the Matsubara sum.

3

Fig. 3.16. I-V characteristic of a Pb-I-Pb junction showing the construction used to find the energy gap. The solid line and open circles are the current in the normal and superconducting states, respectively. Reproduced from [50]

Electron–Phonon Superconductivity

where S and N denote “superconducting” and “normal” state, respectively. The right-hand side of Eq. (3.82) is simply the density of states, computed within the Eliashberg framework (see, for example, [50]). It is not at all apparent what the structure of the density of states is from Eq. (3.82), until one has solved for the gap function from Eqs. (3.77), (3.78) and (3.81). At zero temperature the gap function (!) is real and roughly constant up to a frequency roughly equal to that constant. This implies that the density of states will have a gap, as in BCS theory. At finite temperature the gap function has a small imaginary part starting from zero frequency (and, in fact the real part approaches zero at zero frequency [171]) so that in principle there is no gap, even for an s-wave order parameter. In practice, a very well-defined gap still occurs for moderate coupling, and disappears at finite temperature only when the coupling strength is increased significantly [172,173]. In Figs. 3.16 and 3.17 we show the currentvoltage and conductance plots for superconducting Pb, taken from McMillan and Rowell [50]. These data were obtained from a superconductor-insulatorsuperconductor (SIS) junction, with Pb being the superconductor on both sides of the insulating barrier, so that, rather than directly using Eq. (3.82), the current is given by a convolution of the two superconducting densities of states. Two features immediately stand out in these plots. First, a gap is clearly present in Fig. 3.16, given by 2◦ , where ◦ is the single electron gap defined by ◦ ≡ Re(! = ◦ ),

Fig. 3.17. Conductance dI/dV of a Pb-I-Pb junction in the superconducting state normalized by the conductance in the normal state vs. voltage. Also shown is the twosuperconductor conductance calculated from the BCS density of states which contains no phonon structure. Reproduced from [50]

99

(3.83)

a definition one can use for all temperatures. Secondly, a significant amount of structure occurs beyond the gap region, as is illustrated in Fig. 3.17. McMillan and Rowell were able to deconvolve their measurement, to produce the single electron density of states shown in Fig. 3.18. Since the superconducting density of states is given by the right hand side of Eq. (3.82), the structure in the data must be a reflection of the structure present in the gap function, (!). The structure in the gap function is in turn a reflection of the structure in the input function, ˛ 2 F( ). In other words, Eqs. (3.77) and (3.78) can be viewed as a highly nonlinear

100

F. Marsiglio and J. P. Carbotte

Fig. 3.18. Electronic density of states N(E) vs. E − ◦ for Pb, obtained from the data of Fig. 3.17. The smooth curve is the BCS density of states. Reproduced from [50]

transform of ˛ 2 F( ). Thus the structure present in Fig. 3.18 contains important information (in coded form) concerning the electron–phonon interaction. One has only to “invert”the“transform”to determine ˛ 2 F( ) from the tunneling data. This is precisely what McMillan and Rowell [48, 50] accomplished, first in the case of Pb. The procedure to do this is as follows. First a “guess”is made for the entire function, ˛ 2 F( ), and the Coulomb pseudopotential parameter, ∗ . Then the real axis Eliashberg equations (Eqs. (3.76 and (3.77)) are solved, and the superconducting density of states (Eq. (3.82)) is calculated. The result attained will in general differ from the experimentally measured function (represented, for example, by Fig. 3.18); a Newton–Raphson procedure (using functional derivatives rather than normal derivatives) is used to determine the correction to the initial guess for ˛ 2F( ) that will lead to better agreement. Very often another parameter (for example, the measured energy gap value) is used to fit ∗ . This process is iterated until convergence is achieved. The result for Pb is illustrated by the dotted curve in Fig. 3.13. Once ˛ 2 F( ) (and ∗ ) has been acquired in this way one can use the Eliashberg equations to calculate other properties, for example, Tc . These can then be compared to experiment, and the agreement in gen-

Fig. 3.19. The predicted (solid curve) normalized density of states in Pb as a function of energy ! compared with measured values (open dots) as a function of energy measured from the gap edge. The measured density of states divided by the BCS density of states above 11 meV was not used in the fitting procedure that produced ˛ 2 F(!) and a comparison of theory and experiment in the multiple-phonon region is a valid test of the theory. Reproduced from [50]

eral tends to be fairly good. One may suspect, however,a circular argument,since the theory was used to produce the spectrum (from experiment), and now the theory is used as a predictive tool, with the same spectrum. There are a number of reasons, however, for believing that this procedure has produced meaningful information. First, the spectrum attained has come out to be positive definite, as is required physically. Second, the spectrum is non-zero precisely in the phonon region,as it should be.Moreover,it agrees very well with the calculated spectrum. Thirdly, as already mentioned, various thermodynamic properties are calculated with this spectrum, with good agreement with experiment. Finally, the density of states itself can be calculated in a frequency regime beyond the phonon region, as is shown in Fig. 3.19. The agreement with experiment is spectacular. None of these indicators of success can be taken as definitive proof of the electron–phonon interaction. For example, even the excellent agreement with the density of states could be understood as a mathematical property of analytic functions [174].Also, we have focussed on Pb; in other superconductors this procedure has not been so straightforward. For ex-

3

ample, in Nb a proximity layer is explicitly accounted for in the inversion [170, 175], thus introducing extra parameters. In the so-called A15 compounds (eg. Nb3 Sn, V3 Si, etc.), although the measured tunneling results have been inverted [142], several experiments do not fit the overall electron–phonon framework [9]. More details are provided in [10]. An alternate inversion procedure is also provided there [176], which utilizes a Kramers–Kronig relation to extract (!) from the tunneling result. An inversion of Im(! + iı) then removes ∗ from the procedure. A variant of this, where the imaginary axis quantity (i!m ) is extracted directly from the tunneling I-V characteristic,and then the imaginary axis equations are inverted for ˛ 2 F( ), also works [177], but the accuracy requirements for a unique inversion are very debilitating. Optical Conductivity In principle, any spectroscopic measurement will contain a signature of ˛ 2 F( ). In particular, several attempts have been made to infer ˛ 2 F( ) from optical conductivity measurements in the superconducting state [178–180]. In this section we describe a procedure for extracting ˛ 2 F( ) from the normal state [181]. A common method to determine the optical conductivity is to measure the reflectance [182] as a function of frequency, usually at normal incidence. The reflectance, R( ), is defined as the absolute ratio squared of reflected over incident electromagnetic wave amplitude. The complex reflectivity is defined by r( ) ≡ R1/2( ) exp (i( )) , (3.84) where ( ) is the phase, and is obtained through a Kramers–Kronig relation from the reflectance [182]  ( ) = 

∞ 0

ln R(  ) − ln R( )  d .  2 −  2

(3.85)

The complex reflectivity is related to the complex index of refraction, n( ), r( ) ≡

1 − n( ) , 1 + n( )

(3.86)

Electron–Phonon Superconductivity

101

which, finally, is related to the complex conductivity,  ( ) (using the dielectric function, ( )): 4i ( ) , (3.87) ( ) ≡ n2 ( ) = ∞ +  where ∞ is the dielectric function at high frequency (in principle, for infinite frequency this would be unity). It is through such transformations that the “data” is often presented in “raw” form. Nonetheless, assumptions are required to proceed through these steps; for example, Eq. (3.85) indicates quite clearly that the reflectance is required over all positive frequencies. Thus extrapolation procedures are required at low and high frequencies; a more thorough discussion can be found in [183]; see also [184]. For this review, we will consider both static impurities and phonons as sources of electron scattering. Both contribute to the optical conductivity, and can be treated theoretically either with the Kubo formalism or with a Boltzmann approach [81]. In the Born approximation the result for the conductivity, in the normal state, at zero temperature, is [179]:  ( ) =  !P2 i 1 , (3.88) d! 4   + i/ − £(!) − £( − !) 0

∞

where £(!) =

0

  § − !   d§˛ 2F(§) ln  § + !

|!| − i d§ ˛ 2 F(§)

(3.89)

0

is the effective electron self-energy due to the electron–phonon interaction. The spectral function that appears in Eq. (3.89) is really a closely related function, as has been discussed by Allen [179] and Scher [185]. For our purposes we will treat them identically.The other two parameters that enter these expressions are the electron plasma frequency, !P , and the (elastic) electron-impurity scattering rate, 1/. Equation (3.88) has been written to closely resemble the Drude form as !2 i Dr ( ) = P . (3.90) 4  + i/

102

F. Marsiglio and J. P. Carbotte

The equation could well be recast in this form with a frequency-dependent scattering rate and effective mass (in the plasma frequency) [186]. Equations (3.88) and (3.89) make clear that the optical conductivity is given by two integrations over the electron– phonon spectral function. One would like to “unravel” this information as much as possible before attempting an inversion, so that, in effect, the signal is “enhanced”. To this end one can attempt various manipulations [187–189]. As a first step one can make a weak coupling type of approximation to obtain [181] the explicit result:   1 1 !P2 d 2 ˛ 2 F( ) = . (3.91)  Re 2 4 d 2  ( ) Note that the conductivity data,including a measurement of the plasma frequency, provides us with both the shape and magnitude of ˛ 2 F( ). Equation (3.91) works extremely well, as Fig. 3.20 shows, in the case of Pb. It tells us that, with a judicious manipulation of the conductivity data, the underlying electron– phonon spectral function emerges in closed form. The very simple formula (3.91) introduces some errors — it was derived with some approximations —

Fig. 3.20. ˛ 2 F( ) for Pb (solid curve) vs.  , along with the estimates obtained from (3.91) with an impurity scattering rate, 1/ = 1 meV (dotted) and 10 meV (dot–dashed). These are both qualitatively quite accurate, before they become negative at higher frequencies. Also plotted is the result (dashed curve, indiscernible from the solid curve) obtained from a full numerical inversion, as described in the text. Taken from the second reference in [181]

as can be seen in Fig. 3.20. In fact, a full numerical inversion will also succeed [190, 191]; the first reference requires a Newton–Raphson iteration technique, while the second uses an adaptive method (in the superconducting state). Equation (3.91) was first applied to K3 C60 [181] to help determine whether or not this class of superconductor was driven by the electron–phonon interaction. The result is shown in Fig. 3.21 and provides convincing evidence that the alkali-doped fullerene superconductors are driven by the electron–phonon mechanism. We will return to these superconductors in a later section, and further examine the optical conductivity in the superconducting state in another section. Before leaving this section we wish to discuss some more recent work [195] which utilizes the Memory function M(!) = 1/op (!)−i!op (!).The real and imaginary parts of this function are related by a Kramers-Kronig relation, and one can write, for the optical conductivity,  (!) =

!P2 i . 4 ! + iM(!)

(3.92)

Fig. 3.21. The ˛ 2 F( ) for K3 C60 (solid curve) extracted from the reflectance data of Degiorgi et al. [192], using (3.91). For purposes of analysis we have omitted the negative parts. The neutron scattering results from [193] (dashed curve) are also shown. Clearly the energy scale in ˛ 2 F( ) matches that of the phonons, and some of the peaks even line up correctly. Finally, the dotted curve comes from an analysis of photoemission data [194], where we have arbitrarily broadened the phonon spectrum with Lorentzian line-shapes. Taken from the second reference in [181]

3

Electron–Phonon Superconductivity

103

!2

In this way 1/op (!) = 4P Re(1/ (!)) is interpreted as an optical scattering rate and ![1 + op (!)] = !2

− 4P Im(1/ (!)) is ! times the optical mass ra  tio, m∗ (!)/m op , and op (!) is the optical mass renormalization. Using ordinary perturbation theory to second order in the electron-phonon interaction Allen [179] obtained, at zero temperature, 2 1/op (!) = !

! d§ ˛ 2F(§)(! − §) + 1/imp . 0

(3.93) Note that Eq. (3.91) was derived through different considerations, but is completely consistent with Eq. (3.93). If we consider the case of a single Einstein mode, at frequency !E , we obtain 1 ! − !E for ! > !E , !E 1/op (!) = (3.94) ! 0 for ! < !E . This is to be contrasted with the quasiparticle scattering rate given by Eq. (3.26), −2£2 (! + iı) ≡ 1/qp (!) = !E (|!| − !E ) .

(3.95)

Both the optical and the quasiparticle scattering are zero for frequencies below !E . Beyond this frequency, however, the quasiparticle scattering rate jumps abruptly to a value !E , and remains constant at that value, whereas the optical scattering rate increases continuously from zero and reaches the value !E asymptotically for ! → ∞. Thus, we see through this simple example that boson structure is expected to appear in a more singular fashion in the quasiparticle scattering rate compared to that in the optical scattering rate. This is expected since, according to Eq. (3.89) only the first derivative of the quasiparticle scattering rate with respect to ! is required to give ˛ 2 F(!), whereas Eq. (3.91) indicates that a second derivative is required to extract the spectral function from the optical scattering rate. An application of the Kramer-Kronig relation to Eq. (3.94) gives !op (!) = −!E    2    !E − ! 2   !E + !  !E (3.96)  ;    log  + × log   2 !E − !  ! !E

Fig. 3.22. A comparison of the real part of the quasiparticle self-energy with the corresponding optical quantity, !op (!) vs. frequency. Taken from [195]

this is to be contrasted with the expression for the real part of the quasiparticle self-energy given in Eq. (3.26). In Fig. 3.22 we compare £1 (!) and !op (!) as a function of frequency. The solid curve represents the real part of the self-energy and displays a logarithmic singularity at ! = !E , while the dashed curve for the corresponding optical quantity is much less singular (note that Eq. (3.96) does not contain a singularity, though its slope does). In fact the latter √ quantity has a maximum/minimum at ! = ± 2!E . For this figure, !E = 40 meV and !E /2 = 1 meV.Again we conclude that boson structure that can in principle be identified in both the self-energy and in optical properties has the more pronounced signature in the quasiparticle dynamics [195]. Equation (3.24), valid for any frequency and any temperature, can easily be written for a general spectral function, ˛ 2F(§) [12]. The quasiparticle scattering rate is given by 1/qp (!) ≡ −2£2 (T, !) = ∞ d§ ˛ 2F(§)



(3.97)

0

  ! + § ! − § § − tanh + tanh . × 2coth 2T 2T 2T

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F. Marsiglio and J. P. Carbotte

Shulga et al. [186] have obtained the approximate equation 1/op (!) =

 ! 

∞ d§ ˛ 2F(§) 0

   !+§ § − (! + §)coth (3.98) × 2!coth 2T 2T   ! −§ + (! − §)coth . 2T 

Note that this expression properly reduces to Allen’s equation (3.93) in the limit of zero temperature. The corresponding real parts can be obtained from Eqs. (3.97) and (3.98) through Kramers-Kronig

transformations. Results for both real (top frame) and imaginary (bottom frame) parts are given in Fig. 3.23 for T = 0.1!E (left side) and for T = 0.25!E (right side). Note that temperature does smear out the signature of the boson structure in these quantities; nonetheless, for the Einstein spectrum used here, the structure remains apparent, even at the higher temperature. In all cases the quasiparticle (optical) properties are denoted by dotted (dashed) curves. Figure 3.23 shows that the quasiparticle and optical properties behave in quantitative and even qualitative different ways. Comparing such results obtained from optics with those obtained, for example, from Angle Resolved Photoemission Spectroscopy (ARPES) is therefore difficult. The authors of [195] noted that at T = 0 d ! Allen’s approximation (Eq. 3.93) gives for d! op (!) precisely 1/qp (!) obtained from the self-energy equation (3.89). They therefore defined a quantity (termed  “model” scattering rate) as 1/M (!) ≡ d ! , which can be constructed directly from d! op (!) optical data. This can be compared with 1/qp (!) obtained from ARPES. Of course these are only necessarily equal at T = 0. To test this comparison at finite temperature we can obtain 1/M (!) to high accuracy by using Eq.(3.98); its corresponding real part can be obtained by Kramers-Kronig transformation. The results are plotted as solid curves in Fig. 3.23. These agree remarkably well with the results for the quasiparticle self-energies (dotted curves) at finite temperature. These results are modified when an energydependent bare EDOS is used [143]. Using a generalization of Allen’s formula for the optical scattering rate, Mitrovi´c and Perkowitz obtained 2 1/op (!) = !

Fig. 3.23. A comparison of the real part of the self-energy with the corresponding optical quantity, !op (!) vs. frequency (top frames), along with the corresponding scattering rates (bottom frames). The left-hand side frames are for T = 0.1!E and the right-hand side frames are for T = 0.25!E . The solid curves are for the model scattering rate defined in the text. An Einstein spectrum was used with !E = 40 meV and !E /2 = 1 meV. Taken from [195]

!

!−§ 

d§˛tr2 F(§) 0



d nS ( ), (3.99) 0

 ˜ + N˜ (− ) /2 is the “symwhere nS ( ) ≡ N( ) metrized” renormalized EDOS. Note that we use ˛tr2 F(§) in this expression, which is more accurate than simply using the electron-phonon spectral function. In Fig. 3.24 we show the fit (solid curve) obtained to the data (solid circles) on the suitably normalized transmission through the V3 Si thin film.

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Electron–Phonon Superconductivity

105

Fig. 3.24. Comparison of experiment (solid dots) for a transmission through V3 Si thin film with theoretical fit (solid curve) using Eliashberg theory with a non-constant bare EDOS as discussed in the text. Taken from [143]

Fig. 3.25. The renormalized EDOS (solid curve) along with the unrenormalized EDOS (dashed curve) for V3 Si. See [143] for more details

The fit is very good.Using Eq.(3.99) the symmetrized renormalized EDOS was obtained and plotted in Fig. 3.25 [143]. For the electron-phonon interaction, the electron-phonon spectral function was taken to be proportional to the measured phonon density of states for V3Si with  = 1. The solid curve in Fig. 3.25 is to be compared with the dashed curve (denoted by ns0 ( )) which is the bare EDOS. This latter function is broader than the renormalized one, as expected from the discussion in a previous subsection. Mitrovi´c and Perkowitz also note that inversion of the data in Fig. 3.24 based on a constant bare EDOS leads to an ˛tr2 F(§) which disagrees strongly with the known phonon distribution, F(§) [196] for V3Si.

In particular, a prominent new peak is found near 5 meV that is not observed in the neutron spectrum. This observation constitutes strong evidence for the energy dependence in N( ) in V3Si. Moreover, this is expected both from band structure calculations [132] and from the measured strong degradation of Tc with disorder [137, 197]. Further analysis of other data in the superconducting state provides more evidence for this picture [198]. Finally, returning to tunneling in the superconducting state, the effect of a non-constant bare EDOS can be summarized through the following equations (at T = 0) [139]:

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F. Marsiglio and J. P. Carbotte

∞ (!)Z(!) =

d!  N˜  (!  )

0



× K+ (!, !  ) − ∗ (!c − !  ) ∞ !(1 − Z(!)) =

d!  N˜ Z (!  )K− (!, ! )

0

∞ (!) =

d!  N˜ (!  )K+ (!, ! ),

(3.100)

0

with (!) ≡ (!)/Z(!), K± (!, !  ) =

∞ d§ ˛ 2F(§) 0

 1 1 ± , × !  + ! + § + iı !  − ! + § − iı 

(3.101)

and 1 N˜ Z (!) = − Im 

+∞ N( ) !Z(!) d N( ) D( , !)

−∞

+∞ 1 N( ) + (!) N˜ (!) = Im d  N( ) D( , !) −∞ +∞

1 N˜  (!) = − Im 

−∞

d

N( ) (!) , N( ) D( , !)

(3.102)

with D( , !) ≡ ! 2 Z 2 (!)n− 2(!)−( + (!))2. (3.103) Note that here, following the original reference, we have momentarily dropped the iı in the arguments of the various functions. Using the same parameters as used in the normal state for Nb3 Sn, Mitrovi´c and Carbotte [139] calculated the gap function (!) and obtained the conductance (at T = 0), given by Eq. (3.82). The result is shown in Fig. 3.26 as a function of ! − 0 , where 0 is the single electron gap defined by Eq. (3.83). The top (bottom) frame is for a peak (valley) in the bare EDOS, and the results are

Fig. 3.26. Normalized tunneling conductance (solid curves) for (a) a peak and (b) a valley in the bare EDOS. The dashed curves are obtained assuming a constant bare EDOS equal to the value at the chemical potential. Reproduced from [139]

given by the solid curves.The dashed curves were calculated assuming a constant EDOS with N( ) = N( ) for all . In the inserts of Fig. 3.27 details of the two bare EDOS are shown. While the conductance  (!) vs. ! is significantly affected by the energy dependence of the bare EDOS, it is difficult to differentiate the boson structure (which is also present when a constant bare EDOS is used) from the structure due to the bare EDOS. Mitrovi´c and Carbotte inverted the data in Fig. 3.26 assuming a constant bare EDOS [139]. The results are shown in Fig. 3.27. The solid curves are the results of the inversion and the dashed curves are the actual ˛ 2 F(§) used to compute the conductance in the first place. While these agree reasonably well at low frequencies the solid curve

3

Electron–Phonon Superconductivity

107

Further work in this area can be consulted through the reviews mentioned earlier. We also mention the work of Schachinger et al. [199, 200] on thermodynamic properties based on a generalized free energy formula derived in [139]. A more recent study of the effect of a more realistic bare EDOS on the tunneling characteristics of Nb3 Sn is given in [201]. Photoemission Spectroscopy

Fig. 3.27. Effective electron–phonon spectral functions obtained through inversion of the conductance calculated from Eliashberg theory with non-constant bare EDOS, as indicated. The inversion, however, was carried out assuming a constant bare EDOS. The input ˛ 2 F(§) is indicated by the dashed line, and the insets indicate the input bare EDOS. Reproduced from [139]

shows a negative (positive) tail beyond the maximum phonon frequency for a peak (valley) in the bare EDOS. The lesson learned is that an energy dependent bare EDOS will cause extra structure in the conductance. In one case (positive peak) the structure will lead to an unphysical result, namely a negative tail in the inverted ˛ 2F(§), but in the other one, we have no way of knowing if the inverted ˛ 2 F(§) is reasonable or not (unless, say, neutron spectroscopy indicates definitively that there are no phonons in this energy region).

Angle resolved photoemission spectroscopy (ARPES) has been very successful in providing detailed microscopic information about the electronic properties of the cuprate materials, as described in another chapter of this book. Photoemission spectroscopy has also been successfully applied to several other electron-phonon superconductors (see [202] for a review). Without angular resolution photoemission spectroscopy (PES) measures a convolution of the electron density of states, N(!  ), multiplied by the Fermi function f (!  ), with an instrument resolution function, R(!, ! ), characterized by a peaked distribution with a width. Here, we follow the literature [203] and use a Gaussian with width 2.8 meV. Results from Reinert et al. [203] are reproduced in Fig. 3.28 for a polycrystalline lead sample, taken at T = 4.5 K. The solid black curve is the electron density of states in the superconducting state, N(!) = Re 

|!| ! 2 − 2 (T, !)

,

(3.104)

multiplied by the Fermi function and convoluted with R(!, ! ), where (T, !) is the gap function given by Eliashberg theory. The data (indicated by the symbols) agree remarkably well with this theoretical expectation. The inset highlights the frequency regime where phonon structure occurs, and shows excellent agreement with the calculations. These are based on a tunneling-derived model for ˛ 2 F(§) in Pb shown in the inset of Fig. 3.29 [203]. The main frame of Fig. 3.29 shows data for the electron linewidth as a function of temperature. The curves are from Eliashberg theory, and are given by the imaginary part of the self-energy at zero frequency and at the maximum phonon frequency, 12 meV (they have both been shifted upwards because of an unknown

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F. Marsiglio and J. P. Carbotte

Fig. 3.28. Normalized photoemission intensity in the superconducting state vs. energy. The experimental data are given by the circles, and a BCS-like EDOS convoluted with other functions as described in the text is given by the solid curve. The agreement is very good. Reproduced from [203]

Fig. 3.29. Temperature dependence of the quasiparticle linewidth as a function of temperature. Data is indicated by circles with error bars. Theory has used a fitted impurity contribution to match the magnitude of the data. The inset shows the spectral function for Pb used to achieve such a fit. Reproduced from [203]

3.4 amount of impurity scattering). Chainani et al. [204] have obtained similar data in Pb. They also show the temperature evolution of the PES spectrum in Nb (for 5.3K < T < 10.3K), which illustrates how the gap disappears as the temperature is raised through Tc . These results illustrate that photoemission spectroscopy is being used in much the same spirit as tunneling spectroscopy pioneered by McMillan and Rowell in the 1960s [50]. An earlier study of the A15 compound Nb3Al is given in [205]. Other measurements are for V3 Si [206], MgB2 [207–209], Ba-K-BiO [210] and in the heavy fermion CeRu2 [211].

The Critical Temperature and the Energy Gap

Perhaps the most important property of a superconductor is the critical temperature,Tc .For this reason a considerable amount of effort has been devoted both towards new materials with higher superconducting Tc , and, on the theoretical side, towards an analytical solution of the linearized Eliashberg equations (set  m  to zero, where it appears in the denominator in Eqs. (3.55)–(3.58) ) for Tc (see [10, 12] for reviews); the experimental “holy grail” has enjoyed some success, particularly in the last 15 years; the theoretical goal has had limited success. In fact numerical solutions are so readily available at present, that the absence of an analytical solution is not really debilitating to understanding Tc . In the conventional theory there are two input “parameters”: a function of frequency, ˛ 2F( ), about

3

which we have already said much,and ∗ (!c ),a number which summarizes the (reduced) Coulomb repulsion experienced by a Cooper electron pair. The focus of this chapter will be the effect of size and functional form of ˛ 2F( ) on Tc .

Electron–Phonon Superconductivity

procedures are connected, so they produce the same physical equation in the weak coupling limit. Returning to Eq. (3.108), for large x, (x) ≈ log (x), so, in the weak coupling limit (Tc  !c ), we obtain a BCS-like equation, 

3.4.1

1+ Tc = 1.13!c exp −  − ∗

Approximate Solution: The BCS Limit

The first insight into Tc comes from reducing the Eliashberg theory to a BCS-like theory. This is accomplished by approximating the kernel ∞ (i!m − i!m ) ≡ 0

2

2 ˛ 2 F( ) + (!m − !m )2

(3.105)

by a constant as long as the magnitude of the two Matsubara frequencies are within a frequency rim of the Fermi surface [123], taken for convenience to be !c , the cutoff used for the Coulomb repulsion, ∗ . That is, 1  for both |!m |, |!m | < !c , (i!m −i!m ) = 0 otherwise, (3.106) where  ≡ (0) has already been defined in Eq. (3.21). Then, the linearized version of Eq. (3.55) (with A0 (m ) = 1), for the renormalization function, Z(i!m ), reduces to Z(i!m ) ≈ 1 +  .

(3.107)

Using this and solving the linearized version of Eq. (3.57) for the pairing function yields     1+ 1 1 !c − , (3.108) = + ∗ −

2Tc 2 2 where (x) is the digamma function. The cutoff in these equations is along the Matsubara frequency axis; this procedure is to be contrasted with the BCS procedure, which introduced a cutoff in momentum space. The former is more physical, insofar as the true electron–phonon interaction comes from retardation effects, which occur in the temporal domain; hence the cutoff should occur in the frequency (either real, or imaginary) domain. In practice, the two

109

 .

(3.109)

This equation has essentially summarized all the detailed information contained in the electron–phonon spectral function ˛ 2F( ) into two parameters,  and !c . The mass enhancement parameter, , is a simple moment of ˛ 2F( ) (see Eq. (3.21)), while the parameter !c physically is meant to represent some typical phonon frequency. In more refined treatments [123,212], !c is given by some moment of ˛ 2F( ) as well. For example, in [123], the logarithmic average is used to define !ln (see Eq. (3.71)), a quantity we shall use extensively in the following sections. They modified the McMillan equation [212] to read kB Tc =

  1.04(1 + ) !ln exp − . 1.2  − ∗ (1 + 0.62)

(3.110)

A derivation of this equation is given in Refs. [123,212]. 3.4.2

Maximum Tc , Asymptotic Limits, and Optimal Phonon Spectra

Equation (3.109) (or (3.110)) describes the weak coupling limit of Eliashberg theory reasonably well. It errs in the strong coupling limit; for example, it predicts that Tc saturates as  increases, whereas the Eliashberg equations themselves predict that Tc grows indefinitely with . (Of course, physically we know that Eliashberg theory gets this limit completely wrong; Tc should peak at some point and then decrease to zero as the coupling strength increases. That an approximation to Eliashberg theory improves upon this doesn’t justify that approximation; it simply means two errors have undone one another to get closer to the “right answer”.) Asymptotic results from Eliashberg theory can be obtained correctly and analytically [10,123] through a variety

110

F. Marsiglio and J. P. Carbotte

of arguments. (D. Rainer first obtained the asymptotic result noted in the text in 1973 (private communication).) The methodology based on scaling theorems is particularly powerful, and has been applied to other thermodynamic properties as well [10]. The correct asymptotic result for Tc is [123]: √ (3.111) Tc = 0.183 !E , as  → ∞. In obtaining Eq. (3.111), an Einstein spectrum has been assumed (this is not required), which in turn is characterized by two parameters: the weight, A ≡ !E /2, and the frequency, !E . In writing Eq. (3.111), one tacitly has assumed that the parameter  ≡ 2A/!E is increased while keeping the frequency !E fixed. In reality, the two parameters are not independent — this is the main point of the article by Cohen and Anderson [213]. For example, often phonon softening occurs because the coupling strength increases. In fact, this is made explicit in McMillan’s definition [212] of : ≡

2

N( F )˛ , M!E2

(3.112)

where N( F ) is the electron density of states at the Fermi energy, M is the ionic mass, and ˛ 2 is the electron–ion coupling referred to in the Holstein Hamiltonian, Eq. (3.4) (in a more realistic electron– phonon Hamiltonian, ˛ 2 would be given by a Fermi surface average of the electronic matrix element of the change in crystal potential as one atom is moved [212]). To determine what the optimal phonon frequencies actually are, functional derivatives were introduced [121]. These had already been utilized extensively as an iterative aid in inverting tunneling data with the Eliashberg equations [50]. The most commonly used functional derivative is that of Tc with respect to infinitesimal changes in ˛ 2 F( ), with ∞ 2 fixed area, A ≡ 0 d ˛ F( ), first computed by Bergmann and Rainer [121]. An approximate result, derived in [214], is given by the expression ∞ ¯ ıTc 1  4§ = B , ¯ 2 + 4 2 n2 n ı˛ 2F(§) 1 +  n=1 §

¯ as a function of norFig. 3.30. The universal function G(§) ¯ = §/Tc which enters the curve malized phonon energy § for the functional derivative of Tc with respect to ˛ 2 F(!) in the  ŸŸ model of [214], from which this figure was taken

Bn =

n   1 m=1

2 2 + n 2m − 1 (2m − 1)2



¯ ≡ §/kB Tc , and the Bn are numbers given by where §

2 . (3.114) 4

This function (which is universal) is shown in Fig. 3.30, and reflects well the generic behaviour of the more precise calculation.It illustrates that the optimal phonon frequency lies at some finite frequency (i.e. non-zero, and non-infinite), which is a factor of order 10 times the critical temperature. Thus if one could imagine shifting small amounts of weight in ˛ 2 F( ) around then Tc would increase if spectral weight is shifted either from very high or from very low frequencies towards frequencies near the maximum of the curve shown in Fig. 3.30. The reasoning above leads naturally to the concept of an optimum spectrum, first determined by Leavens [215], and elaborated upon in Refs. [216–220]. In an optimum spectrum calculation, one imagines having a fixed area of ˛ 2 F( ), and asks at what frequency it would best be situated in order to optimize some particular property.An appropriate scaling of the linearized Eliashberg equations for an Einstein phonon spectrum with frequency !E leads to the result ¯ E , ∗) , Tc /A = f (!

(3.113)

−

(3.115)

where A is the area, ! ¯ E ≡ !E /A, and f is a universal function of ! ¯ E , to be determined numerically for each choice of ∗ (a very weak A dependence in the

3

Electron–Phonon Superconductivity

˛ 2G( ) ≡  ˛ 2 F( ) ,

111

(3.117)

and take functional derivatives with respect to this new function. The result is easily obtained from that c in Eq. (3.113), simply by dividing by §. Then ı˛ıT 2 G(§) will peak at zero frequency, and it would seem that it is always advantageous to decrease the phonon frequency. Continuing this process will result in a spectrum for which the calculation which gives Eq. (3.113) is no longer valid, and one would have to self-consistently calculate the functional derivative, numerically. To our knowledge this has not been done for Tc or any other superconducting property. 3.4.3 Fig. 3.31. The constant c( ∗ ) in the relation kB Tc = c( ∗ )A for the maximum Tc associated with a given A as a function of ∗ . Placed on the same figure (solid dots) are the results for Tc /A obtained in the case of many strong coupling superconductors for which ˛ 2 F(!) is known from tunneling spectroscopy. The solid points all fall below the maximum curve as they must. Adapted from [215]

cutoff associated with ∗ has been neglected). The result is a curve with a maximum at ! ¯ E ≈ 1; placing a spectral function at this frequency will yield the maximum Tc . This procedure yields a result, ∗

Tc ≤ Ac( ) ,

(3.116)

where c( ∗ ) is a function of ∗ shown in Fig. 3.31. Also shown are data from many superconductors for which ˛ 2F( ) is known from tunneling spectroscopy, all of which fall below the optimum curve. Interestingly, some superconductors have a critical temperature reasonably close to their optimal value. The last few paragraphs demonstrate the usefulness of functional derivatives in understanding the systematics of Tc .A variant of these results can easily be obtained, which may shed even more light on Tc systematics. As we have already mentioned, the functional derivative discussed involves the moving around of spectral weight, subject to the condition that the area remain constant.However,as Eq.(3.112) suggests,it is not the area which likely remains constant while phonons soften, but rather the area times a frequency. Hence, one can define a different spectral function,

Isotope Effect

As already remarked in the Historical Developments subsection, the discovery of an isotope effect on Tc played an important role in the subsequent development of the theory. In the BCS equation the isotope effect is clear from the prefactor; phonon frequencies for elemental superconductors are inversely proportional to the square root of the ionic mass, and hence the isotope coefficient ˇ is ˇ≡−

d ln Tc 1 = . d ln M 2

(3.118)

The last equality follows from Eq. (3.40), using the fact that , as defined there, is independent of ion mass. In the standard Eliashberg theory,  as defined by Eq. (3.21) remains independent of ion mass, and, with ∗ ≡ 0, we once again obtain ˇ = 1/2. Complications can arise,for example,from a finite electronic bandwidth [221], or from a non-constant density of states near the Fermi level [222–224]. There are two other clear sources of deviation from ˇ = 1/2. One is that in non-elemental superconductors, an isotopic substitution for one of the elements will result in varying changes in Tc ,depending on how the element being substituted contributes to the important phonon modes. One then has to define partial isotope coefficients, defined by [225] ˇi ≡ −

d ln Tc , d ln Mi

(3.119)

the ith element. The where Mi refers to the mass of total isotope coefficient, ˇtot ≡ i ˇi , will sum to 1/2

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F. Marsiglio and J. P. Carbotte

(in the absence of ∗ ). The total isotope coefficient can also be broken down by frequency, with   d ıTc ! 2 ˇ(!) ≡ ˛ F(!) , (3.120) d! 2Tc ı˛ 2F(!) and then

∞ ˇtot =

d!ˇ(!) .

(3.121)

0

Equation (3.121) is useful when phonon modes coming from one of the elements are well separated from those coming from the others, as exists, for example, in the high temperature perovskites, since oxygen is much lighter than the other elements, and hence is chiefly responsible for the high frequency modes.Example calculations can be found in Refs. [10,226]. The second source of deviation from ˇ = 1/2 is because ∗ (!c ) is non-zero. To understand why this causes less of a reduction in Tc (when a heavier mass is substituted) recall that ∗ (!c ) is reduced from some larger value (!B ) through the pseudopotential effect. That is, it is through retardation that a weaker electron–phonon interaction can overcome the stronger direct Coulomb repulsion. In analytical treatments this is often modelled by endowing a mass dependency to the Coulomb pseudopotential through the cutoff [212]. For example, inspection of Eq. (3.79) shows a mass dependency if the cutoff frequency !c is made to correspond to a phonon frequency. Then one can derive, from the McMillan equation for Tc , Eq. (3.110), the following expression for the isotope coefficient (assuming one element):   1 1.04(1 + )(1 + 0.62) ∗2 ˇ= 1− . (3.122)

2 [ − ∗ (1 + 0.62)]2 This result properly reduces to 1/2 when ∗ = 0, and shows that the isotope coefficient is generally reduced when ∗ is finite. In fact it is clear from Eq. (3.122) that the isotope coefficient is reduced for both positive and negative ∗ . The isotope coefficient is reduced for positive ∗ because, when you lower the ionic mass, the increase in phonon frequency will raise Tc , but not as much as would be the case if the Coulomb repulsion were not present. This is because the discrepancy in frequency scales has been reduced

slightly, and the retardation-induced attractive interaction is not as large as before the isotopic substitution.On the other hand a negative ∗ represents some unknown attractive non-phonon mechanism [226], which contributes to Tc .A weakening of the phononinduced attraction (through an isotopic substitution) reduces Tc only partially, resulting again in a reduced isotope coefficient. A more accurate determination of the isotope coefficient can be obtained simply numerically, following the prescription of Rainer and Culetto [225]. In this case a cutoff is imposed on the Eliashberg equations, which is independent of (but much greater than) the maximum phonon frequency. An isotopic substitution results in only a shift in the phonon spectrum, and a subsequent calculation of Tc will yield the isotope coefficient. This is physically more transparent than the analytical approach described above, as an isotope substitution does not alter (at this level of theory) the direct Coulomb repulsion. There is a substantial literature on the isotope effect; much of the older results are summarized in [168]. By 1969 many low Tc superconductors had been found, several of which had very low isotope coefficients. These had, for the most part, been explained through detailed calculations [3,227], due to the physics outlined above.It is worth noting that this explanation of the sometimes low isotope coefficient observed was not universally accepted [228]. The discovery of the high temperature cuprate materials prompted considerable activity concerning the isotope coefficient, as is reviewed in [229]. The isotope coefficient displays some unusual doping dependence in the La2−x {Sr,Ba}x CuO4−ı systems, but is essentially zero in the optimally doped 90 K YBa2 Cu3 O7−y system. The question is, can a realistic (and conventional) electron–phonon interaction give rise to a 90 K superconductor with a near zero isotope coefficient? A qualitative answer can be obtained [230] through the use of the McMillan equation (3.110), (3.122). For a given electron–phonon coupling, , and phonon frequency !E , one can determine the required value of ∗ (!c ) to fix Tc from Eq. (3.110) (assuming !c refers to a cutoff associated with the phonon spectrum). These parameters can then be used in Eq. (3.122) to determine the iso-

3

Electron–Phonon Superconductivity

113

Fig. 3.32. The isotope coefficient, ˇ, vs. Tc , for various values of  and !E . Along each curve Tc changes because the Coulomb pseudopotential ∗ is being varied. These results show that a low value of ˇ is difficult to attain with high-Tc . On the other hand, for low Tc materials, it is not so difficult

tope coefficient, ˇ. Some such results are plotted in Fig. 3.32. To obtain the desired results for optimally doped YBa2 Cu3O7−y (Tc ≈ 90 K and ˇ ≈ 0) would require high frequency phonons !E ≈ 100 meV with very strong electron–phonon coupling ( ≈ 5). That such a coupling strength is unrealistic, particularly for such very high frequency phonons, was discussed much earlier by Cohen and Anderson [213]. The qualitative validity of Fig. 3.32 has been verified by several numerical solutions to the Eliashberg equations [221,230,231]. In particular, in [221] a natural bandwidth cutoff was employed, with similar results. In summary the conventional Eliashberg theory can yield a near-zero isotope coefficient, provided Tc is low. One must go beyond the conventional framework to obtain a zero isotope coefficient with Tc ≈ 90 K. 3.4.4

The Energy Gap

The existence of a single particle energy gap, although not fundamental to superconductivity [232],

nonetheless has played an important role in our understanding of superconductivity. How an energy gap arises in the I-V characteristic of a conventional superconductor has already been discussed in Sect. 4.4.3; there we focussed on extracting detailed information about the mechanism. Here we turn our attention to the gap, a much more prominent feature in the experimental result, and learn what a particular value may imply about the superconductor. The first step is to examine what occurs in BCS theory. The order parameter is then given by a constant, as written in Eq. (3.36). Suitable generalization to the model interaction given by (3.106) yields   1+  = 2!c exp − , (3.123)  − ∗ in the weak coupling limit. The solution at finite temperature is somewhat more complicated; it can be obtained numerically, and shows the typical mean field behaviour near Tc [5]: (T) ∝ (Tc − T)1/2.

(3.124)

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Near T = 0 the order parameter is exponentially flat (see, for example, [233]): (T) ≈ (0) − [2(0)T]1/2 e−(0)/T .

(3.125)

The order parameter is a real (i.e. not complex) number for all temperatures. (This is not to be taken as a statement regarding the phase of the order parameter; we have ignored the phase up to this point, or, equivalently, set it to zero. Rather, as is evident from Eqs. (3.77) and (3.78), in Eliashberg theory the quantity which takes on the role of the order parameter acquires an imaginary part, whose significance is associated with lifetime effects. ) Thus, Eq. (3.82) shows that the dI/dV curve (which provides an image of the density of states) will show an energy gap at (T) at each temperature. An illustration of the temperature dependence of the order parameter is given in

Fig. 3.34. (a) (i!n ) and (b) Z(i!n ) vs !n ,the Fermion Matsubara frequency, for various temperatures, as indicated. Note that the curves are relatively smooth and featureless, and at low temperatures little change occurs, except that more Matsubara frequencies are present. In (a) the units of  are meV. These were produced for Pb

Fig. 3.33. (a) The temperature dependence of the BCS order parameter, and (b) the resulting densities of states at various temperatures below Tc . The only effect of finite temperatures on these latter curves is a reduced gap

Fig. 3.35. Calculated densities of states of Pb for various temperatures. In contrast to the BCS case (Fig. 3.33b), at high temperatures there is considerable smearing

3

Fig. 3.33a, along with the density of states at several temperatures (Fig. 3.33b). Within Eliashberg theory, the calculation of the corresponding property is much more complicated. First of all, a careful distinction between the gap or pairing function (which is now a function of frequency at any given temperature) and the energy gap is required. The energy gap is defined through Eq. (3.83). The gap function is, in general, a complicated and complex function of frequency, that results from a solution of the Eliashberg equations. These, in turn, can be solved either on the imaginary axis (Eqs. (3.55) and (3.58) for the “standard” theory) or the real axis (Eqs. (3.77) and (3.78)). Example solutions for a real electron–phonon spectrum (Pb) are

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shown in Fig. 3.34. The solutions on the imaginary axis turn out to be real; on the real axis they are complex.The corresponding densities of states are shown in Fig. 3.36. The low frequency behaviour of the various functions plotted is not clear on the figures shown. A careful analysis [171] leads to Re(!) Im(!) ReZ(!) ImZ(!)

= = = =

c, 0, d, 0.

(3.126)

at zero temperature, where c and d are constants, whereas at any non-zero temperature, we obtain

Fig. 3.36. The (a) real and (b) imaginary parts of the gap function (in meV) on the real frequency axis, for Pb, for various temperatures, as in the previous figure. Note the considerable structure present on the real axis. Also shown is the (c) real and (d) imaginary part of the renormalization function, Z(!) vs. !

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Re(!) ∝ ! 2 , Im(!) ∝ !, ReZ(!) = d(T), ImZ(!) ∝ 1/!.

(3.127)

The latter result in particular implies that, strictly speaking, at finite temperature there is always “gapless”superconductivity.However,as can be seen from Fig. 3.36, in reality the “finite temperature” density of states at zero energy is generally quite small (except for very close to Tc ). The extent to which this is true depends on the electron–phonon coupling strength; as this increases the zero frequency density of states can be a significant fraction of the normal state value at temperatures near Tc [173]. In the remaining subsections we wish to examine the dependence of the energy gap on coupling strength. Since the electron–phonon interaction is characterized by a spectral function, ˛ 2F(§), we first must decide how to quantify the coupling strength of a particular superconductor. Historically the mass enhancement parameter, given by Eq. (3.21), has played this role. However, depending on the material, the direct Coulomb repulsion, characterized by

∗ (!c ), where !c is some suitable cutoff frequency, can offset the effect of . Another possible parameter is the ratio of the critical temperature to an average phonon frequency, a quantity first advocated by Geilikman and Kresin [234–237]. This approach was further quantified by Mitrovi´c et al. [238]. In this reference (see also [239]), the Allen–Dynes parameter Tc /!ln emerged naturally in the derivation of strong coupling corrections, as an indicator of coupling strength. A large number of superconducting properties were obtained in this way (see Refs. [10, 240] for derivations and more details),and semi-empirical fits were obtained based on accurate numerical solutions. We discuss these further in the next section. 3.4.5

The Energy Gap: Dependence on Coupling Strength Tc =!`n

As we have already emphasized, Tc cannot be reliably calculated at present.The first,perhaps simplest, test for the accuracy of Eliashberg theory is then its ability to properly obtain the gap ratio, 2◦ /kBTc ,

where,by ◦ ,we mean the zero temperature gap edge. In [238] (see also [10]),numerically calculated results were compared to experimental tunneling results for ◦ , obtained for a variety of conventional elemental and alloy superconductors. The deviations of the gap ratio from the BCS universal result, 2◦ /kB Tc = 3.53, are up to 50 %; yet the level of experimental error is about 5 % , with one notable exception (Nb3 Sn). The theoretical results are obtained from a solution of the imaginary axis equations (Eqs. (3.55) and (3.58), with the standard approximations of infinite bandwidth and particle-hole symmetry), followed by an analytical continuation to the real axis. To obtain the gap edge, a Pad´e approximant suffices to get very accurate results [160], as the more systematic continuation [163] verifies. In any event it is desirable to have an analytic form for these corrections. The result of Mitrovi´c et al. [238] is    T c 2 2◦ !ln = 3.53 1 + 12.5 ln ( ) . (3.128) kB Tc !ln 2Tc In obtaining this result the spirit of the McMillan equation was followed, and the coefficients 12.5 and 2 were chosen from fits to the numerical data for a large number of superconductors. These results are plotted in Fig. 3.37. From this Figure it is clear that Eq. (3.128) describes the overall trend very well. As the electron–phonon interaction increases (i.e. becomes more retarded), the gap ratio increases to values exceeding 5.0. Figure 3.37 illustrates that a simple analytic form describes the trend of the gap ratio as a function of Tc /!ln rather well for a variety of conventional superconductors. In each case electron–phonon spectral functions were used, as obtained from tunneling data, or, in some cases, model calculations. On occasion, one sometimes uses a phonon spectrum obtained from inelastic neutron scattering, scaled to give the measured critical temperature. This latter process assumes that the electron–phonon coupling is constant as a function of frequency (seen to be reasonable in the case of Pb), and often assumes a value of the Coulomb pseudopotential, ∗ (!c = 6!max ) ≈ 0.1 (!max is the maximum phonon frequency). Specific references to the sources of these spectra can be found in Refs. [10,240].

3

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where B and b are constants, chosen to span a continuum of values of Tc /!ln . Thus, given some spectral shape, say that of Pb, one can determine a curve of 2◦ /kBTc vs. Tc /!ln . In this way they were able to ascertain, for a given value of Tc /!ln , the shape dependence of the gap ratio. They of course found more significant deviations from the analytical form, Eq. (3.128); nonetheless, the deviations remained small on the scale of Fig. 3.37. Further work, in the weak coupling limit, is given in [242]. Larger deviations were obtained with the use of (somewhat artificial) delta-function model spectra [241,243–245]. Similarly, if the electron–phonon coupling strength is taken to be extremely high, large deviations occur from one spectral shape to another [246]. The net conclusion is that, with physical spectra and physically relevant coupling strength (Tc /!ln < 0.2), the strong coupling corrections are ∼ quasi-universal,and are well described by Eq.(3.128). We explore in the next subsection how this can be used to optimize the gap and gap ratio.

Fig. 3.37. The ratio 20 /kB Tc vs Tc /!n . The solid dots represent results from the full numerical solutions of the Eliashberg equations. Experiment tends to agree to within 10%. In increasing order of Tc /!n , the dots correspond to the following systems: A, V , Ta, Sn, T, T0.9 Bi0.1, In, Nb (Butler), Nb (Arnold), V3 Si(1), V3 Si (Kihl.), Nb (Rowell), Mo, Pb0.4 T0.6 , La, V3 Ga, Nb3 A(2), Nb3 Ge(2), Pb0.6 T0.4 , Pb, Nb3 A(3), Pb0.8 T0.2 , Hg, Nb3 Sn, Pb0.9 Bi0.1, Nb3 A(1), Nb3 Ge(1), Pb0.8 Bi0.2 , Pb0.7 Bi0.3, and Pb0.65 Bi0.35. The drawn curve corresponds to 20 /kB Tc = 3.53[1 + 12.5(Tc /!n )2 n(!n /2Tc )]. The insert shows results for different scaled ˛ 2 F(!) spectra. They all correspond to the same value of Tc and of !n as Pb. They serve to show that some deviation from the general trend is possible. Reproduced from [10]

An important question, particularly when faced with a new superconductor whose phonon characteristics may or may not be “typical”, is to what extent the trend modelled by the semi-empirical analytic form, Eq. (3.128) can be violated, for a given coupling study. This question was considered in [241]. They took existing electron–phonon spectra, ˛◦2 F( ), and scaled them to new spectra, ˛◦2 F( )∗ = B˛◦2 F(b ),

3.4.6

Optimal Phonon Spectra and Asymptotic Limits

A functional derivative analysis similar to that described for Tc yields, for ◦ , an optimum phonon frequency for a given spectral area. One finds that for a delta function spectral function, the zero temperature gap edge obeys a scaling relation just like Tc given by Eq. (3.115): ¯ E , ∗ ), ◦ /A = g(!

(3.129)

where all quantities are as defined following Eq. (3.115). As found there, for a given base spectrum, an optimum frequency ! ¯ E∗ exists whose value is generally lower than the characteristic frequency of the base spectrum — this is particularly clear when the base spectrum itself is a delta function. With Tc one found that shifting the spectral weight to that optimum frequency resulted in an enhancement of Tc . Furthermore, an iteration of this procedure resulted in convergence to the situation where, for a given spectral area, the maximum Tc had been achieved, with a frequency given by ! ¯ E∗ ≈ 1.3 (for ∗

(!c ) = 0.1). The functional derivative of Tc with

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respect to ˛ 2 F( ) using this base spectrum is nonpositive definite [218] with a maximum at ! ¯ E∗ , showing that Tc could no longer be increased. The situation with the gap edge is similar, but differs in the following crucial point. Upon iteration one finds that the optimum frequency continues to decrease, as the Einstein frequency of the base spectrum decreases. Thus, the implication is that the gap edge, and therefore the gapedge ratio, 2◦ /kB Tc , will be maximized in the limit as ! ¯ E∗ → 0. Alternatively, since these calculations are for fixed spectral area, A, this will occur as  → ∞. What is the maximum value of 2◦ /kB Tc allowed within “standard” Eliashberg theory ? Carbotte et al. [217] answered this question through a scaling theorem, and backed up with numerical work. They found that the gap ratio increased monotonically as  increased, finding (numerically) a value close to 10 (recall BCS gives 3.53) for values √ of  ≈ 30. In doing so they proved that ◦ ∝ !E as  → ∞, just like Tc does (Eq. (3.111)). Claims were made to the contrary, but these were definitively put to rest in [173]. By solving a set of Eliashberg equations written specifically for  → ∞, they found a maximum value of the gap ratio equal to 12.7. A variety of other properties were explored in the asymptotic limit,  → ∞, as can be found in the previous references and in Refs. [247–251].

3.5

Thermodynamics and Critical Magnetic Fields

These topics have been amply covered in previous reviews [10]. Nonetheless, we include here for completeness a brief summary of the impact of the electron–phonon interaction on these properties in the superconducting state. 3.5.1

The Specific Heat

To calculate the specific heat one requires the free energy. For an interacting electron system, a practical formulation of this problem was first proposed by Luttinger and Ward [252], and further pursued by Eliashberg [253]. A simpler calculation requires the

free energy difference between the superconducting and normal state, for which an expression due to Bardeen and Stephen [254] is   F 2 2 = −T !m +  (i!m ) − |!m | N(0) m   |!m | × Z S (i!m ) − Z N (i!m )  , (3.130) 2 + 2 (i! ) !m m where, for clarity, we include the two Eliashberg equations from Eqs. (3.55)–(3.58):  Zm = 1 + T (i!m − i!m ) m

(!m /!m )Zm × 2 2 2 !m  Zm + m m = T

(3.131)



 (i!m − i! ) − N(0)Vcoul m

m

×

m 2 2 2 !m  Zm + m

.

(3.132)

These equations ignore band structure effects entirely (except through the electron density of states at the Fermi level, denoted here by N(0)), and we again have adopted the shorthand Z(i!m ) = Zm etc., and used the gap function (i!m ) ≡ (i!m )/Z(i!m ). For the free energy expression we have used superscripts‘S’or‘N’to denote the superconducting or normal state, respectively. In the normal state Z N (i!m ) reduces to the expressions obtained earlier which is easily seen if one uses the relation   £(z) = z 1 − Z(z) , (3.133) where z is a frequency anywhere in the upper half plane, and £(z) is the electron self-energy. Equation (3.130) can easily be evaluated, once the imaginary axis Eliashberg equations (3.131)–(3.132) are solved. From this the specific heat difference, d 2 F , dT 2 and the thermodynamical critical field, √ Hc (T) = −8F , C(T) = −T

(3.134)

(3.135)

3

can be computed. The former displays a jump at Tc , characteristic of a mean field theory, which is the level of approximation of Eliashberg theory. At low temperatures the specific heat in the superconducting state should be exponentially suppressed. This is generally observed [124], and deviations that do occur at very low temperatures can readily be explained by anisotropy in the gap parameter [255]. Because properties like the electron density of states at the Fermi level are difficult to measure or calculate reliably,one would like to focus on observables that are independent of these properties. For the specific heat difference, one way of accomplishing this is to normalize the specific heat to the normal state result, which presumably contains the same electron density of states. The result is then independent of N(0), and can be compared directly to the measured results. A textbook example was provided in the case of Al [256]; the data is reproduced in Fig. 3.38, along with the BCS prediction. The normal state specific heat for a weakly interacting electron gas is given by CN (T) = T ,

(3.136)

where is the Sommerfeld constant given by 2

=  2 kB2 N(0)(1 + ) . (3.137) 3 Here,  is the electron–phonon enhancement parameter, already referred to on many occasions. The electron–phonon interaction alters the low temperature specific heat through the mass enhancement parameter, 1 + . In fact, a more careful treatment [86, 105, 257] yields a temperature-dependent (T) for the specific heat coefficient (which, at very low temperature, reduces to the Sommerfeld ). Besides providing quantitative corrections to the electronic specific heat in the normal state, this correction also provides a properly physical contribution from the low frequency phonon modes, as found in [258]. For a variety of conventional superconductors, like Al, the normal state low temperature specific heat is easily measured by suppressing the superconducting state with a magnetic field. Then the ratio C(T)/ Tc can be determined. At Tc , the BCS result for this ratio is universal, like the gap ratio: it is 1.43. Strong coupling corrections can be derived [239] as

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119

Fig. 3.38. Specific heat of aluminium as a function of temperature in the superconducting state and the normal state (applied field of 300 Gauss). Data taken from [256]. The BCS prediction, given the normal state data, is given by the solid curve

before, as a function of the strong coupling parameter, Tc /!ln . The result is    T c 2 !ln C(Tc ) = 1.43 1 + 53 ln ( ) . (3.138)

Tc !ln 3Tc Again, the coefficients 53 and 3 were determined semi-empirically by fits to numerical data. A plot of this result,along with some of the numerical data, is shown in Fig. 3.39. We already remarked about Al — its calculated value is indicated by the point nearest the ordinate, and agrees very well with experiment. The result for Pb is also shown; the experimental value is 2.65, almost a factor of 2 greater than the BCS result. The theoretical result, based on a numerical solution of Eqs. (3.130)–(3.132), is in good agreement. The result for stronger coupling has also been calculated [246]. In particular, the asymptotic limit can be computed, following standard procedures. The result is [249] C(Tc ) 19.9 , (3.139) =

Tc 

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Fig. 3.40. Specific heat vs. the square of the temperature calculated from Eliashberg theory for various elemental superconductors. From [124]

Fig. 3.39. The specific heat ratio, C(Tc )/( Tc ) vs Tc /!n . The dots represent results from the full numerical solutions of the Eliashberg equations. Experiment tends to agree to within 10%. In increasing order of Tc /!n , the dots correspond to the following systems: A, V , Ta, Sn, T, T0.9 Bi0.1, In, Nb (Butler), Nb (Arnold), V3 Si 1, V3 Si (Kihl.), Nb (Rowell), Mo, Pb0.4 T0.6 , La, V3 Ga, Nb3 A(2), Nb3 Ge(2), Pb0.6 T0.4 , Pb, Nb3 A(3), Pb0.8 T0.2 , Hg, Nb3 Sn, Pb0.9 Bi0.1 , Nb3 A(1), Nb3 Ge(1), Pb0.8 Bi0.2 , Pb0.7 Bi0.3 , and Pb0.65 Bi0.35 . The drawn curve corresponds to C(Tc )/ Tc = 1.43(1 + 53(Tc /!n )2 n(!n /3Tc )). Adapted from [239]

showing that the relative magnitude of the jump decreases for large , and therefore, as is already becoming apparent in Fig. 3.39, the specific heat will have a maximum as a function of coupling strength. Similar results can be derived for other thermodynamic properties as well. These have been summarized in [10] and will be omitted here. In Fig. 3.40 we show the full temperature dependence of the electronic specific heat in the superconducting state, Ces (T), normalized to T for a variety of supercon-

Fig. 3.41. The deviation function D(t) vs. t 2 calculated from Eliashberg theory for various elemental superconductors. Note the agreement of Eliashberg theory of Al with BCS results. From [124]

3

ductors, as calculated in [124], using Eliashberg theory with tunneling-derived electron-phonon spectral functions. A comparison with the BCS result (i.e. Al) is also given. Note the considerable variations from material to material; more retardation generally leads to increased values of Ces (T) near Tc . The curves cross as the temperature is lowered and the exponentially activated region spans a larger temperature range for larger gap values as expected. Another measure of retardation is through the socalled deviation function D(t) ≡ Hc (t)/Hc (0) − (1 − t 2 ),

(3.140)

where t ≡ T/Tc and the two-fluid result is subtracted off from the measured ratio. Results for a variety of superconductors are shown in Fig. 3.41 [124]. These curves are in excellent agreement with the data compiled in [259] (not shown). All of this reinforces the excellent agreement between Eliashberg theory and experiment for a wide variety of so-called conventional superconductors. 3.5.2

Critical Magnetic Fields

In a type-I superconductor, a critical magnetic field (Hc ) exists, given by Eq. (3.135). In a type-II superconductor, a lower critical field, Hc1, and an upper critical field, Hc2, exist; the former signals the departure from the Meissner state to one in which one vortex penetrates the system, while the latter occurs at the normal/superconducting transition. The thermodynamic critical field continues to exist as a thermodynamic property, but not one that can be measured by application of a magnetic field. A theory of Hc1 has been worked out within the BCS approximation in [260,261] (in the dirty limit). This work was extended to the level of Eliashberg theory in [262]. It is traditional to calculate the reduced field, hc1 (T/Tc ) ≡ TcHHc1 (T) as a function of c1 (Tc ) T/Tc . Such a curve has a slope of −1 near Tc , and saturates to some value at T = 0. Rammer [262] found that the low temperature value decreased with coupling strength (there characterized by a particular spectrum). A detailed theory has also been provided for the upper critical field, Hc2. In 1957 Abrikosov essentially

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created the subject of type II superconductivity [263]. Both experimental and theoretical work in this exciting area continued to flourish throughout the 1960’s. Applications of superconductivity in the mixed state require type II superconductivity in order to sustain high magnetic fields. Abrikosov’s solution used the phenomenology of Ginzburg–Landau theory [43]. Further theoretical developments utilized the microscopic theory of Gor’kov [42]. The first of these was by Gor’kov [264] for clean superconductors, followed by five papers by Werthamer and collaborators to include impurity effects [265,266], spin and spin-orbit effects [267], Fermi surface anisotropy effects [268], and retardation effects [269].All of these papers used an instantaneous attractive potential (i.e. as in BCS theory), except for the last. Further developments to include retardation effects were carried out in Refs. [121,122,260,270] and others. Finally, in [271] the Eliashberg theory of Hc2, including Pauli paramganetic limiting and arbitrary impurity scattering, was formulated and solved. Without retardation effects or Pauli limiting, the zero temperature upper critical field,when expressed in terms of the slope near Tc , takes on universal values, dependent only on the elastic impurity scattering rate, given by 1/. For example, the quantity  hc2(0) ≡ Hc2(0)/(Tc |Hc2 (Tc )|) is given by 0.693 in the dirty limit (1/ >> ) and 0.727 in the clean limit (1/ = 0). For intermediate scattering rates the result falls somewhere in between. It is worth mentioning that the absolute value of the upper critical field increases with increased impurity scattering. We often use the ratio because the slope near Tc is measured, and then the zero temperature value is obtained by using the universal number quoted above. The value at zero temperature is of special interest because the Ginzburg–Landau coherence length can then be extracted through Hc2 =

¥0 . 2 2GL

(3.141)

Here ¥0 is the fluxoid quantum, and we have used the subscript‘GL’ to denote the Ginzburg–Landau coherence length, which, at zero temperature, is often close to the BCS coherence length, and gives us an indication of the Cooper pair size [272]. Hence, deviations

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from 0.693 (or 0.727) due to retardation effects are of interest for this reason. For completeness, we quote the equations which govern Hc2 , taking into account electron–phonon interactions in the Eliashberg sense, and Pauli limiting. The gap equation is linear in the order parameter [271]    ∗ ˜ (i! (i! − i! ) −

n ) = T n m m (3.142) ˜ (i! m) × −1 , (!(i! ˜ m )) − 1/2 with !(i! ˜ n ) = !n + T × sgn!m +

 m

(i!n − i!m )

1 sgn!n . 2

(3.143)

The factor (!(i! ˜ n )) is given by (!(i! ˜ n )) =

√2 ˛

∞

2

dq e−q tan−1 √   0 ˛q . × |!(i! ˜ ˜ n )| + i B Hc2 sgn!(i! n) (3.144) Here ˛(T) ≡ 12 |e|Hc2(T)vF2 , with e the charge of the electron and vF the electron Fermi velocity. B is the Bohr magneton. Equation (3.142) can be written as an eigenvalue equation, just like Tc . It is linear because the solution is valid only on the phase boundary between the normal and the superconducting states. We have carried out extensive numerical investigations of hc2(0) as a function of coupling strength. In the conventional regime, the dependence on coupling strength is very weak [273]; in the dirty limit hc2 (0) decreases initially (as a function of Tc /!ln) to about 0.65, and then increases to beyond 0.70. In the clean limit there is first a barely discernible decrease, followed by an increase to values of approximately 0.80. These are all theoretical results, and, in many cases have not been carefully investigated with experiment. On the other hand the expected changes are of order 10% or less, and may well be masked by other effects. A thorough investigation was provided for Nb by Schachinger et al. [274]. The agreement with the available data was excellent (although they

Fig. 3.42. Comparison of experiment (square symbols) with Eliashberg calculations of the upper critical field in Nb with some N impurities. Adapted from [274]

did invoke, in addition to the theoretical framework described here, anisotropy effects). For example, in Fig. 3.42 we show their results for Nb (with 0.3% N) with Tc = 9.1 K. The data is represented by the solid squares, while the solid curve is the result of Eliashberg calculations based on Eqs. (3.142), (3.143) and (3.144) generalized to include anisotropy (Pauli limiting has not been accounted for). The agreement is very good. For further information the interested reader is directed to the aforementioned references. Before leaving this section we should also mention that optimum spectrum analysis [220] and asymptotic limits (3rd reference in [249]) have also been investigated for Hc2 ; the result is very dependent on elastic impurity content, except in the asymptotic limit. In that case, the results approach a universal value, i.e. hc2(0) → 0.57 as  increases.

3.6

Response Functions

In the previous sections we have seen effects due to the inclusion (through the Eliashberg formalism) of the detailed electron–phonon coupling. The result is in many cases a large quantitative correction to the corresponding BCS result. In this way one can

3

infer, from experiment, the necessity of taking into account the dynamics of the electron–phonon interaction. Nonetheless, as we saw in Sect. 4.3 (particularly in the Tunneling and Optical Conductivity subsections) dynamical interactions manifest themselves more clearly in dynamical properties. For this reason we now focus on various response functions. 3.6.1

Formalities

A theory of linear response can be approached from two very different frameworks, the Kubo formalism, and the Boltzmann equation. The two frameworks often lead to the same result; their connection is discussed at length in [81]. Early treatments [5,275] of the various response functions in a superconductor neglected the electron–phonon interaction, except insofar as it provided the mechanism for the superconductivity in the first place. The main interest was the investigation of a new state which apparently had a single electron energy gap, which would manifest itself either directly in spectroscopic methods (optical and tunneling) or more indirectly as a function of temperature (NMR relaxation rate, acoustic attenuation, etc.). Sometime later two seminal papers appeared [105,276], both of which discussed the impact of the electron–phonon interaction on transport in the electron gas. These dealt specifically with the normal state.Work at a similar level but in the superconducting state appeared a little later [277]; this latter work was generalized to apply for arbitrary elastic impurity scattering only much later [278]. These authors used quasiclassical techniques; below we will sketch an alternative derivation based on the Kubo [279] formula. We should preface this work with some remarks about vertex corrections. They are generally ignored in calculations of response functions, so that a particle-hole ‘bubble’, consisting of one single electron Green function and one single hole Green function, requires evaluation (see, for example, [184,189,191,277,278,280–283]; there have been many others in the last ten years especially). Older work [284,285] investigated the need for vertex corrections and found that they contributed very little; later work in the normal state [179, 185] sug-

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123

gested that their contribution could be summarized by substituting a ‘transport’ electron–phonon spectral function, ˛tr2 F( ), for the usual spectral function, ˛ 2 F( ), in the transport equations. This alteration discriminated in favour of back scattering as being particularly effective in depleting the current, as one would expect. Over a large frequency range, however, these spectral functions are not expected to differ substantially; nonetheless, quantitative investigations are currently lacking [81], particularly in the superconducting state. The contribution to the conductivity consists of two components: the paramagnetic and diamagnetic responses. The diamagnetic response is straightforward (Note, in the case of tight-binding, one must exercise care in determining these relative contributions. See, for example, [286].); the paramagnetic response is determined by the evaluation of a current– current response function. A standard decoupling of this function (ignoring, as noted above, vertex corrections) yields i  ne2   ( ) = ¢( + iı) + , (3.145)  + iı m where ¢( + iı) is the paramagnetic response function whose frequency dependence (on the imaginary frequency axis) is given by  ˆ ¢(in ) = N1ˇ k,m Tr(evx )2 G(k, i!m) (3.146) ˆ × G(k, i!m + in ) , ˆ where G(k, i!m ) is actually a matrix in the Nambu formalism [2]. It is given, in terms of functions with which we are already familiar, by i! Z(i! ) + ( k − )ˆ3 + (i!m )ˆ1 ˆ G(k, i!m) = − m 2 m , 2 2 ( k − ) − (i!m Z(i!m )) +  (i!m )

(3.147) where the Pauli spin matrices are given by     10 01 ˆ0 ≡ , ˆ1 ≡ , 01 10     0 −i 1 0 ˆ2 ≡ , ˆ3 ≡ . i 0 0 −1

(3.148)

In Eq. (3.146) the trace is over the Pauli spin space. The presence of the factor (evx )2 shows explicitly that the dressed vertex has been replaced with a bare vertex; vx is the component of the electron velocity in the

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x-direction. The momentum sum is over the entire Brillouin zone; the factors preceding the summations include the total number of atoms in the crystal, N, and the inverse temperature, ˇ ≡ 1/kBT. The diamagnetic piece in Eq. (3.145) contains the electron density n and the electron mass, m. Finally, the single electron energy is denoted by k , and, as before, we use a notation where we explicitly subtract off the chemical potential, . The paramagnetic kernel denoted by ¢(in ) in Eq. (3.146) is a special case of a more general ‘bubble’ diagram. A similar calculation, for example, is required for the NMR relaxation rate [172,287,288], or the phonon self-energy [289–292], except that the vertices are not proportional to ˆ0 (as was the case in Eq. (3.146)), but to some other Pauli matrix. This has the effect that the so-called ‘coherence factors’ will differ, depending on the particular response function; some will result in a cancelation with singularities arising from the single electron density of states, whereas others will result in a potentially singular response, at low frequencies, as we shall see below. An early review outlining these differences in the context of Eliashberg theory is given in [13]. Returning to Eq.(3.146),the standard procedure is as follows; one would like to evaluate the Matsubara sum — only then can one perform the proper analytic continuation to real frequencies required for the optical conductivity.This is straightforward,through the spectral representation, which is the Nambu generalization of Eq. (3.27). The cost is that two new frequency integrals are required, one of which can be done immediately by making use of the Kramers– Kronig-like relation (see Eq. (3.72)) ˆ G(k, z) =

∞

ˆ A(k, !) , z−!

response allows us to make the usual replacement, given already by Eq. (3.19):  1  → d N( ) , N k

where N( ) is the single electron density of states. As in that case N( ) can be taken as constant (= N( F )) and, along with the electron velocity, vx, taken out of the integration as an overall constant, 2N( F )e2vx2 ≡ !P2 ≡ ne2/m, where !P is the electron plasma fre4 quency. However, one would normally like to extend the integration over single electron energy from −∞ to +∞, as is often done within Eliashberg theory. Here, however, one has to be slightly more careful, and first subtract the normal state contribution to the kernel. This makes the integral sufficiently convergent that extension to an infinite bandwidth (effectively) is possible. Then the integral can be readily performed by contour integration. The integration over the normal state contribution alone must be done separately; an integration cutoff ±D must be used, the effect of which is an additional (imaginary) contribution. The final result is [293] ∞   ine2 ˇ!  ( ) = × d! tanh m 2 0   × h1 (!, ! +  ) − h2 (!, ! +  ) 

D d! tanh

+

(3.150)

  × h∗1 (!, ! +  ) + h2(!, ! +  ) , with h1 (!1 , !2 ) =

with z anywhere in the upper half plane. Finally, we would like to perform the Brillouin zone integration analytically; to do so, we note that the only dependence on wavevector k in Eq. (3.146) occurs through k (this is not so for more complicated response functions, such as is required for neutron scattering, for example, where the momentum dependent kernel, ¢(q,  + iı), is required). This feature of the optical

h2 (!1 , !2 ) =

−∞



−

(3.149)

d!

ˇ(! +  ) 2

N(!) = P(!) = (!) =

1 − N(!1 )N(!2 ) − P(!1 )P(!2 ) , 2( (!1) + (!2 )) 1 + N ∗ (!1 )N(!2 ) + P ∗ (!1 )P(!2 ) , 2( (!2) − ∗(!1 )) !(! ˜ + iı) (3.151) (! + iı) , (! + iı) , (! + iı)  ! ˜ 2 (! + iı) −  2 (! + iı) ,

3

where D is the large cutoff mentioned above, to be taken to infinity for large electronic bandwidth, and !(! ˜ + iı) ≡ !Z(! + iı). Various limits can be extracted from these expressions; for example the normal state results of Sect. 4.4.3 can be readily obtained, as well as the simple Drude result, obtained by assuming only elastic scattering characterized by a frequency independent rate, 1/. When inelastic scattering is included (here through electron–phonon scattering), low frequency Drude-like fits can be obtained through simple expansions [187]. We will turn to these later. Equation (3.150) represents the ‘standard’ theory of the optical conductivity with Eliashberg theory. As already mentioned, this characterization includes the caveats discussed above about vertex corrections and ˛ 2 F( ) → ˛tr2 F( ) replacements. It is valid for both inelastic scattering and elastic scattering processes (within the Born approximation). The impact of elastic scattering on the Eliashberg equations has not yet been discussed,so we turn to these now.Equations (3.131) and (3.132),on the imaginary axis,along with Eqs.(3.77) and (3.78),on the real frequency axis, are written for the clean limit (‘clean limit’ is here defined to mean that the elastic scattering rate is zero, 1/ = 0). When elastic scattering is included, new terms appear on the right hand side of these equations. (As an aside, one way of using the existing equations to include elastic scattering is to include a 2 component of ˛ 2 F( ) (called ˛imp F( ) for simplicity) which models the elastic scattering part. At any nonzero temperature it will be given by a zero frequency contribution 2 ˛imp F( ) =

 ı( ). 2T

Electron–Phonon Superconductivity

(! + iı) → (! + iı) +

125

i (! + iı)  , 2 2 ! ˜ (! + iı) −  2 (! + iı)

(3.153) !(! ˜ + iı) → !(! ˜ + iı) +

i !(! ˜ + iı)  . 2 2 ! ˜ (! + iı) −  2 (! + iı)

(3.154) Equations (3.150) and (3.151) remain the same with impurity scattering. This is a consequence of the so-called Anderson’s “theorem” [294]. The modifications are all implicitly contained in the pairing and renormalization functions. Note that the gap parameter, (! + iı) ≡ (! + iı)/Z(! + iı), remains the same, independent of the impurity scattering rate. 3.6.2

BCS Results

The purpose of this chapter is to examine effects specifically due to the electron–phonon interaction. Nonetheless, it is best to first see what occurs in the BCS limit, and then examine the differences. The means for achieving the BCS limit from Eliashberg theory was examined in Sect. 4.4; in general we mean by the‘BCS limit’that limit which corresponds to taking ˛ 2 F( ) to be non-zero only for some very high frequency component (so that the “strong coupling” indicator Tc /!ln → 0). As a result, the renormalization function, Z(! + iı) → 1, and the gap function (!+iı) → ,a constant,as a function of frequency. This allows one to explicitly break up the integrals in Eq. (3.150) into portions involving the BCS gap parameter, , and the electromagnetic frequency,  . A very efficient FORTRAN program has been provided in [295] in this case.

(3.152)

Substitution of this expression into Eqs. (3.131) and (3.132), for example, will yield simple expressions on the right hand side proportional to 1/.) In principle one would think that Eqs. (3.131), (3.132), (3.77) and (3.78) require iteration to a solution for every new value of impurity scattering. In actual fact, however, they need be solved only in the clean limit. Then, the pairing (! + iı), and renormalization !(! ˜ + iı) ≡ !Z(! + iı) functions can be modified by the simple contribution

Finite Impurity Scattering Most results are obtained numerically. However, some can be obtained analytically [184], and so we catalogue them here. Some of these results have been re-derived recently in [296]. Using Eqs. (3.153) and (3.154) in the BCS limit, we get (! + iı) =  +

i sgn! √ 2 ! 2 − 2

Z(! + iı) = 1 +

i sgn! √ . 2 ! 2 − 2

(3.155)

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Using the single electron Green function, whose diagonal component is [13] !Z(! + iı) + k , 2 2 ! Z (! + iı) − k2 −  2(! + iı) (3.156) one can examine the pole, !k ≡ Ek − ik /2 to extract the quasiparticle energy Ek and inverse lifetime k . In the BCS limit the quasiparticle energy is G11(k, ! + iı) =

E = k2 + 2 , as it is without impurities, while the scattering rate is √ 1 E2 − 2 1 | k |  = = . (3.157)  |E|  2 + 2 k

Equation (3.157) shows that at the Fermi surface impurities are ineffectual for electron scattering, in accord with Anderson’s “theorem” [294]. This happens abruptly at Tc ; as soon as a gap develops the scattering rate becomes zero. The single electron Green function (3.156) can be rewritten in the BCS limit [184]: G11 (k, ! + iı) = (3.158)   ! 1 1 1+ √ √ 2 2 2 2 2 ! − ! −  − k + 2i sgn!   ! 1 1 √ − 1− √ 2 2 2 2 2 ! − ! −  + k + 2i sgn!   The coherence factors, 1 ± √!!2 −2 , conspire to make the spectral function, A(k, !) ≡ − 1 ImG(k, !), zero for |!| < , independent of the impurity scattering rate, as can be readily verified explicitly from Eq. (3.158). On the other hand the precise pole of the Green function is given by solving for ! = E − i /2 in the two coupled equations  2 1 2 2 2 2 E − ( /2) =  + k − (3.159) 2 | k | 1 (3.160) |E|  For 1/ = 0 the solution is as before, E = Ek ≡ k2 + 2 , and  = 0. For finite impurity scattering,  =

however, the solution depends on surface ( k = 0):

1 2

at the Fermi

7 E=

2 − (

1 2 ), 2 7

E = 0,

 =

 =0

for

1 <1 2

(3.161) 1 1 >1 ( )2 − (2)2 for  2

(3.162) Further discussion of the meaning of these solutions is provided in [184]. Far-Infrared: Dirty Limit A historically important case is the dirty limit. This is defined by 1/ >> , and was first treated by Mattis and Bardeen [275]. An analytical expression can be obtained at zero temperature [275]:   1 4 2 E(k) − K(k)  > 2 = 1+ n   (3.163) and     2 2 1 1 2  1+ E(k ) − 1− K(k ), = n 2  2  (3.164) 2 where n ≡ nem  is the normal state conductivity (pure real) and the real part of the conductivity is identically zero for frequencies,  < 2. In these expressions   √  2 −     and k = 1 − k2 , k= (3.165)  2 +  and E(k) and K(k) are the complete elliptic integrals of the first and second kind. For other cases (finite temperature and/or lower impurity scattering rate) one must integrate numerically [295]. Figure 3.43 shows (a) the real part and (b) the imaginary part of the conductivity in the zero temperature BCS superconducting state, for various impurity scattering rates. We have used some definite values for the impurity scattering rates and the coupling strength.The latter has been chosen to yield an absorption edge, 2 = 10.4 meV, which, because of the insensitivity of superconductivity to elastic impurity scattering [294],holds for all scattering rates.A well-defined absorption onset is evident in Fig. 3.43a; otherwise

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127

Fig. 3.43. (a) 1 ( ) vs.  in the zero temperature BCS superconducting state for the various impurity scattering rates indicated. The absorption onset at 2(0) remains sharp independent of the scattering rate.A deltafunction contribution (not shown) is also present at the origin. (b) Same as in (a) except for the frequency times the imaginary part of the conductivity. The optical gap is a little less evident in the dirty limit. The conductivity is given in units of ne2 /m ≡ !P2 /4 ). Taken from [184]

the curves simply deviate from what would have been Drude-like curves in the normal state. In Fig. 3.43b the frequency times the imaginary part of the conductivity is shown for the same scattering rates. Such a combination is shown because the zero frequency limit gives a direct measure of the London penetration depth: 1/2 (T) = lim

 →0

4  2 ( ). c2

(3.166)

As is evident from the figure, the penetration depth increases as the impurity scattering rate increases. Another feature stands out in Fig. 3.43b; there is a notable ‘dip’ in  2 ( ) at 2, particularly in the clean limit. Otherwise the curves all approach the Drude limit at high frequency, which, for this property, is unity (conductivities are in units of ne2/m ≡ !P2 /4).

Penetration Depth Before we examine the effects of the electron– phonon interaction on the real and imaginary parts of the conductivity, we first summarize the ‘BCS’ results for the penetration depth as a function of impurity scattering, which can be extracted analytically [189,297] from the zero frequency limit of the 1 conductivity. The result is, with ˛ ≡ 2 , 1 2 (T



= 0)

=

1 cl2 (T

= 0)

 √ 1  −1 2 − √ sin ( 1 − ˛ ) , ˛ < 1 2˛ ˛ 1 − ˛ 2 1 = (3.167) 2 cl (T = 0)   ˛ + √˛ 2 − 1  1  − √ ln √ , ˛ > 1. 2˛ 2˛ ˛ 2 − 1 a − ˛2 − 1

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F. Marsiglio and J. P. Carbotte

Here, the zero temperature London penetration depth in the clean limit is given by cl2 (T = 0) =

mc 2 . 4ne2

(3.168)

In the weak scattering limit Eq. (3.167) reduces to the more familiar form, 1 2 (0)

≈

1 cl2 (0)

1 . 1 + 4 ˛

Fkk = (uk uk ∓ vk vk )2

(3.169)

This expression can be written in terms of the zero temperature coherence length, 0 , and the mean free vF path, , using  =  and vF = /, where vF is the 0 Fermi velocity: 1 1 1 ≈ 2 . 2 (0) cl (0) 1 + 82 0

detailed discussion of the NMR relaxation rate [303] is outside the scope of this review, and the final expression relevant to superconductors is a special case of the microwave conductivity. Within BCS theory the transition probabilities between an initial and final state that enter the expression for various linear response functions are of the form [5,44]

(3.170)

Microwave Regime: Coherence Factors The microwave regime (1–60 GHz) corresponds to very low energies (1 GHz = 0.0041 meV). This energy scale is much lower than that of the superconducting energy gap. Measurements of the microwave response of a superconductor have been used in recent years to determine the penetration depth and optical conductivity in the high-Tc cuprates [298, 299], but, historically, either the real or the imaginary component of the surface impedance was measured,making a determination of the complex conductivity impossible. It is of interest to examine the conductivity in this case, because BCS theory makes a highly nontrivial prediction that the real part of the conductivity shows a so-called coherence peak just below Tc . This coherence peak was almost simultaneously predicted [5] and observed [300, 301] in measurements of the NMR relaxation rate. (Note, BCS acknowledge Tinkham and coworkers (private communication) for finding the increase in absorption in thin films, but we have been unable to track a published reference. Further microwave measurements were carried out for Al [302], but only the real part of the impedance was measured.) We will briefly discuss the source of these coherence factors, and return to a description of the microwave conductivity, since a

,

(3.171)

with uk , vk the amplitudes that relate quasiparticle operators to electron operators 1/2   k −

1 1+ uk = , (3.172) 2 Ek 1/2   k −

1 1− vk = , (3.173) 2 Ek and Ek is the usual quasiparticle energy: Ek = ( k − )2 + 2k , (3.174) where k is the electron band energy, is the chemical potential and k is the gap function. In Eq. (3.171) the upper (lower) sign corresponds to case I (case II) observables. These signs have important consequences for the response, particularly just below Tc . A case in point is the electromagnetic absorption; the temperature-dependent result (derived from Eqs. (3.150) and (3.151)) in the dirty limit (1/ >> ) is [275] 2 1 = n 

∞ dE 0

E(E +  ) + 20 [f (E) − f (E +  )] − +  )2 − 20 )1/2 −0 1 dE (3.175) + ( − 20 )  (E2

20 )1/2((E

0 −

(E2

E(E +  ) + 20 [1 − 2f (E +  )], − 20 )1/2((E +  )2 − 20 )

where 0 ≡ (T) is the temperature-dependent gap function. The second plus sign in E(E + )+0 which appears in this expression is due to the fact that the electromagnetic absorption is a case II observable. In a case I observable this would be a minus sign; it

3

is then readily seen that whereas Eq. (3.175) contains a divergence as  → 0, the corresponding case I observable would not, as the numerator (coming from the coherence factor given in Eq. (3.171)) would then cancel the density of states factors, which are explicit in Eq. (3.175), and which contain square-root divergences. In both cases the ‘freezing out’ of excitations as the temperature is reduced leads to a low temperature suppression of the response function — this is simply a consequence of the gap. On the other hand, near Tc an enhancement is expected for type II observables, while, for type I observables, the response is immediately suppressed as the temperature is lowered below the superconducting transition temperature. In the limit that the frequency is zero, one obtains from Eq. (3.175), 1 =2 n



∞

dE 0

E2 + 20 E2 − 20

 −

@f @E



Electron–Phonon Superconductivity

129

which is formally divergent (at all temperatures). The divergence is in fact eliminated in practice by anisotropy in the gap or retardation effects. As noted by the second equality, this is the expression for the superconducting to normal ratio of the NMR relaxation rate. For a type I observable (like the ultrasonic attenuation) the numerator in Eq. (3.176) has a minus sign, so that numerator and denominator cancel, and the remaining integral is trivial. One obtains ˛s /˛n = 2f ((T)),

(3.177)

where ˛s(n) is the ultrasonic attenuation in the superconducting (normal) state, and f is the Fermi function. This is a monotonically decreasing function as the temperature decreases from Tc to zero.

≡ (1/T1)s /(1/T1)n ,

(3.176)

Fig. 3.44. Frequency dependence of 1 ( ) near the clean limit (1/ = 1 meV) for various temperatures in the BCS superconducting state. The appearance of a gap is evident, even at temperatures close to Tc . (b) Same as in (a), but for  2( ). The appearance of a gap is evident in the imaginary part of the conductivity as well. The conductivity is given in units of ne2 /m ≡ !P2 /4 ). Taken from [184]

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F. Marsiglio and J. P. Carbotte

Fig. 3.45. Frequency dependence of 1 ( ) near the dirty limit (1/ = 25 meV) for various temperatures in the BCS superconducting state. The appearance of a gap is evident, even at temperatures close to Tc . (b) Same as in (a), but for  2( ). The appearance of a gap is evident in the imaginary part of the conductivity as well. The conductivity is given in units of ne2 /m ≡ !P2 /4 ). Taken from [184]

Far-Infrared Regime — Arbitrary Impurity Scattering

3.6.3

Eliashberg Results

The expressions for the optical conductivity provided in the last three subsections apply only in the dirty limit. As already mentioned earlier, a comprehensive expression (for all values of elastic impurity scattering), along with a very efficient FORTRAN program, was provided in [295]. For completeness, we illustrate here the temperature dependence for two extreme cases, close to the clean limit (1/ = 1 meV), and the dirty limit (1/ → ∞), in Fig. 3.44 and Fig. 3.45, respectively. As noted earlier, the optical gap (= 2(T)) is clearly evident in both the real and imaginary part of the conductivity. The evolution from the normal state to the superconducting state is clearly evident as well; note, in particular, that in the real part of the conductivity, the missing area is taken up as a delta function at the origin (not shown).

Within Eliashberg theory, changes occur for two related reasons. First, even in the normal state the self-energy acquires a frequency dependence (no wavevector dependence, because of the simplifying assumptions made at the start); secondly, the gap function in the superconducting state acquires a frequency dependence and acquires an imaginary part. This latter fact tends to smear many of the ‘sharp’ results shown in the last section, a feature which is already evident in comparing the single electron densities of states in Fig.3.36 to those in Fig.3.33b,for example. For this reason, it is important to re-examine the impact of retardation on a variety of observables. NMR Relaxation Rate In the first few years following the discovery of the high temperature superconductors [7], several

3

Electron–Phonon Superconductivity

131

along with data from YBCO [306], and theoretical results obtained using scaled spectra.While the present consensus is that the lack of a coherence peak is not solely due to damping effects,the lesson learned from these calculations is clear: retardation effects damp out the coherence peak in the NMR relaxation rate. It is worth noting here that even within a BCS framework (i.e. no retardation), the coherence peak can be suppressed in the dilute electron density limit [307]. Microwave Conductivity

Fig. 3.46. Nuclear spin relaxation rate vs. reduced temperature. Data points for Indium are indicated by circles and triangles, while data for YBa2 Cu3 O7 are indicated by squares and crosses. The solid curves are calculated with Eliashberg theory for Indium (upper curve) and two model spectra with  = 1.66 and 3.2 (lowest curve). Agreement is good in the case of Indium and the lowest curve. Reproduced from [172]

anomalous features were measured in the superconducting state. One of these was the absence of the coherence peak (the so-called ‘Hebel–Slichter’ peak) in the NMR spin relaxation rate, 1/T1, just below Tc [304]. Motivated by the possibility that this ‘anomaly’ could be explained by damping effects due to retardation, Allen and Rainer [172] and Akis and Carbotte [288] calculated the ratio of the relaxation rate in the superconducting state to that in the normal state with several hypothetical electron–phonon spectra (obtained by scaling known spectra from conventional superconductors). Both groups found that sufficiently strong coupling (as measured by  or Tc /!ln) smears out the coherence peak entirely. An example is shown in Fig. 3.46 (taken from [172]), which shows the theoretical and experimental [305] results for a conventional superconductor (Indium)

A natural extension of this argument applies to the microwave conductivity. In this case, even within BCS theory, a divergence does not occur since the experiment is conducted at some definite non-zero microwave frequency (see Eq. (3.175)). Before discussing retardation effects, however, it is important to realize the amount of impurity scattering (as characterized by 1/) also influences the height and presence of the coherence peak [281, 282]. In Fig. 3.47a we show, within the BCS framework, the conductivity ratio for a small but finite frequency as a function of reduced temperature, for a variety of elastic scattering rates, ranging from the dirty limit to the clean limit. Quite clearly the coherence peak is reduced and then eliminated as a function of 1/. To see how retardation effects also serve to reduce and eliminate the coherence peak (just as in NMR) we focus on the dirty limit (1/ → ∞) where the peak is largest without retardation. In Fig. 3.47b we show results obtained from a Pb spectrum (Fig. 3.13), scaled by varying degrees to increase  from 0.77 to 3.1. For the largest coupling considered the coherence peak has essentially vanished. This is the same effect seen in the NMR relaxation rate. In Fig. 3.47c we illustrate the impact of changing the microwave frequency. Clearly, in the limit of very weak coupling (BCS) one expects the strongest variation, since, as  → 0, the BCS result will diverge logarithmicaly. However, as the coupling strength increases, the damping due to retardation reduces the peak far more effectively than an increase in microwave frequency would, so that the conductivity ratio (at some temperature near where a maximum would occur in the BCS limit) is essentially constant as a function of frequency.This is

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F. Marsiglio and J. P. Carbotte

Fig. 3.47. (a) Conductivity ratio, 1s /1n , versus reduced temperature, T/Tc , in the BCS limit, for various impurity scattering rates. From top to bottom the curves are calculated for 10 = 100, 2, 1, 0.5, 0.1, 0.05, and 0. The frequency used was  /0 = 0.02. In the clean limit (1/ = 0) the coherence peak has disappeared. (b) Same quantity as in (a), but for different coupling strengths,  = 0.77, 1.5, 2.3, and 3.1. (The peak diminishes with increasing coupling strength). These were computed in the dirty limit (1/ = 500 meV) and for  = 0.05 meV. The result for  = 0.77 (largest maximum) is nearly identical with the BCS result. (c) Conductivity ratio versus frequency normalized to the zero temperature gap edge,  /0, for the same coupling strengths as in (b). The curves decrease in magnitude with increasing coupling strength. The maximum apparent in (b) for  = 0.77 and 1.5 is also clear here since the two uppermost curves have magnitude greater than unity. As the coupling strength increases the conductivity ratio becomes independent of frequency. Calculations are in the dirty limit, with T/Tc = 0.85. Taken from [281]

clearly illustrated by the two lowest curves in the figure, representing the strongest coupling situations. A measurement of the coherence peak in the microwave wasn’t actually performed until the early 1990’s, in Pb [308] and in YBCO [309] (although the peaks observed in these latter measurements are now thought not to be the BCS coherence peak [299,310,311]. Several other groups have since examined the microwave response in conventional superconductors.

In [312] Nb was examined in detail. The experiment was performed at 17 GHz, and a prominent coherence peak was observed, as shown in Fig. 3.48. Also shown are theoretical curves obtained from Eliashberg calculations; they all fall significantly below the experimental results. We have also included the BCS result (dotted curve) computed for this frequency; it is not very different from one of the curves obtained using the full Eliashberg formalism. The BCS result represents probably the highest achievable co-

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Electron–Phonon Superconductivity

133

Far-Infrared Regime

Fig. 3.48. Microwave conductivity normalized to the normal state, 1s /1n , as a function of reduced temperature T/Tc . The open squares are the data for Nb. The dotted curve is the BCS result with experimental frequency ! = 17 GHz and impurity scattering rate 1/ = 100.0 meV (dirty limit). The solid and dashed curves are the results of full Eliashberg calculations with two different (˛ 2 F(!)) spectra. None of the theoretical curves can reproduce the data. Taken from [312]

herence peak; other alterations of the standard theory (anisotropy, finite bands, non-dirty limit, etc.) would tend to decrease the theoretical result further. Hence, at present the coherence peak observed in Nb remains anomalous because it is too big. Other measurements in Nb and Pb [313] showed agreement with Eliashberg theory, but they were carried out at a much higher frequency (60 GHz). Another measurement of the electrodynamic response (using simultaneous measurement of the amplitude and phase of the transmission in Nb thin films) [314] supported our results. A more recent measurement of the coherence peak in Nb3 Sn [315] also finds a large discrepancy with Eliashberg theory — the experimental results show a peak which is far too large compared to theory.

While more recent investigations of the far-infrared (and slightly lower Terahertz) regime in superconductors utilize transmission techniques which simultaneously measure amplitude and phase information [310,314], the more conventional Fourier-transform spectroscopy [183] requires Kramers–Kronig relations, as outlined in Sect. 4.4.3. For this reason the entire spectrum needs to be measured, often with an assortment of spectrometers [182]. How do the real and imaginary parts of the conductivity change as a function of the coupling strength  ? In Fig. 3.49 we show real ((a) and (b)) and imaginary ((c) and (d)) parts of the conductivity with 1/ = 2 meV and 25 meV, respectively. In all four figures it is clear that an increased coupling strength decreases the real and imaginary parts of the conductivity, at least in the low frequency regime. In fact, at low temperatures, in the normal state, one can derive a Drude-like expression [187] Drude ( ) ≈

ne2 1/ ∗ m∗  2 + [1/ ∗]2

(3.178)

where m∗ /m = 1 +  and / ∗ = 1/(1 + ). This expression clearly indicates that, while the zero frequency conductivity remains unaffected, the rest of the conductivity is diminished by the electron– phonon interaction [105]. In fact integration of Eq. (3.178) yields the result ∞ d Drude ( ) = 0

 ne2 1 . 2 m∗ 1 + 

(3.179)

This is lower than the Kubo sum rule [279] by the factor of 1/(1 + ), which says that the rest of the area is taken up in the phonon-assisted absorption, which occurs at higher frequency (in the phonon range). Also note that one effect of an increased electron– phonon interaction strength is to decrease the im1 purity scattering rate: 1/ → 1/ 1+ . This occurs because the inelastic scattering reduces the spectral weight of the quasiparticle undergoing the elastic scattering. Further discussion of the Drude-like behaviour at low frequency but for non-zero temperature can be found in [186,187].

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F. Marsiglio and J. P. Carbotte

Fig. 3.49. The real part (a, b) and the imaginary part (c, d) of the conductivity at essentially zero temperature (T/Tc = 0.3) with 1/ = 2 meV (a,c) and 1/ = 25 meV (b, d). In all cases we have used the BKBO spectrum scaled to give the designated value of, , while Tc is held fixed at 29 K by adjusting ∗ . Increased coupling strength suppresses both 1 ( ) and  2 ( ) and broadens the minimum in the latter at 2. Note that 2 increases slightly as the coupling strength is increased. The conductivity is given in units of ne2 /m ≡ !P2 /4 . Taken from [184]

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Electron–Phonon Superconductivity

135

Fig. 3.50. Measured  2 ( ) vs. frequency at T = 9 K and at T = 300 K (solid curves). Also shown are the theoretical fits, using the BKBO spectrum, scaled so that  = 0.2 (dashed curves). Tc is kept fixed to the experimental value with a negative ∗ . Finally, theoretical fits are also shown with  = 1 (dotted curves). The latter curves are clearly incompatible with the experimental results. Adapted from [189]

Returning to Fig.3.49,we note that except for small corrections to the gap edge as  increases (2 tends to increase as well),the occurrence of an abrupt onset of absorption in the real part ((a) and (b)) exists for all coupling strengths. While a cusp remains in the imaginary part ((c) and (d)), its size is clearly diminished as the coupling strength increases. Note that the penetration depth (given by the square-root of the inverse of the intercept in the imaginary part — see Eq. (3.166)) tends to increase as the coupling strength increases. Also note that, while not apparent on the frequency scale shown in (c) and (d), the frequency times the imaginary part of the conductivity approaches unity (in units of ne2/m) as the frequency approaches large values. This fact was utilized in the case of Ba0.6 K0.4BiO3 , which we briefly discuss next. Fig. 3.50 shows the imaginary part of the conductivity obtained from reflectance measurements on Ba0.6 K0.4 BiO3 [189, 316, 317]. A prominent dip occurs near 12 meV, which has been roughly fit by two models as indicated. The occurrence of this dip fully supports the existence of a superconducting state with s-wave symmetry, with a gap value that

is high compared to that expected from BCS theory (2/kB Tc ≈ 5 compared with 3.5). This value is somewhat higher than that obtained previously with infrared [318] or tunneling [319,320] measurements. Nonetheless, a thorough analysis of the temperature dependence of the Drude fits at low frequency [187] and the frequency dependence illustrated in Fig. 3.50 [189] shows that the electron– phonon interaction must be weak in this material,too weak to support 30 K superconductivity. Two model calculations are shown with the data in Fig. 3.50. The data is clearly consistent with an electron–phonon coupling strength  ≈ 0.2 (which requires an additional mechanism to produce Tc = 30 K),and entirely inconsistent with  ≈ 1. As is clear from the preceding paragraph, either the real or the imaginary part of the conductivity contains all the relevant information about the absorption processes in the system. This is due to the fact that they obey Kramers–Kronig relations, which ultimately can be traced to requirements of causality and analyticity [183]. In an effort to make these absorption processes more explicit, one can also favour other functions; a particular example is the effective

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Fig. 3.51. The conductivity–derived scattering rate, !2

1/( ) ≡ 4P Re(1/ ( )) vs. frequency in the normal state for pure elastic scattering (dashed line), combined elastic and inelastic scattering (BKBO spectrum with  = 1), and pure inelastic scattering using a model spin fluctuation spectrum appropriate to YBCO. Because of the difference in spectral function frequency scales, the result for YBCO continues to rise with frequency, even at 300 meV. Reproduced from [184]

dynamical mass, m∗ ( ), and the effective scattering rate, 1/( ), introduced through [186]  ( ) =

!P2 1 , 4 1/( ) − i m∗ ( )/m

(3.180)

where !P and m are the bare electron plasma frequency and mass, respectively. Then, one can define an effective scattering function, 1/( ), which can be extracted (say, from experiment) through 1/( ) ≡

!P2 1 Re , 4  ( )

Fig. 3.52. Conductivity–derived scattering rate, 1/( ) vs. frequency in the BCS s–wave superconducting state for (a) 1/ = 2 meV and (b) 1/ = 25 meV. An abrupt onset of absorption at the optical gap at temperatures near Tc is more apparent in (a) than in (b). The horizontal dashed line indicates the normal state result. Reproduced from [184]

(3.181)

bearing in mind that  ( ) itself has been obtained through Kramers–Kronig relations from, say, reflectance data. This is precisely the function required to invert normal state conductivity data to extract

˛ 2 F( ) (see Eq. (3.91)). A plot of 1/( ) vs. frequency is nevertheless revealing. It tends to illustrate at roughly what energies absorption processes ‘turn on’ [186,321]. For example, we show in Fig. 3.51 the function 1/( ) derived from conductivity results of model calculations for Ba1−x Kx BiO3 and YBaCu3 O7−x [184].The former uses a model phonon spectrum extracted from neutron scattering measurements [322] while the latter uses a model spin fluctuation spectrum [323]. The fact that the YBaCu3 O7−x result continues to rise at 300 meV reflects the frequency scale of the spin fluctuation spectrum. In contrast, the Ba1−x Kx BiO3 result has almost saturated by 100 meV,

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perature), while slightly above the gap the effective scattering rate below Tc is actually enhanced with respect to the normal state value. In Fig. 3.53 we show the effective scattering rate vs. frequency using the model Ba1−x Kx BiO3 spectrum in (a) the clean limit and (b) with significant impurity scattering. The results are qualitatively similar to those in Fig. 3.53. 3.6.4

Fig. 3.53. The conductivity–derived scattering rate, 1/( ) vs. frequency in the s–wave superconducting state for (a) 1/ = 0 meV and (b) 1/ = 25 meV, for temperatures as indicated. In both cases we used the BKBO spectrum with  = 1. In (a) there is no signature for a gap, while one remains at low temperatures in (b)

since the phonon spectrum extends only to 80 meV. More detailed comparisons with self-energy-derived scattering rates have been provided in Refs.[184,186]. The results shown in Fig. 3.51 were obtained in the normal state. In the superconducting state the presence of a gap will modify the low frequency behaviour of the scattering rate, 1/( ). Results within BCS theory (elastic scattering rate only — no inelastic scattering) are shown in Fig. 3.52. At low frequencies the overall scale of the effective scattering rate is set by the elastic scattering rate (2 and 25 meV, respectively). Note that in the gap region (below 32 meV) the effective scattering rate is zero (at zero tem-

Phonon Response

Much of this review has focused on various properties whose determination allows one to infer the degree of electron–phonon coupling that exists in the material under study. The majority of properties that fall in this category refer to a modification of the electronic structure or response due to a coupling with phonons. To a much lesser extent the phonons themselves are modified because of the electron–phonon coupling, and in this section we briefly address a few examples in this category. The impact of the superconducting state on the phonons was first investigated using ultrasound experiments [324]. Sound waves are attenuated due to their absorption in the solid.The absorption requires interaction with electrons with energies very close to the Fermi energy (the phonon energy is typically very low for sound waves — in the 100 MHz = 0.0004 meV range). These electron states are gapped in the superconducting state, so the attenuation is expected to be suppressed to zero as T → 0. The BCS result, given by Eq. (3.177), is valid for an order parameter with s-wave symmetry. A similar law can be derived for other symmetry types, which results in some sort of power law decay rather than exponential at low temperatures. (See, for example, a recent paper which reports measurements on Sr2 RuO4, and references therein [325]. ) Of main interest here is how Eq. (3.177) is modified when retardation effects are accounted for. An early calculation [326] found that retardation effects did not alter the result Eq.(3.177).Therefore,little can be learned about the electron–phonon interaction through ultrasonic experiments; instead, one should examine higher energy phonons. The classic experiment of this type was performed using neutron scattering on Nb and Nb3 Sn [327].The

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idea is simply that the electron charge susceptibility modifies the phonon spectrum. Within the normal state this modification is hardly noticeable in metals over a temperature range of 300 K or so. However, when the material goes superconducting, the electron density of states is profoundly modified at energy scales of order the gap; this in turn will affect phonons whose energy is on the same scale. In particular, a low energy phonon (energy less than 2) that had a finite lifetime because it could decay into an electron-hole pair will be unable to do so in the superconducting state because no states exist at energies below the gap, . Therefore its lifetime will lengthen considerably in the superconducting state, resulting in a narrower lineshape below Tc . Figure 3.54 shows the experimental result from Nb3 Sn [327] where the lineshape has clearly become narrower in the superconducting state. Similarly, if the phonon energy is slightly above 2, then, under the right conditions, the linewidth will increase, since the electron density of states increases in this energy regime in the superconducting state. A detailed theory of these effects was first given in [328], within BCS theory. The theory consists of a calculation of a response function corresponding to a Case I observable. Similar calculations were performed much later by Zeyher and Zwicknagl [289] to understand the frequency shifts and linewidth changes (due to superconductivity) in the q = 0 Raman spectra for various optical modes in

YBaCu3 O7−X . They found, using the BCS approximation, Re ¢(q = 0,  + iı) = (3.182) N(0) ⎧   2 ¯ −1 ⎪ − for ¯ < 1 tan √ √ ⎪ ⎪ ⎨ ¯ 1 − ¯ 2 1 − ¯ 2   √ 1 √ ln 2¯ 2 − 1 + 2¯ ¯ 2 − 1 for ¯ > 1. ⎪ ⎪ ⎪ ¯ ¯ 2 − 1 ⎩ The imaginary part is given for all temperatures by: tanh ˇ /4 Im ¢(q = 0,  + iı) = −(¯ − 1) √ , N(0) ¯ ¯ 2 − 1 (3.183) where ¯ =  /(2(T)). Here, ¢(q,  + iı) is the change in the phonon self-energy between the superconducting state and the normal state. A positive (negative) real part means that phonons harden (soften) in the superconducting state,while a positive (negative) imaginary part means that the phonon linewidths narrow (broaden). Thus, phonons below the gap edge (2) soften while those above harden. Also, above the gap edge they broaden while below their linewidth does not change. The broadening above 2 can be understood as being due to the enhanced scattering with electrons, since the electron density of states now has a square-root singularity in the energy range of , and the phonon self-energy is essentially a convolution of two single electron Green functions (see Eq. (3.146)).

Fig. 3.54. The widths of low energy [0]T1 acoustic phonons broaden appreciably at temperatures above Tc , the superconducting transition temperature. This figure shows the same phonon profile above and below Tc ≈ 18.0 K Reproduced from [327]

3

Fig. 3.55. (a) Real and (b) Imaginary part of ¢( + iı)/ N(0) vs  /(20 ) at zero temperature, for various impurity scattering rates, 1/(0 ) = 0 (solid), 1 (dotted), and 6 (dashed), in the weak coupling (BCS) approximation. Below twice the gap edge the phonons soften; above twice the gap edge they harden in the clean limit and soften in the dirty limit. Note the narrowing that occurs below the gap edge in the presence of impurity scattering. Taken from [290]

Eqs. (3.182, 3.183) have been derived assuming single particle Green functions without impurity scattering. The q = 0 limit is somewhat anomalous in this case, in that the phonon width is already zero in the normal state. Hence, no change can occur in the linewidth in the superconducting state, for frequencies below 2. A calculation with impurities [290] provides a non-zero linewidth in the normal state. Because of the gap in the single electron density of states in the superconducting state, this

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linewidth is reduced to zero when the system enters the superconducting state, so the change in the imaginary part of the phonon self-energy is positive. These results are summarized in Fig. 3.55. Note that the softening below the gap edge is significantly reduced with impurity scattering present, and the phonons above 2 also soften when a significant degree of impurity scattering is present. As Fig. 3.55b shows, phonons whose energy lies below 2 acquire a narrower linewidth in the superconducting state, as noted above. The effects of retardation on the phonon selfenergy are not very significant. The changes that do occur follow the changes already discussed due to including elastic scattering; high energy phonons soften rather than harden, and the broadening that accompanies this softening is reduced compared to the clean BCS case. More detailed changes are documented in Refs. [289,290]. Because these phonon changes can be observed through neutron scattering experiments, it is of interest to examine the phonon self energy at non-zero momentum, q [291,292]. In this case the phonon has a non-zero linewidth in the normal state, and so line narrowing is observed in the superconducting state at low frequencies, due to the development of a single electron gap. The detailed frequency dependence is a function of the band structure; in particular, with two dimensional nesting phonon changes due to superconductivity are enhanced [292].

3.7

Anisotropy and MgB2

Arguably the most recent exciting discovery in superconductivity is in MgB2 with Tc = 39 K [329]. This is a very simple binary compound, although its structure leads to considerable anisotropy in the superconductivity, which is why we combine these two topics here. While preliminary analysis seemed inconsistent with electron-phonon driven superconductivity [330–332], most researchers now appear to believe that MgB2 is an electron-phonon superconductor, albeit with somewhat unusual properties due to the gap anisotropy and the multi-band structure at the Fermi energy. An early review is [333]; a more recent review appears to be lacking.

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We begin with a brief summary of Eliashberg theory with anisotropy.We follow the common approach which confines electronic properties to the Fermi surface, but accounts for multiple band and/or variations in coupling strength as one moves around the Fermi surface(s). We begin with Eqs. (3.50), (3.51) and (3.52) and adopt particle-hole symmetry so that Eq. (3.51) is not required. By assuming that the momentum dependence of various quantities is important only on the Fermi surface, one can perform the energy integration to obtain (k, i!m) = ' 

 ∗ T k,k (i!m − i!m ) − k,k  m

×

(k , i!m )

2 2  2  !m  Z (k , i!m ) +  (k , i!m )



!m Z(k , i!m )/!m

C(Tc ) = 1.43(1 − 4a2 ) ,

Tc

2 2  2  !m  Z (k , i!m ) +  (k , i!m )

(

(3.189)

and is clearly reduced from its BCS value. For temperatures near Tc , we obtain , (3.184)

where the angular brackets denote an average over all directions of k  on the Fermi surface. A popular model to describe the resulting anisotropy in the interaction is the separable model first described in [334], Fk,k = (1 + ak )F(1 + ak ),

(3.188)

while the specific heat jump is given by

Z(k, i!m) = '  1 + T k,k (i!m − i!m ) m

so anisotropy reduces the value of this ’universal’ ratio. Many other properties can be found in [335]. For example, the critical magnetic field at zero temperature, Hc (0), is given by Hc2 (0) 1 = N( ) < k (0) >2 (1 + a2 ) , 8 2

(



where V is the average (attractive) interaction on the Fermi surface. Note the additional factor of (1 + a2 ) compared with Eq. (3.40). This indicates that anisotropy increases the critical temperature.The order parameter at zero temperature, k (0) varies as 1 + ak ; one obtains   3 2 2 < k (0) > (3.187) = 3.53 1 − a , kB Tc 2

(3.185)

where F is any property, and the (small) quantity ak has average zero over the Fermi surface (i.e. < ak >= 0) but has a non-zero average to second order,i.e.< a2k >= a2 ,where a is some small parameter. In this way the precise functional form of ak is not required, and everything is encapsulated in the parameter a2 . In Eqs. (3.184) anisotropy arises through 2 ∗ ˛k,k  F(§) and the Coulomb pseudopotential, k,k  . Following the reduction to a BCS-like model described in an earlier section , we obtain   −1 Tc = 1.13!D exp , (3.186) N( )V (1 + a2 )

< k (T) > = < k (0) > √ 1.74(1 − a2 ) 1 − t[1 − (0.41 − 1.36a2 )(1 − t)] , (3.190) to order (1 − t)3/2 (t ≡ T/Tc ) so that the temperature dependence of the BCS order parameter is modified from its isotropic dependence (see Fig. 3.33, top frame). Finally, to order (1 − t) the specific heat becomes C(T) = 2.43(1 − 2.35a2) − 4.76(1 − 4.79a2 )[1 − t] .

Tc (3.191) At low temperatures we obtain the reduced critical magnetic field hc (t) ≡

Hc (T) = 1 − 1.06t 2 (1 + 2a2 ) − 0.56t 4 (1 + 4a2 ) Hc (0)

(3.192)

and the deviation function Eq. (3.140) D(t) = −(0.06 + 2.11a2)t 2 − 0.56t 4 (1 + 4a2 ) . (3.193)

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Even within the separable anisotropy model one can include more realistic modelling at the level of the electron-phonon spectral function. Using 2 2 ˛k,k  F(§) ≡ (1 + bk )˛ F(§)(1 + bk  ), Zarate and Carbotte [336] found to order b2 Tc = 1.13!D e where

1+

(− −

∗)

 ˛ ≡b 2

2

1 + ˛ 2(

1+  ) ,  − ∗

(1 + ∗ ) (1 + )( − ∗ )

(3.194)

2 .

(3.195) − ∗

Equation (3.194) agrees with Eq. (3.186) with 1+ playing the role of N( )V and ˛ 2 replacing a2 . Note that for a given value of b2 the effective anisotropy parameter ˛ 2 is largest when  is small, i.e. comparable to ∗ . This additional feature is unique to Eliashberg theory, and is not present in BCS theory. The expression for the average gap ratio is modified in two ways. It is  3  − ∗  2 2 < k (0) > + ˛ ) . (3.196) = 3.53(1 − kB Tc 2 1+ A comparison with Eq. (3.187) shows that (i) a2 → ˛ 2 and (ii) the factor 3/2 is increased within Eliashberg theory. These examples should suffice to make it clear that Eliashberg calculations are required for a quantitative analysis of anisotropy effects, even within the separable model for anisotropy. Note, however, that Eqs. (3.189)–(3.193) remain valid with a2 → ˛ 2 . Daams and Carbotte [124] have obtained detailed numerical results for properties of an anisotropic superconductor based on the full Eliashberg equations 2 (3.184) with a separable model for ˛k,k  F(§). The isotropic part was taken from tunneling experiments, as described previously.Their results of D(t) vs.t 2 for several materials are shown in Fig. 3.56. Solid curves are the isotropic case while the dotted curves are for b2 = 0.04 in all cases. Of course in real materials the anisotropy may not be separable. Normal impurity scattering can be included as in Eqs. (3.153) and (3.154), with the appropriately generalized momentum dependent functions, although in general the impurity scattering rate can be dependent on momentum, as well, 1/ → 1/k,k . Here

Fig. 3.56. The deviation function D(t) vs. t 2 calculated from Eliashberg theory for various elemental superconductors with (dotted) and without (solid) anisotropy. From [124]

we assume this rate is independent of momentum. Then,since impurity scattering tends to wash out gap anisotropy, Tc is reduced from its anisotropic pure single crystal value [294], as is clear from Eq. (3.194), for example. Thus, the so-called “dirty limit” is also the isotropic limit. For small impurity content, Tc decreases linearly with impurity scattering, with a coefficient proportional to the mean square anisotropy a2 in BCS theory. Within Eliashberg theory similar results are achieved with a2 → ˛ 2 [337]. Ashraf and Carbotte [338] show that the effect of impurities on Tc of an anisotropic BCS superconductor is qualitatively though not quantitatively correct. Full numerical results are given in [124]. The effect of impurities on the thermodynamic properties within the  model are given in [339]. How large is the typical anisotropy in ˛k2 F(§) in conventional superconductors? In Fig. 3.3 we showed the directional dependence of the electron-phonon spectral function in Al determined through multiple-

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Fig. 3.57. A schematic diagram showing positions on the real Fermi surface for Al at which calculations have been performed. From [340]

plane-wave band structure calculations. In Fig. 3.57 we show a schematic representation of the 62 points (X’s) at which calculations of ˛k2 F(§) were carried out on the irreducible 1/48 of the realistic Fermi surface of Al. The shaded region contains no free states. To have a measure of the electron-phonon anisotropy we examine  ∞ the mass renormalization parameter k = 2 0 d§ ˛k2 F(§)/§; these are shown in Fig. 3.58 [340]. Note the variations of order 30%. Estimates of gap anisotropy based on this calculated spectral function were provided in [340] and are reproduced in Fig. 3.59. Again gap variations are considerable; estimates are that Tc for a pure single crystal will be 9% higher than the isotropic limit. Experiment gives about 5%. Further details and similar calculations for Pb can be found in [150,255]. Equations (3.184) along with impurity contributions can be used to treat an isotropic two-band model, and has been done recently for MgB2. In Eqs. (3.184) the momentum sums in principle can sum over band indices as well. Using this fact we can regard these equations as appropriate to an isotropic two band model, where only 2 indices are now required in the sum. Moreover, 4 discrete values of ˛ij2 F(§) with i, j = 1, 2 are required. For the moment we can regard the direct Coulomb repulsion, ∗ , and the impurity scattering 1/, as isotropic quantities.

Fig. 3.58. The variation of k as a function of  in planes defined by  = constant, calculated in the multiple-planewave approximation. The shaded areas indicate Bragg plane regions. From [340]

3

Fig. 3.59. The variation of the gap function for three constant  arcs. The shaded areas indicate places where there is no Fermi surface. From [340]

If desired, additional anisotropy within a band could be modelled with a separable model. In the case of MgB2 Nicol and Carbotte [341] have reviewed the data and arguments for the simplicity of a minimal model to describe superconducting properties. There is evidence that impurities mainly provide intraband scattering in the case of MgB2 . But intraband scattering does not contribute to Tc or to the thermodynamics. Only interband scattering washes out the anisotropy between bands, and so reduces Tc . Functional derivatives of Tc with respect to all four ˛ij2 F(§) have been computed in [342]; these give some insight into how phonons coupled to the various bands influence Tc . As argued by Nicol and Carbotte [341], there is compelling evidence for two-band,electron-phonondriven superconductivity in MgB2 . However, the field has developed along different lines than that reviewed so far here for older, more conventional superconductors. It has been argued [343] that it may well never be possible to extract from tunneling data detailed information concerning the four spec-

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tral functions, ˛ij2 F(§), i, j = 1, 2. For example, to date tunneling inversion has not been possible [344]. On the other hand these functions have been calculated from first principles band structure calculations extended to include the electron-phonon interaction. Many of these properties have now been accepted as fact, though they are based on calculation only [345–352].Considerable work has also been performed on the determination of the Coulomb pseudopotential parameters [342, 353, 354]. This parameter is most important for the determination of the value of Tc , but less important for determining other superconducting properties. Choi et al. [346, 347] found good agreement with the temperature dependence of the specific heat by solving the multi-band anisotropic Eliashberg equations. They found two distinct sets of order parameters (as a function of momentum) with some significant variation within each group. A more common approach has been to simplify the problem to one which uses two “isotropic” bands. For example, a set of ˛ij2 F(§), i, j = 1, 2 [348–350], also used by Mitrovi´c [342] and Nicol and Carbotte [341] along with the corresponding ij∗ lead to good agreement with the specific heat data. Results are shown in the upper frame of Fig. 3.60. The solid continuous curve is the result of a full two band Eliashberg calculation while the solid points are the data given in [355]. Note in particular the shoulder near the reduced temperature t ≈ 0.3. This comes from the second () band. The dashed curve was obtained by using the same parameters except that the off-diagonal electron-phonon spectral func2 tion, ˛12 F(§), was arbitrarily halved. The deviation from experiment at low temperatures is quite significant. This illustrates that the agreement achieved with parameters obtained from first principles calculations is robust. Also note that the calculation gives a value for the normalized specific heat jump of 1.08, significantly less than weak coupling BCS, and in general agreement with experiment, where values lie in a range 0.82–1.32 [355–358] (see [341] for further details). Moreover, MgB2 is an intermediate coupling superconductor, with Tc /!ln ≈ 0.05, so that C(Tc )/ Tc would normally exceed 1.43 by about 30% [341].As we have seen,anisotropy reduces

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Fig. 3.60. Electronic specific heat (upper frame), penetration depth (middle frame), and the deviation function (bottom frame), for MgB2 . See text for an explanation of the symbols and curves. From [341]

this value; within the separable model discussed earlier, one would require b ≈ 0.3, which is anomalously large. Nicol and Carbotte [341] find that both the anomaly at t ≈ 0.3 and the low specific heat jump can be achieved as follows. Start with two decoupled bands, i.e. ˛ij2 F(§) and ij∗ are both zero for i = j.

In this case there would be two critical temperatures, Tc and Tc for the  and  band, respectively. Moreover there would be two Sommerfeld constants,  and  . At T = Tc (the two dimensional  −band has the higher Tc ) then only the  −band contributes to the jump C(Tc ), but both bands contribute to the normal state =  +  . If the ’s are roughly the same size, the normalized jump would be reduced from a single band case by a factor of 2. Of course the off-diagonal interaction terms are not zero, so these remarks would be modified quantitatively. In MgB2 , however, the main change from the single band picture occurs at temperatures near the lower Tc . Nicol and Carbotte [341] found that very small values of   and  are required (compared to the diagonal components   and  ) to explain the shoulder in the data at low temperatures. Additional comparisons with experimental data are presented in the other frames of Fig. 3.60 [341]. These are the normalized London penetration depth ([L (0)/L (T)]2 )and the deviation function D(t) (see Eq. (3.140)) vs. reduced temperature t = T/Tc . The solid and dashed curves are for the proper Eliashberg calculation and for that with half the off-diagonal coupling, as was done for the specific heat. This illustrates once again that parameters cannot be changed very readily; otherwise the agreement with experiment quickly deteriorates. The penetration depth (deviation function) data is from [348] ( [355,356]), respectively. Agreement with theory is good, except for the one set of penetration depth data (solid triangles). However, impurity scattering has not been included in the calculations (solid curve); this remains as an adjustable parameter which will affect the relative contribution of each band to the superfluid density. Recall that impurity scattering reduces the superfluid density of a given band, so in the limit of decoupled bands,each can be tuned independently by intraband scattering. In Fig. 3.61 the temperature dependence of the two gap ratios 21 /kB Tc and 22 /kB Tc (1 ≡  and 2 ≡ ) are shown [341]. The solid curves are the results of full Eliashberg calculations of the two-band gap equations, while in the dashed curves the offdiagonal elements have been (arbitrarily) reduced by a factor of 2.As can be seen this change has significant

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duced critical field, hc (0) = Hc (0)/(Tc |Hc (Tc )|), for which we find √ hc (0) = 0.576 1 + ˛ ∗ u2 , (3.198) and, with YL (T) ≡ 1/L2 (T), the London penetration depth ratio,

Fig. 3.61. Gap ratios for the larger (1) and smaller (2) gap as a function of temperature in MgB2 . See text for explanations. From [341]

consequences: the  gap has been severely lowered and even its temperature variation is no longer BCSlike (shown by the dotted curves).In the extreme case of no off-diagonal coupling the dashed curve for the smaller gap would simply terminate at its own transition temperature Tc (much lower than Tc ). Some of this tendency is seen even in the solid curve, and is also seen in the data. Any off-diagonal coupling of course leads to one transition temperature. The data are from [359] (open circles) and from [360] (solid circles). The decoupled bands limit is useful for attaining a first understanding of how the two bands will qualitatively affect dimensionless BCS ratios. As we have already mentioned, the normalized specific heat jump, C(Tc )/ Tc = 1.43N1∗ /(N1∗ + N2∗ ), where Ni∗ = Ni ( )(1+ii +i¯i ). Here Ni ( ) is the bare EDOS at the bare chemical potential in the i’th band, and ¯i denotes the complementary band. In the same way 1 + ˛∗

Tc2 = 0.168 , 2 Hc (0) 1 + ˛ ∗ u2

(3.197)

where ˛ ∗ = N2∗ /N1∗ and u ≡ 2 (0)/1 (0) is the ratio of the two gaps at zero temperature; this value can be taken as a convenient measure of the gap anisotropy. A couple of other dimensionless ratios are the re-

1 (3.199) YL (0)/(Tc |YL (Tc )|) = (1 + ˛ˇ) , 2 2 2 where ˇ ≡ (vF2 /vF1 )(1 + 11 + 12 )/(1 + 22 + 21 ) is the ratio of the square of the two Fermi velocities, suitably renormalized by mass enhancement factors. Note that here only ˛ is required, not ˛ ∗ . The factor ˛ is the ratio of the bare EDOs’s at the bare chemical potential, just as for ˛ ∗ , but without the renormalizations due to the electron-phonon coupling. These examples indicate that large changes can occur in these dimensionless ratios because of the presence of two bands. If the factors ˛ ∗ and ˇ in Eq. (3.199) are both unity then that ratio is doubled. This is expected since at Tc only the  −band determines the slope (in the denominator) while at zero temperature both bands contribute to the superfluid density; for the parameters used here, both bands make equal contributions, resulting in a factor of 2. To make a connection with the separable anisotropy model of the older literature, we note that the two band model reduces to the separable model through the correspondence 11 = [1+a]2 /2, 22 = [1 − a]2 /2 and 12 = 21 = [1 − a2 ] and the vf and N( ) the same. Note that this model has only two values for ak and thus corresponds to using two delta-functions for the ak in Eq. (3.185), and then assumes that a2 small.

3.8

Summary

We have examined a variety of ways in which the retarded electron phonon interaction influences the properties of a conventional superconductor. The first and simplest effect is through a renormalization of Fermi Liquid parameters, like the effective mass. While this effect appears in a number of normal state properties (for example, the low temperature electronic specific heat capacity, where the Sommerfeld

is enhanced by 1 +  — see Eq. (3.137)), it also appears in many superconducting properties. The most

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obvious (but least measurable) example is in the Tc equation,Eq.(3.109),where 1+ appears in the exponent.Another (perhaps more detectable) occurrence is in the slope of the upper critical magnetic field. In each of these cases, the renormalization occurs in the normal state — its occurrence in the superconducting state is because the property in question depends on the normal state effective mass, or Fermi velocity, etc.One should also bare in mind that the factor 1+, comes from a weak coupling approach. In a strong coupling approach, an electron–phonon renormalization is still present, but may be much more significant than suggested by the weak coupling approach, and polaron-like physics may dominate [96]. The most important manifestation of the electron–phonon interaction is the superconducting state itself. In fact, according to our present understanding of Cooper pairing, the electron–phononinduced attraction between two electrons would not overcome their direct Coulomb repulsion, except for the fact that the former is retarded whereas the latter is not. This gives rise to the pseudopotential effect; in some sense the pseudopotential effect is the true mechanism of superconductivity, rather than the electron phonon interaction per se. This is perhaps emphasized in the cuprate materials, where presumably the electrons could not utilize the difference in energy (and hence time) scales between the attractive mechanism (whatever it is) and the direct Coulomb repulsion to overcome the latter. Instead the pairing has apparently adopted a different symmetry (dwave) to avoid the direct Coulomb repulsion. Nonetheless a minimal accounting for these retardation effects accounts fairly well for the superconducting ground state. This was accomplished by BCS theory. A more accurate theory with retardation effects (Eliashberg theory) quite clearly accounts for quantitative discrepancies with experiment. Here,Pb and Hg are held up as paradigms for retardation effects, the simplest occurring in a measurement of the gap ratio, for example. The BCS theory predicts a universal number for this ratio, 2/kB Tc = 3.53. With Eliashberg theory a value for Pb is found close to 4.5, in excellent agreement with experiment. We have characterized the discrepancy with BCS theory through a retardation parameter, Tc /!ln.Various

properties have been quantitatively accounted for through simple analytical expressions with this parameter, as given in Sections 4.4.4 and 4.4.5 (see [10] and references therein for many more). Finally, various dynamical properties exhibit ‘signatures’ of the electron–phonon pairing. These tend to manifest themselves as ‘wiggles’ in the data, the most famous of which occurs in the tunneling data, and allows an inversion to extract the electron phonon spectral function, ˛ 2 F( ). As we saw briefly in Sect. 4.4.3, and then again in Sect. 4.4.6, these ‘wiggles’ occur in various two-electron response functions,most prominent of which is the optical conductivity. An accurate measurement of these response functions allows one to infer a significant electron– phonon coupling. We have focussed on very conventional superconductors, and have, for example, avoided any analysis of the high temperature superconductors. Signs of electron–phonon interactions have occurred in these new materials as well, but the relation to the superconductivity in them is yet unclear. Moreover, such effects will no doubt be covered in other chapters. Nonetheless, we wish to add a few remarks about other classes of superconducting materials that have been discovered over the last twenty years. Cubic Perovskites, beginning with strontium titanate (SrTiO3) [59,60], have already been discussed in Sect. 4.4.2. As mentioned there, these compounds (including BaPb0.75 Bi0.25O3 (Tc ≈ 12 K) [61] and Ba1−x Kx BiO3 (Tc ≈ 30 K) [63]) are generally regarded as in a distinct class from the high-Tc cuprates. This has left them, somewhat by default, as electron–phonon driven superconductors. On the other hand, there is strong optical evidence [189,316] that the electron–phonon interaction is very weak in these materials. Hence, as far as we are concerned, the mechanism of superconductivity in these perovskites is not understood at all. Tunneling studies [319,320] are divided on this issue. One- and two-dimensional organic superconductors were discovered in 1979 [361]. The subject had developed sufficiently so that, by 1990, a book devoted to the topic was written [362]. Organic superconductivity represents another interesting idea that was first presented by theorists [110,111], on the ba-

3

sis of a phonon-mediated interaction, but that now is considered by most practitioners not to be due to electron–phonon interactions. Many of the organics abound in physical phenomena, with several containing, on the same phase diagram, charge density wave (CDW) and spin density wave (SDW) instabilities, juxtaposed with superconductivity (see, for example, [363]). The nature of the superconducting state has not really been sharply defined by experiments, to the extent that both singlet and triplet pairing may be present [364], and the presence of a gap has not been unequivocally established. While it is probably fair to say that the electron–phonon interaction has not been ruled out as the mechanism for superconductivity (see, for example, a recent review by [365]), spin fluctuation-mediated pairing seems to be favoured [363]. Heavy Fermion systems were discovered to be superconducting also in 1979 [366]. While Tc has remained low, these compounds have remained of interest because (i) the root cause of the heavy electron mass is not completely understood, and (ii) the superconducting ground state coexists in a number of cases with antiferromagnetic order. It has now been established through thermal conductivity measurements that the order parameter contains nodes [367], and the circumstantial evidence points towards an unconventional magnetically mediated mechanism for superconductivity [368]. There is very little indication that superconductivity in this class of compounds has anything to do with the electron–phonon interaction. Superconductivity in alkali-doped buckminster fullerene (A3 C60, with A = K, Rb, Cs) was briefly mentioned earlier in this chapter. On the basis of optical measurements [192], a sizable electron phonon coupling was inferred, and, in fact ˛ 2 F( ) was extracted by the inversion procedure outlined in Sect. 4.4.3 [181]. Evidence for electron–phonon-mediated superconductivity was also presented in earlier reviews [75]. On the other hand, doubts remain concerning the validity of a weak coupling framework [369]. One would like to understand the ‘bigger picture’, i.e. the progression from insulator with pure C60 through the superconducting phase with A3 C60, and back to insulator with A6 C60. In fact, band structure calculations

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[370] suggest (simplisticly) that A2 C60 should be superconducting with a higher Tc than A3C60 , when, in fact, that compound does not readily form. The possibility of achieving superconductivity in both electron and hole doped field effect transistors utilizing C60 could have led to many more insights into these materials and into superconductivity itself. Unfortunately, these investigations have now been discredited. The borocarbides (RNi2 B2 C, where R denotes a rare earth element) were found to be superconducting in 1993 [371, 372]. In addition to having a sizeable transition temperature (Tc ≈ 20 K), some of these compounds exhibit coexistent superconductivity and antiferromagnetic order, and indeed, share some similarities with the heavy fermion compounds [373]. Nonetheless, tunneling has determined that a well-defined gap exists at low temperatures, and this and other measurements have established these compounds to have very BCS-like properties [374]. A detailed comparison of various superconducting properties with results based on Eliashberg theory (including some small anisotropy) [375] yields excellent agreement. A model spectrum was used for the electron–phonon interaction, and, at present, it remains unclear to what extent this agreement points unequivocally to the electron–phonon mechanism for superconductivity in these compounds. As time goes on MgB2 appears more and more to be an electron-phonon driven superconductor, albeit with two bands (or anisotropy) at the Fermi surface. Nonetheless, the definitive evidence (such as tunneling spectroscopy) that exists for so-called conventional superconductors appears to be lacking, so more (experimental) work is definitely required. Finally, Sulfur has been found to exhibit a high superconducting transition temperature (Tc = 17 K) [376]. Very little work has been carried out regarding the mechanism; a notable exception is [377], where ab initio calculations are performed to estimate the electron–phonon coupling strength for Sulfur. They find that under pressure, in a different structural phase, the electron–phonon coupling is enhanced, consistent with the increase in Tc . As is evident by the foregoing examples, a steady search for new superconductors is being rewarded

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with discoveries of materials with high critical temperatures, now in the same category as those of the high temperature cuprates. The A15 compound record of Tc ≈ 23 K would have been broken many times by now, even if the layered cuprates had not been discovered.Most intriguing is the fact that many of these compounds may be driven to the superconducting state through the electron–phonon mechanism. As far as future developments in this area is concerned, an obvious question to be addressed is the soundness of the original Cohen–Anderson estimate [213] for the maximum electron–phonon mediated critical temperature.It may simply be a matter of quantitative assessment, or perhaps some more exotic effect (within the electron–phonon picture) has been overlooked.An intermediate or strong coupling approach [96] may yet provide new insights. Finally, one can’t help but notice the recent resurgence of investigations in the high temperature cuprates themselves, that indicate strong electron phonon effects (see, for example, [378]). To paraphrase [379], ’The fat lady probably hasn’t yet sung’. Acknowledgements We wish to acknowledge the many colleagues with whom we have collaborated and with whom we have had many discussions over the years. In particular we thank J.E. Hirsch and E. Schachinger. We also acknowledge support from the Natural Science and Engineering Research Council (NSERC) of Canada and the Canadian Institute for Advanced Research (CIAR), for which we are grateful.

Appendix. Microscopic Developments In this Appendix, we will first outline a derivation of Eliashberg theory, based on a weak coupling approach. By this we mean that we start with momentum eigenstates. While other derivations may be given in other chapters, we include one here to keep this chapter somewhat self-contained. Migdal theory follows by simply dropping the anomalous amplitudes in what follows. We will then outline various other attempts to understand electron–phonon

interactions, particularly in the strong coupling regime. A.1 Migdal–Eliashberg Theory We begin with the definition of the one electron Green function, defined in momentum space, as a function of imaginary time [81], † G(k,  −   ) ≡ − < T ck ()ck (  ) >,

(3.200)

where k is the momentum and  is the spin. The angular brackets denote, as usual, a thermodynamic average. With this definition such a Green function can be Fourier expanded in imaginary frequency: G(k, ) =

∞ 1  −i!m  e G(k, i!m) ˇ −∞

ˇ G(k, i!m ) =

dG(k, )ei!m .

(3.201)

0

The frequencies i!m are known as the Matsubara frequencies, and are given by i!m = iT(2m − 1), m = 0, ±1, ±2, . . ., where T is the temperature. Because the c’s are Fermion operators, the Matsubara frequencies are odd multiples of iT. As is evident from these equations, the imaginary time  takes on values from 0 to ˇ (≡ kB1T ). Similar definitions hold for the phonon Green function: D(q,  −   ) ≡ − < T Aq ()A−q (  ) > ,

(3.202)

Fourier transwhere Aq () ≡ aq () + form is similar to that given in Eq. (3.201) except that the Matsubara frequencies are in ≡ iT2n, n = 0, ±1, ±2, . . ., i.e. they occur at even multiples of iT. To derive the Eliashberg equations, we use the equation-of-motion method, taken from [55]. The starting point is the (imaginary) time derivative of Eq. (3.200) a†−q (). The

@ G(k, ) = (3.203) @

 † −ı()− < T H − N, ck () ck (0) > , where, without loss of generality, we have put   = 0. For definiteness, we use the Hamiltonian (3.3), and,

3

in addition, assume, for the Coulomb interaction, the  simple Hubbard model, HCoul = U i ni↑ ni↓ . The sum result is  † H= k ck ck (3.204) k

+

 q

×

 kk  

+

!q a†q aq

1 +√ N

  g(k, k ) ak−k + a†−(k−k ) ck†  ck

U  † † c c c−k +q↓ ck ↑ , N  k↑ −k+q↓ k,k ,q

where the various symbols have already been defined in the text. Working out the commutator on Eq. (3.203) is then straightforward. We obtain   @ + k G↑ (k, ) = (3.205) @ 1  † −ı() − √ gkk < T Ak−k ()ck ↑ ()ck↑ (0) > N k U † < T cp† −k+p↓ ()cp ↓ ()cp↑ ()ck↑ (0) >, + N pp

where for definiteness we are considering the Green function with  =↑. On the right-hand side of Eq. (3.205) various higher order propagators appear; to determine them an equation of motion would have to be written, which would, in turn, generate even higher order propagators, eventually leading to a set of equations with hierarchical structure.This infinite series is normally truncated at some point by the process of decoupling,which is simply an approximation procedure. For example, in Eq. (3.205) the Coulomb term is normally not expanded further; instead a decoupling procedure is employed.Thus,under normal circumstances, the last term would become † < T cp† −k+p↓ ()cp ↓ ()cp↑ ()ck↑ (0) >→ † (0) >→ < T cp† −k+p↓ ()cp ↓ () >< T cp↑ ()ck↑

−ıkpG↓ (p, 0)G↑ (k, ).

(3.206)

The case of the electron–phonon term is a little more subtle, however. In this case we define a Green function,

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† G2 (k, k , , 1 ) ≡< T Ak−k ()ck ↑ (1 )ck↑ (0) >, (3.207) and write out an equation of motion for it. We get

@ G2 (k, k , , 1) = @ † −!k−k < T Pk−k ()ck ↑ (1)ck↑ (0) >,

(3.208)

where Pq () = aq () − a−q (). Taking another derivative yields  2  @  − !k−k G2 (k, k , , 1) = @ 2  2!k−k gk−k (3.209) k   † (0) > . × < T ck† −k+k  ()ck  ()ck ↑ (1 )ck↑

One might be tempted to decouple Eq. (3.209) and thus close the hierarchy that begins with Eq. (3.205). However, retardation effects are properly included only when the phonon propagator is taken into account.While the electron–phonon interaction affects the phonons as well as the electrons, the influence on the phonons occurs most at higher temperatures. For many materials the phonons have reached their ground state configurations by about room temperature. As a result, for low temperatures the phonons remain virtually unaffected by the electron–phonon interaction, and it suffices to disregard the electron– phonon interaction as far as the phonons are concerned provided they have been properly renormalized due to effects which took place at higher temperature. To put this another way, inelastic neutron scattering measurements of the phonon dispersion curves show a dependence on temperature only at temperatures well above room temperature [87,88]. As already mentioned in the text, the phonons are normally taken from experiment, and hence the “calculation” of the phonon propagator is greatly simplified. One simply assumes that the phonons are noninteracting. The equation of motion for the phonon propagator is then  2  @ 2 − !q D(q,  −   ) = 2!q ı( −   ). (3.210) @ 2 Utilizing this expression in Eq. (3.209) then yields

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G2 (k, k , , ) = ˇ

 1  d  gkk D(k − k ,  −   ) N 

(3.211)

k  0

† < T ck† −k+k  (  )ck  (  )ck ↑ ()ck↑ (0) >,

where now 1 has been set equal to  as is required in Eq. (3.205). This can now be substituted into Eq. (3.205), and the whole result can be Fourier transformed (from imaginary time to imaginary frequency). Before stating the result of this exercise, however, we note that the superconducting state is specially characterized by the existence of anomalous amplitudes, attributed to Gor’kov [42] and often referred to as Gor’kov amplitudes. Thus, in the Wick decomposition [81] of the various two-particle Green functions, the anomalous amplitudes also must be taken into account, in addition to the normal amplitudes given, for example, in Eq. (3.206). The anomalous amplitudes take the form F(k, ) ≡ − < T ck↑ ()c−k↓ (0) >

(3.212)

† † ¯ F(k, ) ≡ − < T c−k↓ ()ck↑ (0) > .

(3.213)

and

Now it is necessary to go through the same procedure with F and F¯ as with G. The methodology is the same, so we skip the necessary steps. We then define two self-energies, the usual one (generalized to the superconducting state), denoted by £(k, i!m ), and an anomalous self-energy, often called the pairing function, (k, i!m), and we arrive at Eqs. (3.43)–(3.47). A.2 The Polaron Problem A rather different and less developed approach to the electron–phonon problem focuses on the effect of the phonons on a single electron. A review is provided in [96], and we merely highlight some of the important points here. There are many kinds of polarons, i.e. small vs. large,weakly coupled vs.strongly coupled,Fr¨ohlich vs.

Holstein, etc.As far as we can tell these classifications are merely qualitative, so that, in most cases, distinctions can be readily drawn for extreme parameters only. A case in point is the distinction between an itinerant vs. self-trapped polaron. It seems clear that no such transition exists, but nonetheless a crossover occurs to a regime in which the polaron acquires a very large effective mass. In thinking about the polaron problem, there is the usual competition between kinetic energy (measured by the hopping integral, t, or the bandwidth, D = 2zt, where z is the coordination number for a cubic lattice (z = 2, 4 and 6 in 1,2, and 3 dimensions, respectively)) and the potential energy (measured by g — see Hamiltonian (3.5)). In addition the phonon frequency represents a third energy scale. In the case of the Holstein model, Eq. (3.5), this scale is conveniently represented by a single number, the Einstein oscillator frequency, !E . A dimensionless coupling constant,  ≡ 2g 2 /(D!E ), corresponds roughly to the enhancement parameter introduced in section 4.2 (see Eq. (3.22)). Note that in terms of the parameters of the original Holstein Hamiltonian, Eq. (3.4),  ≡ ˛ 2 /(KD). An increase in  signifies an approach to the strong coupling limit. On the other hand the adiabatic (anti-adiabatic) limit is represented by !E /t → 0(∞). The values of both ratios strongly influence the number of phonons present. An early review that clearly delineates these different regimes is provided by [380]. There have been many approaches to solving the polaron problem (as governed by a Hamiltonian like Eq. (3.4)). Some of the early techniques are amply covered in [52]; these are exemplified by weak and strong coupling perturbation theory, and variational methods. A review of the perturbation approaches is given in Appendix B and C of [381]. Weak coupling follows the Migdal approach, while strong coupling utilizes the celebrated Lang–Firsov transformation. This transformation immediately results in a narrow band, with effective hopping parameter, teff = t exp (−g 2 /! 2 ), along with exponential increases in effective mass and, in the adiabatic regime, a large number of phonons in the ground state [380,382]. With the advent of considerable computing capabilities over the last two decades, exact methods

3

have been used, that, in various cases, can span the entire parameter regime. The first is Monte Carlo for a single electron, pioneered in [383]. Trugman et al. [166, 384, 385] utilized exact diagonalizations based on a variational Hilbert space obtained from repeated applications of the Hamiltonian on a trial state vector; their most recent results are capable of achieving very high precision. In the meantime, Proetto and Falicov [386] and Ranninger and Thibblin [387] used a truncated Hilbert space for a twosite problem, and performed a straightforward numerical diagonalization. This was followed by work on larger (one-dimensional) lattices (for one electron) in Refs. [381, 388, 389]. Most of this work was performed for a specific model — the Holstein model of electron–phonon coupling, already referred to in the text. Further work was carried out also for the BLF (SSH) model, in [390]. Yet another technique utilizes the density-matrix renormalization group (DMRG) [391] method [392], which has also been extended to many electrons [393]. Another variational technique known as the Global–Local variational method [394] also provides very accurate results for the polaron problem. Finally, two new Monte Carlo methods [395, 396] appear to be particularly powerful in obtaining polaron properties. In all cases a clearer understanding is emerging; there is no self-trapping transition, in any dimension, although there is a fairly abrupt (but still smooth) crossover from weak coupling-like to strong coupling-like. This crossover has now been investigated in 1, 2, and 3 dimensions as well as with dynamical mean-field theory, which is exact in infinite dimensions [397]. An actual transition can be observed in higher dimensions in the adiabatic limit (!E = 0) [398]. However, this limit is regarded as somewhat pathological, and not representative of the general case [394]. Finally, some work has been performed on the bipolaron problem, i.e. whether two polarons bind or not.Much of this work is summarized in [96].Various discussions of the supporting evidence and difficulties of these theories can be found in Refs. [399–401]. A related problem has been asked and partially answered in Refs.[166,381,402],which is: to what extent will two electrons interacting through phonon ex-

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change and Coulomb repulsion form a Cooper pair, particularly as the strength of the Coulomb repulsion is increased well in excess of the effective strength of the attractive phonon-induced interaction? In other words, to what degree does the pseudopotential effect play a role in pairing? In [381] one of us found that pairing persists even when the Coulomb interaction strength exceeds that of the electron–phonon attraction, and in [166] this statement was made more precise (see also [403]). In particular, binding persists only up to a point; for sufficiently large Coulomb repulsion, the pair is no longer bound. While more work is required, this finding implies that the usual pseudopotential reduction, given by (Eq. 3.79), may be too strong. Equation (3.79), for example, achieves a large reduction in the limit (EF ) → ∞, whereas the result of [166] says that for two electrons, at least, the binding is lost in this limit. A.3 Many Electrons on a Lattice The problem of many interacting electrons is, in many ways, significantly more difficult than that of one or two electrons. The dimension of the Hilbert space grows exponentially, so that exact diagonalizations become prohibitive. A review of methods and results can be found in [404]. As far as the electron– phonon problem is concerned, there is some limited work which utilizes direct diagonalization, usually in the context of the t − J model [405, 406]. Mainly, however, this problem has been approached through Monte Carlo methods, and a variety of (somewhat uncontrolled) Green function techniques. Monte Carlo methods have an illustrious history [407].While they are not formally exact (because, for example, of a Trotter [408] breakup), the error introduced by such a decomposition can be controlled. Hence, in principle, and even in practice through extrapolations, one can obtain results which are exact to within some known error. Some of the first papers to utilize Monte Carlo methods in many body fermion problems (in the condensed matter context) addressed the electron– phonon problem [409–411]. This particular methodology integrated out the fermion degrees of free-

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dom analytically, leaving the boson degrees of freedom to which Monte Carlo algorithms were applied. Various modifications immediately arose, and were used to address the same electron–phonon problem [412–415] as well as electron–electron problems [416]. Much of this work is reviewed in [66]; a more comprehensive review of the many variants of the Monte Carlo method (in condensed matter) is provided in [67]. Studies in two dimensions became more feasible in the late 1980’s; an immediate question that was addressed was the competition between superconductivity and the charge density wave (CDW) instability [93,94,417]. At half-filling (where simulations are easiest) the CDW instability overwhelms the tendency towards superconductivity,in part because the tight-binding model with nearest neighbour hopping exhibits nesting at half-filling. Veki´c et al. [418] explored the impact of next nearest neighbour hopping (to remove the nesting) but found it was difficult to discern whether an incommensurate CDW instability or superconductivity dominates. Another means of eliminating the CDW is through doping; again most of the work is inconclusive. A third means is through the use of a Hubbard U . A study [419] of the so-called Hubbard–Holstein model in two dimensions found that both the CDW and the superconductivity susceptibilities are suppressed as U grows. To our knowledge, however, the pseudopotential effect (where the U would essentially cancel the electron–phonon interaction as far as the CDW was concerned, but not as far as superconductivity was concerned) has never been detected in manyelectron Monte Carlo studies. One of the reasons for exact studies of these lattice models (on small lattices) is for use as a benchmark to which diagrammatic methods can be compared. Thus, for example, the conclusion in [94] was that the Migdal formalism, without vertex corrections, described the Monte Carlo results fairly accurately,

provided phonon renormalization was taken into account. A model system is required to determine this, since, in real systems, the phonons are often taken from experiment, and already contain renormalization effects. This conclusion was confirmed in [419], as well as in [420]. In this latter reference, the authors developed the formalism even further to accommodate a CDW gap, and found good agreement with Monte Carlo results. Nonetheless,it is probably safe to say that a reliable formalism has not yet been developed to investigate low temperature properties of electron–phonon systems, particularly slightly away from half-filling. A number of attempts have been made, particularly in the case of electron–electron interactions [421,422], although a comprehensive treatment has not yet been achieved (the many-body approaches are also becoming almost as numerically intensive as the Monte Carlo methods, and so one of their advantages is diminishing). Attempts have also been made to incorporate specific kinds of corrections to the Migdal–Eliashberg formalism. One of these categories is the inclusion of vertex corrections. Many feel that they may be necessary because the adiabatic ratio !D /EF is not small in some cases (eg. high-Tc cuprates, and doped buckyballs). In the cuprate materials two-dimensional effects may enhance vertex corrections as well. Vertex corrections may or may not enhance Tc , depending on the degree of anisotropy in the interactions [423]. In [424], a different tack is taken; Tc is kept fixed, and calculations with vertex corrections included can mimic those without through an adjusted ∗ (except for the isotope effect). In [425] a two-dimensional tight-binding model is used and once again, the conclusion is that vertex corrections enhance the pairing interaction. To our knowledge, however, these effects have never been observed in exact or controlled calculations.

3

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4

Coexistence of Singlet Superconductivity and Magnetic Order in Bulk Magnetic Superconductors and SF Heterostructures

M. L. Kuli´c Brookhaven Natl. Lab., Upton, New York, USA and Max Planck Institute – Physics of Complex Systems, Dresden, Germany A. I. Buzdin Inst. Univ. de France and Univ. Bordeaux, Talence, France

Abstract. The basic physics of bulk magnetic superconductors (MS) related to the problem of the coexistence of singlet superconductivity (SC) and magnetic order is reviewed. The interplay of the exchange (EX) and electromagnetic (EM) interaction between conduction electrons and localized magnetic moments in the formation of the coexistence phase is discussed.It is shown that the singlet SC and uniform ferromagnetic (F) order practically never coexist in bulk materials.In the case of their mutual coexistence the F order is modified into a domain-like or spiral structure depending on magnetic anisotropy. It turns out that the domain-like structure is realized in several superconductors such as ErRh4 B4 , HoMo6 S8 , HoMo6 Se8 with electronic order and (probably) the spiral structure in AuIn2 with nuclear magnetic order. The latter problem is also discussed. The coexistence of SC with antiferromagnetism (AF) is more favorable. A rich phase diagram is realized in AF systems with SC and weak-ferromagnetism. A number of properties of magnetic superconductors in the magnetic field are very peculiar, especially near the (ferro)magnetic transition temperature, where the upper critical field becomes smaller than the thermodynamical critical field. The Josephson junctions based on MS with spiral magnetic order is discussed, in which the induced triplet pairing amplitude gives rise to  -contact (with the  -phase difference in the ground state). The interplay of the superconducting and the magnetic phase, ' , , respectively, in such a contact renders possibilities for an interplay of the charge and spin current, and for a new type of coupled Josephson-qubits in a single Josephson junction. The coexistence of superconductivity and ferromagnetism may be easily achieved in artificially fabricated superconductor/ferromagnet heterostructures. Due to the proximity effect, the Cooper pairs penetrate into the F layer and one has the unique possibility to study properties of superconducting electrons under the influence of a large exchange field. Moreover, varying in a controllable manner the thickness of the ferromagnetic and superconducting layers it is possible to change the relative strength of two competing orderings. The Josephson junctions with ferromagnetic layers reveal many unusual properties that are quite interesting for applications, in particular the recently fabricated so-called  -Josephson junction.

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.2 Ferromagnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.2.1 Spiral Magnetic Order in Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.2.2 Sinusoidal Magnetic Structure Due to SC for T  Tm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171 4.2.3 Domain Magnetic Structure Due to Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.2.4 Domain Magnetic Structure in Thin Superconducting Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.2.5 Coexistence of Nuclear Magnetism and Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.3 Antiferromagnetic Superconductors (AFS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.3.1 Weak Ferromagnetism in Antiferromagnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.4 Magnetic Superconductors in the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183 4.4.1 Ferromagnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.4.2 Antiferromagnetic Superconductors in the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.5 Josephson Effect with Bulk Magnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

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4.6 Superconductor/Ferromagnetic Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.6.1 Fulde–Ferrell–Larkin–Ovchinikov State and Proximity Effect in S/F Structures . . . . . . . . . . . . . . .190 4.6.2 Oscillatory Superconducting Transition Temperature in S/F Multilayer and Bilayers . . . . . . . . . . 193 4.6.3 Superconductor-Ferromagnet-Superconductor -Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 A.1 Gor’kov Equations for Magnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 A.2 Eilenberger-Larkin-Ovchinikov Equations for Magnetic Superconductors . . . . . . . . . . . . . . . . . . . . 197 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

4.1

Introduction

The physics of magnetic superconductors (MS) is a very intriguing subject due to the pronounced competition of the ferromagnetic order and singlet superconductivity in bulk materials. Two fundamental properties of the singlet superconducting state are zero electrical resistance and the Meissner– Ochsenfeld effect; expulsion of the magnetic field from the volume of a superconductor due to screening currents. On the other hand, superconductivity can be destroyed by applying a magnetic field that exceeds some critical value and acts on the orbital motion of superconducting electrons. A magnetic order is characterized by magnetization that creates a dipolar magnetic field Bd in the sample.This field induces the screening (the Meissner–Ochsenfeld effect) superconducting current jscr and the corresponding magnetic field Bscr making the total field (magnetic induction) in the sample B = Bd + Bscr = ∇ × A. This type of interaction between the orbital motion of superconducting electrons and (ferro)magnetic order is called the electromagnetic (EM) interaction, the EM model. The question regarding the coexistence of SC and ferromagnetism was first addressed theoretically by Vitalii Lazarevich Ginzburg in 1956 [1] in the framework of the EM model. The latter is schematically shown in Fig. 4.1. In [2] it was found that in type-I SC with 4M < Hc the coexistence of ferromagnetic order and superconductivity is possible in the Meissner phase, where the superconducting order parameter  and the magnetization M are con-

Fig. 4.1. The Meissner superconducting-ferromagnetic phase. Schematic mechanism of electromagnetic (EM) interaction in ferromagnetic superconductors. Dipolar field Bdip , created by the localized moments with the magnetization M(r), induces screening current jscr (r) and field Bscr . The total induction is B = Bdip + Bscr

stant in the sample. The magnetic induction B = 0, which is due to the surface screening current. In the case when 4M > Hc the coexistence is possible only in a very thin film where the orbital effect of the magnetic field is suppressed, or by applying an external field in the opposite direction to M. After the advent of the microscopic theory of superconductivity, by Bardeen, Cooper and Schrieffer in 1957 [3], it became clear that the singlet superconductivity may be also destroyed by the Zeeman (paramagnetic) effect of the effective magnetic field,

4

Coexistence of Singlet Superconductivity and Magnetic Order

which is due to the exchange interaction (EX) of electronic and localized spins; the EX model. The resulting effective exchange field, which acts on conduction electrons in the ferromagnetic state, tends to align spins of Cooper pairs in the same direction, thus preventing pairing. The BCS theory predicts that a Cooper pair is formed from two electrons with opposite momenta and also opposite spins, singlet superconductivity, and when the gain in the energy | EZ |= B H ( B is the Bohr magneton) due to the electronic spin reorientation becomes comparable to the Cooper pair superconducting coupling energy (which is of the order of the superconducting critical temperature Tc ), then the singlet superconductivity is destroyed. One obtains the following estimate for the paramagnetic limit Hp ≈ 1.24Tc / B (or in Tesla Hp [T] ≈ 1.8 Tc [K]) above which the singlet superconductivity is destroyed. While the orbital effect in non-magnetic superconductors is more important for superconductivity destruction by the magnetic field, i.e. the EM effects dominate, the situation is quite different in ferromagnetic superconductors where the EX effects usually dominate over the EM ones. The internal magnetic induction (Bi ≈ 4M) due to the spontaneous magnetization M does not exceed several kOe in the rare earth (RE) ternary magnetic superconductors that are discussed in this chapter, and it is smaller than the typical upper critical field Hc2 in type II superconductors. Therefore the EM interaction is usually not harmful for superconductivity in magnetic superconductors as it is the case with the EX one.. On the other hand, the effective internal exchange field hex (which acts on electronic spins and plays the role of the Zeeman energy B H in non-magnetic superconductors) is very large. Typically, if expressed in the energy units hex ≈ (100−1000) K, which substantially exceeds the superconducting critical temperature Tc . This makes the exchange interaction the main destruction mechanism of the superconductivity in most magnetic superconductors. The strong exchange interaction is at the origin of very strong suppression of superconductivity by magnetic impurities. Early experiments [4] demonstrated that even very small (several percent) concentration x of magnetic atoms is enough to completely destroy superconductivity.The

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Fig. 4.2. The critical temperature variation versus the concentration n of the Gd atoms in La1−x Gdx Al2 alloys. points: experiment [6]. Line: the A-G theory. Tc0 = 3.24 K and ncr = 0.590 atomic percent Gd

influence of magnetic impurities on superconductivity is described by the Abrikosov–Gorkov theory [5], see Fig. 4.2. This theory predicts that the slope of the curve Tc (x) is given by (dTc /dx) ≈ −ex , where ex ≈ N(0)h2ex and N(0) is the density of states per electron spin. So, if one puts even a very small concentration of Fe atoms, they destroy SC since hex and ex are very large in Fe. However, the allowed small concentration of magnetic atoms in superconducting alloys is not enough to produce ferromagnetic ordering and there are no possibilities to investigate its competition with SC. The breakthrough in the physics of MS came with the discovery of rare earth (RE) ternary compounds, such as borides (RE)T4 B4 with transition elements T = Rh, Ir, chalcogenides (RE)Mo6X8 (X = S, Se), silicides (RE)2 T2 Si5 and stannides (RE)Tx Sn5 [6–8]. In most of these type-II superconductivity is realized and in all of them the localized RE magnetic moments are regularly distributed in the crystal lattice. The basic crystallographic structure, which is shown in Fig. 4.4 for RERh4 B4 , contains localized magnetic moments (LMs) placed far a way from the Rh and B blocks which deliver conduction electrons. Due to this spatial separation the conduction electrons rarely jump onto magnetic ions, thus making the effective direct exchange interaction (EX) Jsf much smaller than in transition metallic magnets,i.e.

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Fig. 4.4. Crystal structure of tetragonal ferromagnetic superconductor ErRh4 B4 phase [8] Fig. 4.3. Schematic mechanism of the indirect RKKY exchange (EX) interaction HRKKY in rare earth superconductors. The spin Si at Ri polarizes spins of conduction electrons with the spin susceptibility (Ri − Rj ). The polarization is transferred to the spin Sj at Rj

Jsf ( 104 K).In the rare earth compounds the 4f shell is responsible for localized moments,since the f -level lies much below the Fermi energy, i.e. Ef  EF . In these compounds various magnetic ordering are realized. Since the distance between the neighboring magnetic ions is large and the direct exchange interaction of localized moments is negligible, this means that the magnetic ordering in these system must be due to some other mechanism. In RE metals this mechanism is provided by conduction electrons in the following way. The magnetic moment SRi at place Ri polarizes the spins of conduction electrons (via the direct exchange interaction Jsf ) and this polarization is transferred to electrons surrounding the spin SRj at Rj . The latter interacts via the direct exchange interaction Jsf with the polarization cloud of conduction electrons. Since this is a second order process the characteristic effective exchange interaction between localized moments at distance Ri − Rj is proportional to Jsf2 and to the spin susceptibility (Ri − Rj ) of the conduction electrons. This type of indirect exchange between localized spins, which is shown in Fig. 4.3, is called the RKKY interaction with the Hamiltonian

  Hex = −Jsf2 Ri − Rj SRi · SRj .

(4.1)

In the normal state the susceptibility can give rise either to a ferromagnetic order or to a non-uniform one such as antiferromagnetic or spiral ordering, depending on the topology of the Fermi surface. In the superconducting state the susceptibility is significantly changed by the Cooper pairs, what shall be discussed below. In a number of the RE earth ternary compounds, such as (RE)Rh4 B4 (RE = Dy, Sm, . . .), the coexistence of superconductivity and antiferromagnetism has been realized; they are called antiferromagnetic superconductors (AFS). In most of them the Ne´el (AF) transition temperature TN is smaller than the superconducting one Tc , i.e. TN  Tc . The reason for the peaceful coexistence of superconductivity and antiferromagnetism is well understood and is based on the fact that the averaged exchange and orbital fields of the AF order is zero on the superconducting coherence length 0 . This will be discussed in the following sections. Since the late 1970s much experimental and theoretical research has been devoted to MS systems in which ferromagnetic (F) order and singlet SC compete due to their strong antagonistic characters. These systems are called ferromagnetic superconductors (FS) and they are one of the main subjects of this review. It will be demonstrated below that superconductivity modifies the F order into a modulated structure such as spiral or domain-like one,

4

Coexistence of Singlet Superconductivity and Magnetic Order

depending on the type and strength of magnetic anisotropy in the system, thus giving rise to the coexistence of SC and magnetic order. This was first pointed out in 1959 by Anderson and Suhl [9] for the case when the exchange field is small (h/  1) and later elaborated on by the present authors and collaborators by including the exchange and electromagnetic interaction for any ratio of h/ [8, 10]. In the RE ternary compounds the competition of ferromagnetism and superconductivity is rather strong and therefore these two ordering coexist in a limited temperature interval Tc2 < T < Tm < Tc1 only; the re-entrant behavior. For instance, in ErRh4 B4 the coexistence region is narrow with Tm ≈ 0.8 K,Tc2 ≈ 0.7 K. The appearance of the modulated (non-uniform) magnetic structure with the period L = 100 Å observed in neutron scattering experiments [11], while the superconductivity appears in the paramagnetic state at much higher temperature Tc1 = 8.7 K. By further cooling the first order phase transition into the ferromagnetic phase occurs at Tc2 ≈ 0.7 K with simultaneous destruction of superconductivity. The transition at T = Tc2 is called re-entrant transition because the superconductor goes back into the normal phase. The temperature dependence of the resistivity and magnetic susceptibility in the re-entrant superconductor ErRh4 B4 is shown in Fig. 4.5, where the hysteresis is realized around and slightly below T = 1 K, which is a sign of the first order phase transition. In HoMo6S8 the coexistence interval of the non-uniform magnetic structure with the period L = 200 Å and superconductivity is even narrower, with Tm ≈ 0.74 K, Tc2 ≈ 0.7 K, while Tc1 = 1.8 K [12]. It turns out that the exchange interaction in HoMo6Se8 is weaker (than in ErRh4 B4 and HoMo6S8 ), with Tc1 = 5.5 K, Tm ≈ 0.8 and the coexistence of SC and modulated ferromagnetism persists down to T = 0 K; see the discussion below and in [6,8,10]. The theory of magnetic superconductors in systems with localized magnetic moments, such as the RE ternary compounds, was developed in the last 30 years to a such level that nowadays there is a rather good qualitative and quantitative understanding of a number of relevant properties of these systems [8]. It turns out that in the RE ferromagnetic ternary superconductors the exchange interaction dominates over

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Fig. 4.5. A.C. susceptibility and resistance vs. temperature in ferromagnetic superconductor ErRh4 B4 [13]

the EM one in the competition of superconductivity and magnetic order. The role of the electromagnetic (EM) interaction in this competition is to make the magnetic structure transverse. The reason is that the rather large wave vector Q of the spiral or domain structure reduces the effect of the non-uniform magnetic field (due to the magnetization) on SC much more than the effect of the non-uniform exchange field. This will be discussed in the forthcoming sections. A new research field in the physics of ferromagnetic superconductors was opened in 1997 by Pobell’s group in Bayreuth [14], who discovered the coexistence of superconductivity and nuclear magnetic order in the type-I superconductor AuIn2 . In this system SC appears at Tc1 = 0.207 K and the magnetic ordering at Tm = 35 K. Although in this system there is a tendency to the nuclear ferromagnetic order in absence of SC, the theory [15] predicts that below Tm the superconductivity enforces appearance of a spiral or domain-like nuclear magnetic ordering in the SC state, depending on the strength of the nuclear dipole–dipole interaction. Experimental studies of the Josephson effect in systems with bulk magnetic superconductors are rather scarce. However, recent theoretical studies of Josephson junctions based on MS with the spiral ordering show that these junctions, if realizable, might have fascinating properties, such as the existence of

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-contacts [16], the combination of a magnetic analog of the Josephson effect for spin current with the ordinary Josephson effect for charge current [17]. In such a system the realization of two qubits (a superconducting and magnetic one) in a single Josephson junction is possible in principle. Furthermore, all these novel properties originate from the crucial fact that in systems with non-uniform magnetization the triplet amplitude F↑↑ =  ↑ ↑  (and F↓↓ =  ↓ ↓ ) of Cooper pairs is finite. It turns out that this model is in fact a paradigm for superconductor/ferromagnet superlattices with non-collinear orientation of magnetization. The importance of the triplet amplitude was recently rediscovered in the SF1F2 S weak links with rotating magnetization [18]. The latter induces the triplet amplitudes giving rise to so-called giant proximity effects, and in thicker films the Josephson current is dominantly transported via the triplet amplitude. Contrary to bulk materials, the coexistence of superconductivity and ferromagnetism may be easily achieved in artificially fabricated superconductor/ferromagnet heterostructures. Due to the proximity effect the Cooper pairs penetrate into the F layer giving the unique possibility to study properties of superconducting electrons under the influence of the huge exchange field. Moreover, by varying the thicknesses of the ferromagnetic and superconducting layers in a controllable manner it is possible to change the relative strength of the two competing orderings. The Josephson junctions with ferromagnetic layers reveal many unusual properties that are quite interesting for applications, in particular the so-called -Josephson junction (with the -phase difference in the ground state, [18–21]) which was recently fabricated [22]. This interesting aspect of physics is also discussed in the following sections. We would like to point out that our discussion is restricted to magnetic superconductors in the ternary rare earth compounds with localized moments as well as to superconductor/ferromagnet heterostructures, while the coexistence problem of SC with the bulk itinerant magnetism is discussed only briefly. However, one expects that the results discussed here hold also for itinerant magnetism if it is treated in the mean-field approximation.

There are a number of other magnetic superconductors that will be discussed here only briefly, if at all. The quaternary borocarbides (RE)Ni2 B2 C show the coexistence of superconductivity and antiferromagnetic order, which is a phenomenon understood in principle, although details of related to particular structures await further elaboration. Very interesting compounds are heavy fermion superconductors, URu2Si2, UPd2Al3 , UBe13 , etc., which show the coexistence of superconductivity and (itinerant or localized) (anti)ferromagnetic order. While in the RE ternary compounds and quaternary borocarbides the superconductivity is due to the electron–phonon interaction, in heavy fermions it is probably due to spin fluctuations and with (possible) triplet pairing [23]. The question related to the interplay between magnetism and superconductivity in high Tc superconductors is well beyond the scope of this chapter. The mechanism of high temperature superconductivity (HTSC) still remains mysterious though a number of theoretical approaches have been proposed. From the theoretical point of view the electron–phonon interaction (EPI) and spin fluctuations are the most serious candidates for pairing in HTSC, while the former interaction is favored by the recent ARPES experiments; see the reviews [24]. In (RE) Ba2 Cu3O6+x and La2−x Srx )CuO4+ı the oxygen concentration is a very important parameter controlling the magnetism of Cu ions. For small x and ı the compounds are insulators and Cu ions order antiferromagnetically with TN ≈ (300–500) K (see, for example, [25]). For optimally doped compounds the Cu antiferromagnetic transition is absent and Tc is around 40 K and 95 K for (La2−x Srx )CuO4−ı and (RE)Ba2 Cu3 O6+x , respectively. The trivalent rare earth magnetic ions (RE) have very little effect on the superconducting properties of (RE) Ba2 Cu3 O6+x compounds. The (RE) spins order antiferromagnetically at low temperature TN ≈ (0.5−2) K [25], which means that the exchange interaction between (RE) spins and superconducting electrons (mainly moving in the Cu-O2 planes) is small. Note that in these systems the magneto-dipole and RKKY mechanisms give comparable contributions to the antiferromagnetic ordering energy. The condition

4

Coexistence of Singlet Superconductivity and Magnetic Order

TN  Tc ensures quite weak interplay of the AF order and superconductivity in (RE) Ba2 Cu3O6+x . However, the coexistence of SC and itinerant antiferromagnetism in high-temperature superconductors have been realized for some doping. For instance in (La2−x Srx )CuO4−ı one has: Tc = 42 K, TAF ≈ 42 K – for x = 0, ı = 0.14; Tc = 12 K, TAF ≈ 25 K (or Tc = 29 K, TAF ≈ 25 K depending on the synthesis); for x = 0.12, ı = 0, there is a pronounced Bragg peak at Q ≈ QAF = (, ) and the static correlations are developed on the length scale m > 400 Å. Since m  ab ≈ (20−25) Å (the SC coherence length in the ab-plane) this means that these static correlations can be considered approximately as a static AF order that competes with SC. Similar results hold for YBa2 Cu3 O6.5 (Tc = 55 K, TAF ≈ 310 K ) and in YBa2 (Cu1−y Coy )3 O7+ı (Tc = 93 K, TAF ≈ 330 K for y = 0.013); see [26] and references therein. The neutron scattering measurements show that the effective magnetic moment in all these superconductors were small < 0.1 B in the La family and ≈ (0.02−0.05) B in the Yfamily. This small moment points to a rather small spin-fluctuation coupling constant in these systems, which is apparently insufficient for the pairing mechanism. Based on the theory of weak (  B ) itinerant antiferromagnetism this coupling is extracted in [26] to be gsf < 0.1−0.2 eV. This value is much smaller than the ad hoc assumed value gsf ≈ 0.65 eV in the spin-fluctuation theories of pairing in high-temperature superconductors [27], thus giving strong arguments against the spin-fluctuation theories of pairing. Nowadays, several type of layered compounds, where superconducting and magnetic layers alternate, are known. For example in Sm1.85 Ce0.15 CuO4 [28], which reveals superconductivity at Tc = 23.5 K, the superconducting layers are separated by two ferromagnetic layers with opposite orientations of the magnetic moments and the Neel temperature is TN = 5.9 K. Several years ago, a new class of magnetic superconductors based on the layered perovskite ruthenocuprate compound RuSr2GdCu2 O8 , comprising CuO2 bilayers and RuO2 monolayers was synthesized; see the review [29]. In RuSr2GdCu2 O8 the magnetic transition occurs at Tm ≈ 130−140 K

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and superconductivity appears at Tc ≈ 30−50 K. Though the magnetization measurements give evidence for the small ferromagnetic component, the neutron diffraction data on RuSr2GdCu2 O8 revealed the dominant antiferromagnetic ordering in all three directions. Later, the presence of the in-plane ferromagnetic component of about (0.1−0.3) B was confirmed by neutron scattering on iso-structural RuSr2GdCu2 O8 ; see [30]. Finally, let us mention that the ferromagnetic order is much less detrimental in superconductors with triplet pairing, since the parallel electronic spins are weakly affected by the exchange field. In this case these two orderings can coexist in bulk materials.The first truly ferromagnetic superconductors UGe2 [31] and URhGe [32] were discovered only recently and have attracted a lot of interest. Apparently these systems display the triplet superconductivity with parallel spin orientation of electrons in Cooper pairs, thus withdrawing the paramagnetic limit. However, the physics of these materials is far from understood and therefore it is not the subject of this chapter.

4.2

Ferromagnetic Superconductors

In the ferromagnetic superconductors the magnetic ordering of the localized 4f moments (LM) is due to the indirect exchange interaction (RKKY) going via the conduction electrons. As will be demonstrated below the effective (characteristic) exchange energy (ex ) between localized moments is of the order of ex ≈ N(0)h20 , where N(0) is the density of states at the Fermi level (per LM) and h0 (= (gJ − 1)nJsf (0)Jˆz ) is the maximal exchange field acting on conduction electrons. Here, gJ is the Lande factor, n is the density of localized magnetic moments (LMs), Jsf (0) is the direct (local) exchange energy between conduction electrons and LMs, Jˆz  is the averaged total angular moment of localized moments. Note that the exchange field acting on electrons is hex (r) = h0S(r) (note that very often we use h instead of hex ), where S(r)(= Jˆz /J) is the localized relative spin normalized to one. Let us mention in advance that in most RE ternary superconductors the maximal exchange field h0 is rather large, h0 ∼ 102 K and it is much

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larger than the bare superconducting gap 0 , i.e. h0  0  10 K. We shall see below that in spite of the fact that h0 is much larger than the Clogston paramagnetic field hp , i.e. h0  hp ≈ 0.70 , the coexistence of SC and modified ferromagnetic order can be realized. The energy parameter ex is a measure of the gain in the exchange energy (per spin) in the ferromagnetic ground state, i.e. | Em |∼ ex . In the ferromagnetic superconductors that will be discussed here one has ex ∼ 1 K. On the other hand the gain in the superconducting condensation energy per one electron is very small, i.e. | Es |≈ N(0)2 /2 ∼ Tc (Tc /EF ) ∼ 10−3 Tc , where EF is the Fermi energy. This means that even in the case Tm  Tc one has | Es || Em |, i.e. the ferromagnetism is an energetically very robust phenomenon in comparison with the singlet superconductivity. Therefore, the singlet superconductivity cannot prevent the ferromagnetic ordering but may only modify it. On the contrary, ferromagnetism can easily destroy superconductivity, as will be discussed below. As we have already mentioned, in addition to the exchange mechanism the electromagnetic (EM) interaction of superconductivity and ferromagnetism is always present in real compounds. It is related to the Meissner screening of the magnetic induction created by the magnetization. Namely, the magnetization M(r) = n S(r) creates a dipolar magnetic field B(r) = rotA(r), which on the other hand induces the screening current js of conduction electrons; the Meissner effect. The Fourier transformed js is related to the vector potential A by the kernel Ks (q), i.e. js (q) = −Ks (q)A(q). This mechanism also favors a non-uniform magnetic structure instead of a ferromagnetic one [33]. In the study of the RE ternary superconductors the mean-field approximation for both the SC and the magnetic subsystem will be assumed. It is a good approximation for SC and for ferromagnetism it is discussed below. This means that in the Hamiltonian we replace Sˆ (r) → S(r) =< Sˆ (r) > and g ˆ↑ (r) ˆ↓(r) → (r) = g < ˆ↑ (r) ˆ↓(r) > where ˆ↑,↓ are spin up and down annihilation operators. Here, (r) is the singlet superconducting order parameter and g is the electron–phonon coupling constant. The total Hamiltonian of the system is given by

 ˆ BCS + Hˆ imp + Hˆ CF + Hˆ = H [−B(ri )g B ˆJi ] , i     e  Hˆ BCS = d3 r ˆ † (r) ˆ pˆ − A − ˆ (r) c 2 | (r) | + ˆ † (r)Vˆ ex (r) ˆ (r) + gepi

(4.2)

 1 ∗ 1 † † + (r) ˆ (r)iy ˆ (r) −  (r) ˆ (r)iy ˆ (r) . 2 2

ˆ p− ec A) is the band energy of electrons in the Here, (ˆ magnetic field B = ∇ × A,  = {x ,y ,z } are Pauli matrices, while the matrix of the exchange field in the acting mean-field approximation is given by %  z hex (r) h⊥ ex (r) . (4.3) Vˆ EX (r) = ⊥,∗ (r) −hzex (r) hex For the moment we proceed in advance by informing the reader that in the SC phase the ferromagnetic order is modified into a spiral or domain-like structure with the wave vector Q. As to which magnetic structure is realized depends on the magnetic anisotropy contained in the operator Hˆ CF . If it is small (or if it is of the easy-plane type) then the spiral structure is realized with h⊥ (r) = heiQz and hz (r) = 0, while in the opposite case with an easy axis anisotropy one has h⊥ (r) = 0 and hz (r) = hz (r + L), L = 2/Q. The effect of non-magnetic impurities is described by Hˆ imp , whose effect is characterized by the mean-free path l and time . 4.2.1

Spiral Magnetic Order in Superconductors

The question is: how is the ferromagnetic order modified in the presence of SC and do they coexist at all? The first hint was given by Anderson and Suhl [9] who studied the case h   and calculated the spin susceptibility (q) of conduction electrons. They found that (q) is peaked at Qm ≈ (a2 0 )−1/3 , where the length a ∼ kF−1 and kF is the Fermi momentum and 0 is the superconducting coherence length, see Fig. 4.6. This magnetic order, called the cryptoferromagnetic order by Anderson and Suhl, has the wave-length L where one has a  L  0 . The problem of the coexistence of the spiral magnetic order

4

Coexistence of Singlet Superconductivity and Magnetic Order

and superconductivity for any ratio of h/ was studied in [10] and solved exactly. Because this solution is a paradigm not only for the competition of superconductivity and magnetic order but also for S-F superlattices, we discuss some relevant results of this theory by assuming (i) that the bare superconducting transition temperature is much larger than the bare magnetic critical temperature, i.e. Tc0  Tm0 and (ii) large exchange energy h0  . In an isotropic system with negligible magnetic anisotropy and electromagnetic interaction the natural choice for magnetic ordering is the spiral (helicoidal) spin structure with hzex = 0 and h⊥ ex = 0

 h⊥ (4.4) ex (r) = h ex cos(Q · r) + ey sin(Q · r) . Here, ex and ey are unit vectors along the x-axis and the y-axis, respectively. In isotropic systems in the absence of EM interaction there is no preferable direction of Q. Later we show that EM makes the structure transversal, i.e. S · Q = 0. By solving the Gorkov equations (given in Appendix A) one obtains the normal and anomalous Green’s functions G1 2 and F1 2 , respectively. We shall not give details here (for details, see [10]) because of simplicity but stress only some important properties. First, besides the singlet pairing amplitude, which is responsible for the singlet pairing  = gF↑↓(r, r), the triplet amplitudes F↑↑ =  ↑ ↑  and F↓↓ =  ↓ ↓  appear in the presence of the spiral exchange field F!n ,↑↑ (k + Q/2, k  − Q/2) = i!n − k , (4.5) −ı(k − k )2h 2 2 2 [!n + E1,k ][!n2 − E2,k ] with F↓↓ = F↑↑ (Q → −Q, h → −h). Curiously, F↑↑ (F↓↓ ) is an odd function of the combined inversion k → −k, ! → −!, F↑↑ (k, !) = −F↑↑ (−k, −!). Moreover, the quasiparticle spectrum is given by 2 E1,2,k = "k2 + k2 + h2 + 2 ∓ 2 "k2 (h2 + k2) + 2 h2 ,

(4.6)

where "k = (k+Q/2 + k−Q/2 )/2 and k = (k+Q/2 − k−Q/2 )/2. The important fact is that for h >  the

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branch E1,k = 0 is gapless for "k2 = h2 − 2k and k = 0. In the case of the parabolic band the gapless spectrum is realized on the line that is the intersection of the plane vF · Q = 0 with the Fermi surface. Near this line the spectrum is approximated by k E1,k ≈ k2 + 2g,k ; g,k = , (4.7) h2 + k2 which leads to the linear density of states N(E) h E N(E) = , N(0) 2vF Q 

E  .

(4.8)

The latter gives rise to the power law behavior of a number of transport and thermodynamics properties. For instance, the specific heat is quadratic in T Cel (T) =

hQ 2 T . vF Q

(4.9)

The model with the spiral order predicts a generic phase diagram for ferromagnetic superconductors that is qualitatively correct also for systems with magnetic anisotropy, see Fig. 4.8 below. For, instance at Tc1( Tc0) there is a second order phase transition into the superconducting state with paramagnetic magnetic phase, while at Tm = Tm0 (1 − ˛a/) ≈ Tm0 there is a second order transition to the coexistence phase; SC with the spiral magnetic order with the wave vector Qm ≈ (a2 0 )−1/3 . By lowering T the exchange field h grows, reaching the value h  0 , and at Tc2 the superconductivity disappears in the first order transition where the ferromagnetic phase is realized. At this point one has Qc2 ≈ Qm (0 /EF )1/6(ln(0 /EF ))1/3, i.e. Qc2 < Qm . The phase diagram resembles that in Fig. 4.8. The effect of impurities is similar to those in the sinusoidal and domain structures, which will be discussed below. 4.2.2

Sinusoidal Magnetic Structure Due to SC for T  Tm

In the case when there is a large easy axis magnetic anisotropy D, S(r) is parallel to the z-axis. Below but near Tm the magnetic order parameter is small, S(r)  1, and the non-uniform sinus structure S(r) ≈ S(T)ez sin qr is realized. In this case

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hex (T) = h0 S(T)  h0 ,  and the magnetic freeenergy can be calculated by the perturbation theory. The general structure of the free-energy is given by F{S(r), (r), A(r)} = Fm {S} + Fs {}  (B − 4M)2 . (4.10) + Fint {S, , A} + d3 r 8 Here, Fm (= Em − TSm + Eanis ) and Fs is the magnetic and SC free-energy functional in the absence of mutual interaction, respectively. The  energy of the magnetic anisotropy is Eanis = − q D | Sz (q) |2 , while the magnetic energy Em and the entropy are obtained straightforwardly from the microscopic theory based on the Hamiltonian in Eqs. (4.2)–(4.3). For Em it gives Em {S} = −

ex  n,q | Sq |2 +Ed {S} , 2 q n,0

(4.11)

where S(q) is the Fourier transform of S(r). The first term is the lowering of the energy due to the indirect RKKY exchange interaction via the spin polarization of the conduction electrons in the normal state. The RKKY coupling is proportional to the spin susceptibility n (q) of the conduction electrons. Since ex  1 K, the dipole-dipole energy Ed between moments must inevitably be taken into account. It has the form [8]  ˜em  Ed {S} = (˛d a2 q2 − 1) | Sq |2 2 q  (qSq )(qS−q ) . + q2

(4.12)

Here, the characteristic dipole–dipole energy ˜em =  (em /3) + em , where em = (B2 /8n) = 2n 2 and  em characterize the long-range and short-range part of the dipole–dipole interaction between LMs, respectively. Their values are given in Table 4.1. The parameter ˛d characterizes the stiffness of the dipole– dipole interaction [34]. It is evident from Eq. (4.12) that Ed is minimized when the structure is transverse qSq = 0, i.e. q ⊥ Sq . The length (magnetic stiffness) a, which is of the order of the lattice constant, depends on the lattice structure and the Fermi surface of conduction electrons. The bare (in the absence of

superconductivity) ferromagnetic critical tempera ture is Tm0 = ˜ex + ˜em with ˜ex = ex + ex , where  ex and ex characterize the long-range and shortrange parts of the EX interaction [8]. Proceeding in advance we inform the reader that both Em and Ed are renormalized (screened) in the superconducting state. Since Dz > 0, the magnetic anisotropy orients spins along the z-axis. The last term in Eq. (4.10) is the magnetic energy for a given magnetization M(r) = n S(r) and in the equilibrium the magnetic induction in absence of SC is given by B = 4M. At temperatures near Tm ( Tc1) one has | S(r) | 1 and the exchange field is small, h  . In this case the SC free-energy density Fs is weakly temperaturedependent and is given by 1 e2 Fs {(r)} = − N(0)2 ln 20 . (4.13) 2  It is minimized for  ≈ 0 since at T near Tm SC it is weakly affected by the ferromagnetism. Therefore, near Tm the SC order parameter is determined solely by the pairing mechanism. The part Fint = Fint,ex + Fint,em describes the EX and EM interaction between  SC and the magnetic order. Since Fint,em = − jsq Aq /2 and by using the relation jsq = −Ks,q Aq one obtains    ex n,q − s,q 1 2 2 Fint = | Sq | + Ks,q | Aq | . 2 ,0 2 q (4.14) The first term, which is positive, describes the increase of the free-energy due to the decrease of the spin susceptibility s,q in the SC state,see Fig.4.6.The EM kernel Ks,q describes the Meissner screening of the dipole–dipole interaction by the superconducting electrons. After minimization of F{S(r), (r), A(r)} with respect to A(r) one obtains F{S(r), (r)} in the following form [8] 1 F{S, } = [(T − Tm0 ) + a2 q2 ] | Sq |2 2 q ex n,q − s,q | Sq |2 +Fs 2 n,0  | Sq |2 +(qSq )(qS−q ) . (4.15) q2 + 4Ks,q

− Dz | Szq |2 + +

em 4Ks,q 2

4

Coexistence of Singlet Superconductivity and Magnetic Order

173

Fig. 4.6. Schematic spin susceptibility in the normal (continuous line) and SC state (dotted line) n,s (q): (a) for the ferromagnetic order in the normal state, peak at Q = 0; (b) for the antiferromagnetic order, peak at Q0  Here,  = ex + ˛ex ex + ˛d d = ex ; for more details see [8,34]. Due to the singlet SC pairing the spin susceptibility s,q is reduced significantly at small wave vectors q < 0−1, where 0 is the SC coherence length. In the clean limit (l → ∞) and at T = 0 one has s,0 = 0, which means that the ferromagnetic order cannot coexist with the singlet superconductivity. In Fig. 4.6 we show s,n,q schematically for the cases when the ferromagnetic (a) or antiferromagnetic order (b) is realized in the normal state. It is seen from Fig. 4.6 that the singlet superconductor behaves as a normal metal at large momenta, i.e. s,q≈kF is weakly affected by SC. Therefore AF competes with SC much less than the ferromagnetic order does. We stress that in the case of the singlet s-wave SC s,0 = 0 and falls off exponentially at finite temperatures, i.e. s,0 ∼ exp(−/T), while in d-wave SC one has s,0 ≈ 1.4 n,0(T/0 ). In the presence of the spinorbit (SO) scattering s,0 is also finite. The general expression for s,q in this case is [35]

s,q = 1 − T

 !n

1 (1 +

u2! )(P!,q

− 1/21)

, (4.16)

where u! = !n / and P!,q =

2 arctan

qvF

qvF  1 2

+ u2! + 1/2−

.

(4.17)

Here !n = T(2n + 1) and 1−1 = −−1 − (4/3)so−1 , −−1 =  −1 + so−1 , where  −1 and so−1 is the relaxation rate due to non-magnetic impurities and spin-orbit scattering, respectively. Later, we shall discuss effects of the SO interaction on the coexistence phase. The effect of the exchange scattering is similar, i.e. s,0

is finite for finite exchange scattering time s . Concerning this point, as well as other theoretical questions related to magnetic superconductors see the review [7]. In the following we shall also study the competition between SC and ferromagnetism at low temperatures and in the clean limit where the EM kernel has the form  xqv  1 3ne  1 − x 2 arcsh 2F Ks,q = dx  xqv 2 . (4.18) mcqvF x 1 + 2F 0 The expression for Ks,q in the case of finite meanfree path l is rather complicated and is omitted here, see [7,8].However,some limiting expressions for Ks,q , which are relevant for real magnetic superconductors with impurities, will be given below. By knowing s,q and Ks,q one can minimize F{S(r), (r)} with respect to the wave vector q,whose equilibrium value Qm depends on microscopic parameters a, 0 , L (the London penetration depth), ex , em . From Eq. (4.15) we conclude that the EM interaction is minimized for q · S(q) = 0, i.e. the magnetic structure in SC is transverse. We give explicit expressions for s,q and Ks,q in the interesting range of parameters. In the clean limit and for q0  1 one has   (q0 )2 1 , 1 − Ks,q = ; − = n,q s,q n,0 30 4L2 (4.19) while for q0  1 Ks,q =

3 4L2 q0

;

n,q − s,q = n,0

 . (4.20) 2q0

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M. L. Kuli´c and A. I. Buzdin

The minimum of F if Eq. (4.15) gives the transition temperature into the sinusoidal magnetic structure Sz (r) ≈ S sin(Qmr) (with Qm ⊥Sz )

 Tm = Tm0 1 − 3(ex a/40 )2/3 . (4.21) Since a  0 and ex ∼  one gets that Tm ≈ Tm0 , i.e. the spiral order is very favorable since it lowers the exchange energy and critical temperature very little. In the case when 02  aL , the wave vector Qm is determined by the EX interaction only and for ex /em  (a/0 )2 it is given by [34]  Qm =

 ex 4 a2 0

1/3 .

(4.22)

For the extremely low exchange energy, i.e. for ex /em  (a/0 )2 , the EM interaction prevails with Qm = (1/aL )1/2. In the opposite limit 02  aL , the EX interaction dominates for ex /em  (a2 0 /L3 )2/5, which again gives Qm = (ex 4a2 0 )1/3. For ex /em  (a2 0 /L3 )2/5 the EM interaction dominates, which gives Qm ≈ (1/a2 0L2 )1/5 [34]. The microscopic parameters of some rare earth ferromagnetic superconductors are given in Table 4.1. The latter gives evidence that the wave-vector Qm of the oscillatory structure is determined dominantly by the EX interaction,i.e.it is independent of the EM parameter L . It turns out that the EM interaction is dominant only for an extremely small EX interaction (ex  em (a/0 )2 or ex  em (a2 0 /L3 )2/5), i.e. for (ex /em )  10−4 − 10−5 . However, in typical ferromagnetic superconductors, such as ErRh4 B4 , HoMo6S8 , HoMo6Se8 and AuIn2 the EX interaction dominates since ex > 0.1 em and a  0  L . In real compounds non-magnetic impurities are always present and the knowledge of s,q (l) and Ks,q (l) as a function of the mean-free path l is needed. The corresponding calculations [34] show that if (l5 /a2 0 L2 )  1 and for ex /em  a2 0 /l3 the EX interaction dominates and one has Qm = (ex /4a2 0)1/3 , while for (a2 0 /l3 )  ex /em  (l2 /L2 ), a2 l/03 one obtains Qm ≈ (ex /)(1/la20 )1/4 . In the case when (ex /em )  (a2 0 /l3 ), or (ex /em )  l2 /L2 , the EM interaction dominates and Qm ≈ (l/a2 0 L2 )1/4.

Let us stress some interesting theoretical predictions for ferromagnetic superconductors. (i) The ferromagnetic critical temperature Tferro is strongly reduced in the presence of SC due to the formation of Cooper, i.e. one has Tferro = Tm0 [1 − (ex + em )/]  Tm0 . Note that for the spiral state one has Tm ≈ Tm0, which means that the oscillatory magnetic structure is more favorable than the uniform ferromagnetic one. In fact, this result is more general and holds also for the coexistence of SC and itinerant ferromagnetism (F) in the mean-field approximation (MFA). Namely, in systems with the BCS-like pairing and when the EX interaction dominates the EM one the singlet superconductivity and ferromagnetism do not coexist in the MFA. In this sense a number of recent papers claiming that the itinerant F and SC coexist in the MFA should be abandoned [36]. However, in some itinerant ferromagnets such as Y9 Co7 (with TF = 4.5 K) the microscopic parameters favor the spiral or domain-like magnetic structure in the SC state with Tc1 = 2.5 K,as was proposed in [37].Unfortunately, this interesting compound has not been investigated sufficiently. (ii) In isotropic magnetic systems near the critical temperature Tm the inverse scattering time due to magnetic fluctuations (at small q) can diverge and thus destroy SC. However, this effect is suppressed in the real RE ternary superconductors due to the longrange dipole–dipole interaction. The interaction of SC with magnetic fluctuations is described by the free-energy contribution Fsc,fl =

n,q − s,q ex  Sz,q Sz,−q  , 2 q n,0

(4.23)

where Sz,q Sz,−q  ∼



+ a2 q2

1 , + (em /)q2z /q2

(4.24)

with  = (T − Tm0 )/. Due to the large dipole–dipole temperature with em ∼  these fluctuations look four-dimensional, thus giving a rather small value for the inverse scattering time m−1 ∼   Tc1 ;

4

Coexistence of Singlet Superconductivity and Magnetic Order

(iii) the relative strength of the EX and EM interactions is controlled by the parameter r r=

(EM) FInt (EX) FInt

=

em 1 . ex Q2 L2

(4.25)

In the RE ternary compounds the case r  1 is always realized, due to the large value of Q2 L2  1. Therefore, practically in all RE ternary superconductors the EX interaction dominates in the formation of the magnetic structure in the superconducting state, while the EM interaction, which contributes significantly to Tm , makes the structure transversal; see the exception in the case of weak ferromagnets below. (iv) In the RE ternary superconductors the ferromagnetism is stronger phenomenon than SC, since the gain in the ferromagnetic energy (per LM and at T = 0 K) Em ≈ N(0)h20 is larger than the gain in the SC condensation energy Ec ≈ N(0)20 , since h0 (∼ 102 K)  0 ( 10 K). Nevertheless, the ferromagnetic order is more “generous” since it varies spatially in the SC state, while the SC order parameter stays practically homogeneous. The reason for this peculiar phenomenon lies in the fact that the (ferro)magnetic stiffness (∼ a) is much smaller than the superconducting stiffness (∼ 0), since a  0 . We stress that the sinusoidally modulated magnetic structure in the SC state perfectly resolves the antagonism between superconductivity and ferromagnetism. Indeed, from the “point of view of superconductivity” such a magnetic structure is like the antiferromagnetic one since Qm L  1 (for the EM interaction) and Qm 0  1 (for the EX interaction). On the other hand, it is like a ferromagnetic one from the “point of view of ferromagnetism”, since Qm a  1. Then it is really a compromised structure. 4.2.3

175

h0 S(T) by lowering T, but if hex (T) <  the mutual interaction of the magnetic order and SC can be treated by the perturbation theory. In such a case the free-energy density is completed by the energy density of domain walls, QEW /, where EW is the wall-energy per unit surface. In the case of the sufficiently large magnetic anisotropy, Dz > , the rotation of magnetic moments in the wall is unfavorable and the linear domain wall with Sz (x) = Sth(x/l W ), √ Sx = Sy = 0 is realized. Here, lW = a/  is the domain-wall thickness [8]. The domain wall energy per unit surface is given by  √  EW = 4 2/3 nS2 a 1/2 ≡ nS2 a˜ , (4.26) where  = (Tm0 − T)/, see [8]. The free-energy density F˜ DS in the DS phase is given by   Q 1 2 b 4 7(3) S2 . F˜ DS = n − S + S + EW + nex 2 4  2 Q0 (4.27)

Domain Magnetic Structure Due to Superconductivity

By lowering T the higher order term in the freeenergy, ∼ S4 (r), makes the change of the modulus of S(r) unfavorable.As a result, the sinusoidal-structure is transformed into the striped domain structure (DS), see Fig. 4.7(b). The exchange field grows hex =

Fig. 4.7. (a) Spiral structure with the period Ls . (b) Striped domain magnetic structure S(x) = Sz (x)ez with the period LD = 2 /QDS. The wave vector of the DS structure QDS is along the x-axis

176

M. L. Kuli´c and A. I. Buzdin

F˜ DS depends on S and Q only, since SC is practically unaffected by the magnetic order near Tm. Note that if the anisotropy energy is small, i.e.  > 2Dz /, the rotating domain wall is realized with the wall thickness lW ≈ a(/Dz )1/2 and the wall energy is EW ∼ nS2 a(Dz )1/2.In the case of an extremely small magnetic anisotropy (Dz /) < (a/0) ∼ 10−2 − 10−3 the spiral structure is favorable. Minimizing FDS in Eq. (4.27) with respect to Q one obtains the equilibrium wave vector of the striped DS phase 

QDS

ex 1 2  a˜ 0

1/2 .

(4.28)

By comparing Eqs. (4.22) and (4.28) one concludes that the period of the DS structure (LDS = 2/QDS ) is larger than that of the sinus one. We point out that (i) the striped domain structure is due to SC and it is a property of the bulk material, and (ii) the structure of the free-energy of the DS phase is mathematically similar to the problem of the domain structure in a normal ferromagnetic plate with the magnetization perpendicular to the surface of the sample; see also Sect. 4.2. In this case the role of Fint plays the magnetic energy dissipated out of the plate.Generally,the domain structure is realized when the wall thickness √ a/  is much smaller than the domain thickness /Q, thus implying that   (a/0 )2/3 ∼ 10−2 . At lower temperatures when the exchange field is large, hex (T) > , the problem appears to be non-perturbative. Since the period LD of the domain structure is much larger than the atomic length a, i.e. LD  a, the problem is studied by the quasiclassical ELO equations, see the Appendix A.2. In the presence of non-magnetic impurities these equations are solved for the striped domain structure with the domain-like exchange field hz (r) = h0Sz (r) where the relative magnetization in the DS phase is given by ∞

4S(T)  sin(2k + 1)Qr  Sz (r) = ≡ Szq eiqr .  2k + 1 q k=0

(4.29) Since S · Q = 0, it follows that Q is along the x-axis. The solution for the quasi-classical Green’s functions f (v, x), (x) and g(v, x) is searched in the form

f (x, v) = f0 (v) +



fk (v)eikx ,

(4.30)

gk (v)eikx ,

(4.31)

k

g(x, v) = g0(v) + (x) =  +



 k

k eikx ,

(4.32)

k

where k = (2m + 1)Q and | fk || f0 |, | k ||  | and | gk || g0 |. In order to solve the problem the self-consistent equations for the SC order parameter are needed. The calculations were done in [8,34] and here we present only the final result in the dirty limit (l  0). It turns out that in this case the quasiclassical equations are reduced to those for f¯0 = f0 (v) and g¯0 = g0 (v) (. . . is the average over the Fermi surface) which are the solutions of the following equations f¯02 + g¯02 = 1, ! f¯0 − g¯0 = −2f¯0 g¯0 m−1 . (4.33) The structure of these equations is the same as those for SC in the presence of magnetic impurities with the inverse scattering time m−1 . In the case of the magnetic domain (due to SC) one obtains   hz,q hz,−q −1 L1 (ql) m = vF q q (4.34)  3Bq · B−q + L2 (ql) , 16L2 nN(0)vF q3 where hz,q = h0Sz,q and L is the London penetration depth. The functions L1 and L2 are given by [34] 2y arctan y (4.35) L1 (y) = (y − arctan y) and L2 (y) =

2 

   1 1 1 + 2 arctan y − . y y

(4.36)

The minimization of F with respect to the vector potential A gives the magnetic induction Bq which is related to Sq by [34] Bq =

4n [q2 Sq − q(qS q )] q2 + Ks,q (1 − 4/3m)

,

(4.37)

where the EM kernel Ks (q) in the dirty limit has the form 3 Ks,q = L2 (ql) . (4.38) 16vF L2 q

4

Coexistence of Singlet Superconductivity and Magnetic Order

177

In the RE ternary superconductors the case m  > 1 is realized and in this case F˜ DS is given by   b 1 ˜FDS =  S2 + S4 + QEW 2 4   2 N(0) 2 e0 (4.39)  ln − 2 2   2  . 1− − m 3m  Here, N(0) is the electronic density of states per electron spin and per magnetic atom. Based on the free-energy in Eq. (4.39) we can study the coexistence problem in the whole temperature region and for various Ql; for more details see [34]. We summarize the main results. (a) At T = Tm the sinusoidal magnetic order appears with the wave vector Qm ∼ (1/a20 )1/3 where Qm is perpendicular to S, i.e. the structure is transverse. (b) By lowering the temperature the striped domain structure appears with QDS ∼ (1/a0)1/2, which is also transverse and persists down to the temperature of the first order phase transition Tc2 where the DS phase passes into the normal ferromagnetic state. At Tc2 one has FDS {SDS , , QDS,c2, } = FFN {SF , 0, 0}, where QDS,c2 ≈ 1.8(˜a(Tc2)0 )−1/2 ∼ (a0 )−1/2 , (Tc2 ) = 0.850 and 2 (Sc2 /Qc2) ≈ 0.07(0 vF /h20); (c) The DS phase is stable down to T = 0 K if SDS (Tc2) > 1; this situation is realized in systems with small EX interaction (which still dominates over the EM one),i.e.for ex < 3 c ex ∼ (Tc1 /h20 ). (d) In dirty SC with (h)2  1 there is a gap in the quasiparticle spectrum for E <  in the whole range of the domain phase existence. (e) The calculations [8,10] show for clean SC in the DS phase that the spectrum is gapless for h(T)   and E N(E) h E 4 = ln , (4.40) N(0) vF Q  E (f) The spin-orbit interaction decreases the value of n,0 − s,0 (this also holds for small q) and this effect is detrimental for the existence of the DS phase. However, the analysis in [38] shows that the spin-orbit scattering can destroy the peak in s,q in unrealistically dirty systems (l ∼ a) only. We would like to point out that a lot of studies of ferromagnetic superconductors based on the phe-

Fig. 4.8. Phase diagram of ferromagnetic superconductors with easy plane anisotropy. Full lines separate N – paramagnetic normal phase, S – superconducting phase, DS – coexistence phase and FN ferromagnetic normal phases. Tc1 and TM are second order transition lines. Dash-dotted line:  is the second order line that separates N and FN phases in absence of the S phase. The dashed lines indicate overheating and supercooling. Tc2 is the first order transition line from DS to ferromagnetic phase [8]

nomenological Ginzburg–Landau (G-L) theory take into account the EM interaction only [33, 39]. Although very interesting,this phenomenology is inadequate in describing real materials, such as the above cited RE ternary superconductors, where the EX interaction prevails in the formation of the oscillatory structure (with Q  0−1 , L−1 ) in the SC state. Finally, based on the above theory the schematic phase diagram for RE ternary superconductors is calculated and shown in Fig. 4.8. Experimental Situation in the Coexistence Problem The important microscopic parameters of the RE ternary ferromagnetic superconductors with the coexistence phase are extracted from various experiments and are given in Table 4.1. Based on the proposed theory and by using the microscopic parameters from Table 4.1 one concludes

178

M. L. Kuli´c and A. I. Buzdin

Table 4.1. Basic parameters of some ferromagnetic superconductors. The parameters are defined in the text ErRh4 B4

HoMo6 S8

HoMo6 Se8

n[cm ]

∼ 10

∼ 4 × 10

∼ 4 × 1021

[ B ]

5.6

−3

22

21

9.1

a˜ [Å]

∼1

2.5

L (0)[Å]

900

1200

0 [Å]

200

1500

0 [K]

15.5

3.2

10

N(0)−1 [K · spin]

1850

3600

1754

vF [cm · s−1 ]

1.3 × 107

1.8 × 107

1.8 × 107

ex [K]

0.5–0.8

0.2

0.1 < ex ∼ 10

2.7 470

h0 [K]

40

24

m−1 [K]

3

0.9

em [K]

1.8

1.3

∼ 1.3

Tc1 [K]

8.7

1.8

5.5

Tm [K]

0.8–1

0.7–0.74

0.53

Tc2 [K]

0.7

0.65

no

L = 2 /Q[Å]

90–100

200

70–100

Fig. 4.9. Temperature dependence of the intensity of satellite peaks and resistance in ErRh4 B4 single crystal [11]

that in ferromagnetic superconductors HoMo6S8 , ErRh4 B4 , HoMo6Se8 the superconductivity and oscillatory magnetic order coexist in a narrow temperature interval. The unambiguous experimental evidence of the existence of the non-uniform magnetic structures was obtained from the neutron diffraction measurements in ErRh4 B4 [11], HoMo6S8 [12] and HoMo6Se8 [40]. For example, in Fig. 4.9 it is demonstrated that in ErRh4 B4 [11] just below the Curie temperature Tm in the superconducting phase the satellite neutron Bragg scattering appears in the narrow temperature interval around (0.1–0.2) K below the magnetic transition. Some peculiarities in ErRh4 B4 , such as the appearance of the ferromagnetic peak in the neutron scattering also in the coexistence phase (Tc2 < T < Tm ), are explained by the asperomagnetic ferromagnetic structure in the normal ferromagnetic state; see the extensive discussion in [41]. The behavior of the first satellite in polycrystalline samples of HoMo6S8 is well described by the proposed theory, which also predicts that the intensities of higher harmonics are very small and cannot be seen experimentally [34]. Further lowering of the temperature provokes the transition into the ferromagnetic state with the simultaneous destruction of the superconductivity, which is in agreement with the theoretical predictions. The period L of the oscillatory magnetic structure (either sinus or domain-like) in all three compounds does not exceed the value L(= 2/Q) < 200 Å, i.e. 100 Å in HoMo6S8 and 200 Å in ErRh4 B4 . This important result means that the energetics of the coexistence phase in the bulk sample is predominantly due to the EX interaction, while, as we said above, the EM interaction makes the structure transverse (Q · S = 0). The latter property is due to the fact that in this case the density of “magnetic charges” is zero, i.e. divM = 0, and the corresponding magnetic energy is also zero.In this sense the recent proposal [42] of the coexistence of SC and the longitudinal (with Q · S = 0) modulated (by SC) ferromagnetic order should be abandoned, since the energy of this structure is higher than that of the transversal one. It can be seen from Table 4.1 that the compound HoMo6Se8 is different from the other two, since in it SC and the oscillatory magnetic order coexist down to T = 0.

4

4.2.4

Coexistence of Singlet Superconductivity and Magnetic Order

Domain Magnetic Structure in Thin Superconducting Films

In the above calculations we have assumed that the thickness L of the sample is very large, i.e. L  −1 0 , QDS , so that the dissipated magnetic dipole energy (stray field) is negligible.In the case of thin films with L ∼ 0 the stray magnetic energy Est , which exists around the domain walls and near the surface of the sample, must be added to the free-energy FDS in Eq. (4.39). The total free-energy Ftot = FDS + Est is given by [43] F˜ tot /n = (F˜ DS /n) + Est = (F˜ DS /n) + 0.85em

S2 (T) . QL

(4.41)

(EM) (EX) In the case when r(= FInt /FInt )  1 the minimization of Ftot w.r.t. Q gives 2 2 Qtot = QDS + QF2 .

(4.42)

QDS is the wave vector of the DS phase without the stray magnetic energy and QF ≈ 1.6(em / a˜ L)1/2 is the wave vector of the striped domain structure in the normal ferromagnetic state. From Eq. (4.42) it is seen that in a thin film of the ferromagnetic superconductor the period of the DS phase (d = 2/Qtot ) is decreased due to the stray field. The theory based on Eqs. (4.39) and (4.41) gives that the transition

179

temperature Tc2 (for the first order phase transition DS → FN (with domains) can be pushed to zero 4 4 when L < Lc = 30 (em Sc2 (L = ∞))/ex (1 − Sc2 2 (L = ∞)) ; see Fig. 4.10. For the parameters from Table 4.1 one obtains Lc < 100 in HoMo6S8 and Lc < 20 in ErRh4 B4 . Some experiments in thin films of HoMo6S8 show such a thickness dependence of Tc2, where Tc2(L) < Tc2(∞) holds. Let us mention that even in the normal ferromagnetic state, which is realized for T < Tc2 , there is a possibility that SC exists in domain walls as it was shown in [44,45] and in more detailed calculations in [8,46]. It seems that this situation is realized in some pseudoternary compounds in which h0  0 . 4.2.5

Coexistence of Nuclear Magnetism and Superconductivity

In 1997 Pobell’s group from Bayreuth made an important discovery [14] by observing that superconductivity and nuclear magnetism coexist in AuIn2 , which is type-I superconductor with Tc1 = 0.207 K and Tm = 35 K. At first glance this is not too surprising, having in mind the smallness of the hyperfine interaction between conduction electrons and nuclear spins. However, the above exposed theory of ferromagnetic superconductors is very well applicable also to this problem [15]. A surprising result was obtained: the effective nuclear “exchange” field (in fact, the hyperfine contact interaction) is rather large hhyp ≈ (0.6 − 1) K. At the same time the superconducting gap is 0 ≈ 0.4 K, i.e. hhyp  0 . We point out that the hyperfine interaction has the same (mathematical) structure as the exchange interaction between the 4f LMs and conduction electrons   Hˆ hyp = d3 r Ahyp ı(r −Ri ) ˆ † (r) Iˆi ˆ (r) . (4.43) i

Fig. 4.10. Qualitative behavior of the DS-FN transition temperature Tc2 on the film thickness L of ferromagnetic superconductor. For L = Lc one has Tc2 (Lc ) = 0 and (dTc2 /dL)Lc = ∞ [43]

Here, Ahyp is the hyperfine interaction and the “hyperfine exchange field” is given by hhyp = nAhyp  ˆIi , where ˆIi is the nuclear spin. It turns out that the nuclear magnetism in AuIn2, which shows a strong tendency toward the ferromagnetism in absence of SC, competes rather strongly with SC. The estimation from the experiment [14] gives

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M. L. Kuli´c and A. I. Buzdin

em (= 2nn n2 ) ≈ 1 K and ex (≈ N(0)h2hyp ) ≈ 35 K, 0 ≈ 105 Å,L ≈ 105 Å,l ≈ 3.6×104 Å (l < 0 ).This set of parameters implies that the “EX” (hyperfine contact) interaction is much stronger than EM (dipole– dipole). The theory, which was originally invented for the RE ternary compounds [8], is also applicable to this problem. It predicts that if the nuclear magnetic anisotropy (which is due to the dipole–dipole interaction) is small, i.e. (D/ex ) < 10−3 , the spiral magnetic structure should be realized. In the opposite case (D/ex ) > 10−3 the striped domain structure is preferable. The experiments in magnetic field [14] give indirect evidence that SC and a modulated magnetic order coexist up to T = 0 K.This case resembles c HoMo6Se8 , since in both cases the condition ex < ex for Tc2 = 0 is realized. Unfortunately, to date there were no nuclear scattering measurements on AuIn2 that can precisely resolve the nuclear magnetic structure below Tm = 35 K. We stress that the subject of the coexistence of SC and nuclear magnetic order is of enormous importance also for fundamental physics. These systems provide an opportunity to study the coexistence problem in cases when the electronic temperature (Te ) is different from the nuclear one (Tn ), i.e. Te = Tn. However, probably the most interesting problem is the coexistence of SC and nuclear magnetism for negative nuclear temperatures (Tn < 0 K). Unfortunately, the well-known low-temperature Bayreuth laboratory closed several years ago and further progress in this fascinating field has stopped.

4.3

Antiferromagnetic Superconductors (AFS)

Most RE ternary borides and borocarbides show the coexistence of superconductivity and antiferromagnetic order (AF). In most of them the antiferromagnetic N´eel temperature TN is smaller than Tc , as can be seen from Table 4.2. In all of these superconductors the RE earth ions have localized 4f electrons (responsible for localized magnetic moments) that interact with conduction electrons via the direct (local) exchange interaction Jsf < 103 K. This fact makes the theoretical analysis of their coexistence much easier than in the case of the AF order in itinerant magnets.

Table 4.2. Superconducting and antiferromagnetic critical temperature Tc and TN , respectively, of various antiferromagnetic superconductors Tc (K)

TN (K)

NdRh4 B4

5.3

1.31

SmRh4 B4

2.7

0.87

GdMo6 S8

1.4

0.84

TbMo6 S8

2.05

1.05

DyMo6 S8

2.05

0.4

ErMo6 S8

2.2

0.2

GdMo6 Se8

5.6

0.75

ErMo6 Se8

6.0

1.1

DyNi2 B2 C

6.2

ErNi2 B2 C

10.5

6.8

TmNi2 B2 C

11

1.5

HoNi2 B2 C

8

1.3

HoNi2 B2 C

8.7

5

11

An evident experimental fact in RE ternary compounds is that SC coexists with the antiferromagnetic (AF) order much more easily than with the modified ferromagnetic order. The reason for their weak competition lies in the fact that in the AF phase magnetic moments change their directions rapidly over the atomic distances. Therefore, superconductivity influences the AF order very weakly and the spin susceptibility at the AF wave vector QAF (∼ a−1 ) is practically the same as in the normal state  n (QAF ) − s (QAF ) Tc ≈ ∼  1. n (0) vF QAF EF

(4.44)

(EX) their interThis implies that in the free-energy FInt action part is very small and SC influences AF order very weakly. However, the AF order affects SC more strongly mainly via two basic effects. (a) The spin splitting of electronic levels by the exchange field and producing partial gapping of the Fermi surface in the normal state. This causes a decrease in the density of states. In the SC phase the AF order can make SC gapless on the part of the Fermi surface [47–49]. (b) The magnetic scattering of conduction electrons on spin fluctuations above the N´eel temperature TN and

4

Coexistence of Singlet Superconductivity and Magnetic Order

on spin waves below TN is pair-breaking for Cooper pairs [50]. (a) Let us briefly analyze the AF superconductor in the clean limit with the mean-field Hamiltonian   † † ˆ AFS = H k ck, c, +  hQ ck+Q, ck, k,

+



(ck,+ c−k,− + c.c) +

k

2 , g

(4.45)

where Q is the AFwave vector and the SC order parameter is  = g k < ck,+ c−k,− >, g > 0. The spin projection is  = ± and hQ is the Fourier component of the AF exchange field. With a good accuracy one has  ≈ const. since the theory shows [8,49] that its spatial change is small,i.e.ı(r)/ ∼ (hQ /vF Q)  1. This means that in the RE ternary AF superconductors the pairing effects with non-zero momentum is negligible. That is the reason that the results based on the uniform pairing (with (r) = ) [51] and those in [47, 48], where the non-uniform pairing is also included, are practically the same [8,49]. By taking into account that   hQ  vF Q, Hˆ AFS can be approximately diagonalized  ˆ AFS ≈ En,k b†n,k bn,k + small , (4.46) H n,k

where the spectrum En,k is given by Eq. (4.36). The term “small” accounts interband pairings, which are negligible in the considered systems. As it is seen the superconducting gap in the AF state is g,k = ık / ık2 + h2Q . It turns out that the exact results obtained in [10] for the spiral order, given in II.A, reduce to those in Eq. (4.46) for   hQ  vF Q. This is understandable since the AF order is in many respects similar to the spiral order with large Q. In that case Fint ≈ ex (Q0 )−1 ∼ (a/0)ex which allows a peaceful coexistence of SC and AF. By knowing the spectrum En,k it is easy to calculate thermodynamic and other properties of clean AF superconductors. For instance, in the case when Tc < TN the critical temperature is given by −1/˜

Tc ≈ 1.14!D e  ı( n,k − EF ) ˜ = k,n=1,2

ık2 . ık2 + h2Q

(4.47)

181

It is seen that the AF exchange field affects Tc by changing the density of states and by modulating the electron wave functions ∼ ık2 . The effect of hQ on the SC order parameter in clean systems is given by ı/0 ≈ (hQ /vF Q)ln(hQ /0 )  1. In dirty systems one obtains ı/0 ≈ TN /Tc  1 since TN  Tc  hQ  vF Q and 0  hQ . These results are confirmed in a number of the RE ternary compounds in which the Ne´el temperature TN (≈ N(0)h2Q ) is in most cases (much) smaller than Tc [6].Due to the fast oscillation of the magnetization in the AF state the EM interaction is in general very small in AFS since the change of K(Q) due to AF is very small, i.e. ıKs (Q) ∼ a3 / (EM) (EX) ( FInt ). In clean sys(L2 0 )  1 and also FInt tems with hQ   there is a gapless line on the Fermi surface, which gives the liner density of states N(E) = N(0)(hQ /vF Q)E and the quadratic electronic specific heat Cel (T) = (hQ /vF Q)T 2 [10,49]. Because the pre-factor (∼ hQ /vF Q) is very small in AFS, the gapless effects are small in these systems. (b) The magnetic scattering, although pair-breaking, is not very harmful for SC in real AF ternary compounds, since the inverse life time m−1 is small, i.e. m−1 ∼ TN  Tc . However, the effect of the magnetic scattering on the upper critical field Hc2 may be much more pronounced, especially at temperatures near TN . In the case when m−1 ∼ TN  Tc , which is, for instance, realized in TmRh4 B4 , the Hc2 curve is weakly affected by the magnetic (exchange) scattering. In cases where m−1 ∼ Tc , for instance in SmRh4 B4 , this scattering changes Hc2 significantly; see more in IV.B and [7,48,49]. Concerning the role of the non-magnetic scattering, already the above analysis on the decrease of  tells us that non-magnetic impurities (characterized by the life-time ) increase the depairing effect of the exchange field, which means a breakdown of the Anderson theorem [48,52]. In cases when TN  Tc the effect of non-magnetic impurities is like that of magnetic ones with the inverse scattering time m−1 =

h2Q  . 2vF QAF 1 + (hQ )2

(4.48)

When hQ   1 one obtains m−1 ∼ TN ≈ N(0)h2Q  Tc , since (1/vF Q) ≈ N(0). This means that in

182

M. L. Kuli´c and A. I. Buzdin

this case the pair-breaking effect of impurities is rather small [52]. A very interesting situation is realized in systems with TN  Tc . Even in such a case the exchange field does not suppress Tc significantly, since the theory [8, 49] predicts that (ıTc /Tc0) ∼ (h/EF )(ln h/EF )  1. However, in the presence of non-magnetic impurities Tc is renormalized appreciably and SC disappears for the meanfree path l < lc ≈ 100(h/vF QAF ) ∼ 0 TN /h. In that respect there is one very interesting AF superconductor Tb2Mo3S4 with TN = 19 K and Tc = 0.8 K. In this case one expects (naively) that SC should disappear due to the strong magnetic scattering. However, it turns out that in this compound the magnetic anisotropy, in conjunction with the large momentum J = 9, strongly suppress this pair-breaking effect, thus giving rise to superconductivity. 4.3.1

Weak Ferromagnetism in Antiferromagnetic Superconductors

In the case of the competition of SC and the ferromagnetic order in the RE ternary compounds the theory predicts that in the presence of an appreciable EX interaction SC can coexist only with spiral and DS (or sinus) order, depending on the strength of the magnetic anisotropy D. The realization of other phases are improbable. It turns out that in AF superconductors with weak ferromagnetism (WF), of

Fig. 4.11. Phase diagram of weak FS in the T, ˇ plane.FS: the Meissner phase with weak ferromagnetism; VS: the spontaneous vortex phase; DS: superconducting phase with domain magnetic structure [53]

the Moriya–Dyalozhinski type, the phase diagram is much richer than in the case of ferromagnetic superconductors. For instance, the Meissner phase (M = 0, B = 0) and the spontaneous vortex state [53] can be realized in these systems; see Fig. 4.11. We discuss this problem briefly by studying the simplest AF order with two sublattices, in which case l = S1 − S2 is the AF order parameter. In systems that allow WF there is an additional term in the freeenergy FWF = D[S1 × S2 ], which is responsible for the spin canting. If, for instance, l is along the xyplane and D is so oriented that it allows the appearance of the weak ferromagnetism, then m = S1 + S2 (M = n m) also lies in the xy-plane but l ⊥ m. In this case FWF is given by F˜ WF = ˇnex (mx ly + my lx ) .

(4.49)

Since in most systems m ∼ 10−3 l, this immediately implies that the parameter ˇ  1. In this case and when TN  Tc the interaction part Fint of the total free-energy (F = Fm + Fs + Fint ) is given by Eq. (4.14), while the magnetic system is described by Fm    c Fm = d3 rex al l2 + (l2 )2 + bm2 + a2 (∇l)2 4    (B − 4M)2 3 . + d r ˇex (mx ly + my lx ) + 8 (4.50) The minimization of F with respect to A, l, m and q gives an very rich phase diagram for AFS with WF [8,49,53]. We stress that the resulting freeenergy F is similar to the case of ferromagnetic superconductors in Eq. (4.15) with an effective magnetic stiffness aeff = (ab/ˇ)  a. It turns out that if ˇ  a/0, the EX interaction dominates in the formation of the magnetic structure, and the sinus structure (l ∼ sin Qr and m ∼ sin Qr) is realized at and below TN , while for (a/L ) < ˇ  a/0 the EM interaction prevails in the formation of the si√ nusoidal structure. If ˇ < (a/L ) 2em /ex , then the non-uniform structure is unfavorable and the so-called Meissner state (first proposed by Vitalii Ginzburg in 1956) is realized. It is characterized by M = const. and by the averaged (over the sample cross-section) magnetic induction < B > = 0 in the bulk sample (Note that B = 4M exp{−z/L }),which

4

Coexistence of Singlet Superconductivity and Magnetic Order

is due to the SC screening current on the surface of the sample. By further lowering the temperature the sublattice magnetization | S1,2 | grows and it is necessary to take into account higher orderterms in F. As a result one obtains that for ˇ  a/0 the EX interaction dominates again  and the striped  DS phase is realized, while for a/L  ˇ  a/0 the striped DS phase is realized due to the EM interaction. However, by lowering the temperature the domain wall energy grows and it my happen that a spontaneous vortex state, with 4M >  Hc1 , the lower critical field, is realized for ˇ  a/0 and for the AF vector l > lc ∼ (Hc1/M(0))(ˇL2 /a˜ 2)1/3 , a˜ = a[(TN − T)/TN ]1/2, see Fig. 4.11. From the known RE ternary compounds a good candidate for such a behavior is the body centered tetragonal (b.c.t.) system ErRh4 B4 .

4.4

Magnetic Superconductors in the Magnetic Field

4.4.1

Ferromagnetic Superconductors

There are a number of interesting effects of the magnetic field H either in the coexistence phase or above the magnetic transition temperature Tm where S(T > Tm ) = 0. We discuss some of them briefly and for more details see [8,38]. DS Phase in the Magnetic Field It is known that in the bulk sample the applied magnetic field penetrates only on the length L , thus affecting the surface of the sample only. However, in thin films the paramagnetic effect of the field is more important than the orbital one [8]. This problem was studied in the case of a thin (along the y-axis) film with the thickness Ly < 0, when the magnetic field is parallel to the striped domains, i.e. H = Hez . As a result the magnetization Sz (x) contains, besides the odd harmonics, also the zeroth-one as well as the even harmonics Sz (x) = ∞  2S [1 − (−1)k cos(kı)] sin(kQx) Sı + k

k=1 + (−1)k sin(kı) cos(kQx) , (4.51)

183

with ı = H/2Sex . This change of harmonics in Sz (x) can be observed in magnetic neutron diffraction experiments. Equation (4.51) tells us that domains with M parallel to H increase their thickness, i.e. d → d(1 + ı), while the thickness of antiparallel domains is decreased, i.e. d → d(1 − ı). In the case when the zeroth component of the exchange ¯ h0 Sı) > h¯ c = field is sufficiently large, i.e. when h(= [1 − (1/m)2/3 ]2/3 the DS phase is destroyed by the Zeeman effect and SC disappears, i.e.  = 0. For h¯ < h¯ c the parameters of the DS phase are renormalized, for instance one has Q(H) < Q(0). In the case when H = Hey (i.e. the field is orthogonal to the z-axis), all domains have the same thickness and there is no redistribution of intensities of neutron peaks. However, there is only a decrease of intensities of (2k + 1)Q peaks by the factor (1 − ı⊥2 ) where ı⊥ = H/S(ex + Dz ) and Dz is the magnetic anisotropy. FS in magnetic field at T > Tm The effect of the exchange field on SC in magnetic field is negligible for T  Tc1, since for Tm  Tc1 the magnetic susceptibility m is very small near Tc1 . However, at T near Tm there is a pronounced increase of m and accordingly an increase of the paramagnetic effect. This means that at temperatures T ∼ Tm the applied field strongly affects the superconductivity. (i) Thermodynamic critical field Hc (T). We illustrate this effect by analyzing the change of the thermodynamical field Hc (T) (for the transition N → MS) in magnetic superconductors. In this case the Gibbs energy density of the paramagnetic normal phase is equal to that of the SC phase, G˜ N (Hc ) = G˜ SC (Hc ) where H2 G˜ SC (Hc ) = F˜ n (0) − c0 , (4.52) 8 G˜ N (Hc ) = Fn (0) −

Hc2 . 8

(4.53)

This gives the critical field Hc (T) = 

Hc0(T) 1 + 4 m (T)

,

(4.54)

184

M. L. Kuli´c and A. I. Buzdin

2 where (Hc0 /8) = N(0)2 /2 is the SC condensation energy and the magnetic permeability is = 1 + 4 m . (Note that we neglect the conduction electron susceptibility e since in MS one has e  m .) In ferromagnetic superconductors for T > Tm one has m (T) ≈ (em Tm0/4)/(T − Tm0 ) and  Hc (T) ∼ T − Tm0 . (4.55)

It is seen from Eq. (4.55) that Hc (T) is drastically reduced near Tm0  Tc , due to divergence of m (T). Very near Tm non-linear effects of the magnetic field start to dominate. (ii) Upper critical field Hc2(T). In the presence of the external field He and at temperatures near and above Tm0 the superconductivity is suppressed by the orbital effect of the field B = Hi (1 + 4 m ) and by the pronounced paramagnetic effect of the exchange field h (since the Zeeman effect of B is much smaller). Here, the internal magnetic field is Hi = He + HD where HD is the demagnetization field. The upper critical field is calculated by the same formula as for non-magnetic SC, see [54], where B is replaced by ˜ B = B + h0M/n Hi and the electron charge e by e˜ = e(1+4M/Hi ).In the clean limit,and for T  Tc1, one obtains the modified Gruenberg– G¨unther formula for Hc2(T) [55] √ f (˛) 2 H ∗ (0) , (4.56) Hc2(T) = 1 + 4 m (T) c2 ˛ ∗ where Hc2 (0) us the upper orbital critical field in absence of magnetic moments. The function f (˛) is

calculated numerically in [55] and the parameter ˛ describes the relative role of the orbital and paramagnetic effect ˛=

∗ (0)h0 m (T) 2Hc2 . (1 + 4 m (T))n 0

(4.57)

In the RE ternary magnetic superconductors one usually has h0  0 and 4M = 4n , which is ∗ one order of magnitude smaller than Hc2 (0). This gives ˛  1 in the region where T  Tc1. It is known that in pure superconductors for ˛ > 1.8 [55] the Larkin–Ovchinikov–Fulde–Ferrell (LOFF) phase (due to paramagnetic effects) is realized [56,57]. In the LOFF state the SC order parameter oscillates spatially, also being zero at some points. For ˛  1 one has f (˛) ≈ 1 and Hc2(T) ≈ 1.5

0 Tm0 (T − Tm0 ) . h0 

(4.58)

It is seen that Hc2(T) depends linearly on T − Tm0 near Tm0 and falls off much faster than Hc (T) by reaching the value Hc2 < Hc near Tm0. This leads to a very interesting effect: at Hc (T) the system goes into the Meissner or vortex state by the first order phase transition,depending on the relation between Hc and Hc1. Various phases can be realized in the H − T phase diagram depending on the demagnetization factors and sample purity as is shown in Fig. 4.12. For instance, the theory predicts the existence of the Larkin–Ovchinikov–Fulde–Ferrell (LOFF) state in clean samples (l > 0 ) with finite demagnetization factors as shown in Fig. 4.12a,c.

Fig. 4.12. (H, T) phase diagram for ErRh4 B4 above TM with demagnetization factors: (a) Nz = 1/3, (b) Nz = 0, (c) Nz = 1. H is along the aaxis. MS: Meissner phase, VS: vortex phase, LOFF: Larkin–Ovchinikov– Fulde–Ferrel phase [8]

4

Coexistence of Singlet Superconductivity and Magnetic Order

Fig. 4.13. Experimental upper critical field Hc2 vs. temperature for c-axis and a-axis directions in ErRh4 B4 . The lower critical field Hc1 along the a-axis is also shown; that for the c-axis is similar [58]

The magnetization and Hc2 measurements in ErRh4 B4 [58] give that the samples were clean and that the experimental curve, shown in Fig. 4.13, resembles strongly the theoretical ones in Fig. 4.12. The conclusion is that the behavior of the critical field Hc2 in ErRh4 B4 uniquely proves the dominant role of the EX mechanism in the destruction of superconductivity near the magnetic transition temperature Tm0 . (iii) Hc2 in polycrystalline FS.Most ferromagnetic superconductors were synthesized as polycrystals and are also characterized by the rather large anisotropy of the magnetic susceptibility near Tm0 , i.e. Mk = kl Hl . For instance in ErRh4 B4 with the easy plane anisotropy one has z / ⊥ ≈ 1/64. Since the effective exchange field is large in these compounds it influences superconductivity strongly. If the size of the crystallites is larger than the superconducting coherence length 0, then Hc2 in each crystallite depends critically on the direction of the applied field H. If the orientations of the crystallites

185

are random, then the decrease of the magnetic field increases the number of crystallites that are in the superconducting state. This causes the decrease of the sample resistance. When the concentration of crystallites, where the superconductivity is set in, reaches the critical concentration Ccr = 0.25 in 3D systems, a continuous superconducting path appears throughout the sample. In this case the resistance of the sample is significantly decreased. Therefore, we are here dealing with a typical percolation phenomenon in ferromagnetic superconductors as it was recognized and studied in [59]. min max In the field Hc2 < H < Hc2 the concentration C(H) of crystallites in the superconducting state is C(H) = 1 − cos 0 (H), where 0 (H) is the boundary angle between the applied field H and the axis of anisotropy up to which superconductivity is preserved. This angle is determined from the condition that H = Hc2(0 ), i.e. the given H is the critical field for the angle 0 . To calculate Hc2 (0 ) one should take into account: (i) the exchange field which acts on electronic spins, hex = ( ⊥ H⊥2 +  H2 )1/2 with = Jsf (gJ − 1)/n B2 and (ii) the orbital field | B = H + 4M |. Since the studied polycrystals of ErRh4 B4 where in the dirty limit Hc2() is then given by    T 1 1 DeH ln + = ( ) − Re Tc 2 2 2T × (1 + 4 ⊥ )2 sin2  + (1 + 4  )2 cos2  

2 2 2 2 H ⊥ sin  +  cos  . (4.59) + i 2T Hc2 was experimentally determined from the condition Rm(Hc2 )/RN = 0.5, where Rm is the measured resistance and RN is the normal resistance.The percolation theory gives the relation between the effective medium resistance Rm and the concentration C(H) of crystallite in the superconducting state by [61] Rm (H) = RN [1 − 3C(H)] .

(4.60)

If one uses the experimental values for ⊥ ≈ 0.087 and ⊥ ≈ 64  and for the coefficient of diffusion D obtained from the slope of Hc2 at Tc1 one obtains very nice agreement for Rm(H) at T = 4.35 K between this

186

M. L. Kuli´c and A. I. Buzdin

terizes the renormalization of the Meissner screening in the presence of the exchange field [8]. Here m (T) = ˜ 2 /(T − Tm0 ) is the magnetic susceptibility. By minimizing G in Eq. (4.61) with respect to A one obtains Hc1 in the standard way Hc1 =

Fig. 4.14. Resistance Rm (H) and upper critical field Hc2 vs. temperature of polycrystalline ErRh4 B4 . Line: theory [59]; (x): experimental results [60]

theory and experiments; see Fig. 4.14. In such a way the determined experimental curve Hc2(T) is very well described by this theory [59] as can be seen in Fig. 4.14. (iv) Lower critical field. Hc1 is very weakly affected by the exchange field. This can be easily demonstrated by adding to Eq. (4.10) the magnetic field term − B· HdV /4. The minimization of the Gibbs free-energy with respect to the relative magnetization S(r) one obtains the Gibbs-energy for the single vortex   2  ¥02 B G = dV pex + 8 32 3 L2 (4.61)   2 & B·H 2 − . × ∇' − A ¥0 4 Here ' is the phase of the SC order parameter, ¥0 the flux quantum and pex = 1 − em (em + ex + ˜ 2 /

2 m (T))−1 is the“screening”parameter that charac-

eff ¥0 , ln 2  4L

(4.62)

√ where eff = L p. Note that in the theory, which neglects the EX interaction, i.e. with ex = 0, one obtains pex → 0 for T → Tm0. This odd result means that the effective penetration depth eff tends to zero and the Ginzburg–Landau (G-L) parameter also goes to zero, i.e.  = (eff /) → 0. If this assumption were realized in the RE ternary superconductors, then at temperatures near Tm0 we would have a change from type-II to type-I superconductivity. However, this cannot be realized in the RE ternary superconductors, since one has ex ∼ em , thus making p finite and the G-L parameter  stays practically unchanged. So, the change of the type of transition near Tm0 is not due to the change of  but to the much faster temperature fall-off of Hc2(T) than of Hc (T) [8]. (v) Surface energy of ferromagnetic superconductors. A very interesting result is obtained for the surface (FS) energy ns of ferromagnetic superconductors (FS) placed in magnetic field at T > Tm. In this case in the normal state the magnetization is M = m H and decays in FS on the distance eff . The surface energy of FS was calculated in [62] and it was obtained that in systems with large Ginzburg–Landau parameter (FS) eff = eff / the surface energy ns (T) changes sign at some temperature Tns > Tm, as shown in Fig. 4.15. The reason for such a behavior lies in the strong temperature dependence of the thermodynamic critical field Hc (T)(< Hc0(T)) given by Eqs. (4.54)–(4.55). (S) The surface energy for non-magnetic( ns ) and fer(FS) romagnetic (ns ) superconductors has the following approximate forms, respectively ( = /) H2 (S) ns (4.63) =  c0 (1 − ) , 8   H2 H2 (FS) (4.64) ns =  c0 1 − eff 2c . 8 Hc0 The first term in Eqs. (4.63) and (4.64) denotes the loss in the condensation energy on the length ,while

4

Coexistence of Singlet Superconductivity and Magnetic Order

187

One can see that the vortex energy Ev(FS) depends on 2  2 and eff while the surface energy is linear in  and eff . This means that the vortex line defect is energetically stable for Tv < T < Tns where Tv is obtained from the condition Ev(FS) (Tv ) = 0.

4.4.2

Fig. 4.15. The temperature dependence (qualitative) of: (a) critical fields Hc2 (T), Hc (T), Hc1 (T) and (b) surface (FS) energy ns [62]

the second term is the gain in the magnetic energy due to the penetration of magnetic field inside the non-magnetic (∼ ) and ferromagnetic superconductors (∼ eff ), respectively. Since Hc (T) < Hc0(T) this leads to an increase of the surface energy in FS with respect to the non-magnetic superconductor.So, below some temperature Tns but near Tm the surface (FS) energy of FS becomes positive, i.e. ns (T < Tns ) > (FS) 0. However, in spite that ns (T < Tns ) > 0 the vortex line is thermodynamically stable object in FS for these temperatures! This is contrary to the case of non-magnetic superconductors where the vortex (S) line is unstable when the surface energy ns > 0.The order of magnitude of the energy (per unit length) of the vortex line defect is given by Ev(S) =  2

Ev(FS)

=

2 Hc0 (1 −  2 ) , 8

2 2 Hc0

8

 1−

2 eff

Antiferromagnetic Superconductors in the Magnetic Field

IAs a rule the appearance of the AF ordering below TN is at the origin of an anomalous temperature dependence of the upper critical Hc2 (T) in antiferromagnetic superconductors.Some typical examples of the Hc2(T) curves measured in the RE ternary compounds Tb1.2Mo6 S8 , Dy1.2 Mo6 S8 , in the RE ternary compounds Tb1.2Mo6 S8 , Dy1.2 Mo6 S8 , have in many respects similar properties. For instance, the increase of Hc2 below TN in SmRh4 B4 (and also in ErNi2 B2 C), presented in Fig. 4.16, may be related to a decrease of the magnetic scattering below TN . On the other hand the decrease of Hc2(T) immediately below TN is particulary pronounced in Dy1.2Mo6S8 ; see Fig.4.16.This effect is due to the internal exchange field hex ∼ m H, which is created by the polarized magnetic atoms and which is proportional to the susceptibility m . The later has maximum at TN .

(4.65) Hc2 2 Hc0

 .

(4.66)

Fig. 4.16. Upper critical field Hc2 (T) in antiferromagnetic superconductors: (a) NdMo6 S8 and (b) SmRh4 B4 [49]

188

4.5

M. L. Kuli´c and A. I. Buzdin

Josephson Effect with Bulk Magnetic Superconductors

Before discussing the coexistence of superconductivity and ferromagnetism in heterostructures we consider a theoretical prediction of the Josephson effect in magnetic superconductors with spiral ordering of magnetic moments; the MS/I/MS junction [16]. Besides a number of potentially very interesting effects the MS/I/MS junction is a paradigm for some novel phenomena in heterogeneous SF superstructures. For instance, recent theoretical proposals [18] on the giant proximity effect in the SFFS hybrid systems is based on the existence of the triplet amplitude which appears in superconductors with non-uniform orientation of magnetization. The triplet amplitude F↑↑ =  ↑ ↑  (and F↓↓ =  ↓ ↓ ) was discovered in 1979 in studying magnetic superconductors with spiral magnetic ordering [10]. Their importance in Josephson junctions with magnetic superconductors was recognized in 2001 [16]. (i) -contact in the MS/I/MS junction. We briefly discuss the tunneling Josephson junction based on magnetic superconductors, where the left-L and right-R bulk magnetic superconductors are with spiral magnetic ordering [16], see Fig. 4.17. The spiral magnetic order is characterized by the wave vector QL,R and the exchange fields hL(R) = hL(R) eiL(R ) , respectively, while the superconductivity in the banks is described by the order parameter L,R = | L,R |

ei'L,R . For simplicity reasons one assumes | L | = |R | = , hL = hR = h, |QL | = |QR | = Q. The spiral wave vectors QL,R = L,R Qˆz are orthogonal to the surface of the tunneling barrier where the spiral helicity L(R) = ±1 for QL,R along (+) or opposite (–) to the zaxis. Note that such a junction is characterized by the standard superconducting phase ' = 'L −'R = 0 and additionally by the magnetic phase  = L − R = 0. The magnetic phase  is a novel property giving rise to very interesting physics. As was mentioned above, besides the singlet amplitude F↑↓(F↓↑ ) the induced (by the non-uniform spiral ordering) triplet pairing amplitudes, F↑↑ and F↓↓ ,play an important role in the Josephson effect [16]. In this case F↑↑ and F↓↓ give rise to an additional, and in some cases dominant, channel for tunneling of superconducting electrons. It turns out that the Josephson current JJ (' , ) contains two contributions t JJ (' , ) = (Jcs − J− cos ) sin ' ,

(4.67)

where the standard term Jcs ∼ 2 is due to the left and † † right singlet amplitudes F↑↓ (kL , !n ), F↑↓ (kR, !n ), t while the “triplet” current J− is due to the triplet amplitudes  † t J− ∼ −T | TkL ,kR |2 {F↑↑k (!n ) L kL ,kR ,!n † † (−!n )]∗ + F↓↓k (!n ) × [F↑↑k R L

(4.68)

† × [F↓↓k (−!n )]∗ } . R

Fig. 4.17. The Josephson junction with the insulating contact. SL and SR are superconductors with spiral magnetic order. The exchange fields h L,R at the surface make angles L,R with the y-axis. Q L,R are along the z-axis [16]

4

Coexistence of Singlet Superconductivity and Magnetic Order

t The calculations give J− ∼ 2 h2 [f1 +( L R )f2(, h)], where the functions f1,2(, h) are calculated in [16]. The Josephson current also depends on the total helicity = L R . It was shown in [16] that in some parameter region the effects of the triplet amplit tude dominate since one has | J− |> Jcs , thus giving rise to the -Josephson junction. If such a junction is placed in a superconducting ring with the sufficiently large inductance L, i.e. with L > Lc , then a spontaneous current flows in the ring producing the half-flux quantum in the hole of the ring [63]. From Eq. (4.67) one can see that by changing the magnetic phase  and helicity one can tune the system from the 0-junction to the -junction. These new degrees of freedom in the junction, the magnetic phase  and chirality, first proposed in [16], open a new possibility for switching elements and quantum computing. From the physical point of view the above model is a paradigm for the S/F0/F /S structures with rotating magnetization. In this case  is the angle between rotating magnetization in neighboring layers [17].

(ii) Charge and spin Josephson currents in MS/I/MS contacts. In ferromagnetic singlet superconductors with rotating magnetization (e.g. as spiral) besides the standard Green’s function G↑↑ (G↓↓ ), F↑↓ (F↓↑ ), the non-diagonal functions in spins G↑↓ (G↓↑ ) and the triplet amplitudes F↑↑ (F↓↓ ) are of immense importance [16,17], since they produce the spin current Jspin through the junction even in absence of voltage Jspin = JG + JF = ˛h2(1 + ˇ2 cos ' ) sin  ,

(4.69)

where JG ∼ h2 sin  ∼



  | T |2 G↑↓,L G↓↑,R − G↓↑,L G↑↓,R

and JF ∼ h2 2 cos ' sin   ∗  † † ∼ | T |2 F↑↑k (! ) F (−! ) n n ↑↑kR L

† † ∗ − F↓↓k (! )[F (−! ) ] . n n ↓↓kR L The part JG ,which is a magnetic analog of the Josephson junction was also discussed thoroughly in [64], while the term JF describes the spin current due to

189

superconducting triplet amplitude. The physics of the MS/I/MS (and also S/F0 /F /S) junction is more transparent if one writes the energy E(' , ) of this combined magnetic and superconducting Josephson junction. Since E(, ' ) must be an even function on ' and  it has the form E(, ' ) = −Ah2 cos  − 2 (B + C h2 cos ) cos ' . (4.70) A(, h), B(, h), C(, h) are a cumbersome expression, which is not given here. Note that both, the spin Jspin (, ' ) ∼ @E/@ and the charge JJ (' , ) ∼ @E/@' Josephson current, depend on ' and . Thus, by tuning  and ' one can tune these currents. For instance, one can obtain the superconducting junction, with spontaneous charge currents in the superconducting ring. Analogously, the magnetic contact can be realized and accordingly the spontaneous spin current flows in the ring from magnetic superconductors. Another interesting aspect of the two-phase Josephson junction might be realized in small Josephson contacts. In such a case the smallness of the charge and “spin” capacitance brings the system to the quantum regime, thus providing a possibility for a novel Josephson qubit. In fact, the latter consists from two qubits, the superconducting and magnetic one,which might be also of a potential interest for applications in hardware for quantum computing [17].

4.6

Superconductor/Ferromagnet Heterostructures

The coexistence of singlet superconductivity with ferromagnetism was not observed in bulk materials, but it may be easily achieved in artificially fabricated layered ferromagnet/superconductors (F/S) systems. Due to the proximity effect,the Cooper pairs can penetrate into the F layer and induce superconductivity there. This provides the unique possibility to study the properties of superconducting electrons under the influence of a large exchange field. Furthermore, it is possible to study the interplay between superconductivity and magnetism in a controlled manner. For instance, by varying the layer thickness and magnetic moment of F layers the relative strength of two

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competing orderings is changed. The behavior of the superconducting condensate under these conditions is quite peculiar. This field is covered in several excellent reviews articles [18,19,21,65]. 4.6.1

The Fulde–Ferrell–Larkin–Ovchinikov State and the Proximity Effect in S/F Structures

A long time ago Fulde and Ferrell [56] and Larkin and Ovchinnikov [57] demonstrated that in a pure superconductor at low temperature the paramagnetic effect leads to a non-uniform superconductivity (r) (this is the so-called FFLO or LOFF state). The modulation of the superconducting order parameter in the FFLO state is related to the Zeeman’s splitting of electronic levels under magnetic field that acts on electron spins. To elucidate this effect qualitatively, we consider the simplest case of the 1D superconductor. In the absence of the field, the Cooper pair is formed by two electrons with opposite momenta +kF and −kF and opposite spins (↑) and (↓), respectively. The resulting momentum of the Cooper pair is zero, kF + (−kF ) = 0. In the magnetic field, because of the Zeeman’s splitting, the Fermi momentum of the electron with spin (↑) will shift from kF to k1 = kF + ıkF , where ıkF = BH/vF and vF is the Fermi velocity. Similarly, the Fermi momentum of an electron with spin (↓) will shift from −kF to k2 = −kF +ıkF , see Fig. 4.18. Then, the resulting momentum of the Cooper pair will be k1 + k2 = 2ıkF = 0, which just implies the space modulation of the superconducting order parameter with the resulting wave-vector 2ıkF . Due to the incompatibility of ferromagnetism and superconductivity it is not easy to verify this prediction on experiment. Moreover the electron scattering on the impurities destroys the FFLO state very quickly and its observation is possible only in the clean limit [66]. It turns out that in a ferromagnet in contact with a superconductor the Cooper pair wave function has damped oscillatory behavior [20], which may be considered in some sense as an analogy with the decaying non-uniform FFLO superconducting state. Indeed, when a superconductor is in a contact with a normal metal the Cooper pairs penetrate across the interface at some distance inside the metal. A Cooper pair in a superconductor comprises

Fig. 4.18. (a) Energy band of 1D superconductor near the Fermi energy in the presence of Zeeman splitting. In the FFLO state electrons (−kF + ıkF , ↓) and (kF + ıkF , ↑) are paired with total momentum ıkF = h/vF ; see the text. (b) The (h, T) phase diagram for the 3D superconductor. At T < T ∗ the second order transition occurs between normal and FFLO state. The bold line is the first order transition from FFLO to uniform superconducting state [21]

two electrons with opposite spins and momenta. In a ferromagnet the up spin electron (with the spin orientation along the exchange field) decreases its energy by h , while the down spin electron increases its energy by the same value. To compensate this energy variation, the up spin electron increases its kinetic energy, while the down spin electron decreases its kinetic energy. In the result, the Cooper pair acquires a center of mass momentum 2ıkF = 2h/vF , which implies the modulation of the order parameter with the period vF /h. The direction of the modulation wave vector must be perpendicular to the interface, because only this orientation is compatible with the uniform order parameter in the supercon-

4

Coexistence of Singlet Superconductivity and Magnetic Order

ductor. This phenomenon, however, is quite general and must be present in both the clean and dirty limits [67]. It results in many interesting effects: the spatial oscillations of the electronic density of states, the non-monotonous dependence of the critical temperature of S/F multilayers and bilayers on the ferromagnet thickness, the realization of the Josephson -junctions in S/F/S systems. The interplay between superconductivity and magnetism in S/F structures occurs at the nanoscopic range of layer thicknesses and the observation of these effects became possible only recently due to the great progress in the preparation of high-quality hybrid S/F systems. Let us consider the question of the proximity effect in a weak ferromagnet (hex ∼ ) in contact with a superconductor (x < 0). To get an idea of the proximity effect in S/F structures, as well as of the FFLO state, we may start from a description based on the Ginzburg–Landau functional, which takes into account the paramagnetic (Zeeman) effect in the magnetic field. As is known the standard Ginzburg– Landau functional F , which describes the behavior of the superconducting order parameter ¦ (r) near Tc , is given by b (4.71) F = a | ¦ |2 + | ∇¦ |2 + | ¦ |4 . 2 The orbital effect of the magnetic field is taken into account by the replacement ∇ → ∇ − (2ie/c)A. Near Tc the coefficients a, > 0, b > 0 do not depend on B(= ∇ × A). For T < Tc and B = 0 one has a < 0 and the uniform superconductivity occurs with | ¦ |2 = −a/b. However, when the paramagnetic effect becomes dominant for B B ∼  this approximation fails, since near the point (B∗ , T ∗ ) the coefficient changes sign. This means that the in the presence of the paramagnetic effect the minimum of F is realized for the oscillating ¦ (r). To describe this situation it is necessary to add higher derivatives in the expansion of Eq. (4.71). This is also confirmed by the microscopic derivation in the presence of the paramagnetic effect [68–70]. It turns out that in such a case the generalized G-L functional is given by [70] F = a(B, T) | ¦ |2 + (B, T) | ∇¦ |2 ×

(4.72)

(B, T) b c | ∇ 2 ¦ |2 + | ¦ |4 + | ¦ |6 , 2 2 3

191

where a(B, T) = a (T −Tcu (B)) and Tcu(B) is the critical temperature into the uniform (¦ = const.) superconducting state. In fact in the FFLO state the coefficient b can be small and even negative and therefore the sixth order term ∼ c must be added. The effect of the orbital motion is accounted for by the replacement ∇ → ∇ −(2ie/c)A; for more details see [69,70]. The critical temperature of the second order phase transition and the non-uniform structure is obtained by solving the linearized equation (4.72) a¦ − ¦ +

 2  ¦ = 0. 2

(4.73)

If we seek the solution in the form ¦ = ¦0 exp(iq · r), then after the minimization of F with respect of q one obtains the modulation vector q0 and the critical temperature of the non-uniform (FFLO) state Tci (> Tcu) q20 = −



a ≡ a (Tci − Tcu) =

2 . 

(4.74)

This means that the FFLO state is obtained simply by changing sign of in the phenomenological G-L equation. The more detailed microscopic theory [69] gives that the order parameter near the point (B∗ , T ∗ ) is not the plane-wave but the one-dimensional sinusoidal structure ¦ ∼ sin q · r. Let us apply this theory to the proximity effect in the S/F metal surface which is perpendicular to the x-axis. It is known that in the case of the S/N (superconductor-normal) interface Cooper pairs can penetrate into the normal metal at some distance (T), which is (D/T)1/2 for diffusive electron motion and vF /T for pure normal metal. In the framework of the standard G-L theory (with > 0) for the S/N ideal contact and by assuming perfect transparency of the interface, one obtains that the superconductivity is induced in the normal metal and decaying exponentially ¦ = ¦0 exp[−x/(T)]; the proximity effect. On the other hand, the leakage of the SC electrons weakens SC near the surface (inverse proximity effect) as can be seen in Fig. 4.19. In the case of the S/F interface being the solution of Eq. (4.73) one obtains that the exponential decay of the induced su-

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M. L. Kuli´c and A. I. Buzdin

perconducting order parameter in the ferromagnet is accompanied by its oscillation ¦ = ¦0 exp(−x/1) cos(x/2) , | | √ ( 1 +  − 1) , 2 | | √ ( 1 +  + 1) , = 2

1−2 = 1−2

(4.75)

where  = (T − Tci )/(Tci − Tcu). We stress that the oscillation of the order parameter in the ferromagnetic metal F is the fundamental difference between the S/F and S/N systems. In real ferromagnets the exchange field may be much larger than the superconducting gap, which causes that the gradients of the superconducting order-parameter variations are also large. The latter cannot be treated by the generalized G-L functional and we need to use a microscopic approach, for instance Bogoliubiv–de Gennes equations or the quasiclassical Eilenberger and Usadel equations; see [21] and references therein. If the electron scattering mean free path l is small (which is usually the case in S/F systems), the most natural approach is to use the Usadel equations for the Green’s functions averaged over the Fermi surface; see Appendix A.2. The linearized Usadel equation for the anomalous Green function Ff in the ferromagnet reads (note in Appendix A.2 we use f for Ff ) 



Df @2 Ff = 0, 2 @2x (4.76) where !n = (2n + 1)T are the Matsubara frequencies, Df = (1/3)vF l is the diffusion coefficient in the ferromagnet and h(≡ hex ). The parameter 1/s describes the magnetic scattering in the F layer. Note that this form of the Usadel equation in the ferromagnet implies a strong magnetic uniaxial anisotropy when the magnetic scattering in the plane (xy) perpendicular to the anisotropy axis is negligible [21].In the F region, we may neglect the Matsubara frequencies compared to the large exchange field (h  Tc ). Also assuming first that the magnetic scattering is weak, we readily obtain the decaying solution for Ff 1 | !n | +ih · sgn(!n ) + s

Ff −

Fig. 4.19. Schematic behavior of the superconducting order parameter ¦ near (a) superconductor-normal and (b) superconductor-ferromagnet interface. The continuity of the order parameter at the interface is assumed [21]

  1+i Ff (x, !n > 0) = A exp − x , (4.77) f  where f = Df /h is the characteristic length of the superconducting correlation decay (with oscillations) in the F-layer. In systems with h >> Tc , this length is much smaller  than the superconducting coherence length s = Ds /h (where Ds is the diffusion constant in the superconductor), i.e. f  s . In a ferromagnet, the role of the Cooper pair wave function is played by ¦ , which decays and oscillates as      x x ¦ ∝ cos . Ff (x, !n ) ∝  exp −   f f !n (4.78) The different behavior of the superconducting order parameter in S/F and S/N systems is illustrated schematically in Fig. 4.19. As was said before,the damping oscillatory behavior of ¦ is the fundamental difference between the proximity effect in S/F and S/N systems, and it is at the origin of many peculiar characteristics of S/F heterostructures. In the absence of magnetic scattering the scale for the oscillation and decay of the Cooper pair wave function in a ferromagnet is the same. If

4

Coexistence of Singlet Superconductivity and Magnetic Order

193

we take into account the magnetic scattering, then the decaying length f 1 becomes smaller than the √ oscillating length f 2 , i.e. f 1 = f /( 1 + ˛ 2 + ˛)1/2 √ and f 2 = f /( 1 + ˛ 2 − ˛)1/2, where the parameter ˛ = 1/s h characterizes the relative strength of the magnetic scattering. The damped oscillatory behavior of the order parameter may lead to the electronic density of states oscillations in a ferromagnet in contact with a superconductor [71]. This prediction has been confirmed experimentally [72], and up to now this remains the only experimental observation of the density of states oscillations in the F layer. The magnetic scattering effect complicates this type of experiments strongly reducing the amplitude of the oscillations.

4.6.2

Oscillatory Superconducting Transition Temperature in S/F Multilayers and Bilayers

The damped oscillatory behavior of the superconducting order parameter in ferromagnets may produce the commensurability effects between the period of the order parameter oscillation (which is of the order of f 2 ) and the thickness of the F layer. This results in the striking non-monotonous superconducting transition temperature dependence on the F layer thickness in S/F multilayers and bilayers. Indeed, for the F layer thickness smaller than f 2 , the pair wave function in the F layer changes a little and the superconducting order parameter in the adjacent S layers must be the same. The phase difference between the superconducting order parameters in the S layers is absent and we call this state the 0-phase. On the other hand, if the F layer thickness becomes of the order of f 2 the pair wave function may go trough zero at the center of F layer providing the state with the opposite sign (or shift of the phase) of the superconducting order parameter in the adjacent S layers; the so-called -phase. The increase of the thickness of the F layers may provoke the subsequent transitions from 0 to -phases, which superposes on the commensurability effect and results in a very special dependence of the critical temperature on the F layer thickness [67,73]. The experimental observation [74] of this unusual dependence in Nb/Gd was the first

Fig. 4.20. Oscillation of Tc of Nb/Gd multilayers vs. thickness of Gd layer dG for (a) dNb = 600 Å and (b) dNb = 500 Å [74]

strong evidence in favor of the -phase appearance; see Fig. 4.20a. For the S/F bilayers, the transitions between 0 and -phases are impossible; nevertheless the commensurability effect between f 2 and F layer thickness also leads to the non-monotonous dependence of Tc on the thickness of the F layer. Processes of the normal quasiparticle reflection at the free F layer bound-

Fig. 4.21. Variation of Tc of Nb/Cu0.43Ni0.57 bilayer vs. F layer thickness [75]. The theoretical fit for h ≈ 130 K [76]

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M. L. Kuli´c and A. I. Buzdin

ary and Andreev reflection at SF-interface interfere and this results in Tc minima being reached when the thickness of the F layer is close to a quarter of the spatial oscillation period. The dependence of Tc of the Nb/Cu0.43 Ni0.57 bilayer on the thickness of the F layer [75] is presented in Fig. 4.21. 4.6.3

Superconductor-Ferromagnet-Superconductor -Junction

The experiments on the critical temperature of the S/F multilayers and bilayers attracted a lot of interest in the proximity effect in S/F systems but their interpretations were controversial due to the very small value of the characteristic length f 2 (only several nanometers). The most direct proof of the -phase observation would be the observation, following the theoretical predictions [20,67], [73] of the vanishing of the critical current at the 0 to -phase transition. The first experimental evidence of the 0 to  transition in S/F/S (Nb-Cux Ni1−x -Nb) Josephson junction was obtained in [77] from the measurements of the temperature dependence of the critical current. The 0 to  transition was signalled by the vanishing of the critical current with the temperature decrease. Such a behavior is observed for a F layer thickness df close to some critical value dfc . In fact, this simply means that the critical thickness dfc slightly depends on the temperature.The temperature variation serves as a fine tuning and permits the study of this transi-

Fig. 4.22. jc vs. the F-layer thickness dF for Nb/Cu0.47 Ni0.53 / Nb junction at T = 4.2 K. Open circles experimental data and dashed line model calculations [80]

tion in detail. The damped oscillations of the critical current as a function of the thickness of the F layer were observed later in Nb/Al/Al2 O3 /PdNi/Nb [78] and Nb/Cu/Ni/Cu/Nb junctions [79]. Very recent experiments ( [80] have enabled the observation of the two-node thickness dependence of the critical current in Josephson SFS junctions with a ferromagnetic interlayer, i.e. both direct transition into the -state and the reverse one from  into the 0-state, as shown in Fig. 4.22. The complete quantitative analysis of the S/F/S junctions is rather complicated, because the ferromagnetic layer may strongly modify superconductivity near the S/F interface. In addition, the boundary transparency and electron mean free path, as well as magnetic scattering, are important parameters affecting the critical current. In the case of small conductivity of the F layer or small interface transparency, the “rigid boundary” conditions [81] are applied and the influence of the ferromagnet on the superconducting order parameter in the electrodes may be neglected. This solution of Eq. (4.76) describes the F(x) behavior near the critical temperature and gives the sinusoidal current-phase dependence Is (' ) = Ic sin(' ), where ' is the phase difference on the junction. The critical current Ic passes through zero and change its sign with the increase of the thickness of the F layer. In the most interesting limit from the practical point of view, when the F layer thickness df > f 1 the universal expression for the Ic (df ) dependence is [21]     df df sin Ic ∝ exp − +ˇ , (4.79) f 1 f 2 where the angle /4 < ˇ < /2 depends on the magnetic scattering amplitude and the boundaries transparencies. In [82] it was pointed out that the -junction incorporated into a superconducting ring (with large inductance L > Lc ) would generate a spontaneous current and a corresponding magnetic flux would be half of the flux quantum ¥0 = hc/2e. The appearance of the spontaneous current is related to the fact that the ground state of the -junction corresponds to the phase difference  and thus this phase difference will

4

Coexistence of Singlet Superconductivity and Magnetic Order

generate a supercurrent in the ring, which short circuits the junction.Naturally,the spontaneous current is generated if there is an odd number of -junctions in the ring. This circumstance has been exploited in an elegant way in [77] to provide unambiguous proof of the -phase transition. The observed half-period shift of the external magnetic field dependence of the transport critical current in triangular S/F/S arrays was observed on the 0– transition occurring with temperature variation. The current-phase relation for a metallic S/NS/ Josephson junction is sinusoidal Is (' ) = I1 sin(' ) only near the critical temperature Tc . At low temperature the higher harmonic terms appear. However in the diffusive limit at df > f 1 they are very small and in the usual junctions their presence is hardly observed. In S/F/S junctions in general in the dirty limit the first harmonic contribution is I1 ∼ exp(−df /f 1 ) and the second harmonic contribution is very small and positive ∼ exp(−2df /f 1 ). The peculiarity of the situation with the 0- transition is that in the transition region the first harmonic term changes its sign passing trough zero and then the role of the second harmonic contribution becomes predominant. To study the scenario of the 0– transition let us address to the general current-phase relation j(' ) = I1 sin(' ) + I2 sin(2' ) ,

(4.80)

which corresponds to the following phase dependent contribution to energy of the Josephson junction EJ (' ) =

 ¥0

−I1 cos(' ) − I2 cos(2' ) . 2

(4.81)

If we neglect the second harmonic term, then the 0 state occurs for I1 > 0. Near the 0– transition I1 → 0 and the second harmonic term becomes important.The critical current at the transition jc =| I2 | and if I2 > 0, the minimum energy always occurs at ' = 0 or ' = . In the opposite case (I2 < 0) the transition from the 0 to the -state is continuous and there is region where the equilibrium phase difference takes any value 0 < '0 < . The characteristics of such a “' -junction” are very peculiar but at the moment there is no experimental evidence of its observation.

195

F/S/F Spin-Valve Sandwiches The strong proximity effect in superconductormetallic ferromagnet structures may lead to the spinorientation-dependent superconductivity in F/S/F spin-valve sandwiches. A long time ago de Gennes [83] considered theoretically the system consisting of a thin superconducting layer between two ferromagnetic insulators. He argued that the parallel orientation of the magnetic moments is more harmful for superconductivity because of the presence of the non-zero averaged exchange field acting on the surface of the superconductor. This prediction has been confirmed experimentally in an In film sandwiched between two Fe3 O4 films [84] and on an In film between oxidized FeNi and Ni layers [85]. A similar effect has been predicted for metallic F/S/F sandwiches [86, 87] and observed on experiment in CuNi/Nb/CuNi [88] and Ni/NbNi [89]. The spin-valve effect was proposed also for semiconducting F/S/F sandwiches in [90], which could be also used to determine the spin structure of Cooper pairs in triplet superconductors. In the diffusive regime the proximity effect in S/F structures with a spatial rotation of the magnetization may be rather special [18]. As was shown in [10] spiral rotation of spins (magnetization) in superconductors generates the triplet component of the anomalous Green’s functions F↑↑ =< ¦↑ ¦↑ > and F↓↓ =< ¦↓ ¦↓ >. The importance of the triplet amplitudes in the Josephson junction based on magnetic superconductors was pointed out in [16]. In these systems F↑↑ and F↓↓ were responsible for the -contact, for the spin current and the dependence of the current on the spiral helicity. In a metallic S/F structure with rotating magnetization F↑↑ and F↓↓ penetrate into the ferromagnet at distances much larger than f . However, it is not the triplet superconductivity itself because the corresponding triplet order parameter is zero in these systems, unlike the superfluidity in He3 , for example.An important finding [18] was that the decay length-scale of triplet component is insensitive to the exchange field and can generate the triplet long-range proximity effect  (with f ,↑↑ ∼ vF /T  f ,↑↓ ∼ vF /h, Df /h. The special long range triplet proximity effect was pre-

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dicted to exist in the dirty limit. In the clean limit it disappears [91], the spin-orbit and magnetic scattering also destroy this effect. Up to now there have been no reports on its experimental observation. The proximity effect is related to the passing of electrons across the superconductor/ferromagnet interface.In addition to this effect,if the magnetic field, created by the ferromagnet, penetrates into the superconductor, it switches on the orbital interaction between the superconductivity and ferromagnetism. This situation is naturally realized in the case when the ferromagnet is an insulator or the buffer oxide layer separates the superconductor and the ferromagnet. In such a case the ferromagnet plays the role of an additional source of the local magnetic field. In particular the nucleation of superconductivity in the presence of a domain structure occurs near the domain walls [92]. Hybrid S/F systems have been intensively studied in connection with the problem of the controlled flux pinning. Enhancement of the critical current has been observed experimentally for superconducting films with arrays of sub-micron magnetic dots and antidots. There is an excellent review on these questions related to the electromagnetic interaction of superconductivity and magnetic dots [65]. During the last five years an enormous progress in the controllable fabrication of the superconductorferromagnet heterostructures has been achieved. The peculiar effects predicted theoretically were observed experimentally resulting in the qualitative and semiquantitative understanding of the mechanism of the superconductivity and ferromagnetism interplay in S/F systems.Now,this domain of research enters into a period when the design of the new types of the devices becomes feasible and we may expect a lot of interesting finding in the near future.

4.7

Conclusion

The rare earth ternary compounds are rich physical systems that allow the study of the coexistence of singlet superconductivity and various magnetic orders, such as ferromagnetic, antiferromagnetic, weak ferromagnetism. It turns out that in bulk materials

superconductivity and ferromagnetism practically never coexist.However,the singlet superconductivity modifies the ferromagnetic order into a spiral or domain structure (depending on magnetic anisotropy), thus allowing their coexistence. This is realized not only in rare earth ternary compounds but also in AuIn2, where a modified nuclear ferromagnetism, spiral or domain-like structure, coexists with singlet superconductivity. Although the antiferromagnetic order and superconductivity coexist much easier,these systems show a peculiar behavior in the presence of non-magnetic impurities, which surprisingly act as pair-breakers. In the case when the antiferromagnetic order is accompanied by the weak ferromagnetism new coexistence phases appear; the Meissner and spontaneous vortex states. Magnetic superconductors show peculiar behavior in the magnetic field.Near the magnetic critical temperature the upper critical field tends to zero faster than the thermodynamical field, thus giving rise to the first order transition. Various phases are possible in the H − T diagram depending on the purity and demagnetization effects of real samples. The lower critical field is weakly affected by the exchange field (which is due to localized moments). A specificity of ferromagnetic superconductors is that the surface energy can be positive but the vortex state is still thermodynamically stable. The Josephson junctions based on bulk ferromagnetic superconductors with spiral order are characterized by the superconducting and magnetic phase. This opens possibilities for a new kind of coupled qubits in a single Josephson junction. The triplet pairing amplitude, arising in systems with rotating magnetization, gives rise to the -Josephson junction and spin current, which can be tuned by changing the magnetic phase and chirality. The coexistence of singlet superconductivity with ferromagnetism may be easily achieved in artificially fabricated layered ferromagnet/superconductors systems.Due to the proximity effect,the Cooper pairs can penetrate into the F layer and induce superconductivity there. The peculiarity is that in the ferromagnetic metal the induced superconducting amplitude oscillates spatially giving rise to a number of novel effects (not existing in S/N systems) such as

4

Coexistence of Singlet Superconductivity and Magnetic Order

-contact, oscillation of Tc with the thickness of the F-layer, etc. This provides the unique possibility to study the properties of superconducting electrons under the influence of a huge exchange field acting on the electron spins. Acknowledgments We would like to thank L.N. Bulaevskii for fruitful collaboration in formulating and solving a number of problems in the physics of magnetic superconductors. We are thankful to Vitalii Lazarevich Ginzburg and Aleksei Alekseevich Abrikosov for a number of inspiring and useful discussions as well for support. M.L.K. thanks Karl Bennemann for support and fruitful collaboration in solving some interesting problems in the field, and Ivan Boˇzovi´c for support. This work was supported in part by DOE (Contract No. DE-AC02-98CH10886) Brookhaven National Laboratory, Department of Condensed Matter Physics and Material Science.

197

where the pairing interaction V (r i − r j ) is responsible for the singlet pairing in absence of magnetic order. Note that in the presence of any (in)commensurate magnetic order  should be nonuniform, i.e. (r 1 , r 2 ) = (r 1 − r 2 , r), r = (r i + r j )/2. However, it was shown in [10] that when h  vF Q, which is realized in the rare earth ternary superconductors, the non-uniform part of  is small and of the order (h/vF Q). This is the reason for our assumption ∗ (r i , r j ) = ∗ (r i − r j ). In the absence of magnetic field (A = 0) the set of equations for the Green’s functions is given by   i!n − ˆ0 (pˆ ) − Vˆ ex (r) Gˆ !n (r, r  )  ˆ x)Fˆ !† (x, r  ) = ı(r − r  ) , (4.83) + d3 x (r, n   tr i!n + ˆ0 (pˆ ) + Vˆ ex (r) Fˆ !† n (r, r  )  − d3 x ˆ ∗ (r − x)Gˆ !n (x, r  ) = 0 , (4.84) where

Appendix

ˆ = 

The Hamiltonian of magnetic superconductors is given by Eqs. (4.2)–(4.3). The coexistence of SC and magnetic order with a wave vector Q can be studied in principle by using Gorkov equations for any Q. However, in systems where the atomic length a  Q−1 is fulfilled the quasiclassical Eilenberger– Larkin–Ovchinikov (ELO) equations are more suitable and efficient. We first start with Gorkov’s equations and then introduce the ELO method. A.1 Gor’kov Equations for Magnetic Superconductors These equations contain normal and anomalous ˆ ˛ˇ (x, y) = − < Tˆ ˛ (x) † (y) > Green’s functions (G) ˇ and (Fˆ †)˛ˇ (x, y) =< Tˆ † (x) †(y) >, where x ≡ ˛

ˇ

(r, ) and ˛, ˇ =↑, ↓. The superconducting order parameter  is defined by 1 V (r i − r j )[< ↑† (r i ) 2 − < ↓† (r i ) ↑† (r j ) >] ,

∗ (r i − r j ) = −

† ↓ (r j )>

(4.82)



 0 1 , −1 0

Vˆ ex =



 hz (r) h⊥ e−i(Qr+) . (4.85) h⊥ ei(Qr+) −hz (r)

In the case of the spiral magnetic ordering (with spatial rotation of magnetization) besides the sin† glet pairing amplitude F↓↑ ∼  there is also a triplet † † pairing amplitude F↑↑ ∼  · h∗ (and F↓↓ ), which is due to the rotating magnetization. The latter mixes † spin up and down, [10]. F↓↑ is responsible for singlet pairing in the rare earth magnetic superconductors giving rise to the order parameter ↑↓ (r), while the † triplet amplitude F↑↑ does not give rise to the triplet pairing since it is assumed from the very beginning that ↑↑ = ↓↓ = 0. A.2 Eilenberger–Larkin–Ovchinikov Equations for Magnetic Superconductors In problems where the exchange field h(r) = h(r)ez changes slowly on the atomic length a generalization of the standard quasiclassical ELO equations is possible [8, 34, 93]. In this case the Gor’kov equaˆ 1 , x2 ) = tions contain 4 × 4 Green’s functions G(x −Tˆ ¦ˆ (x1)¦ˆ † (x2 ) with the four-component spinor

198

M. L. Kuli´c and A. I. Buzdin

¦ˆ † (x1 ) = ( ˆ↑† ˆ↓† ↑ ↓ ). Here ˆi are Nambu matrices in the electron-hole space and ˆ i are Pauli matrices in the spin-space and x ≡ (r, ). The SC order parameˆ ter is (r) = iˆ+ ˆ 2(r) − iˆ− ˆ 2 ∗ (r) with

ˆ (r) = iVTr ˆ+ ˆ 2 G(x, x) /2 . (4.86) By integrating out unimportant degrees of freedom, i.e. changes at small distances, one defines the quasiclassical Green’s function gˆpF ,!n (R, )   gˆpF ,!n (R) = dp d(r1 − r2) (4.87) ˆ 1 , r2, !n ). × e−ip((r1 −r2 ) ˆ3G(r In the presence of non-magnetic impurities the ELO equations for the quasiclassical Green’s functions read [34]

ˆ ivF ∇R gˆ!n = i!n ˆ3 + ˆ3 (R) + h(R)ˆ 3  (4.88) imp − ˆ3 £pF ,!n (R), gˆ !n , in the Born apwhere the impurity self-energy proximation is given by 9 : £imp ˆ pF −pF gˆpF ,!n (R)ˆupF −pF p . pF !,n (R) = ci N(0) u F (4.89) If the impurity potential (matrix) contains both the non-magnetic and spin-orbit scattering then (4.90)

with the vector matrix ˛ˆ = [(1 + ˆ3 ) + (1 − ˆ3 ) 2  2 ]/2 and pˆ F = pF / | pF |. The matrix gˆpF !n (R) is given by [8,34,93] ⎛

g+ 0 ⎜ 0 g− ⎜ gˆpF ,!n (R) = −i ⎝ 0 −if−† −if+† 0

0 if− −g− 0

⎞ if+ o ⎟ ⎟. 0 ⎠ −g+

1 [! ˜ n,± + ievF · A(R) ± ih(R) − vF · ∇R ]f±pF (R, !n) 2 (4.92) = ˜ ±pF (R, !n )g±pF (R, !n) , 1 † [! ˜ n,± + ievF · A(R) ± ih(R) + vF · ∇R ]f±p (R, !n) F 2 ∗ (4.93) = ±pF (R,!n )g±pF (R, !n) . The normalization condition and the self-consistency equation are given by g±2 + f±† f± = 1 , (pF , R) =

(4.94)

T  V (pF , pF ) (4.95) 2  !n ,pF = > × f+ (R, pF , !n ) + f− (R, pF , !n ) .

! ˜ n,± and ˜ ± (pF , R, !n) are defined by

imp £!n

uˆ pF −pF = uˆ3 + iuso [ˆpF × pˆ F ]˛ˆ ,

By including also orbital effects of the magnetic field B(R) =∇ × A(R) the ELO equations for the components g± and f± read

(4.91)

1 g±pF (R, !n )pF (4.96) 2 : 3 9 + g∓pF (R, !n) sin2( −   ) p , F 2so

! ˜ n,± = !n +

˜ ±pF (R, !n ) = pF (R) + +

1 f±pF (R, !n )pF 2

(4.97)

: 1 9 f∓pF (R, !n ) sin2 ( −   ) p , F so

where 1/ = ci N(0) | u |2 and 1/so = ci N(0) | uso |2 . The microscopic theory of magnetic superconductors, which takes into account spin (exchange) and orbital (electromagnetic) effects of magnetic order, as well as of non-magnetic impurity scattering, has been developed by using the above generalized ELO equations [8,34].

4

Coexistence of Singlet Superconductivity and Magnetic Order

199

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5

Theory of Superconducting Alloys

L. P. Gor’kov

Florida State University, National High Magnetic Field Laboratory, Tallahassee, USA

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2 Averages of Green Functions over Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.2.1 “Cross-Technique” in Normal State Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202 5.2.2 Generalization to Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.2.3 Properties of Homogeneous Superconducting Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.3 Superconducting Alloys with a Small Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.3.1 The Superconducting Transition Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.3.2 Ordinary Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 5.3.3 Spin-Orbit Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 5.3.4 Paramagnetic Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.3.5 Impurities and Crystalline Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 5.3.6 Ginzburg–Landau Equations for Superconducting Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.3.7 Upper Critical Field for “Dirty Alloys” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.4 Paramagnetic Alloys and Gapless Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 5.5 Eilenberger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218 5.5.1 Quasi-Classical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.5.2 Derivation of the Eilenberger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.5.3 “Dirty” Alloys: Usadel Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.6 Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5.1

Introduction

Various lattice defects (dissolved foreign atoms, vacancies, atomic substitutions, etc.) in normal metals determine the so-called residual resistance. The latter is characterized by the corresponding mean free path, , and for many physical phenomena studied in the physics of normal metals the longer the mean free path is, the more pronounced are most of the physical effects. For instance, to observe galvanomagnetic phenomena in normal metals very pure and perfect 1

crystal samples are necessary. In other words, defects are important for studying normal phase properties, but their role is often a negative one.The role of static defects in superconductors can be quite diverse. On the one hand, electron scattering by randomly positioned defects breaks down the spatial correlation of electrons forming a Cooper pair. The characteristic length scale for the latter is 0  vF /2Tc 1 . Thus one expects that the role of defects becomes important for  ≤ 0 . On the other hand, such scattering does not always have a destructive character and

Mostly we use units with  = 1, c = 1 ( is Planck constant, c is speed of light). However, in some cases, especially when it is desirable to emphasize the gauge invariance of a result obtained, c and sometimes  appear explicitly in formulae.

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L. P. Gor’kov

may result in important changes of superconducting characteristics and even in the appearance of qualitatively new phenomena which may not be present in the pure superconductor. Forexample, an increase of the critical magnetic fields may occur in alloys. Also one may have the so-called “gapless” state in superconductors with paramagnetic impurities.Therefore, the theory of superconductors with impurities, called “superconducting alloys”, or “dirty superconductors”, constitutes an essential part of our understanding of the phenomenon of superconductivity. The theoretical methods and results of this chapter were mainly developed for superconductors of the “old generation”, the so-called “ordinary, or classical superconductors”. In other words, they utilize ideas and approaches of the microscopic weak coupling BCS theory (for the phonon mechanism, this can be extended to the strong coupling regime by making use of the Migdal–Eliashberg approximation). Since the discovery of high temperature superconductivity in cuprates by Bednorz and M¨uller in 1986, other new superconductors have been found some of which may also not belong to the same class as the “oldfashioned” superconductors. Undoubtedly cuprates are strongly correlated electronic systems; although there is no consensus yet regarding a theory explaining their superconducting properties. In spite of this, the treatment of defects in such superconductors remains based on the use of the classical techniques. While various mechanisms of superconductivity in the new materials are still being debated, the basic idea of forming the Cooper pair condensate apparently remains correct in most of the recently discovered superconductors. For this reason many theoretical predictions and results known for ordinary superconductors continue to be of interest for studying the physics of High-Tc and other new materials. This chapter is not intended to provide an exhaustive coverage of all the results accumulated since the publication of the microscopic BCS-theory in 1957. The disorder inevitably present in the doped HighTc cuprates, has resulted in an enormous amount of theoretical activity on the role played by defects in anisotropic superconductivity such as the d-wave superconducting order parameter now commonly accepted for cuprates. In what follows we describe

the basic theoretical approaches and methods developed so far for treating defects. The classical theories have a well defined range of applicability within the framework of the old microscopic theory. Whether it is still true for superconductivity in cuprates or in heavy-fermion/mixed valence materials is an open question. Nevertheless, it should be mentioned once again that, so far, there is no alternative for the old tools. Below we begin with a discussion of the diagrammatic “cross” technique. This is the method by which most of the qualitative theoretical results were first obtained. We then introduce the more powerful and efficient approach of the so-called quasi-classical Green functions followed by a summary of the principal results.

5.2

Averages of Green Functions Over Impurities

5.2.1

``Cross-Technique´´ in Normal State Metals

In the case of normal metals one is mainly interested in the role the impurities play for kinetic (transport) properties. In the framework of the Landau Fermi Liquid (FL) theory the familiar transport equations for free quasi-particles provide an adequate mathematical description. The applicability of the latter rests on the quasi-classical character of electronic motion in metals for pF   1, where pF is a scale for the Fermi momentum and  is the electronic mean free path. In the superconducting phase excitations have a more complex character. The Cooper channel pairing introduces a new energy scale, Tc , the temperature of the phase transition, or equilivantly the energy gap, (0), and the spatial coherence length, 0 = vF /2Tc . Therefore, to account for the existence of these new scales different approaches become necessary. Diagrammatic methods turn out to be the most convenient tool to include scattering by impurities into the theory of superconductivity. To begin with, let us first consider the so-called “crosstechnique” for static defects in normal metals (A.A. Abrikosov and L.P. Gor’kov, 1958 [1]; S.F. Edwards, 1958 [2]. It goes without saying that both methods are completely equivalent). The diagrammatic tech-

5

nique reproduces all the well-known results obtained by the standard kinetic equation approach. Let U (r−ra ) be the potential energy for an electron in the field of a defect positioned at the point ra . The Hamiltonian of electrons interacting with impurity atoms in the second-quantized form is  ˆ Hint = U (r − ra ) ˆ˛† (r) ˆ˛ (r)d3 r . (5.1) a

The summation in Eq. (5.1) runs over all defects. For the positions ra of the defects we assume the random (Poisson) distribution, i.e. there are no correlations between different impurity positions. In what follows we employ the diagrammatic methods associated with the thermodynamic Green function    G˛ˇ (x, x ) = − Tˆ ˆ˛ (x), ˆˇ†(x  ) (5.2) expressed in terms of the field operators, ˆ˛ (x), ˆˇ† (x ), written in the so-called Matsubara representation:     ˆ ˆ˛ (r) exp −Hˆ  , ˆ˛ (x) = exp H     ˆ ˆ ˆˇ† (x) = exp H ˆˇ† (r) exp −H , (5.3) where (x = r, ) and , the “imaginary time”, varies in the interval  ∈ (0, 1/T). Notice that the ˆ (x), ˆ † (x)-operators are no longer Hermitian conjugates. However, the notation Eq. (5.3) causes no misunderstanding below.The symbol Tˆ implies the chronological ordering of operators with respect to (,   ),while . . . denotes the thermodynamic (Gibbs) average. ˆ is routinely sub-divided into The Hamiltonian, H, the free-electron term, Hˆ 0 − N with and N standing for the chemical potential and the total number of particles, respectively, and an interaction Hamiltonian Hˆ int . We have ˆ 0 − N + Hˆ int . Hˆ = H ˆ int is given by Eq. (5.1). In our case H

(5.4)

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The diagrammatic methods, including the thermodynamic technique, are described in a number of good textbooks [3,4]). It is assumed that the reader is familiar with them. In this section, we recount only some details relevant for the impurity problem. The Green function Eq. (5.2) in the momentum and frequency representation is taken in the form   −3 G˛ˇ (p, p; !n ) (5.5) G˛ˇ (x, x ) = (2) T !n ip·r−ip ·r  −i!n  3

×e

e

d pd3 p

with !n = (2n + 1)T,  ≡  −   . G˛ˇ (x, x ) depends both on r and r in the presence of impurities. The methods developed provide recipes for calculating the physical characteristics averaged over random positions of impurities. The Green function G˛ˇ (p, p; !n ) can be formally expanded in powers ˆ int Eq. (5.1), as shown in Fig. 5.1. Each cross of H in Fig. 5.1 symbolizes the Fourier component of the impurities potential Eq. (5.1). The diagrammatic expansion in Fig. 5.1 can be rewritten as an integral equation for G˛ˇ (p, p; !n ) = ı˛ˇ G(p, p; !n ): G(p, p; !n ) = ı(p − p)G 0 (p, !n ) +

 ×

U (p − p )ei(p−p



)ra

1  0 G (p; !n ) (2)3 a

G(p , p; !n )d3 p

(5.6)

with G 0 (p, !n ) = (i!n − )−1

(5.7)

( = vF (p − pF )). We consider here the simplest case of scattering potentials without spin dependence. To average the Green function G(p, p; !n ) = G(p; !n )ı(p − p ) ,

(5.8)

consider a few diagrams in Fig. 5.1. For the single cross in Fig. 5.1 the matrix element is

Fig. 5.1. Formal expansion of the Green function in powers of the impurity potential

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1   U (p − p )ei(p−p )ra . 3 (2) a Since the impurity atoms are distributed randomly, one has to average any expression over the positions of each impurity site. We assume that the sizes of defects are of atomic dimensions and that the concentration of defects is low enough to allow averaging to be carried out over distances much greater than inter-atomic distances. In metals the characteristic values of momenta in Eqs. (5.6)–(5.8) are given by the Fermi momentum, i.e. electronic wavelengths are of the order of an atomic distance. Averaging gives then the constant nU (0) , where n is the impurity concentration. This term can be interpreted as a correction to the chemical potential. The last diagram in Fig. 5.1 contains two matrix elements of the same form with the sum over (ra, rb ). If a = b, the same averaging procedure over ra and rb results in the product (nU (0))2. However, for a = b the resulting average is  d3 p n |U (p − p)|2 G 0 (p, !n ) = (2)3  dd§p nmpF . (5.9) |U (p − p)|2 3 (2) i!n − 

Convergence of the integration over the energy variable, , is of course guaranteed by the momentum dependence of U (p − p ). However, the contribution to the integral which comes from an integration over the values of momenta far from the Fermi surface gives a constant. This, like the expression nU (0), can be included in the definition of the chemical potential. A nontrivial (imaginary) contribution arises only in the vicinity of the Fermi surface in spite of a formally divergent term of the integral in Eq. (5.9). Thus, one gets   1 −isign !n , 2 where

1 nmpF =  (2)2

 |U ()|2d§

is the inverse collision time resulting for the impurity potential in the Born approximation. Diagrams with more crosses do not produce any structurally new terms. Besides the contributions which come from different impurities for each“cross” involving the matrix elements, and all possible pairs. There are of course higher order corrections to the Born scattering amplitudes. Also interference terms occur. The latter ones represent contributions due to scattering by two or more different defects. As an example, consider a few of the fourth order diagrams

Fig. 5.2. Different contributions to the “self-energy” parts resulting from an averaging of the fourth order terms in Fig. 5.1

5

shown in Fig. 5.2. The dashed lines connecting different crosses denote possible ways of selecting pairs of atoms in the product of the four matrix elements. Thus, Fig. 5.2(a,b) give nothing but corrections to the Born expressions. The contribution to the matrix element of the form shown in Fig. 5.2(c) is proportional to n2 and describes mixed (interference) scattering processes by two different (a = b) defects. Diagrammatically the latter can also be viewed as a “vertex” correction to the “self-energy” part of Fig. 5.3. Although Fig. 5.2(c) describes the matrix element which contains three G 0 lines integrated over two independent momenta q1 , q2 , it can be neglected because the denominators of the three internal Green functions are only small when the values of all momenta lie close to the Fermi surface. This may happen only inside a narrow fraction of the phase space. At the same time contributions such as shown in Fig. 5.2(d,e) formally provide an expansion in powers of 1/T and must be accounted for in the averaged Green function for T ∼ 1/. A summation over all “non-intersecting” diagrams is given by the following nonlinear equation for the averaged function G(p; !n ): n G(p; !n ) = G 0 (p; !n ) + G 0 (p; !n ) (2)3    2  3  × |U (p − p )| G(p ; !n )d p × G(p; !n ) .

(5.10)

The same arguments as we have already used for the evaluation of the “self-energy” allow one to write down for G(p; !n ) the solution G(p; !n ) = [i!n −  + isign!n /2] . −1

It is easy to verify that for instance the “self-energy” part shown in Fig. 5.2(d) is in our approximation identically zero. Equation (5.10) in its diagrammatic form is shown in Fig. 5.4. The dotted line connecting

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205

Fig. 5.3. “Self-energy” part in the Dyson equation for the average electronic Green function

two crosses and carrying the momentum q corresponds to the factor n|U (q)|2 in Eq. (5.10). Having developed the above diagrammatic technique for calculating the averaged Green functions, one may apply it to other correlators as well. We will not describe here its applications to physical phenomena in normal metals. As was emphasized from the very beginning the technique is applicable for quasi-classical electrons (i.e. pF   1), but it does not produce any results which could not be obtained by the standard methods.

5.2.2

Generalization to Superconductors

The diagrammatic cross technique can easily be extended to superconductors. As discussed in Chapter 2, the superconducting phase is characterized by the normal, G(x, x ) and the two anomalous Gor’kov Green functions (L. P. Gor’kov, 1958 [5])    F˛ˇ (x, x ) = Tˆ ˆ˛ (x) ˆˇ (x ) ,    † F˛ˇ (x, x ) = Tˆ ˆ˛† (x) ˆˇ† (x ) .

(5.11)

In the absence of impurities (and any external fields) we will assume that all the functions depend only on the coordinate differences (x − x  ) and can be expressed in the momentum and frequency representation in the standard manner. Scattering by static defects breaks only the spatial homogeneity. For the Green functions we again write

Fig. 5.4. Diagrammatic illustration of the Dyson equation (5.10) for the average electronic Green function in the normal phase

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L. P. Gor’kov

G˛ˇ (x, x ) = T



G˛ˇ (p, p; !n )

(5.12)

g(r − r  ) = gı(r − r )

(5.16)

!n 





× eip·r−ip ·r e−i!n (− )

d3 pd3p (2)3

(and similarly for the two anomalous functions). If we neglect the effects of the spin-orbit interactions and assume non-magnetic defects, the spin indices can be omitted; the ordinary Green function is then diagonal in (˛, ˇ), G˛ˇ ∝ Gı˛ˇ , while the functions † F˛ˇ ,F˛ˇ are proportional to the spinor metric tensor Iˆ˛ˇ . Thus, F˛ˇ = −F Iˆ˛ˇ ;

† F˛ˇ = F † Iˆ˛ˇ ,

where Iˆ˛ˇ ≡ i(y )˛ˇ =



0 1 −1 0

(5.13)

 ˛ˇ

is nothing but the antisymmetric (singlet) spin wave function of the two electrons forming the Cooper pair (if not said otherwise, we assume s-wave pairing below). Let us consider again a random, homogeneous distribution of impurities. The full equations for G(r, r; !n ) and F (r, r; !n ) are of the form 1 2  ˆ0 + − i!n − H U (r − ra ) G(r, r ; !n ) a

+ (r)F † (r, r; !n ) = ı(r − r ) , 1 2  −i!n − Hˆ 0 + − U (r − ra ) F † (r, r; !n ) a

− ∗ (r)G(r, r ; !n ) = 0.

(5.14)

Here, Hˆ 0 is the free electron operator. As we plan to comment below on the role of crystalline anisotropy, Hˆ 0 is left unspecified. In the weak coupling limit one may write (r) and ∗ (r) as   g(r − r )F (r, r; !n )d3 r , (r) = T ∗ (r) =

!n

 T



g(r − r )F † (r, r; !n )d3 r . (5.15)

!n 

Here, g(r − r ) is a short-ranged interaction. For the isotropic model

(if crystalline anisotropy is not specifically mentioned, we always assume g(r − r  ) has the “contact” form, Eq. (5.16)). Equations (5.14) must be supplemented by equations determining F (r, r; !n ). For completeness we shall later write down explicitly the whole set of Gor’kov equations. However, at least in the absence of an external magnetic field, both anomalous functions can be expressed in terms of each other using their definitions and their symmetry properties. Before proceeding to discuss the averaging of the Green functions G(r, r; !n ), F (r, r; !n ) and F † (r, r; !n ) by means of the cross-technique, it is important to discuss whether one can use the averaged values ¯ and ¯ ∗ for (r) and ∗ (r). When an impurity is introduced into a superconductor, (r) is expected to experience local changes. This change of course depends on the strength of the impurity potential. However, local variations of (r) rapidly decrease away from an impurity center. Therefore, we anticipate that the order parameter, (r), is a selfaveraging quantity, at least at low enough impurity concentrations. It will be seen from the results below that the assumption (r)  ¯ indeed turns out to be valid. Taking again the impurity Hamiltonian Eq. (5.1) as the perturbation for the field operators Eq. (5.3), or  merely expanding the equations in terms of a U (r − ra ), one obtains an expansion for each of the two Green functions which is formally similar to the one shown in Fig. 5.1. The major difference is that for the Green functions G 0 (r − r ; !n ) connecting the crosses in Fig. 5.1 can now alternate with the bare functions F 0 (r − r; !n ) and F †0(r − r ; !n ) of the pure superconductor. The above line of argument (including the neglect of crossing diagrams such as in Fig. 5.2(c)) can be repeated in writing the equations for the averaged G(p; !n ), F (p; !n) and F † (p; !n ). One then obtains the two equations shown schematically in Fig. 5.5(b), where double lines stand for the averaged (exact) functions G, F and F † (Fig. 5.5(a)). Writing them in a more convenient and compact form with the help of the Gor’kov equations in the absence of impurities leads to

5

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207

Fig. 5.5. (a) Directions of arrows identify the ordinary and anomalous Green functions; (b) equation for all three functions generalizing the Dyson equation in Fig. 5.4 for the superconducting phase

  ¯ n) G i!n −  − G(!   +  + F¯ (!n ) F †  †  ¯ i!n +  + G(−! n) F   + ∗ + F¯ † (!n ) G

(p; !n ) (p; !n ) = 1 ,

!˜n

2

i! ˜n + , ˜ n )|2 +  2 + |(!

˜ n) (! , F (p; !n ) = − 2 ˜ n )|2 !˜n +  2 + |(!

(p; !n )

(5.19)

†

(p; !n ) = 0 , (5.17)

with the notations

where

 n |U (p − p )|2G(p ; !n )d3 p , (2)3  n † ¯ |U (p − p )|2F † (p; !n )d3 p . F (!n ) = (2)3 (5.18) (and similarly for F (!n )). ¯ n) = G(!

5.2.3

G(p; !n ) = −

Properties of Homogeneous Superconducting Alloys

In the absence of external electromagnetic fields and currents the order parameter  can be chosen to be real, F (p; !n ) = F † (p; !n ), and the averages for the isotropic model do not depend on p. The solution of the system of equations (5.17) is then given by

¯ n) ; i! ˜ n = i!n − G(!

˜ n ) =  + F¯ † (!n ) , (!

¯ ¯ and where we used the property G(−! n ) ≡ −G(!n ). ¯ n) There are again contributions present in G(! which are due to integration over  far from the Fermi surface (the latter is defined by  = 0). These contributions do not depend on whether the system is in the normal or superconducting state and as before will be included as corrections to the chemical potential.With this in mind we integrate over  in Eq.(5.18) in the symmetric interval  ∈ (−L, L) around  = 0. This gives ¯ n )/i!n = F¯ † (!n )/ −G(! and

(5.20)

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L. P. Gor’kov

i! ˜ n = i!n !n , with !n given by: !n = 1 +

!n 2

˜ n ) = !n , (!



d ˜ n )|2  2 + (! ˜ n )2 + |(!

.

Integration over  gives !n = 1 +

1 1  2 2 !n + 2

.

(5.21)

The expression for the function F † (r − r ; !n ) in the coordinate representation (|r − r| = R) can be written in terms of the above as i (5.22) (2)2 R  ipR   2 e − e−ipR !  n × pdp 2  2 2  + !n + ||2 ! n

F † (R; !n ) = −

(after integration over the variables). Rewriting p as p ≡ pF + /vF and integrating over  we easily obtain   cos pF R m F † (R; !n ) =  2R !n2 + ||2   R × exp − !n !n2 + ||2 . vF From the expression for !n it follows that the only change in the spatial variation of F † (R; !n ) (and of the Green function G(R; !n )) is the appearance of the multiplicative factor exp(−R/2)

(5.23)

(where  = vF  is the mean free path). In particular, it follows from the isotropic model for the average gap that ¯ (r) ≡ 0 , (5.24) i.e., the gap remains the same as for a superconductor without defects or impurities. Since the thermodynamics of a superconductor depends only on the value of the gap, one has the important result that the thermodynamics of homogeneous superconductors including the temperature of the superconducting transition Tc is not changed by the presence of defects,at least for low enough defect concentrations.

This result was first obtained by A.A. Abrikosov and L.P. Gor’kov in 1958 [1] and independently by P.W. Anderson in 1959 [6] and is referred to in the literature as “the Anderson theorem”. Strictly speaking, the statement is correct for the isotropic model only. For an anisotropic metal, as will be discussed below, one may expect variations of Tc upon alloying. Experimentally, in most superconductors of the “old generation”, i.e. ordinary low temperature superconductors such as elemental metals, the Anderson theorem is satisfied with such a high accuracy that it is often considered as an experimental test of whether a superconductor belongs to the ordinary type. One last comment is necessary to emphasize the range of applicability of the above results: “low enough concentration” in practice means that one neglects any terms of order 1/"F  1 or higher.

5.3

Superconducting Alloys with a Small Gap

In this section we consider various applications of the cross technique developed above. The technique has its advantages and drawbacks. Among the latter one may mention some awkwardness in carrying out the calculations, especially for the intermediate concentrations where the mean free path  is of the same order as the coherence length 0 = vF /2Tc (i.e., for  ∼ 0). On the other hand, in many cases this technique allows one to use simple symmetry considerations and to analyze different problems. Recall that we neglect all diagrams with crossing dashed lines as in Fig. 5.2(c). This fact will also allow us to derive the so-called Eilenberger equations. The cross technique is especially simple and transparent when applied to problems where the superconducting order parameter  is small. 5.3.1

The Superconducting Transition Temperature

For a first application let us return again to the question of whether the superconducting transition Tc depends on the concentration of impurities n. This time, however, we study a more general case, i.e. we allow the impurity potential to depend on the spin variables. Thus, in addition to the ordinary potential energy U (r) of a single defect or impurity, for the

5

1

heavier elements the spin-orbit interactions would also contribute to the scattering potential. We may write this contribution as Uˆ so (r) = ˛(ˆ × ∇U (r)) · pˆ ,

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$ −i!n − H0 + −



209

&∗t 2 ˆ − ra ) U(r

a

ˆ r ; !n ) = 0 . × Fˆ † (r, r; !n ) − ˆ ∗ (r)G(r,

(5.25)

where pˆ = −i∇ is the momentum operator and ˆ stands for the Pauli spin matrices.This interaction,of course, preserves the time-reversal invariance for the electronic system as follows directly from Eq. (5.25) upon substitution p → −p,  → − . At the same time scattering by so-called paramagnetic impurities, i.e. by magnetic centers with fixed local moments (spins) S, breaks the time-reversal symmetry for the conduction electron subsystem (ˆ → −ˆ at t → −t). One may describe such scattering by a potential Uˆ p (r) = Up (r)(S · ˆ ) . (5.26) The difference with respect to time reversal symmetry between Eq. (5.25) and Eq. (5.26) has important consequences. While spin-orbit interactions only affect paramagnetic properties in the superconducting state, the presence of paramagnetic centers turns out to be detrimental to superconductivity causing new and rather nontrivial changes in the very nature of the superconducting state such as the emergence of so-called “gapless superconductivity” (A.A. Abrikosov and L.P. Gor’kov, 1960 [7]). In the momentum representation the potential describing a spin-orbit interaction Eq.(5.25) adds to the scattering amplitude fˆ (p, p) a term which we shall write below in the following normalized form (for the isotropic model) 

[p × p ] · ˆ fˆso (p, p) = iUso . p2F To treat properly the spin-dependent contributions, Eqs. (5.25) or (5.26), it is necessary to restore the spin variables in all Green functions: 1 2  ˆ i!n − H0 + − U(r, ra ) a

ˆ r ; !n ) + (r) ˆ Fˆ † (r, r; !n ) = ı(r − r ) , × G(r,

The notation [. . .]∗t in the second equation denotes the hermitian conjugation together with permutation of the spin indices. As far as the value of the transition temperature is concerned the problem is linear in the gap . One is left with the second equaˆ r; !n ) can be tion only where the normal state G(r, † † used. Assuming again ˆ (r) ≡ ˆ together with the self-consistency equation (5.15), one may write ˆ † (r) =



  ˆ r  ; !n ) g(r, r) Gˆ t (r, l; −!n )G(l,

imp

× ˆ † (l)d3 ld3 r ,

(5.27)

where . . .imp stands for the product of two Green functions averaged over impurity positions. The interaction kernel g(r, r) in most cases below is chosen in its simplest “contact” form, with the exception of Sect. 5.3.5, where the role of crystalline anisotropy in the critical temperature Tc is studied.

5.3.2

Ordinary Impurities

As a demonstration of the usefulness of Eq. (5.27), we rederive the result that the temperature of the superconducting transition does not change upon alloying with ordinary impurities. Simple analysis quite similar to that we have done previously shows that in averaging the product of two Green functions on the right hand side of Eq. (5.27) one has to keep only diagrams of the type shown schematically in Fig. 5.6(b,c). Diagrams of the first type result in the “exact” (normal state) Green functions as given previously, while the “ladder” diagrams in Fig. 5.6(c) must be summed up with the same expression for the internal Green functions.In the absence of spin effects the , † matrices in Eq. (5.27) are given as above . The self-consistency equation (5.27) acquires the form shown in Fig. 5.7 with the vertex  (p; !n ) satisfying the equation for the sum of all ladder diagrams in Fig. 5.6(c). Thus,

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Fig. 5.7. The vertex part arises as a contribution from the “ladder” diagrams in Fig. 5.6(c)

The sum over frequencies !n in Eq. (5.31) diverges which means that it must be supplemented by a cutoff ! ˜ as usual. Fig. 5.6. (a) Diagrammatic form of the linearized gap equation; (b) corrections to each of the two Green functions; (c) “vertex” corrections

n  (p; !n ) =  + (2)3



|U (p − p)|2

(5.28)

× G(p ; −!n )G(−p ; !n ) (p ; !n )d3 p and  = gT

 !n

3

dp G(p; −!n ) (2)3

(5.29)

× G(−p; !n ) (p; !n ) . Performing the integrations over p and p we obtain for  (p; !n )   sign!n , (5.30)  (p; !n ) = 1 + 2!n which exactly cancels the factor appearing in the expression in the denominator after integration of the product of two Green functions. We thus return to the familiar self-consistency equation which defines Tc0 of a pure material    1 mpF 2Tc0 1=g . (5.31) 2 2 !n n>0

5.3.3

Spin-Orbit Scattering

Consider now the contribution of the spin-orbit term to the scattering processes shown in Fig. 5.6(b,c). Repeating all the arguments which have led us through the analysis of the diagrams in Fig.5.2 and to the final expression for the normal metal Green Function, it is easy to verify that the spin-orbit interaction results only in the substitution 1 1 1 ⇒ + , (5.32)   so    F where 1/so = 23 nmp |Uso (p − p )|2 d§p (for the (2)2 isotropic model).In the ladder diagrams in Fig.5.6(c) the spin-orbit term contributes to the dashed line surrounding the matrix ˆ † as  n d3 p|Uso (p − p )|2 (5.33) (2)3     ∗t . × iˆ · [p × p] ˆ † iˆ · [p × p] With the help of the identity gˆ ˆ t gˆ = −ˆ ,

(5.34)

where gˆ is the metric tensor, the matrix equation summing up all the ladder diagrams in Fig. 5.6(c) reduces to the same equation (5.28) with the only change being 2 |U (p − p)|2 ⇒ |U (p − p )|2 + |Uso (p − p )|2 , 3

5

i.e., to the renormalization of Eq. (5.32). Therefore, the same cancellation takes place again as for defects without the spin-orbit term. Thus, adding a spinorbit scattering amplitude does not change the expression for the the superconducting transition temperature. Actually, reviewing our analysis above one may easily verify that this result has a general character: the use of the identity Eq. (5.34) allows one to rewrite a new equation (5.17) with the spin-orbit contribution in the form of Eq. (5.32). This results in the appearance of the multiplicative factor Eq. (5.23) with the new mean free path . Therefore, the spinorbit contribution to scattering by defects does not change the thermodynamics of the superconducting phase, although it may change dramatically the spin susceptibility in the superconducting state (A.A. Abrikosov and L.P. Gor’kov, 1962 [8]). 5.3.4

Paramagnetic Impurities

Let us now consider the concentration dependence of the transition temperature in the presence of paramagnetic impurities. In what follows we treat local spins S at impurity sites such that the so-called Kondo-effect (screening of the impurity spin by conduction electrons) is neglected; for a discussion of the Kondo effect [9]. The exchange interaction in Eq. (5.26) is usually weaker than scattering by the ordinary potential (electrons forming local spins occupy an internal shell on the impurity center). As for the Kondo effect, it is characterized by an energy scale TK which is exponentially small in terms of the exchange coupling Up (r). Therefore, TK is assumed to be small compared to the temperature of the superconducting transition, TK  Tc . A straightforward calculation of the self-energy diagram in Fig. 5.3 for the averaged normal Green function again gives

−1 G(p; !n ) = i!n −  + isign!n /21 , where now 1/1 = 1/ + 1/s and   n m pF S(S + 1) 1 |Up (p−p)|2 d§p . (5.35) = s (2)2 3 (we have used for the local spins: Si Sk = S(S+1)ıik /3). However the exchange term Eq. (5.26) contributes

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differently to the “ladder” diagrams in Fig. 5.6(c) as follows from the presence of the Green function Gˆ t (r, l; −!n ) in Eq. (5.27). Transferring the metric tensor gˆ which defines the matrix structure of the gap, ˆ † across (Sˆ )t and again making use of the identity Eq. (5.34) we arrive at a modified equation (5.28) for the vertex function  p (p; !n ) in which, instead of |U (p − p)|2 , one has |U (p − p )|2 ⇒ |U (p − p )|2 (5.36) S(S + 1) |Up (p − p )|2 . − 3 As a result, in an isotropic model where  p (p; !n ) =  p (!n ) we have  p (!n ) =

!n + 1/21 . !n + 1/s

(5.37)

Substitution of Eq. (5.37) into Eq. (5.29) leads instead of Eq. (5.31) to a new equation for the superconducting transition temperature Tc :    1 gmpF 2Tc . (5.38) 1= 1 2 2 ! n + s n>0 Combining Eq. (5.38) with Eq. (5.31), the latter defining the transition temperature Tc0 of the pure material, it is not difficult to write down the following equation for the dependence of the critical temperature Tc on the concentration of paramagnetic centers (A. A. Abrikosov and L. P. Gor’kov, 1960 [7])     Tc0 1 1  ln + − , (5.39) = Tc 2 2 2 where  = (Tc s )−1 , (x) is the derivative of the logarithm of the  -function, and     1 1  + − = 2 2 2   1 1 − . n + 1/2 n + 1/2 + /2 n>0 (5.40) At low concentrations (  1) Tc decreases linearly with concentration: Tc = Tc0 − /4s .

(5.41)

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L. P. Gor’kov

On the other hand, assuming that Tc is small,   1, and using the asymptotic form for the -function in Eq. (5.40), one gets Tc2 = (6/ 2s2 )ln(s Tc0/2 ) .

(5.42)

Equation (5.42) determines the critical concentration: Tc0 (1/s )cr = . 2 Above this concentration superconductivity is fully destroyed by impurities (where ≈ 1.78 is the Euler constant). 5.3.5

Impurities and Crystalline Anisotropy

The above non-trivial result for the concentration dependence of the critical temperature in the presence of paramagnetic centers, was obtained under the assumption of s-wave symmetry in calculating averages over impurities in the Green function and the vertex  (p; !n ) in Fig. 5.6(c). This immediately raises the question about the effect of impurities on a superconducting state with a symmetry of the order parameter other than the so-called “s-wave” pairing. So far we have focused on the isotropic model. In this section we consider a more general case of an anisotropic metal. In a crystal of a given the symmetry density of states  (p) is defined as  (p)dd§p d3p p2F (p) ⇒ dd§p ≡ 3 3 (2) vF (p)(2) 4 and depends on the angular (or other) variables used to define the position on the Fermi surface (, as usual is the energy variable). At the same time  (p) remains invariant under all the transformations of the point symmetry group. The same is true for the pairing kernel g(r, r ), see Eq. (5.27). We already know that in a weak coupling scheme superconductivity affects only states close to the Fermi surface. Therefore, for the short ranged potential g(r, r) it is more convenient to use its corresponding form in momentum space g(p, p). The latter can be written in the general form    ∗ g(p, p) = g i i,n (p)i,ni (p ) , (5.43) i i,ni

where in the expansion Eq. (5.43) the functions i,ni (p) represent a complete set of orthogonal eigenfunctions for the kernel g(p, p) describing its dependence on the Fermi surface variables. Indices i denote functions transforming according to a representation of the crystal symmetry group, while ni stands for the number of eigen functions belonging to this representation in cases where a representation is degenerate. An essential difference relative to the isotropic model, where for all ’s in Eq. (5.43) we can use spherical harmonics, lies in the fact that for a finite (point) crystalline group the number of irreducible representations is also finite. Therefore, in Eq. (5.43) functions i,ni (p) and j,nj (p) with i = j transform according to the same representation of the group. Returning to the discussion of our impurity problem, we assume that in a given superconductor pairing with the highest Tc0 corresponds to the choice of one of these constants, say g i0 , so that the order parameter (p) is of the form (p) = 0 (T)i0 (p) .

(5.44)

Its transformation properties are determined by those of i0 (p). For calculating the averaged normal Green function one can readily repeat all the steps which have led to the previous equations with the only difference that crystalline anisotropy is explicitly present in the momentum dependence of all parameters  1 n = |U (p, p)|2  (p )dSp , (5.45) 1 (p) (2)2 where the integration is over the Fermi surface. The function  (p ) under the integral sign transforms according to the identity representation and so does 1/1(p). Turning now to the calculation of the “ladder” diagrams, one sees that generally speaking a new “scattering time” appears due to the integral in Fig. 5.6(c), which has the form  1 n = |U (p, p)|2 (p )i0 (p)dSp . 2 (p) (2)2 Thus, for instance, if i0 (p) does not belong to the identity representation (as with the so-called “dwave” in cuprates), 1/2(p) differs drastically from

5

1/1(p) in Eq.(5.45).Hence,the results of our analysis for paramagnetic impurities apply: superconductivity for the case of any phase with a non-trivial order parameter is already reduced by the presence of ordinary defects. Note, in addition, that even if i0 (p) belongs to an invariant (identity) representation, the above two integrals are not equal. From this point of view the “Anderson theorem” is an approximation which works better for a reasonably weak anisotropy of (p).Experimentally it is fulfilled surprisingly well in most of traditional superconductors. The isotropic BCS model also displays the well known square-root singularity in the density of states, S (E), in the superconducting state 1 0, |E| <  S (E) = . (5.46) 2 2 1/2 N (E) /(E −  ) , |E| >  This singularity is seen explicitly in tunneling experiments and was crucial for the explanation of the well-known Hebel–Slichter peak (L.C.Hebel and C.P. Slichter, 1957 [10]) in the NMR-relaxation time, T1−1 near Tc in ordinary superconductors. Observation of this phenomenon by Hebel and Slichter in 1957 was initially taken as the ultimate proof of the correctness of the BCS-theory. Since then, the nature of the superconducting order parameter is often judged by whether the Hebel–Slichter peak is observed in a given material or not, or whether the critical transition temperature is sensitive to the presence of defects. Nevertheless, one should be aware that excessive anisotropy, both in the normal properties and in the gap, (p), may smear out the BCS singularity Eq. (5.46) and be responsible for the sensitivity of Tc to impurities, even if the gap has no nodes. 5.3.6

Ginzburg–Landau Equations for Superconducting Alloys

Among other situations involving a small order parameter ((p) and † (p)), we will briefly discuss the derivation of the Ginzburg–Landau (GL) equations for superconducting alloys (L. P. Gor’kov, 1960 [11]) and the dependence of the critical field, Hc2 (T), on temperature in the so-called “dirty-limit” (  0 ) (K. Maki, 1964) [12].

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Equations for the order parameter in the presence of magnetic fields and the GL-functional near Tc were obtained microscopically for pure superconductors by L. P. Gor’kov in 1959 [13]. The derivation in the presence of defects or impurity atoms begins again by expanding the anomalous Green function F † (r, r; !n ) in the self-consistency equation (5.15) in powers of (r), ∗ (r). This is done with the help of equations (5.14),and applying the cross-technique to average different terms over defects. To maximally simplify our discussion, the expansion of  ’s is first shown schematically in Fig. 5.8(a). Recall that the order parameter is self-averaged. Therefore, the terms containing (r), † (r) in Fig. 5.8(a) only the slow spatial variation caused (near Tc ) by the presence of weak external fields, (i.e a non-zero A(r), where A(r) is the vector potential of the magnetic field). The magnetic field dependence of each of the normal Green functions in Fig. 5.8(a) can be explicitly written by making use of the quasi-classical character of electron motion in metals (pF 0 (T)  1, pF   1) (L. P. Gor’kov, 1959, 1960 [13,14]): G(r, r ; !n ) ⇒ exp

⎧  ⎨ ie r ⎩c

r

⎫ ⎬

A(l)dl G(r, r  ; !n ) . (5.47) ⎭

While Eq. (5.47) has a general character and is applicable at all temperatures (L. P. Gor’kov, 1960 [14]) near Tc it can be expanded in the phase factor, since the critical magnetic field is small (H ∝ 1 − T/Tc ). The spatial dependence of the Green function, on R = |r − r |, is governed either by 0 = vF /2Tc0 or by the mean free path . Recall that the GL-equation for the order parameter (r), which is proportional to  (or † for †) (L. P. Gor’kov, 1959), is a second order differential equation, quadratic in the operator @ˆ = (−i∇ − 2ec A). The phase factors guarantee a gauge-invariant form of the operator @ˆ (note we have a charge 2e for the wave function of the Cooper pair!). After these preliminary remarks, it is a straightforward task to determine the coefficients in the GL expansion for alloys. First, since the equations must be gauge-invariant, i.e., include the gradient and the vector potential A only in the combination defined ˆ for the purpose of determining by the operator @,

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L. P. Gor’kov

Fig. 5.8. Expansion of the self-consistency equation for † (r) up to third order terms; (b) spatial dependence, † (r) comes through the k-dependence in the momentum representation

the coefficients one may omit the vector potential A and consider only the spatial variation of (r) (or † (r)). It is more convenient to study the Fourier transform, (k) († (k)). Before proceeding further with the cross-technique, let us recall that for a constant  (and † ) the averaged anomalous functions, F(r, r; !n ) and F † (r, r; !n ) coincide at r = r with their values in the absence of impurities. The transition temperature does not change and the third order term in  and † in Fig. 5.8(a) (corresponding to the bi-quadratic terms in the GL-functional) is the same as for a GL-functionalof the pure superconductor. In other words, almost everything is already known except that it is necessary to recalculate the diagram in Fig.5.8(b) (and the vertex,  † (p, k; !n)) through second order in k. As in Fig. 5.7 the vertex  † (p, k; !n) satisfies the equation  |U (p − p )|2 d3 p  † (p, k; !n) = † (k) + n (2)3 × G(p; −!n )G(−p + k; !n) † (p, k; !n) . Expanding with respect to k we obtain for the GL equation (L. P. Gor’kov, 1960 [11]): 1

 2 2  2ie  7(3) Tc − T @ 2 − A(r) + − | (r) | Tc 8(Tc )2 2m @r c

× (r) = 0 ,

(5.48)

where  =

2"F 3 2 Tc2  ×

1 

(5.49) 

1 2 + 8 2

and =

   1 − 2



1 + 2

 ,

1 0 , = 2Tc tr 

(here tr is the transport “collision” time). The first two terms in Eq. (5.48) are the same as for a pure superconductor.As for the third term, it significantly changes the behavior of an alloy in a magnetic field. As an illustration let us consider the problem of finding the so-called “upper-critical field” Bc2 near Tc for superconductors of the second type (V. L. Ginzburg and L. D. Landau, 1950 [15]; A. A. Abrikosov, 1957 [16]). To do this one may choose the Landau gauge for the vector potential, A = (−By, 0, 0), in the linearized equation (5.48).The latter then reduces to the familiar harmonic oscillator equation defining the shape of the nucleation center for the second-order transition from the normal to superconducting state in the presence of a magnetic field B    Tc − T  d 2 4e2 2 2 (y) = 0 . + − 2 By Tc 2m dy2 c

5

Solving this equation we obtain   1 Tc − T .

B Bc2 = Tc 2

(5.50)

The conclusion drawn from Eq. (5.50) is of a great practical importance. Critical magnetic fields increase dramatically for “dirty” alloys, i.e., for short mean free path,   0 . Using the limit of  in Eq. (5.49) for  → ∞ and taking T = 0 in Eq. (5.50), one can get an estimate for the limiting value of the critical fields at low temperatures   6 1

BBc2 = Tc . (5.51)  pF  Equation (5.51) makes sense, of course, only before the Ioffe–Regel criterion pF  ∼ 1 signals the onset of localization of the conduction electrons. Note, however, that close to this limit the critical field Bc2 that is due to the mechanism of diamagnetic currents reaches a point where one can no longer ignore the paramagnetic effects (L. P. Gor’kov, 1963) [17]: the spins of the two electrons comprising a Cooper pair are oriented in opposite directions, while the magnetic field, via the Zeeman energy term BBˆ , tends to align them along the field. Thus, the Zeeman energy itself results in breaking electron pairs. Therefore,this leads at T = 0 to the so-called paramagnetic limit (A.M. Clogston, 1962 [18]; B.S. Chandrasekhar, 1962) [19].

B B∗cr  1.25Tc . Hence, in sufficiently “dirty” superconducting alloys both the paramagnetic and diamagnetic effects must be treated simultaneously (K. Maki and T. Tsuneto, 1964) [20].Let us also mention that the anisotropy inherent in some layered or otherwise low-dimensional superconductors, in turn, may enhance the significance of the Pauli mechanisms (L.N.Bulaevskii,1974) [21]. This subject, however, lies beyond the scope of this chapter. 5.3.7

Upper Critical Field for ``Dirty Alloys´´

Returning to expression Eq. (5.50) for the critical field, Bc2, near Tc and to its estimate Eq. (5.51) for dirty alloys at low temperature, we can show that in

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215

the case of the isotropic model and of large concentrations of defects, impurities,   0, the critical field Bc2(T) can actually be obtained in the framework of the same technique for all temperatures T < Tc (K. Maki, 1964 [12]). For this let us again consider Fig. 5.7(b). In the product G(p; −!n )G(−p + k; !n ) one has for G(−p + k; !n ) G(−p + k; !n ) = (i!n −  + vF k + i/2)−1 . Near Tc , vF k is small compared to !n ∼ Tc and 1/. However, in the “dirty limit” case it is vF k  1/ everywhere below Tc .This according to Eq.(5.51) yields an estimate for the characteristic “magnetic length”, √ aH = c/eB  (0 )1/2 . Therefore, for vF k one has vF k ∼ (/0 )1/2  1 .

(5.52)

As for the vertex  † (p, k; !n) (formally it is of order of unity) it becomes strongly renormalized (∼ 1/Tc according to Eq. (5.30)). To proceed further with the calculations, let us first integrate G(p; −!n )G(−p + k; −!n ) over the energy variable  (d 3p ⇒ mpF dd§p ; the integral over  converges rapidly and one may assume that  † (p, k; !n) depends only on the momentum at the Fermi surface). Then   nmpF i † †   (p, k; !n ) =  (k) + (2)2 2  × |U (p − p )|2d§p (5.53) ×

 (p , k; !n)sign!n i!n +

i 2 sign!n

+

vF k 2

.

Expanding using Eq. (5.53) in |!n |1, vF k  1/ in the denominator we arrive at the following equation:  nmpF †  (k) = |U (p − p )|2  † (p , k; !n) (5.54) (2)2 = > × −2!n + i(v  F k)sign!n −  2 (v  F k)2 d§p . For the isotropic model Eq. (5.54) can be easily solved in terms of the first two spherical harmonics  † (p , k; !n)   †0 (p, k2 ; !n ) + (vF k) †1 (p; !n ) . (5.55)

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L. P. Gor’kov

5.4

Thus,  †1 (p; !n ) = isign!n

2 †    (p ; !n ) , tr 0

(5.56)

where the “transport time” tr is given by nmpF 1 = · tr (2)2



|U (p − p)|2 (1 − cos )d§p . (5.57)

After a short calculation we finally find  −1 † (k) tr k 2vF2 |!n | + . 2 6 (5.58) The next step would be to return to the real-space representation using the correspondence k↔ − i∇ and the arguments given above in the derivation of the GL-equation (5.48) with regards to the gauge invariance. Correspondingly, we get k2 ⇒ (−i∇ − 2e A(r))2 . The eigenvalues of the latter operator in c the presence of a homogeneous magnetic field B are well-known. Its lowest value, (2eBc2/c), determines the upper critical field Bc2 (T) after substituting equation (5.58) into the self-consistency equations (5.27). Manipulations of formally divergent sums over !n , which are quite analogous to those leading to equation (5.39), give the following equation from which the temperature dependence Bc2(T) can be obtained (K. Maki, 1964 [12]):  †0 (p, k 2; !n ) =

Tc0 ln = T



 1 +z − 2

  1 , 2

(5.59)

In this section we discuss in more detail some nontrivial properties of a superconductor containing paramagnetic impurities. In the (T, 1/s)-plane, where 1/s ∝ n characterizes the dependence on impurity concentration, n, the phase boundary between normal and superconducting states is given by equation (5.39). The critical concentration (1/s )cr  0.88Tc0 limits the allowed concentration range for the existence of superconductivity in such an alloy. To analyze the thermodynamic properties below the transition temperature, Tc (n), it is necessary to know the energy spectrum of the system. This information is contained in the Green functions of the system. One can repeat all the steps in the derivation of the equations for the averaged Green functions given before. The same equations result for G(p; !n ) and F † (p; !n ) except that in the definitions ¯ n ) and ˜ † = † + F¯ † (!n ) one of i! ˜ n = i!n − G(! needs to account for the different “collision” times which according to Eq. (5.35) and Eq. (5.37) appear now upon averaging two Green functions. Thus, 1 un  , 21 u2n + 1 1 1  , ˜ n =  + 22 u2n + 1

! ˜ n = !n +

z=

tr vF2 eBc2(T) 6T

.

(5.60)

For |T − Tc |  T we reproduce the result Eq. (5.50) (  0). Then at T → 0:   0.42  Tc0 (0tr )−1 , 

etr v2 e

(5.61)

in order-of-magnitude agreement with our estimate Eq. (5.51) (note Eq. (5.60) and Eq. (5.61) are written in units:  = 1, c = 1).

(5.62)

where 1/1 −1/2 = 2/s in accordance with Eq. (5.35) we defined the new variable un =

where

3 Bc2 (0) = 2

Paramagnetic Alloys and Gapless Superconductivity

! ˜n ˜

(5.63)

which is an implicit function of the Matsubara frequency %  1 1 !n . (5.64) = un 1 − ·  s  u2n + 1 In a pure material the poles of the Green function on the real frequency axis would directly determine the quasi-particle energy spectrum and its dependence on the momentum p.While the latter is not preserved in the presence of scattering by defects,the important

5

information regarding quasi-particles and the den¯ n ) if ansity of states is contained in G(p; !n ) or G(! alytically continued from the upper Matsubara axis i!n into the whole complex plane i!n → z [3]. We will not elaborate on the thermodynamics of a superconducting paramagnetic alloy but focus only on a new fundamental feature of its energy spectrum: the appearance of the so-called“gapless” regime. For this purpose let us consider the spectral representation of the Green function G(p, z) (a similar representation exists for the anomalous functions, F and F † ) ∞

(p, x)dx . x − i!n

G(p, z) = −∞

(5.65)

Being analytically continued from the upper (lower) complex plane z = i!n , the resulting function coincides with the retarded (advanced) functions G R(A) (p; !) on the real axis z = ! ± iı. At T = 0 the imaginary part of the latter is of the form ImG R(A) (p; !) = ±



l (p)ı(! − !l0 ) ,

l

with !l0 = El −E0 being the excitation energies of the system from its ground state, E0 , into an excited state, El (for the details of the derivation see [3]). The frequency ! ∗ at which an imaginary part first ocurs in ¯ n ) after being analytically continued G(p; !n ) or G(! to the physical frequency axis (z = i!n → ! ±iı), is the threshold, or the gap in the excitation spectrum. With this in mind, we return to Eqs. (5.62)–(5.64) and formally replace !n by −i!, and un by −iu (with ˜ Equation (5.64) is then written ˜ n → (!)).   1 ! 1 =u 1− . √  s  1 − u 2 For low impurity concentrations, i.e. beginning with small (1/s ), it is clear, that in order to have real values for the square root on the imaginary axis, u → iun as it is given by the equations, one √ must choose the branch of the real positive radical 1 − u2 for −1 < u < 1; this analytical continuation procedure fully defines the function u(z) in the whole z-plane [3].

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217

Let us now return to the retarded (advanced) Green functions on the physical axis. With ! > 0 on the real axis one sees that a maximum occurs at

1/2 u0 = 1 − (1/s )2/3 . (5.66) This defines the position of the branching point for u(!) and, hence, the appearance of an imaginary component in G R(A) (p; !) or F †R(A) (p; !). The energy gap ! ∗ according to Eq. (5.66), is:

3/2 . ! ∗ =  1 − (1/s )2/3

(5.67)

This is equal to zero for s  = 1. We still have to determine the corresponding concentration value for which this happens. We again use the self-consistency condition Eq.(5.15) and integrate first over d in the expression for the averaged F † (p; !n ) . However, the new definitions for ! ˜ n and ˜ n are taken into account. At T = 0 the sum over frequencies, T n , becomes the inte gral (1/2) d! along the Matsubara axis, z = i!. Thus, mpF 0 = |g| 3 4

!˜ 0

d!  . u2 (!) + 1

(5.68)

To handle the cutoff, !, ˜ we add and subtract under the integral on the right hand side the proper compensating term 0 (! 2 + 20 )−1/2 and make use of the corresponding definition for the gap, 00 , of the pure superconductor (we also distinguish 0 , the gap of an impure superconductor at T = 0, from  00 = / Tc0, the gap value for the parent material). After some simple transformations one arrives at the equation 0 0 ln = 00

1

∞ d! 0

1

0

− √ u2 + 1 ! 2 + 20

2 . (5.69)

To calculate 0 as a function of 1/s , it is convenient to convert the integration over ! into an integration over u, making use of Eq. (5.64). We do not discuss the details of the calculations. (A helpful comment, however, is that depending on 1/s 0 the lower limit, ! = 0, in the integral Eq. (5.69) changes from 0

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L. P. Gor’kov

 at 1/s  < 1 to (s 0 )−2 − 1 at 1/s 0 > 1). For 1/s 0 = 1, where the energy gap disappears, we find ln

  0 =− ≡− , 00 4s 0 4

(5.70)

or in terms of the concentration (i.e., 1/s ) 1   = e− 4 00 = e− 4 s

   Tc0 .

(5.71)

Since the critical concentration ncr at which superconductivity is destroyed is given by (1/s )cr = (/2 )Tc0, one sees from Eq. (5.71) that the energy gap disappears at the smaller concentration 

n = 2e− 4 ncr  0.91ncr .

(5.72)

Therefore, in superconductors with paramagnetic impurities there exists a gapless regime in which the specific heat Cp (T) at low T has a linear slope (A.A. Abrikosov and L.P. Gor’kov, 1960 [7])

1/2 Cs (T) = 1 − (s 0 )2 T.

(5.73)

The non-zero superconducting order parameter in the absence of an energy gap in the excitation spectra for the s-wave superconductivity is due to broken time reversal symmetry in the electronic sub-system (in the Hamiltonian) Eq.(5.26).A gapless regime may also appear in the presence of a magnetic field which also breaks t ⇒ −t invariance (K. Maki, 1964 [12]; P.G. de Gennes and M. Tinkham, 1964) [22]. For a dtype or other non-trivial order parameter a gapless spectrum is intrinsic due to the presence of nodes in the gap. In addition, the nodes can be easily smeared by ordinary defects. To complete the discussion of the superconducting gapless regime Eq. (5.72) caused by the presence of paramagnetic defects, we give without derivation a generalized GL equation for (r) and the current j(r). Near the critical concentration, ncr , (Tc  Tc0) one has



 ||2  (T − ∗ + (5.74) 2   2ie 2 ∗ 2v2 tr A  = 0, ∇+ + F s c   4e2 ie (∇∗ − ∗ ∇) − A||2 . j = Ns tr m mc (5.75) 2

2

Tc2 )

(A. A. Abrikosov and L. P. Gor’kov, 1960 [7]).

5.5

Eilenberger Equations

5.5.1

Quasi-Classical Approximation

In the preceding Sections we obtained some basic results for the theory of superconducting alloys. So far, for each of these problems an application of the cross-technique provided a natural and straightforward approach which led directly to a final answer. However, when considering more complicated problems, such as nonlinear behavior of a superconductor in a strong magnetic field and at temperatures far from the critical temperature Tc , a solution of the Gor’kov equations, even in the absence of impurities, becomes a difficult task that requires numerical methods.When considering alloys the crosstechnique becomes also rapidly more tedious. Thus, while for a homogeneous superconductor one needs to average only the Green functions themselves, spatial variations of the order parameter (r) and † (r) introduce other correlators of the Green functions, for example,such as an average of the sum of the“ladder” diagrams associated with  † (p, k; !n) which we considered for the linearized gap equation in the presence of an external magnetic field. Fortunately, at least for ordinary (i.e. the BCS-like) superconductors, further simplifications turn out to be possible for the Gor’kov equations. Using the method of quasi-classical Green functions one can derive the less-complicated Eilenberger equations. The new formalism significantly simplifies the calculations for most problems,especially for superconducting alloys (G. Eilenberger, 1968 [23]; A.I. Larkin and Yu.N. Ovchinnikov, 1968 [24]; G.M. Eliashberg, 1971 [25]). The applicability of the quasi-classical

5

approach, as follows from its very name, is based on the fact that in metals the typical wavelength of an electron, p−1 F , is on the atomic scale and, hence, is much shorter then any other length scale under consideration: the mean free path, l = vF , the magnetic field penetration depth, ı, or the coherence length (the “size” of the Cooper pair), 0 = vF /2Tc (P.G. de Gennes, 1964 [22]; E.A. Shapoval, 1964, 1965 [26]; for the review see J.W. Serene and D. Rainer, 1983 [27]). The possibility of neglecting all crossing diagrams, such as shown in Fig. 5.2(c), is also due to the inequality, pF l  1. This very important observation allows one to write down the Gor’kov equations for alloys, and as a consequence the Eilenberger equations, in closed form. 5.5.2

Derivation of the Eilenberger Equations

To begin with let us return again to equations (5.14) and rewrite them in a more compact form for a general case where, in addition to the impurity potential, a superconductor experiences an inhomogeneous magnetic field B(r) = [∇×A(r)] (where A(r) is the vector potential). Indeed, the Gor’kov equations can be presented in matrix form [3]):   ˆ − x ), ˆ Gˆ 0−1 − Vˆ · G(x, x ) = ı(x (5.76) where



ˆ G(x, x) = and Vˆ is given by Vˆ ≡ Vˆ (r) ≡

G(x, x ) −F (x, x ) F † (x, x ) G(x , x) 



% ,

(5.77)

% U (r − ra ) · eˆ

,

(5.78)

a

(Vˆ is a sum over all impurity potentials multiplied by the two-by-two unit matrix eˆ .) The operator for the inverse (“bare”) Green’s function is 

Gˆ 0−1 =

ˆ 0 − ) −@/@ − (H −† (r)

(r) ˆ − ) @/@ − (H 0

%

. (5.79)

As in Sect. 5.2, Hˆ 0 and Hˆ 0 are single particle Hamiltonians including the vector potential associated with

Theory of Superconducting Alloys

219

an external magnetic field. For the isotropic model one has 2 e 1  −i∇ − A(r) , Hˆ 0 ≡ 2m c  2 e 1 −i∇ + A(r) . Hˆ 0 ≡ (5.80) 2m c It was suggested that (r) and † (r) are selfaveraging variables. Local distortions produced in the vicinity of an impurity decay rapidly as does the magnetic field. Therefore, both the order parameters (r), † (r) and the vector potential A(r) are slowly varying functions of the coordinates with characteristic scales (l) or ı(l), respectively. These depend on impurity concentration through the mean-free path l. These scales are much longer than the effective distances over which the averaging actually takes place. In what follows we restrict ourselves to stationary problems only. Therefore, the dependence on the “imaginary time” coordinates ,   again reduces to the “time” difference ( −   ). Correspondingly, we can rewrite Eq. (5.76) in terms of the Fourier freˆ r; !n ). This can be easily achieved by quencies, G(r, the substitution: @/@ ⇒ −i!n in expression (5.79) for the operator Gˆ 0−1 . We further transform the matrix equation (5.76) to the momentum representaˆ r; !n ) ⇒ G(p, ˆ tion: G(r, p; !n ). Without going into details, let us recall that in all Green functions the characteristic scale for the values of two momenta p and p is given by the Fermi momentum pF . As usual within the quasi-classical approximation we can make in Eq. (5.80) the approximation e Hˆ 0 p2F /2m − vF · A(r) , c e Hˆ 0p2F /2m + vF · A(r) . c The fact that both A(r) and (r), † (r) depend on r needs to be taken into account for all equations written in the momentum representation. This is, however,a trivial matter that does not affect the averaging over impurities and will be discussed later. After these preliminary remarks it becomes clear that one can apply the same averaging procedure (see the diagrammatic cross-technique of Sect. 5.2.1) to the matrix equation (5.76). Indeed, let us formally

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L. P. Gor’kov

ˆ the impurity potential, to obexpand Eq. (5.76) in V, tain the perturbation series. This is shown in Fig. 5.1. The transform matrix expression (5.78) is given by 

%

1  U (p − p) · exp i(p − p) · ra · eˆ . (2)3 a

Analysis of the relative significance of the different diagrams in Fig. 5.2 can be applied without any changes. The new Dyson equation for the averaged ˆ matrix function G(p, p; !n ) shown in Fig. 5.9 by the line (without arrows) looks like that in Fig.5.4,except that both Gˆ 0(p, p; !n ), the Green function for the ˆ pure superconductor, and the average G(p, p; !n ), depend on two momenta. This is due to the spatial dependence of the gap parameters and of the field. In Fig. 5.9 the three vectors, k, l and s which come about due to such a dependence are small compared to p and p. The equations describing Fig. 5.9 are: ˆ G(p, p − k; !n) = Gˆ 0(p, p − k; !n) (5.81)   ˆ  , p − s; !n ) + Gˆ 0(p, p − l; !n )G(p ˆ − l − s, p − k; !n ) × G(p

 2 d3 l d3 s d3 p     . × U (p − p + l) (2)3 Let us again present this equation in a symbolic form:

ˆ Gˆ = eˆ . (Gˆ 0−1 − £)

(5.82)

The notations in Eq. (5.82) have the following simple meaning. According to the previous equations, Gˆ 0−1

in addition to its trivial part is given by  0 i!n − vF · (p − pF ) 0

% .

−i!n − vF · (p − pF )

A(m), (m) and † (m) contribute to Eq. (5.82) as e %  (m) d 3m c v F · A(m) (2)3 −† (m) − ec v F · A(m) ˆ − m, p − k; !n). × G(p Here we use the conventional definition of the Fourier transform  d3 m y(r) = y(m)eim·r . (5.83) (2)3 For the “self-energy” £ˆ one has  ˆ£(p, m; !n ) = n |U (p − p)|2 ˆ , p − m; !n ) × G(p

(5.84) d3 p . (2)3

The dependence on m corresponds to the spatial variation (in the absence of an external magnetic field, £’s may depend only on the direction of the momentum p for an anisotropic scattering). It is convenient to change slightly the notations so that   ˆ 1, r2 ; !n ) =  G(r (5.85) Gˆ (p, p − k; !n ) × eip·(r1 −r2 )+ik·r2

d3 p d3 k . (2)3 (2)3

Note, the Fourier transform is symmetric with respect to each of the two variables, r1 and r2 . The use

Fig. 5.9. Schematic representation of the Dyson equation for the averaged matrix Green function

5

of  Gˆ instead of Gˆ in Eq. (5.84) allows us to write the spatial dependence of the £’s on an equal footing with that of A(r), (r) and † (r). In practice the transition from Gˆ to  Gˆ involves nothing but an additional factor 3 (2) in the old definitions of the Green functions. For a homogeneous superconductor one has ˆ !n)ı(k) .  Gˆ ≡ (2)3 G(p;

!n

The current density j(r) is   ie (∇r − ∇r )r  →r j(r) = 2T ei!n ı 2m !

(5.88)

n

 e2 × G(r, r; !n) − A(r) G(r, r; !n) . m

Here the factor two arises from the two spin directions. For the model with a “contact” interaction one would have  (r) = gT F (r, r; !n )(r → r  ) . (5.89) !n

At r1 = r2 we can write  d3 p ˆ r; !n) =  . G(r, Gˆ (p, r; !n) (2)3

 ˆ £(p, r; !n ) = n |U (p − p )|2

(5.90)

One may write down the self-energy Eq. (5.84) in the coordinate representation to show that these can also be expressed in terms of  Gˆ (p, r; !n):

221

(5.91)

d3 p × . Gˆ (p, r; !n ) (2)3 We can write the integration over p as mpF d§p d3p dp . = (2)3 2 2 4

(5.86)

From the previous calculations it is seen that the dependence on  = vF (p − pF ) can be integrated out in the self-energy parts . All integrals rapidly converge at small . This mathematical trick comprises the essence of the Eilenberger formulation in terms of “quasi-classical Green functions”. Since the latter is defined only in the vicinity of the Fermi surface. There is no need for a solution of the more general equations for the Gor’kov Green functions. Before writing the equations, let us make a few other helpful comments. First we note that many physical properties are defined by the Green functions at r1 = r2 = r. Thus, the density of electrons n(r) is given by [3]  n(r) = 2T ei!n ı G(r, r; !n ) . (5.87)

Theory of Superconducting Alloys

(5.92)

Convergence of all the integrals over p has already been discussed in Sect. 5.2. The contribution from the momenta lying far from the Fermi surface is the same as in the normal phase and can be included in the definition of the chemical potential. The remaining integrations over p in Eq. (5.91) converge rapidly. It is helpful for the following to introduce the definition W (p − p ) =

mpF |U (p − p)|2 . (2)2

(5.93)

One can immediately verify that Eq. (5.93) is nothing but the differential probability for the elastic scattering between the two states with momenta p and p lying on the Fermi surface and calculated in the Born approximation. With this in mind we define the“quasi-classical Green functions” (G. Eilenberger, 1968; A. I. Larkin and Yu. N. Ovchinnikov, 1968) as:  dp ˆ  gˆp (r; !n ) = (5.94) G(p, r; !n) . i The self-energy parts Eq. (5.91), after making use of Eqs.(5.92) and (5.93), acquire the form  i ˆ£(p, r; !n) = nW (p − p )ˆgp (r; !n ) d§p . (5.95) 2 Let us now return to the Gor’kov equation (5.82). We will compare the “left-side” of equation (5.82) with its “right-side”. We assume that all operators, ˆ 0, Hˆ¯ 0 , etc., apply to the variable x  of the Green @/@, H ˆ functions G(x, x ) in Eq. (5.77). Correspondingly, instead of Eqs.(5.76) and (5.79) one uses  ∗ −1  ˆ G(x, x ) · Gˆ0 − Vˆ = ı(x − x  ) , (5.96) where

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L. P. Gor’kov ∗ −1 = Gˆ0



@/@  − (Hˆ¯0 − ) †

%

(r  ) −@/@ − (Hˆ0 − )





− (r )

From Eq. (5.101) it follows

.

(5.97) There is no need to repeat all the steps that are similar to those that led us to equations (5.82). One obtains  ∗ −1  Gˆ Gˆ0 − £ˆ = eˆ . (5.98) Here, the matrices  i!n − p + vF · k

%

−i!n − p + vF · k

0 and 

0

d3 m ˆ G(p, p + m − k; !n ) (2)3 ⎞ ⎛e v F · A(m) (m) ⎠ × ⎝c e −† (m) − v F · A(m) c

appear. Upon subtracting equation (5.98) from Eq. (5.82) one sees that p drops out. We write down the resulting equations for gˆp (r; !n ) directly in the coordinate representation using as always for k the correspondence k ⇒ −i∇ r . Thus,   ˆ p (r; !n ) , gˆp(r; !n ) = 0 , (5.99) vF ·∇r gˆp(r; !n )+ M ˆ B] ˆ denotes the matrix commutation, and where [A, ˆ for Mp (r; !n) one has ⎛

ie

−!n + vF · A(r) c ˆ p (r; !n ) = ⎜ M ⎝ −i† (r)

ˆ − £(p, r; !n) .

⎞ i(r) ⎟ ⎠ ie !n − vF · A(r) c

(5.100)

Taking the trace of the matrix equation (5.99) one obtains   (5.101) (vF · ∇r ) Tr gˆp (r; !n ) = 0. 2  Similarly, for the trace of gˆp (r; !n ) one finds the equation 2  (5.102) (vF · ∇r ) Tr gˆp (r; !n ) = 0.

gp(r; !n ) + gp (r; −!n ) = 0 , and from Eq. (5.102) we get gp2 (r; !n ) − fp (r; !n ) fp† (r; !n ) = 1. So, as one can see, we used the results for a homogeneous superconductor (and the boundary conditions for Eqs (5.101) and (5.102)). One also concludes that there are actually only two independent functions. It is instructive to write the equations for fp (r; !n ) and fp† (r; !n ):   2e vF · −i∇ − A(r) fp (r; !n ) c − 2i!n fp(r; !n ) − 2(r)gp (r; −!n ) 

+ (i/2) n W (p − p ) gp (r; !n )fp (r; !n )  − fp (r; !n )gp (r; !n ) d§p = 0 and

 2e vF · i∇ − A(r) fp† (r; !n ) c − 2i!n fp† (r; !n ) − 2† (r)gp(r; −!n )   + (i/2) n W (p − p) gp (r; !n )fp† (r; !n )  − fp† (r; !n )gp (r; !n ) d§p = 0. 

In this form the equations bear some resemblance to the kinetic equation for the elastic scattering from impurities and can be treated similarly (see the next section). One also sees explicitly the gauge invariance of the quasi-classical theory. For the reader’s convenience we will write down the self-consistency equation and the expression for the electromagnetic currents in terms of the quasi-classical Green functions:     d§p gmpF (r) = fp (r; !n ) , (5.103) iT 2 2 4 n   mpF  d§p v F gp(r; !n ). j(r) = −i(2e)2T 2 2 4 ! >0 n

(5.104)

5

5.5.3

``Dirty´´ Alloys: Usadel Equations

The limiting case of a short mean-free path, l  0 , is of special interest from a practical point of view. As we have seen the critical field is then considerably larger. One also expects some simplifications in the mathematical apparatus as well. From the formal standpoint, the “collision terms” are of the order of 1/ and are therefore large. This allows one to write an expansion for the matrix gˆp (r; !n ) in the form gˆp (r; !n )  gˆ0 (r; !n ) + vF · gˆ (r; !n ) ,

(5.106)

Averaging Eq. (5.99) over momentum directions gives the following equation  2  v F ˆ ˆ = 0, @ · gˆ + i(hˆ gˆ0 − gˆ0 h) (5.107) 3 where we introduce the matrices   ∇ ·g −(∇ − 2ieA(r)) · f ˆ , @g = −∇ · g −(∇ + 2ieA(r)) · f † (5.108) and   −i!n −(r) . (5.109) hˆ = † (r) i!n Multiplying Eq. (5.99) by vF and integrating over the p-direction one arrives at     ˆ g0 + i hˆ gˆ − gˆ hˆ + 1 gˆ0 gˆ − gˆ gˆ0 = 0 . (5.110) vF @ˆ tr Finally multiplying Eq. (5.110) by gˆ0 one obtains with the help of Eq. (5.106) gˆ = −ltr gˆ0 @ˆ gˆ0

(5.111)

223

(Here, small terms are omitted).After simple algebra the equation becomes 

iD (∇ − 2ieA) · g0(∇ − 2ieA)f0 − f0 ∇g0 = 2g0 + 2i!n f0 , and   iD (∇ + 2ieA) · g0 (∇ + 2ieA)f0† − f0† ∇g0 =

(5.105)

with gˆ ∼  gˆ0 . Indeed, the isotropic part, gˆ0 , cancels the “collision” term. With this in mind, it is easy to write down the Eilenberger equations in the dirty limit (K. D. Usadel, 1970 [29]). From the normalization condition gˆ 2 = 1 one obtains gˆ0 gˆ + gˆ gˆ0 = 0.

Theory of Superconducting Alloys

2∗ g0 + 2i!n f0† .

(5.112)

Here, D is the diffusion coefficient, D = vF ltr /3. In the limit , ∗ → 0, the equations for the temperature dependence of the upper critical field,Hc2 (T) are found.These Equations were first derived by K.D.Usadel (K.D. Usadel, 1970 [29]).An extended discussion of the Eilenberger formalism and the Usadel equations and their applications can be found in [30,31].

5.6

Final Remark

As was emphasized from the very beginning, this chapter is not intended to provide a summary of the known results for superconducting alloys or the influence of defects of various kinds on superconducting properties. The literature on the subject is enormous. Many applications are concerned with vortices and their pinning, different phases of the vortex lattice, etc. With the advent of the era of nonconventional superconductors, for which the role of defects is of utmost importance, the number of publications on the subject has increased dramatically. In some of the new superconductors, such as many cuprates,superconductivity itself arises due to a doping process (which inevitably produces a disorder). Nevertheless, the methods described above remain a major tool to study impurity effects theoretically.

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References 1. A.A. Abrikosov and L.P. Gor’kov, Zh. Exp. Theor. Fiz. 35, 1558, (1958); Sov. Phys. JETP 8, 1090 (1959); ibid. 9, 220 (1959) 2. S.F. Edwards, Phylosoph. Magazine 3, 1020 (1958) 3. A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinski, In: Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, NJ, 1963) 4. E.M. Lifshitz and L.P. Pitaevskii, In: Statistical Physics, Part 2 (Pergamon Press, Oxford, 1986) 5. L.P. Gor’kov, Sov. Phys. JETP 7, 505 (1958) 6. P.W. Anderson, Jour. Phys. Chem. Solids 11, 26 (1959) 7. A.A. Abrikosov and L.P. Gor’kov, Sov. Phys. JETP 12, 1243 (1960) 8. A.A. Abrikosov and L.P. Gor’kov, Sov. Phys. JETP 15, 752 (1962) 9. A.A. Abrikosov, In: Fundumentals of the Theory of Metals (Elsevier, North-Holland, Amsterdam, 1988) 10. L.C. Hebel and C.P. Slichter, Phys. Rev. 107, 901 (1957); ibid., 113, 1504, (1959) 11. L.P. Gor’kov, Sov. Phys. JETP 10, 998 (1960) 12. K. Maki, Physics 1, 21 (1964) 13. L.P. Gor’kov, Sov. Phys. JETP 9, 1364 (1959) 14. L.P. Gor’kov, Sov. Phys. JETP 10, 593 (1960) 15. V.L. Ginzburg and L.D. Landau, Zh. Exp. Teor. Fiz. 20, 1064 (1950) 16. A.A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957) 17. L.P. Gor’kov, Sov. Phys. JETP 17, 518 (1963) 18. A.M. Clogston, Phys. Rev. Lett. 9, 266 (1962) 19. B.S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962) 20. K. Maki and T. Tsuneto, Progr. Theor. Phys. 31, 945 (1964) 21. L.N. Bulaevskii, Sov. Phys. JETP 38, 634 (1974) 22. P. G. de Gennes and M. Tinkham, Physics 1, 107 (1964) 23. G. Eilenberger, Z. Physik 217, 195 (1968) 24. A.I. Larkin and Yu.V. Ovchinnikov, Sov. Phys. JETP 28, 1200 (1969) 25. G.M. Eliashberg, Sov. Phys. JETP 34, 668 (1972) 26. E. A. Shapoval, Sov. Phys. JETP 20, 675 (1965); ibid. 22, 647 (1966) 27. J.W. Serene and D. Rainer, Phys. Rep. 101, 222 (1983) 28. L.D. Landau and E. M. Lifshitz, In: Quantum Mechanics: Part 1 (Pergamon Press, London, 1985) 29. K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970) 30. G.L¨uders and K.D. Usadel, In: The methods of the Correlation Function in Superconductivity Theory (Springer, Berlin, 1971) 31. N. Kopnin, In: Theory of Nonequilibrium Superconductivity (Clarendon Press, Oxford, 2001)

6

Impurity Nanostructures and Quantum Interference in Superconductors

D. K. Morr University of Illinois at Chicago, Illinois, USA

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.2 Review: Single Impurity Effects in Conventional and Unconventional Superconductors . . . . . . . . . . . 228 6.2.1 Impurities in s-wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6.2.2 Resonant Impurity States in dx2 −y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.2.3 Impurities and First Order Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .230 6.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 ˆ 6.3.1 T-matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231 6.3.2 Bogoliubov–de Gennes Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.3.3 Real-Space Dyson Equation for Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233 6.4 Quantum Interference and Quantum Imaging in s-wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . 234 6.4.1 Quantum Interference Induced Quantum Phase Transitions in s-wave Superconductors . . . . . . . 234 6.4.2 Quantum Corrals and Quantum Imaging in s-wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . .240 6.5 Quantum Interference Phenomena in dx2 − y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.6 Molecules on the Surface of dx2 − y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

6.1

Introduction

The study of impurities in superconductors has attracted significant experimental and theoretical interest since the mid 1950s. Impurities were first studied in the context of disorder effects that are induced by a finite concentration of impurities in the physical properties of s-wave superconductors. Motivated by experiments on “dirty” s-wave superconductors, Anderson [1], as well as Abrikosov and Gorkov [2] proposed that since non-magnetic impurities do not perturb the pairing of time-reversed states in an s-wave superconductor, the bulk Tc as well as the average bulk superconducting gap are unaffected by weak non-magnetic impurity scattering, a result now known as the Anderson theorem. Subsequently, Abrikosov and Gorkov (AG) [3] in-

vestigated the effects of magnetic impurities on the bulk properties of s-wave superconductors. They showed that magnetic impurities are pairbreaking, since they lift the degeneracy of the spin-↑ and spin-↓ states. This pairbreaking effect leads to a reduction of the bulk superconducting gap, the bulk critical temperature, Tc , and to modifications in the disorder-averaged density of states. The predictions of the AG theory were subsequently confirmed in tunneling experiments by Woolf and Reif [4] who measured the dependence of the density of states on the impurity concentration. Abrikosov and Gorkov also predicted the possibility for gapless superconductivity in s-wave superconductors, an effect which arises since with increasing impurity concentration the superconducting gap is suppressed faster than the superconducting order parameter.

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D. K. Morr

In contrast to disorder-averaged properties,the effect of a single magnetic impurity on the local electronic structure of an s-wave superconductor were first studied in the mid 1960s by Lu [5], Shiba [6], and Rusinov [7]. They demonstrated that the pairbreaking nature of a magnetic impurity leads to the formation of a fermionic bound state,whose spectroscopic signature in the local density of states (LDOS) are peaks at energies below the superconducting gap. Experimental advances in scanning tunneling microscopy (STM) since the early 1990s, made it possible to directly test these predictions by measuring the LDOS in the vicinity of single magnetic and nonmagnetic impurities. In high-resolution STM experiments, Yazdani et al. [8] showed that when a magnetic Gd atom is placed on the surface of the s-wave superconductor Nb, a fermionic bound states is induced in its vicinity whose energy lies inside the superconducting gap. In contrast, the LDOS around a non-magnetic Au atom does not exhibit the signature of a fermionic bound state. These results confirmed the predictions by Anderson [1] and Abrikosov and Gorkov [2,3], as well as those by Lu [5], Shiba [6] and Rusinov [7] on an atomic length scale. The discovery of the high-temperature superconductors has spurred a significant theoretical [9–24] and experimental [25–28] interest in the study of single impurity effects in these materials. The reasons for this profound interest are twofold. First, the unconventional dx2 −y 2 -wave symmetry of the cuprate superconductors implies that magnetic as well as non-magnetic impurities are pair-breaking, thus induce fermionic resonance states and hence possess a dramatic effect on the properties of the superconducting state. In addition, due to the dx2 −y 2 -wave symmetry of the superconducting order parameter, the spatial form of the resonance states is highly anisotropic. This spatial anisotropy in turn is an important experimental tool for identifying the symmetry of unconventional superconductors in general. Second, by perturbing the local electronic structure of a superconductor, impurities can provide insight into the still elusive microscopic mechanism of hightemperature superconductivity, which is a topic of ongoing scientific debate. For example, it was argued that impurities can provide insight into the nature

of collective excitations in the cuprates [29–31], and might even be able to reveal the nature of the bosonic pairing mode giving rise to high-temperature superconductivity [32]. Moreover, it was recently shown that quantum interference effects due to scattering of multiple impurities yield information on the spatial and frequency dependence of the superconducting correlations [33–37] (for a more comprehensive discussion of impurity effects in superconductors, we refer the reader to the excellent review by Balatsky, Vekhter and Zhu [24]). In contrast to the type of impurities studied in superconductors, molecules possess internal vibrational or rotational modes whose effects on metallic surfaces have been extensively investigated in the context of inelastic electron tunneling spectroscopy (for a recent review, see [38]) Starting with the study of trapped (i.e., localized) phonon modes at interfaces [39, 40], it was shown that molecular modes lead to a step-like feature in the LDOS, and a peak in its first derivative, at the mode frequency [41–43]. Recently, Gross et al. [44] demonstrated that the spatial pattern of the LDOS around a large Lander molecule is determined by quantum interference between electrons that are scattered by different parts of the molecule [45]. Such a molecule therefore represents a molecular nanostructure in which spatially separated localized bosonic modes interact with conduction electrons giving rise to novel quantum interference effects. Over the last few years, a new research direction has emerged in which ordered nanoscale impurity structures, so-called quantum corrals, are used to reveal the wavelike nature of electrons in condensed matter systems [46–51]. Nanostructures are suitable for this task since their length scale is of the same order as the typical wavelength of electrons,  ∼ 1/kF (kF is the Fermi wave-vector). In a groundbreaking experiment,Manoharan et al.[48] demonstrated that electronic waves, similar to light waves, can be used to form quantum images. In this experiment, an elliptical quantum corral consisting of magnetic Co atoms, was placed on a metallic Cu(111) surface. An additional Co atom was placed in one of the foci of the ellipse, while the other focus was left empty. At low temperature, Manoharan et al. observed Kondo-

6

Impurity Nanostructures and Quantum Interference

screening of the Co atom located in the focus of the corral through the formation of a Kondo resonance in the Cu LDOS. In addition, they found that a quantum image of the Kondo resonance was formed in the empty focus. This projection of the Kondo resonance from the occupied into the empty focus utilized the specific spatial form of the corral’s low energy electronic eigenmodes. As shown in detail in a number of interesting theoretical studies [52], these eigenmodes were created by quantum interference of electronic waves that are scattered off the corral wall’s Co atoms (for a recent review, see [53]). This experiment beautifully demonstrated that quantum interference of electronic waves can be utilized for “custom-designing” exciting new quantum phenomena. Further examples of nanostructures which exhibit a variety of intriguing quantum effects include triangular [49] and ferromagnetic [50, 51] quantum corrals, optical quantum corrals [54–56], and interacting Kondo impurities [57–59]. In this chapter, we discuss a series of exciting quantum phenomena at the intersection of the three fields discussed above by studying the effects of ordered nanoscale impurity structures and molecules on the local electronic structure of superconductors. It is the interplay between quantum interference and the nature of the superconducting correlations that is responsible for the emergence of novel quantum effects ranging from unconventional quantum imaging,“optical” selection rules and the reversal of pairbreaking effects to a zero-bias conductance peak, the screening of impurity states, quantum interference induced quantum phase transitions, and new possibilities for manipulating a superconductor’s local electronic structure. Conventional and unconventional superconductors each possess unique properties that are essential for the emergence of these effects. For example, the formation of a fermionic bound state around a magnetic impurity in an s-wave superconductor provides a new “quantum candle” (i.e.,a characteristic signature in the LDOS viz. peaks inside the superconducting gap) for quantum imaging. In addition, the qualitatively different effect of magnetic and non-magnetic impurities on the local electronic structure permits the study of quantum interference effects separately from the formation of

227

fermionic impurity states. Furthermore, the possibility to tune an s-wave superconductors through a first order phase transition in which its ground state spin polarization changes [11, 14, 24, 60] opens the possibility to study quantum interference induced phase transitions [61,62]. On the other hand, unconventional superconductors give rise to novel types of quantum interference effects due to the momentum dependence of their order parameter. The interplay between this momentum dependence and the geometry and orientation of a nanoscale impurity structure leads to destructive or constructive quantum interference and the possibility to screen impurity states and to identify the symmetry of unconventional superconductors in general. The first observation of quantum interference effects in the high-temperature superconductor YBa2 Cu3 O6+x was recently reported in STM experiments by Derro et al. [63] who studied the LDOS of the CuO chains. They observed two important results below Tc : first, a gap exists in the LDOS of the CuO chains, and second, several fermionic resonances exist inside the gap. Specifically, they found two hole-like and two particle-like peaks in the LDOS at symmetric energies, ±6 meV and ±13 meV. Since a single impurity in general gives rise to one particlelike and one hole-like peak in the LDOS only, it was suggested [61] that the observed two pairs of peaks arise from quantum interference between two oxygen defect sites in the chains. This simple (two impurity) nanostructure gives rise to a hybridization of the impurity states and the formation of bonding and antibonding resonance with a resulting four peak structure in the LDOS. Quantum interference effects were also recently investigated by McElroy et al. [64], Howald et al. [65] and Vershinin et al. [66] who studied Friedel-like oscillations in the LDOS of the cuprate superconductor Bi2Sr2 CaCuO8+ı . These oscillations arise from quantum interference due to spatially random and weakly scattering impurities. It was shown that the combined momentum and frequency dependence of these oscillations in the superconducting state provides insight into the electronic structure of this complex material, and in particular, into the form of its normal state Fermi surface [64]

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as well as the presence of collective modes or unconventional ordering [65,66]. Finally, quantum interference phenomena are not only of great fundamental interest, but might also possess important applications in the field of spin electronics [67] and quantum information technology [68], as these phenomena might lead to the creation of new coupling mechanisms between quantum qubits. The remainder of this chapter is organized as follows: in Sect. 6.2 we briefly review some of the salient effects of single impurities in conventional and unconventional superconductors.In Sect.6.3 we present the theoretical formalism for the study of impurity nanostructures in superconductors. In Sects. 6.4 and 6.5 we discuss a series of novel quantum interference effects in s-wave and dx2 −y 2 -wave superconductors, respectively. Finally, in Sect. 6.7 we summarize our results and conclusions and discuss future directions for the study of impurity nanostructures and quantum interference phenomena.

6.2

Review: Single Impurity Effects in Conventional and Unconventional Superconductors

Before discussing novel quantum interference phenomena in impurity nanostructures, we briefly review some of the salient effects that are induced by single impurities in the local electronic structure of conventional and unconventional superconductors. Disorder effects arising from a finite concentration of impurities are by themselves of great current interest, but their discussion is beyond the scope of this brief review. 6.2.1

Impurities in s-wave Superconductors

One of the most important properties of impurities in s-wave superconductors is that the nature of their scattering potential (magnetic versus non-magnetic) determines their qualitative effects on the local electronic structure of the superconductor. The interaction of a point-like impurity with the electronic degrees of freedom of the superconductor is described by the scattering Hamiltonian

Hscatt =



 †  cR,˛ U ı˛,ˇ + JS ·  ˛ˇ cR,ˇ ,

(6.1)

˛,ˇ=↑,↓ † , cR,˛ are fermionic operators that create where cR,˛ and annihilate an electron with spin ˛ at the impurity site, R, respectively. U and J are the impurity’s non-magnetic and magnetic scattering potential, respectively, S is the spin operator of the impurity with spin S and  are the Pauli matrices. As was pointed out by Anderson [1] and Abrikosov and Gorkov [2], the pairing of time-reversed states in an s-wave superconductor is unaffected by a non-magnetic scattering potential, which therefore does not induce a fermionic bound state in its vicinity. As a result, a sufficiently low concentration of non-magnetic impurities (with sufficiently weak scattering potential) does not reduce the bulk superconducting order parameter from its value in a clean system. Note, however, that the local superconducting order parameter around non-magnetic impurities is gradually suppressed with increasing scattering strength [11]. Since the scattering off magnetic impurities lifts the degeneracy of the spin-↑ and spin-↓ electrons forming a Cooper-pair, magnetic impurities are pairbreaking and hence exert a dramatic effect on the properties of s-wave superconductors, as first pointed out by Abrikosov and Gorkov [3]. The effects of single isolated magnetic impurities on the local electronic structure of the superconductor were systematically studied by Lu [5], Shiba [6] and Rusiˆ nov [7] within the T-matrix formalism (a more detailed discussion of this approach is provided in Sect. 6.3). They showed that a magnetic impurity induces a fermionic bound state inside the superconducting gap whose energy is given by

§b = 0

1 − [JS N(0)]2 1 + [JS N(0)]2

,

(6.2)

where N(0) is the host system’s density of states at the Fermi energy in the normal state. For a magnetic impurity with Sˆz , particle-hole mixing in the superconducting state leads to a wave-function of the induced bound state that possesses a particlelike and a hole-like component, denoted by |p, ↑ and |h, ↓, respectively (↑, ↓ represent the opposite spin quantum numbers, Sz = ±1/2 of the two com-

6

Impurity Nanostructures and Quantum Interference

ponents). The spectroscopic signature of these components are two peaks in the LDOS at energies ±§b . With increasing scattering strength, the bound state shifts to lower energies, and crosses zero energy at (JS)cr = 1/N(0). Sakurai [60] showed that this crossing is the signature of a first order quantum phase transition in which the spin polarization of the superconductor’s ground state changes, as discussed in more detail in Sect. 6.2.3. The spatial form of the superconducting order parameter in the vicinity of a magnetic impurity was studied by a number of different analytical [69,70] and numerical [11] approaches. These studies find that the pairbreaking effect of a magnetic impurity leads to a local suppression of the superconducting order parameter which is the largest at the impurity site.The deviation of the order parameter from its bulk value, ı = (r)−0 , decays as ∼ (kF r)−2 for r   ( beeing the superconducting coherence length) as one moves away from the impurity site. Since typically kF ∼ a−1 0 with a0 being the lattice constant, while  ∼ 102 − 103a0 , one finds, that the order parameter recovers its bulk value over distances from the impurity of a few lattice constants, and not as expected, on a length scale set by the superconducting coherence length (an exponential dependence of ı on  exists for only r >  [69, 70], a limit, which for the reasons given above is irrelevant). Finally, Satori et al. [71] using the numerical renormalization group (NRG) reported that Kondoscreening of a magnetic impurity in an s-wave superconductor is only possible for values of J exceeding a critical value Jc (for which Tk /0  0.3), in contrast to metallic systems where Jc = 0 [72]. As mentioned above, the predictions by Lu [5], Shiba [6] and Rusinov [7] for the form of the fermionic bound state in the vicinity of a magnetic impurity were confirmed by Yazdani et al. [8] via high resolution scanning tunneling spectroscopy experiments on the s-wave superconductor Nb. Moreover, these experiments did not find any evidence for Kondo screening in the superconducting state, in agreement with the results by Satori et al. [71].

6.2.2

229

Resonant Impurity States in dx 2 −y 2 -wave Superconductors

The discovery of the cuprate superconductors led to a number of theoretical [9–24] and experimental [25–28] studies focusing on the interplay between the symmetry of the superconducting order parameter and impurity effects (for other unconventional superconductors, such as Sr2 RuO4 , see, for example, Refs. [73–75]). These studies showed that the dx2−y 2 wave symmetry of the high-temperature superconductors possesses two important consequences for the nature of impurity induced fermionic states.First, since  the on-site superconducting gap vanishes (i.e., k (k) = 0) not only magnetic impurities, but also non-magnetic impurities are pairbreaking in a dx2 −y 2 -wave superconductor, incontrast to s-wave superconductors. Second, the LDOS of the clean system vanishes linearly with frequency in the low energy limit. Thus any impurity state at non-zero frequency is resonant in that it is coupled to the particle-hole continuum which leads to a finite lifetime of the state. The properties of a resonant fermionic state induced by a non-magnetic impurity were first studied by Balatsky, Salkola and Rosengren [9] within the ˆ T-matrix formalism. They showed that in the unitary scattering limit, the frequency, §res , and inverse lifetime,  , of the resonant state (for a particle-hole symmetric system) are given by   −1 §2  8 §res = c0 ln ;  = res , (6.3) 2 c c0 where c = [UN(0)]−1 . Since the superconducting order parameter with dx2 −y 2 -wave symmetry vanishes along the nodal (110) direction, the resonant state is delocalized along this direction, leading to its finite lifetime. An important consequence of this delocalization is that the resonance state decays algebraically as ∼ 1/r 2 along the nodal directions, but decays exponentially along the antinodal (100) direction [10]. Moreover, using the BdG-formalism, several theoretical studies [11,76] showed that while the superconducting order parameter is locally suppressed by non-magnetic impurities, it relaxes back to its bulk value within a few lattice spacings from the impurity site.

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D. K. Morr

Over the last few years, a series of high resolution STM experiments have studied the form of resonance states around non-magnetic impurities [25– 27],specifically Au [26] and Zn [27] atoms on the surface of the cuprate superconductor Bi2Sr2 CaCu2O8+ı . These experiments find that the frequency and lifetime, as well as the 1/r 2-decay of the resonance along the nodal directions are in good agreement with theoretical predictions. However, the experimentally measured large intensity of the resonance at the Zn site was quite unexpected since the closed orbital shell configuration of Zn suggests that it is a unitary scatterer, which should lead to a low intensity of the resonance on-site. In order to resolve this contradiction,three different proposals were made.First,it was argued that the experimentally measured LDOS corresponds to a spatially averaged density of states due to the tunneling path of the electrons from the STM tip through the BiO and SrO-layers into the CuO2 plane. Thus a direct comparison with experiment requires that the theoretical results for the LDOS be convoluted with a so-called filter function [20, 77], which accounts for the orbital overlap between the Bi, O and Cu atoms. Second, it was suggested that the discrepancy can be explained by considering spatially extended scattering centers [78]. Third, it was proposed [21–23] that Zn impurities induce local magnetic moments that are Kondo-screened at low temperatures and thus induce Kondo resonances in the LDOS. The possibility for Kondo-screening of a magnetic impurity in a dx2 −y 2 -wave superconductor was first considered by Withoff and Fradkin [79] within the so-called pseudo-gap model (for a more detailed discussion, see [80]). This model considers a system with linearly vanishing LDOS (i.e., N(!) = C|!| for |!| < D), but does not account for superconducting long range order with dx2 −y 2 -symmetry. Withoff and Fradkin showed that a magnetic impurity in such a system can undergo a Kondo-effect if its scattering strength, J exceeds a critical value, JK = (CD)−1 . At the moment, no consensus has been reached on the correct theoretical explanation for the experimentally measured LDOS near Zn atoms. In order to shed further light on the properties of magnetic impurities in dx2−y 2 -wave superconductors,

Hudson et al. [28] studied the LDOS in the vicinity of magnetic Ni impurities located on the surface of Bi2Sr2 CaCu2 O8+ı . The results of their STM measurements are in good agreement with theoretical results ˆ of the T-matrix formalism [11]. Since this formalism assumes that the spin of the magnetic impurity can be treated as a static (classical) variable, these experiments suggest that magnetic Ni impurities are not Kondo-screened in the superconducting state. Finally, it was recently argued that quantum interference effects between electrons that are scattered by two or more impurities directly reflect the real space structure of superconducting correlations, and thus provide novel spatially resolved information on the pairing mechanism in the cuprate superconductors [33–37]; we will return to a discussion of this topic in Sect. 6.5. 6.2.3

Impurities and First Order Quantum Phase Transitions

It was first pointed out by Sakurai [60] that an s-wave superconductor can be tuned through a first order quantum phase transition with the help of magnetic impurities. As discussed in Sect. 6.2.1, when the scattering strength, J, of a single magnetic impurity is increased, the energy of the bound state that is induced by the impurity decreases, and eventually crosses zero energy at a critical value Jcr . Sakurai realized that such a zero-energy crossing is the signature of a first order quantum phase transition, in which the ground state of the superconductor changes from a state with spin polarization Sz  = 0 to a state with Sz  = ±1/2, where the sign depends on the orientation of the magnetic impurity. The spin polarization of a d-dimensional superconductor is defined as 1 Sz  = 2



∞ d

d r

d!

(6.4)

−∞

 × N↑ (r, !) − N↓ (r, !) nF (!) , where N (r, !) is the LDOS of the electrons with spin  =↑, ↓, and nF is the Fermi distribution function. This phase transition arises from a level crossing in the superconductor’s free energy, F , resulting in a discontinuity of @F /@J at the transition;

6

Impurity Nanostructures and Quantum Interference

hence the first order nature of the transition (for a more detailed discussion see [24]). It was subsequently shown that for a single magnetic impurity, the quantum phase transition possesses a characteristic signature in the on-site superconducting order parameter which decreases discontinuously and changes sign at Jcr , thus undergoing a -phase shift in comparison to its bulk value. In a particle-hole symmetric dx2 −y 2 -wave superconductor, this type of quantum phase transition cannot occur since the real part of the local fermionic Greens functions vanishes linearly with frequency at low energies. As a result, the resonance state cannot cross zero energy with increasing J [9–11]. If the particle-hole symmetry is broken, a quantum phase transition becomes possible at Jcr < ∞. However, for large values of Jcr (i.e., small particle-hole asymmetry), this transition competes with the onset of Kondo-screening.

6.3

Formalism

The theoretical study of impurities in superconductors has primarily used two complementary theoretˆ ical approaches: the T-matrix [5–7, 24, 81] and the ˆ Bogoliubov de-Gennes [82] formalism. While the Tmatrix approach allows one to study a finite number of impurities in an infinitely large host system, it does not account for the local suppression of the superconducting order parameter around impurities. In contrast, within the BdG formalism, one can selfconsistently calculate the local superconducting order parameter, but is confined to consider finite-size host systems. By using both theoretical approaches for the study of impurity effects, it is possible to investigate the importance of spatial fluctuations of the superconducting order parameter, and to eliminate spurious finite size effects. Both approaches assume that the impurities under consideration do not possess an internal degree of freedom and can thus be treated as static,classical variables.While this restriction is obviously satisfied for non-magnetic impurities, it requires that for the study of magnetic impurities, one considers the classical (static) limit S → ∞ with J → 0 and JS = const. On the other hand, a description of Kondo screening of a magnetic impurity

231

requires that the full quantum dynamical nature of the impurity spin be taken into account. The Kondo effect thus corresponds to the opposite limit from the classical case considered in the BdG and Tˆ -matrix formalism.Hence,the use of the BdG or Tˆ -matrix formalism for the study of magnetic impurities is only justified in those cases in which magnetic impurities do not undergo a Kondo effect. For both s-wave [71] and dx2 −y 2 -wave superconductors [79, 80] it was argued that a Kondo-effect can only occur if the coupling between the magnetic impurities and the delocalized electrons exceeds a critical value, Jc . Indeed, STM experiments measuring the LDOS around magnetic Gd atoms in the s-wave superconductor Nb [8], and around magnetic Ni atoms in the cuprate superconductor Bi2Sr2 CaCu2 O8+ı do not find any evidence for Kondo-screening below Tc . Finally, in order to study the effects of molecular nanostructures on the surface of dx2 −y 2 -wave superconductors, we have developed a theoretical formalism in which the Dyson equation is solved in real space [83–85] by computing the spatially dependent fermionic self-energy and vertex corrections. In what follows,we present a brief derivation of the ˆ T-matrix and BdG formalisms for impurity nanostructures as well as of the real-space Dyson equation for molecular nanostructures. 6.3.1

Tˆ -matrix Formalism

ˆ Starting point for the derivation of the T-matrix formalism for impurity nanostructures is the Hamiltonian H = Hhost + Hint where  † Hhost = k ck, ck, (6.5) k,

+

  † † k ck,↑ c−k,↓ + H.c. ; k

Hint =

   ˛,ˇ

 Uri ı˛,ˇ + Jri Sri ˛,ˇ cr†i ,˛ cri ,ˇ . (6.6)

i

Here, Hhost describes the electronic degrees of freedom of the superconducting host system and Hint represents the interaction of the host electrons with the impurity nanostructure. Specifically, cr†i, cri , are

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D. K. Morr

the fermionic creation and annihilation operators for a (host) conduction electron at site ri with spin  , and k is the host system’s normal state tightbinding dispersion. k is the superconducting gap with k = 0 for an s-wave superconductor, and  k = 0 cos kx − cos ky /2 for a dx2 −y 2 -wave superconductor. The primed sum runs over all sites ri (i = 1, . . . , N) of the N impurities forming the nanostructure. Sri is the spin-operator of the impurity at site ri with spin quantum number Sri , and Uri and Jri are the impurity’s non-magnetic and magnetic scattering potential, respectively. Since one assumes that a spin impurity can be treated as a classical (i.e., static) variable, one replaces the operator Sri in the limit Sri → ∞ by the classical vector Sri (sin i , 0, cos i ). The fermionic Green’s function of the superconductor in the presence of the impurity nanostrucˆ r,  −   ) = −T ¦r ()¦r† (  ) with ¦r† = ture, G(r, † † , cr,↑, cr,↓), in Matsubara frequency space is (cr,↑ , cr,↓ given by [6,11,81,86,87]  ˆ r , !n ) = Gˆ 0 (r, r, !n ) + G(r, Gˆ 0 (r, ri , !n ) i,j

ˆ i , rj , !n )Gˆ 0 (rj , r, !n ) . × T(r

(6.7)

ˆ The T-matrix is obtained from the Bethe–Salpeter equation ˆ i , rj , !n ) = Vˆ ri ıri ,rj T(r +Vˆ ri

N 

ˆ l , rj , !n ) , Gˆ 0 (ri , rl , !n )T(r

(6.8)

l=1

 1 N(r, !) = − Im Gˆ 11 (r, ! + iı)   + Gˆ 22(r, ! + iı) , (6.11) ˆ with ı = 0+ . Note that within the T-matrix formalism, any interaction between the magnetic impurities is only taken into account to the extent that it determines the angles, i , of the impurity spins. 6.3.2

Bogoliubov–de Gennes Formalism

The Bogoliubov–de Gennes formalism [82] allows us to treat quantum interference between electrons that are scattered by the impurity nanostructure and a spatial variation of the superconducting order parameter on equal footing. In order to derive the BdG equations, one considers a finite size (host) system that is described by the real space version of the Hamiltonian presented in Eqs. (6.5) and (6.6). In order to obtain the eigenenergies of this finite size system, as well as the electronic wave-functions, one solves the eigenvalue equation    H + ij  uj,n  u ij = En i,n , ∗ij −Hij− vj,n vi,n

with Hij± = tij + (Vi± − )ıij , and the sum runs over all sites j of the system. tij is the hopping integral between sites i and j, and is the chemical potential. The scattering potential, Vi± =

and i and i are the Pauli matrices in spin and Nambu space, respectively. It follows from Eq. (6.9) that the effective magnetic scattering strength of an impurity is given by Jri Sri . The Greens function of the unperturbed (clean) superconductor in momentum space is Gˆ 0−1 (k, i!n ) = [i!n 0 − k 3 ] 0 + k 2 2 . (6.10) Finally, the LDOS, N(r, !), is obtained from Eqs. (6.7)-(6.10) via

 

 Urn ± Jrn Srn ıi,rn ,

(6.13)

n

where

 Vˆ ri = Uri 0 + Jri Sri (3 cos i + 1 sin i ) 3 (6.9)

(6.12)

j

with non-magnetic and magnetic scattering potential Urn and Jrn , respectively, is only non-zero at the impurity sites rn (n = 1, . . . , N). Srn is the impurity’s spin quantum number and we assume in what follows that all impurity spins are aligned along the zˆ -axis. In an s-wave superconductor, the superconducting order parameter ij = i ıij is local and given by i = −Vp

 n



En ui,n vi,n tanh 2kB T

 ,

(6.14)

6

where Vp is the effective pairing interaction and the sum runs over all eigenstates of the system. Eqs. (6.12) and (6.14) are solved self-consistently in order to obtain the energy, En , of all eigenstates of the system, the local superconducting order parameter, i , as well as the Bogoliubov coefficients ui,n , vi,n of state n at site i. Finally, the LDOS at site r is obtained via 

N(!, r) = u2r,nı(! − En ) n



2 ı(! + En ) . + vr,n

6.3.3

(6.15)

Real-Space Dyson Equation for Molecules

A nanostructure consisting of N molecules on the surface of a dx2 −y 2 -wave superconductor is described by the Hamiltonian H = Hhost + Hb + Hint [83,85], where Hhost is given in Eq. (6.5) and  Hb = !0 a†ri ari i

 a†ri + ari cr†i , cri , .

the case of two interacting molecules √ considered be+ 4 with  ≡ low, we have §1√= !0 , §2 = !0 1 √ (2) J/!0 , gr(1) = g/ 2 and g = ±g/[ 2(1 + 4)1/4 ]. r1,2 1,2 The full fermionic Green’s function is given by ˆ r, !n ) = Gˆ 0 (r, r, !n ) G(r, (6.18)  ˆ i , rj , !n )Gˆ 0 (rj , r, !n ) , + Gˆ 0 (r, ri , !n )£(r i,j

where £ˆ is the fermionic self-energy obtained from the Bethe–Salpeter equation ˆ i , rj , !n ) = £ˆ 0 (ri , rj , !n ) £(r (6.19)  ˆ q , rj , !n ) , + £ˆ 0 (ri , rp , !n )Gˆ 0 (rp , rq , !n )£(r p,q

with p, q = 1, . . . , N and  3 0gr(l)i £ˆ 0 (ri , rj , !n ) = T

(6.20)

m,l

The retarded form of the bosonic propagator, Dl (m ) is given by

i,j

 

233

× Gˆ 0 (ri , rj , !n + m )Dl (m )3 0 gr(l)j .

2   a†ri + ari − a†rj − arj , +J Hint = g

Impurity Nanostructures and Quantum Interference

(6.16)

i,

Hb describes the vibrational modes of the molecules with frequency !0 , and the primed sum runs over all locations ri (i = 1, .., N) of the molecules. These modes, which are represented as bosons [88] with a† being the bosonic creation operator, are coupled by a quadratic potential (second term in Hb with (J ≥ 0)) which is the quantum mechanical analog of two harmonic oscillators coupled by a spring [89]. Hint represents the interaction between the superconductor and the molecules and g is the boson-fermion scattering vertex. Diagonalizing Hb via a Bogoliubov transformation to operators b†l , bl (l = 1, . . . , N),  one obtains Hb = l §l b†l bl and    gr(l)i b†l + bl cr†i , cri , , (6.17) Hint = i,l,

where gr(l)i is a site and mode dependent interaction vertex,and §l is the bosonic excitation spectrum.For

DlR( + iı) = [ + i ( ) − §l ]−1

(6.21)

− [ + i ( ) + §l ]−1 . Finally, the LDOS N(r, !) = A11 + A22 where Aii (r, !) = − 1 Im Gˆ ii (r, ! + iı) is obtained from Eq. (6.18) with ı = 0+ . Let us briefly discuss the effect of vertex corrections [83]. The lowest order vertex corrections scale as ıg/g ∼ (gkF /4vF )2 F where F is a function of O(1) for bosonic and fermionic frequencies smaller than 0 . For the parameter range considered below, gkF /4vF  1 and vertex corrections can thus be neglected.Moreover,the interaction,Hint ,not only leads to changes in the fermionic LDOS, but also affects the bosonic excitation spectrum twofold. First, Hint shifts the unperturbed bosonic frequencies, a shift that we take to be included in the effective values for §l . Second, the bosonic modes acquire a finite lifetime,  (!) = 0. The lowest order bosonic selfenergy correction yields  (!) = g 2 ! 3 /(6vF2 v2 ),

234

D. K. Morr

where vF (v ) is the Fermi (superconducting) velocity at the nodal points perpendicular (parallel) to the Fermi surface. Since for all cases considered below  (!0 )  !0 , which is satisfied for g  500 ,  (!) can be neglected and we take for simplicity  (!) = 0+ .

6.4

Quantum Interference and Quantum Imaging in s-wave Superconductors

We begin by studying quantum interference effects in conventional s-wave superconductors. In order to avoid complications arising from the coupling of bulk and surface states,we consider two-dimensional (2D) superconducting host systems. While many conventional superconductors are intrinsically not 2D, it has become possible to grow thin superconducting films that are only a few layers thick and practically exhibit the same superconducting properties as the bulk systems. These superconducting films are therefore the candidate host systems to explore the quantum effects discussed in the following. The results presented below are obtained using ˆ both the T-matrix and the BdG formalism. For ˆ our calculations within the T-matrix formalism we take the normal state dispersion to be given by k = k2/2m − ( = 1) with chemical potential

= kF2 /2m,kF = /2,m−1 /0 = 15,and ı = 0.2 meV. For the solution of the BdG equations, we considered a normal state dispersion with nearest-neighbor hopping integral t, next-nearest neighbor hopping integral t  /t = 0.2, chemical potential /t = −2, yielding an almost circular Fermi surface (FS) with kF ≈ /2. We chose the pairing strength to be given by Vp /t = 2.5, such that in the clean system, the translationally invariant superconducting order parameter is 0 /t ≈ 0.112, resulting in a superconducting coherence length of c = vF /0 ≈ 25a0 , where vF is the Fermi velocity, and a0 is the lattice constant (in the following, we set a0 = 1). Unless otherwise stated, we consider a system with N = Nx × Ny sites where Nx = 28 and Ny = 18. This choice of Nx , Ny accounts for the spatial anisotropy of the impurity structures considered below.

6.4.1

Quantum Interference Induced Quantum Phase Transitions in s-wave Superconductors

The simplest impurity nanostructure that exhibits interesting quantum interference effects consists of two magnetic impurities. For simplicity, we first consider impurities with parallel spins that are aligned along the z-axis, i.e., Sˆz. If the impurities are infinitely far apart (with interimpurity distance r = ∞), the two bound states with components |p, ↑, i and |h, ↓, i (i = 1, 2) are degenerate. This degeneracy is lifted for r < ∞, since there exists a non-zero probability that an electron scattered by one of the impurities is also scattered by the second impurity. Quantum interference between the scattering events off both impurities leads to a coupling and hence hybridization of the induced bound states. In order to gain insight into the form of the hybridized impurity states, we use a toy model in which the unhybridized bound states of energy E0 are coupled by a hopping term D(r), which depends on the distance, r, between the two impurities. This hopping term is responsible for the hybridization of the impurity states, and the formation of even and odd bound states with energies Ee,o = E0 ± D(r). The particlelike components of the even and odd wavefunctions √ are given by |p, ↑e,o = (|p, ↑, 1 ± |p, ↑, 2)/ 2, and similarly for the hole-like components. As a result of the hybridization, the bound state peaks in the LDOS are split by an energy E = Ee − Eo = 2D(r). This picture is confirmed by the numerical solution of the ˆ T-matrix equations, as shown in Fig. 6.1(a) where we present the LDOS at r1 for the case of two impurities, located at r1 = (0, 0) and r2 = (2, 0) (r = 2) with scattering potential JS/0 = 15 [62]. As expected, the LDOS exhibits four mid-gap peaks with peak (1), (2) representing the particle-like components |p, ↑e,o at energies §e,o ↑ , respectively, and peak (3), (4) representing the hole-like components |h, ↓e,o at energies §e,o ↓ , respectively. Similar results were also obtained by Flatt´e and Reynolds [16]. In order to determine which peaks in the LDOS correspond to the even and odd states, we plot in Fig. 6.1(b) the spatial dependence of the particlelike states (1) and (2) along r = (x, 0) (the locations of the impurities are indicated by arrows). By sym-

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Fig. 6.1. (a) LDOS at r1 for two magnetic impurities with JS/0 = 15, located at r1 = (0, 0) and r2 = (2, 0) (dotted line). Also shown is the LDOS for an unperturbed system (solid line), and for a single impurity (dashed line) at r1 . (b) Spatial dependence of peak (1) and (2) in the LDOS shown in (a) along r = (x, 0) (the positions of the impurities are indicated by arrows). (c) Same as (b) but for two impurities located at r1 = (0, 0) and r2 = (4, 0). Results are obtained ˆ within the T-matrix formalism

metry, the LDOS of the odd states vanishes at the midpoint between the two impurities, and we can therefore identify peak (2) as the particle-like component of the odd state, while peak (1) represents the particle-like component of the even state. Besides their even/odd symmetry, the spatial depen-

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dence of these two states exhibits another interesting difference: while the odd state exhibits LDOS oscillations well beyond the two impurity region, the even state is primarily confined to the region between the two impurities. This qualitative difference is a direct consequence of the (kF r)-oscillations that are exhibited by the wave-functions of the impurity states. Specifically, since kF = /2 and r = 2, the components |p, ↑, 1 and |p, ↑, 2 are shifted by a phase  = kF r =  outside the two-impurity region (x > 2 in Fig. 6.1(b)) and hence are out-of-phase. As a result, |p, ↑, 1 and |p, ↑, 2 interfere destructively for |p, ↑e and only weak spatial oscillations exist for x > 2. In contrast, |p, ↑o shows constructive interference of |p, ↑, i (i = 1, 2) and its spatial oscillations are enhanced for x > 2. By changing the inter-impurity distance to r = 4 , with  = 2, the interference pattern between the even and odd states is exchanged. |p, ↑e (|p, ↑o) now exhibits strong (weak) oscillations beyond the two impurity region, x > 4, as shown in Fig. 6.1(c). The question naturally arises whether the suppression of the superconducting order parameter in the vicinity of the magnetic impurities, which is not ˆ taken into account within the T-matrix formalism, qualitatively changes the form of the LDOS presented above. In order to answer this question and to consider the spatial variation of the order parameter, we solve the BdG equations for the case of two magnetic impurities located at r1,2 = (∓2, 0) (r = 4) with JS/t = 2, and present the resulting LDOS at r1 in Fig. 6.2(a). A comparison of these results with those of the Tˆ -matrix formalism (see Fig.6.1(a)) shows that the suppression of the order parameter in the vicinity of the impurities leaves the qualitative features of quantum interference in the LDOS unchanged [90]. In order to understand this robustness of the LDOS towards the suppression of the local order parameter, we present in Fig. 6.2(b), the spatial dependence of the order parameter for the case considered in Fig. 6.2(a). As expected, we find that the order parameter is reduced from its bulk value in the vicinity of the impurities, with the largest suppression at the impurity sites (indicated by arrows in Fig. 6.2(b)). Note that the order parameter recovers its bulk value within a few lattice spacing from the impurities. Sur-

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Fig. 6.2. (a) LDOS at r1,2 as a function of energy for two impurities located at r1,2 = (∓2, 0) with JS/t = 2. (b) Superconducting order parameter as a function of position along the line connecting the two impurities at r1,2 = (∓2, 0).The positions of the impurities are denoted by arrows. Results are obtained within the BdG formalism

prisingly enough, even for the relatively small interimpurity distance of r = 4 the superconducting order parameter comes close to its bulk value in the region between the two impurities.We therefore conclude that the superconducting order parameter relaxes back to its bulk value on a length scale r of a few lattice spacings. It is important to note that this lengthscale r is much shorter than the superconducting coherence length, c = vF / ≈ 25a0. This conclusion is in full agreement with numerical [11] and analytical [69, 70] studies of the superconducting order parameter near a single impurity, in which

r was identified as the Fermi wavelength F . In our case, F ≈ 4a0  c . With increasing scattering strength, JS, of the two magnetic impurities, the frequency of the bound states moves to lower energies,as follows from a comparison of the LDOS for JS/t = 2.0 [see Fig.6.2(a)] with that for JS/t = 2.5 [see Fig. 6.3(a)], and eventually crosses zero energy. As discussed in Sect. 6.2.3, this zero-energy crossing is the signature of a first order quantum phase transition, in which the ground state of the superconductor changes from a state with spin polarization Sz  = 0 to a state with Sz  = −1/2 (assuming for the impurity spin S  zˆ ). For a single magnetic impurity, only one quantum phase transition occurs at a critical value, (JS)cr (for the band dispersion considered here, one finds (JS)cr /t = 3.09). In contrast, for the case of two magnetic impurities, quantum interference leads to the emergence of two quantum phase transitions at (JS)cr,1 and (JS)cr,2,corresponding to the transitions Sz  = 0 → −1/2 and Sz  = −1/2 → −1, respectively, of the ground state’s spin polarization. Specifically, the even bound state crosses zero energy at (JS)cr,1 /t ≈ 2.54, such that for JS > (JS)cr,1 , the spin-↑ (spin-↓) peak of the even bound state is hole-like (particle-like), as shown in ˆ Fig. 6.3(b) for JS/t = 2.7. However, while in the Tmatrix approach, the frequency of the even bound state, §e↑,↓ evolves continuously as a function of JS and reaches zero at the transition [62], within the BdG formalism, §e↑,↓ crosses zero-energy discontinuously at (JS)cr,1 , similar to the transition induced by a single magnetic impurity [11, 24]. Specifically, for JS = (JS)cr,1 − 0+ one finds §e↑,↓/t = ∓0.005, while for JS = (JS)cr,1 + 0+ one has §e↑,↓ /t = ±0.001 [the upper (lower) sign corresponds to the spin-↑ (spin-↓) peak]. Interestingly enough, we find that the energy of the odd bound state does not exhibit a discontinuity at (JS)cr,1. A similar discontinuous zero-energy crossing of the bound state energies is found at all phase transitions discussed below. The second quantum phase transition (with Sz  = −1/2 → −1) occurs at (JS)cr,2/t = 3.69 when the odd bound state crosses zero energy, such that for JS > (JS)cr,2 , all spin-↑ (spin-↓) peaks are now located at positive (negative) energies [see Fig. 6.3(c) for JS = 4.0t > (JS)cr,2].

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Fig. 6.4. On-site superconducting order parameter as a function of JS for two impurities located at (a) r1,2 = (∓2, 0) with r = 4, and (b) r1 = (−1, 0) and r2 = (2, 0) with r = 3. Results are obtained within the BdG formalism

Fig. 6.3. LDOS at r1,2 as a function of energy for two impurities located at r1,2 = (∓2, 0) with (a) JS/t = 2.5, (b) JS/t = 2.7, and (c) JS/t = 4.0. Results are obtained within the BdG formalism

The first order phase transitions discussed above are accompanied by a discontinuous change of the (on-site) superconducting order parameter, r1,2 , at r1,2 as shown in Fig. 6.4(a) where we plot the on-site

order parameter as a function of JS for two impurities located at r1,2 = (∓2, 0). While the on-site order parameter changes discontinuously at (JS)cr,1, we find that in contrast to the case of a single magnetic impurity [11, 24, 60], it remains positive and does not exhibit a -phase shift. Moreover, for JS > (JS)cr,1 the superconducting order parameter decreases continuously with increasing JS, and crosses zero at JS/t ≈ 3.15. Interestingly enough, this zero-crossing of the on-site order parameter does not lead to any signature in the LDOS. Finally, we note that the second first order transition at (JS)cr,2 leads to an increase in the magnitude of the superconducting order parameter, in contrast to the transition at (JS)cr,1 , where the magnitude of the superconducting order parameter decreases. For all interimpurity distances we considered, the dependence of the LDOS and the superconducting order parameter on the scattering strength is similar to the one discussed above. However, the splitting between (JS)cr,1 and (JS)cr,2 depends strongly on the interimpurity distance. For example, for r = 3, the two phase transitions occur at (JS)cr,1 /t = 2.87 and (JS)cr,2/t = 2.9, as shown in Fig. 6.4(b). This split-

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Fig. 6.5. (a) Frequency of the even and odd bound state peaks as a function of interimpurity distance r for JS/t = 2.5. (b) Superconducting order parameter, r1,2 at r1,2 as a function of interimpurity distance, r, for several values of JS/t. Results are obtained within the BdG formalism

ting between the two critical values is thus considerably smaller than that for the case r = 4 shown in Fig. 6.4(a). In order to gain further insight into the dependence of the critical values on the interimpurity distance, r, we plot in Fig. 6.5(a) the frequency of the even and odd bound state peaks as a function of r. As the distance between the two impurities is changed, the frequency of the even and odd bound state peaks, as well as the splitting between them oscillates [16,62]. The strength of the coupling, D(r), between the impurity states can be directly

obtained from the frequency splitting between the even and odd bound states, E = Ee − Eo = 2D(r). The results shown in Fig. 6.5(a) demonstrate that the effective coupling between the impurity states does not only vary in magnitude, but also changes sign as the distance between the impurities is changed. Specifically, we find from Fig. 6.5(a) that the splitting between the even and odd bound states is significantly smaller for r = 3 than for r = 4, implying |D(r = 3)| < |D(r = 4)|. The smaller value of D for r = 3 also explains the smaller splitting between the critical values (JS)cr,1 and (JS)cr,2 . Note that the dependence of D(r) on r is determined by scattering processes involving both impurities. Consider, for example, the wave function of an electron that is scattered by the impurity at r1 . The sign and magnitude of its wave function at r2 (and hence those of D(r)) depend (a) on the distance between the impurities due to the (kF r)-oscillations of the electronic wave-function, and (b) the scattering phase shift at r1. Since the latter depends on JS, the relative splitting between the even and odd bound states also varies with JS, as further discussed below. Note in particular that if the electron’s wave function vanishes at r2 , the impurity bound states associated with each of the two impurities cannot hybridize and hence remain degenerate [16,62]. It also follows from Fig. 6.5(a) that when the interimpurity distance is changed from r = 3 to r = 2, the odd bound state crosses zero energy, signaling that the superconductor undergoes a first order quantum phase transition in which its ground state spin polarization changes from Sz  = 0 to Sz  = −1/2. Hence, changing the spatial location of one of the impurities gives rise to a phase transition of the entire superconductor. This transition is accompanied by a drastic change in the on-site superconducting order parameter at r1,2, as shown in Fig. 6.5(b) where we plot the order parameter as a function of interimpurity distance for several values of JS. For JS/t = 2.5, the on-site order parameter changes from r1,2 = 0.036t at r = 3 to r1,2 = 0.003t at r = 2. It is interesting to note that when the interimpurity distance is decreased even further from r = 2 to r = 1, another first order transition occurs, in which the superconductor’s ground state polarization changes

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from Sz  = −1/2 to Sz  = 0. The change in the superconductor’s ground state directly reflects the dependence of (JS)cr,1 on the interimpurity distance. For r = 3, one finds (JS)cr,1/t > JS/t = 2.5, implying that the spin polarization of the superconductor is Sz  = 0. In contrast, for r = 2 one has (JS)cr,1/t < JS/t = 2.5 < (JS)cr,2 /t,yielding a ground state with Sz  = −1/2. Thus as the interimpurity distance is monotonically decreased (or increased), the superconductor can be tuned through several first order transitions. This is the first example of a quantum interference induced quantum phase transition in impurity nanostructures. If JS is too small, however, (as, for example, in the case with JS/t = 2), no first order transition occurs,and the order parameter exhibits only weak Friedel-like oscillations when the distance between the impurities is varied. For larger JS (see, for example, JS/t = 2.7), the number of first order transitions that occur by changing the interimpurity distance increases. In all of these transitions, the superconductor’s ground state changes between states with Sz  = 0 and Sz  = −1/2. In contrast, for JS/t = 3.0, the superconductor still undergoes a series of phase transitions as r is varied, but now the transitions occur between states with Sz  = −1/2 and Sz  = −1. When JS is increased even further, the number of phase transitions decreases again (see, for example, JS/t = 4.0), since eventually JS > JScr,2 for any r ≥ 1, implying that the superconductor is in a ground state with Sz  = −1. Finally, we note that the superconductor can also be tuned through a quantum phase transition by changing the relative angle, , between the two impurity spins, as shown in Fig. 6.6, where we plot §e,o ↑,↓ as a function of  for two impurities with JS/0 = 20 located at r1 = (0, 0) and r2 = (1, 0). Since (JS)cr,1 < JS < (JS)cr,2 , the ground state for  = 0 possesses a spin polarization Sz  = −1/2. As  increases from zero, the frequency of the even bound state decreases and crosses zero at c ≈ 0.27 (indicated by an arrow in Fig. 6.6). This crossing is accompanied by a first order phase transition in which the spin polarization of the superconductor’s ground state changes from Sz  = −1/2 (for  < c ) to Sz  = 0 (for  > c ). The frequency separation between the even and odd states decreases with in-

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Fig. 6.6. §e,o ↑,↓ as a function of the angle, , between the direction of the impurity spins for r = 1 and JS/0 = 20. ˆ Results are obtained within the T-matrix formalism

creasing  and vanishes at  = . This is expected since for antiparallel impurity spins ( = ), the bound state for impurity 1 (consisting of |p, ↑, 1 and |h, ↓, 1) and impurity 2 (|p, ↓, 2 and |h, ↑, 2) possess different quantum numbers; thus they cannot hybridize and remain degenerate. However, since the bound state of one impurity is subjected to the repulsive potential of the second impurity, their resonance frequencies are larger than those of a single impurity with the same JS (indicated by the arrows on the right). This repulsion leads to the disappearance of all bound states for r → 0. Finally, first quantum interference effects in superconductors have recently been reported in STM experiments by Derro et al. [63] in simple nanostructures consisting of oxygen defect sites in the one-dimensional (1D) CuO chains of the hightemperature superconductor YBa2 Cu3 O6+x . We recently argued [61, 91] that these effects are similar to the quantum interference phenomena discussed above for bulk s-wave superconductors. In particular, we showed that the superconducting gap which emerges in the chains due to a proximity coupling to the CuO2 planes, possesses an s-wave symmetry. This symmetry arises since due to the reflection symmetry of the 1D chains one has (kFc ) = (−kFc ), where kFc is the Fermi momentum in the chains.We demonstrated that quantum interference effects between magnetic defects leads to the formation of antibonding and bonding impurity states in the chains,a result which is fully consistent with the experimentally ob-

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served splitting between the bound state peaks in the LDOS. 6.4.2

Quantum Corrals and Quantum Imaging in s-wave Superconductors

As discussed in the introduction, Manoharan et al. [48] recently demonstrated that quantum imaging via electronic waves can be achieved in quantum corrals that are placed on the surfaces of metals. In this section, we investigate the possibility for new quantum phenomena in quantum corrals located on the surface of an s-wave superconductor. In general, we expect that the non-trivial electronic correlations arising from particle-hole mixing in the superconducting state affect the quantum imaging properties of the quantum corral. In what follows, we again use the fermionic bound state induced by a single magnetic impurity as the quantum candle whose spectroscopic signature in the LDOS can be quantum imaged. In order to study quantum interference effects separately from the formation of fermionic impurity states, we consider quantum corrals that consist of non-magnetic impurities. We begin our study of quantum imaging in superconductors by considering an elliptical corral with √ semi-axes a = 20,b = 15 (eccentricity e = 7/4) that consists of 100 non-magnetic impurities. A magnetic impurity is located in the corral’s focus at f+ = (13, 0), while the other focus at f− = (−13, 0) is left empty (the center of the corral is located at (0, 0)). The fermionic bound state induced by the magnetic impurity possesses a particle-like and hole-like peak in the LDOS at frequencies §(1,2) /0 = ∓0.4 as b shown in Fig. 6.7(a). Note that in the absence of magnetic impurities,no eigenmodes exist for frequencies |!| < 0 , i.e., inside the superconducting gap. However,the bound state excites an eigenmode of the corral inside the superconducting gap, leading to spatial oscillations in the LDOS, as shown in Fig. 6.7(b) for (2) ! = §b .This eigenmode,in turn,creates a quantum image (in the LDOS) of the bound state peaks in the empty focus at f− (Fig. 6.7a) [86]. The presence of two peaks in the LDOS at §(1,2) permits us to study eigenb modes at different excitation energies. It was noted earlier [52], that eigenmodes can only be excited if

Fig. 6.7. (a) LDOS in the occupied (dashed line) and empty focus (dotted line) for a non-magnetic corral with a magnetic impurity located at f+ . Solid line: LDOS of the clean system. (b) Spatial plot of the LDOS inside the quantum corral (the filled yellow circles represent the corral impurities) at §(2) b /0 = 0.4. Results are obtained within the ˆ T-matrix formalism

the excitation (via the impurity bound state) takes place at a position where the spectral weight of the eigenmode is large, and if the excitation energy, i.e., (1,2) §b , is close to the eigenmode’s energy (note that due to the corral wall’s porosity and its finite scattering potential, the eigenmodes possess a substantial frequency width [52]). While two eigenmodes exist (±) outside the superconducting gap at §m /0 = ±1.1, (+) the mode’s amplitude at §m in the vicinity of the foci is considerably larger than that at §(−) m , yielding a weaker eigenmode, a weaker quantum image and (1) a concentration of spectral weight around f+ at §b .

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We thus conclude that an impurity bound state can excite eigenmodes inside the superconducting gap, which in turn can be used for creating quantum images of the bound state peaks [86]. The ability of the corral to form a quantum image of the quantum candle strongly depends on the  ratio of the decay length, d = c / 1 − (§b /0 )2 , and the corral’s semi-axes, a and b. For d  a, b the bound state wave-function at the position of the corral wall is exponentially suppressed, no eigenmodes can be excited, and the spatial form of the LDOS at the frequency of the impurity bound state remains unaffected by the presence of a quantum corral. In contrast, for the case shown in Fig. 6.7, one has d = 16.4  a, b and the spatial LDOS pat(2) tern at §b is strikingly different for a single magnetic impurity at f+ with and without a corral. Note also that the presence of a corral shifts §(1,2) from b (1,2) (1,2) §b /0 = ±0.575 (no corral) to §b /0 = ±0.4 which is a qualitatively new feedback effect of the corral on the impurity bound state. As discussed in Sect. 6.4.1 quantum interference effects between two magnetic impurities in an s-wave superconductor lead to a frequency splitting of the bound state peaks through the formation of bonding and antibonding bound states [16,62]. This splitting, which is a direct measure of the interaction between the impurities, can be enhanced if the magnetic im-

Fig. 6.8. Splitting of the bound state peaks in the LDOS at f− = (−13, 0) with and without a quantum corral for two magnetic impurities with parallel spins. Results are ˆ obtained within the T-matrix formalism 1

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purities are placed in the foci of a corral, as shown in Fig. 6.8.We consider the corral of Fig. 6.7 and assume for definiteness that the impurity spins are parallel, however, qualitatively similar results are obtained for arbitrary angle between the spins. In the absence of a quantum corral, the splitting of the bound state peaks is small, ı§b /0 = 0.1, due to the large distance between the impurities.However,when the two impurities are located inside a quantum corral, the splitting increases to ı§b /0 = 0.65, implying that the interaction between the two magnetic impurities is enhanced. This splitting therefore provides a direct probe for the strength of the interaction between magnetic impurities [86]. Our results thus demonstrate that quantum corrals provide new possibilities for manipulating the interaction between magnetic impurities. Based on the results shown above, the question naturally arises whether quantum corrals can also be employed for the simultaneous creation of multiple quantum images. To study this question, we consider a triangular equilateral quantum corral consisting of 90 non-magnetic impurities with Ui = 4 eV,as shown in Fig. 6.9 (the impurities, represented by filled yellow squares, are separated by r = 2). Motivated by recent experiments by Braun and Rieder [49], who studied triangular quantum corrals on the surface of Ag, we consider a 2D host system with a triangular lattice1 (lattice constant a0 = 1) and dispersion k = k2 /2m − ( = 1) with = −65 meV and kF = 0.24. In Fig. 6.9(a)-(h), we present a spatial plot of the LDOS at the frequencies of the eight lowest energy eigenmodes in the normal state [87]. In the unitary scattering limit considered here, the eigenmodes (i.e., their spatial structure and ordering in energy) are well described by the eigenstates, lm , of an infinitely deep triangular potential well (TPW) [92] [the corresponding quantum numbers (m, l) are shown in the upper left corner of Figs. 6.9(a)-(h)]. It is the spatial form of the corral’s low-energy eigenmodes that opens the possibility to project multiple quantum images in the superconducting state. To demonstrate this novel quantum effect, we assume that the host system is an s-wave superconductor with gap 0 = 4 meV, yielding a supercon-

All coordinates given in the following are those of a triangular lattice.

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Fig. 6.9. (a)–(h) Spatial plot of the LDOS at the frequencies of the eight lowest energy eigenmodes of a triangular corral (eigenmode energy is shown in the upper right corner) in the unitary scattering limit (Ui = 4 eV). (m,l) are the quantum numbers of the corresponding TPW eigenstate. Results are obtained within the ˆ T-matrix formalism.

Fig. 6.10. (a) LDOS as a function of frequency. (b),(c) Spatial plot of the LDOS at the bound state energies §(1,2) = ∓2.4 meV, respectively. Results are obtained b ˆ within the T-matrix formalism

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Fig. 6.11. (a) Spatial plot of the LDOS for |!| > 0 . (b) LDOS at the site of the magnetic impurity. Results are ˆ obtained within the T-matrix formalism

ducting coherence length c = kF /(m0) = 135, and place a single magnetic impurity (JS = 1.0 eV) in the center of the triangular corral at r1 = (0, 0). The particle-like and hole-like peak in the LDOS of the impurity induced bound state are located at ener(1,2) gies §b = ∓2.4 meV, as shown in Fig. 6.10(a). In Figs. 6.10(b) and (c) we present spatial plots of the LDOS at §(1,2) (the location of the magnetic impurity b is shown as a filled red circle). The formation of the impurity bound state is accompanied by the excitation of the (2, 4)-eigenmode [Fig. 6.9(c)] and by the emergence of three images of the bound state peaks inside the corral. As already discussed above, only eigenmodes that possesses sufficiently large spectral weight at the impurity site and are close in energy (1,2) to §b are relevant for the formation of quantum images. Since the energy of the (2, 4)-eigenmode, (2) E(2,4) = −4 meV,is closer to §(1) b than to §b ,the spectral weight of the quantum images is larger at §(1) b than at §(2) b , as follows directly from a comparison of Figs. 6.10(b) and (c). Note that the spatial pattern of the quantum images can be “custom designed” by shifting the eigenmode energies via changes in the corral’s size [92]. For example, in a triangular corral consisting of 117 impurities,the spatial pattern of the quantum images in the superconducting state is determined by the (1, 5)-mode [Fig. 6.9(f)] since in this case E(1,5) = 0 meV. The above results demonstrate that by choosing an appropriate geometry and size of a quantum corral, it is possible to custom design a corral’s imaging properties,and,as discussed here,to create multiple quantum images at specific locations. As discussed above,magnetic impurities in s-wave superconductors are pairbreaking, and thus lead to the formation of impurity bound states inside the superconducting gap [3,5–7].We find,however,that this

fundamental property of a magnetic impurity can be reversed inside a quantum corral,leading to the complete suppression of the impurity bound state [87]. To demonstrate this effect, we place the magnetic impurity at a node [r1 = (−5, −5)] of the (2, 4)and (1, 4)-eigenmodes, as shown in Fig. 6.11(a) (the LDOS is shown at |!| > 0 ). We find that in the superconducting state, no bound state is induced by the magnetic impurity, as follows from the plot of the LDOS at r1 shown in Fig. 6.11(b), and hence, no image is observed anywhere inside the corral. This complete suppression arises from the incompatibility of the bound state with the boundary conditions provided by the corral’s wall. In other words, an impurity bound state can only be formed if it can couple to one of the corral’s eigenmodes. The importance of this result lies in the fact that while non-magnetic impurities cannot induce a fermionic bound state in an s-wave superconductor, they can suppress its formation and thus reverse the pair-breaking effect of a magnetic impurity. To further study the interplay between the corral’s imaging properties, its geometry and the locations of the quantum candles, we next consider the case of three magnetic impurities with parallel spins that are located at the corners of an equilateral triangle at r1 = (−10, −10), r2 = (20, −10), and r3 = (−10, 20). Since the degeneracy of the impurity bound states is lifted via quantum interference, we expect to find six peaks in the LDOS. Instead, the LDOS exhibits only four peaks, as shown in Fig. 6.12(a), corresponding to the presence of only two non-degenerate impurity states. This reduction to two impurity states arises from a new type of optical selection rule that is based on the interplay between the corral geometry and the location of the quantum candles. Under a rotation of 2/3 around

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Fig. 6.12. (a) LDOS as a function of frequency. (b)–(d) Spatial plot of the LDOS at the frequency of the impurity bound state with n = 2 [(b) and (e)] and n = 0 [(c) and (d)]

the corral’s center, lm , and hence the corral eigenmodes, transform as lm → exp[i2(m + l)/3]lm . Due to the impurities’ geometry the non-degenerate impurity bound states possess the same transformation properties. Hence, the formation of three hybridized bound states requires that they couple to eigenmodes with n = (m + l)mod3 = 0, 1, 2. However, the eigenmodes with n = 1 are at energies |E(m,l) |  0 , thus preventing the creation of the hybridized bound state with n = 1. As a result, only the bound states with n = 0 [Figs. 6.12(c) and (d)] and n = 2 [Figs. 6.12(b) and (e)] are formed via their coupling to the (2, 4)- [Fig.6.9(c)] and (1, 4)-eigenmodes [Fig. 6.9(d)], respectively. To summarize,we find that the combination of impurity nanostructures and s-wave superconductors leads to the emergence of a series of novel quantum phenomena, ranging from the creation of multiple quantum images and novel optical selection rules, to the suppression of pairbreaking effects and quantum interference induced first order phase transitions.

6.5

Quantum Interference Phenomena in dx 2−y 2 -wave Superconductors

In dx2 −y 2 -wave superconductors, novel quantum interference phenomena arise from the interplay between the momentum dependence of the dx2 −y 2 -wave superconducting gap,the geometry of the nanostructure and its orientation with respect to the lattice. For the results presented below, we take the normal state tight-binding dispersion of the cuprate superconductors [93] to be of the form   "k = −2t cos kx + cos ky − 4t  cos kx cos ky − , (6.22) with t = 300 meV, t  /t = −0.4, and /t = −1.18, and the maximum superconducting gap to be given by 0 = 25 meV. Before studying more complex impurity structures, we start by investigating quantum interference phenomena in nanostructures consisting oftwo nonmagnetic impurities. This simple system allows us to

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Fig. 6.13. LDOS at r1,2 as a function of frequency and r for two non-magnetic impurities with U0 = 700 meV located at r1 = (0, 0) and r2 = (r, 0). Results are obtained within ˆ the T-matrix formalism

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gain first insight into the dependence of quantum interference effects on the interimpurity distance and the relative orientation of the two impurities with respect to the underlying lattice. To investigate the former, we consider two impurities with scattering potential U1,2 = U0 = 700 meV (corresponding to the unitary limit [9,12]) that are located along the crystal xˆ -axis at r1 = (0, 0) and r2 = (r, 0). In Fig. 6.13, we present the LDOS at the impurity sites r1,2, as a function of r [33]. For comparison, we note that for a single impurity with U0 = 700 meV, the resonances are located §res = ±1.5 meV. As the interimpurity distance r is varied, the number of observable low-energy resonances changes, the frequency of the resonances oscillates and at the same time, the width of these resonances in energy, which is a measure of the resonance’s inverse lifetime, changes. For example, for r ≤ 6, only two sharp low-energy resonances can be clearly identified. Moreover, for a single impurity, the resonance frequency and width are correlated, such that as |§res | decreases, the frequency width decreases as well [9,12]. In the case of two impurities, we find that |§res | and the lifetime of the resonances are not necessarily correlated. For example, the resonance frequency for r = 2.0 and r = 3.5 is §res = ±4.0 meV, but the width of the resonances are considerably larger in the second case. Moreover, for some values of r, all resonances are very weak and, for example, for r ≈ 1 disappear almost completely. Note, that even for a rather large interimpurity distance of r ≈ 10, the LDOS is still affected by quantum interference.

Fig. 6.14. (a) Spatial plot of the LDOS at §res = 0 meV for two non-magnetic impurities located at r1 = (0, 0) and r2 = (1, 1). Non-magnetic impurities are represented by filled yellow circles. (b) LDOS as a function of frequency at (1, 0). Results are obtained within the ˆ T-matrix formalism

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The relative orientation of the two non-magnetic impurities (the impurity dimer) with respect to the underlying lattice possesses a strong effect on quantum interference and the LDOS, and in particular, on the mere existence of an impurity state.If the two impurities are located at r1 = (0, 0) and r2 = (1, 1), and thus are aligned along the (110)-direction (case I), a resonance state exists at §res = 0 meV with the spatial structure of the LDOS shown in Fig. 6.14(a). In contrast, if the impurities are located at r1 = (0, 0) and r2 = (0, 1) (case II) and the dimer is aligned along the (100)-direction, the LDOS exhibits only Friedel-like oscillations, but no impurity resonance [33,37]. Similar results were also obtained in Refs. [34–36]. This striking difference is particularly apparent when one plots the LDOS at (1, 0) as shown in Fig. 6.14(b). This dependence of the resonance on the dimer’s orientation is similar to that of the zero bias conductance peak (ZBCP) observed near one-dimensional surface edges in the cuprate superconductors [94]. Only if the electrons that are specularly scattered along the impurity dimer experience a sign change in the superconducting order parameter (and hence pairing potential), a ZBCP-like impurity resonance emerges, such as the one shown in Fig. 6.14(a) for case I. This orientational dependence is an important feature of nanostructures, as further discussed below. How does the ZBCP-like state evolve when the length of the impurity line along the (110)-direction is increased? In Fig. 6.15(a) we show a spatial plot of the LDOS for a line of N = 10 impurities that are aligned along the (110)-direction. A characteristic signature of the ZBCP-like state is that it extends spatially perpendicular to the impurity line. Due to the hybridization of the ZBCP-like states on both sides of the impurity line, and the resulting formation of bonding and antibonding resonances, we find that the ZBCP-like state is shifted away from zero energy to §res = ±2 meV. The hybridization is mediated by the next-nearest neighbor hopping term (the t  -term), which permits the exchange of electrons between the two sides of the impurity line without a scattering process. When this hybridization is suppressed, as for example, in the nanostructure shown in Fig. 6.15(b), the ZBCP-like state is again located at zero-energy. In this nanostructure,

the ZBCP-like states are spatially separated and the absence of an impurity state between the two lines prevents the coupling of the states and thus their hybridization. A similar effect is also observed for other nanostructures, such as squares (see below). The same nanostructure also induces impurity states with other distinct spatial patterns, such as the“froglike” resonance shown in Fig. 6.15(c). Quite interestingly, the global spatial pattern of this resonance extends along the (100)-direction, but consists of lines with increased LDOS along the (110)-direction. As the length of the impurity lines is further increased, as in the nanostructure shown in Fig. 6.15(d) with N = 68 impurities, the spatial extent of the ZBCPlike state increases as well. Moreover, the spatial pattern of the eigenmodes inside the nanostructure becomes more distinct, as shown in Figs. 6.15(e) and (f). Note that the intensity of the ZBCP-like states decreases algebraically as ∼ 1/r ˛ along the (110)direction with distance from the nanostructure. For the resonance at §res = 0 meV of a single impurity, one has ˛ = 2 ( [10,25]). In contrast, we find ˛ < 2 up to distances, r, from the nanostructure that are of order of its lateral size, l (note that for r  l, one recovers ˛ = 2). For example, for the √ nanostructure shown in Fig. 6.15(b) with l = 10 2a0 , we find ˛ ≈ 1.18 up to a distance r ≈ 20a0 perpendicular to the lines. This decrease of ˛, which is expected since for a truly one-dimensional ZBCP one has ˛ = 0,provides an explanation for the increasing spatial extent of the ZBCP with increasing length of the impurity line. To further explore the interplay between geometry and orientation, we next consider square-like nanostructures, such as the one shown in Fig. 6.16(a) whose sides are parallel to the (110)-direction [37]. This nanostructure induces a ZBCP-like resonance at ! = 0 meV [Fig. 6.16(a)], which extends in all four equivalent (110)-directions. The ZBCP-like resonances associated with each side are spatially separated and thus do not hybridize. Impurity states at non-zero frequencies exhibit different characteristic spatial patterns. The impurity resonance shown in Fig.6.16(b) extends along the (100)-direction outside the square with a“V-like” fine structure consisting of branches that run along the (110)-direction. In con-

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Fig. 6.15. Spatial plot of the LDOS for (a) N = 10 impurities aligned along the (110)-direction; (b), (c) two lines with N =√ 10 impurities each separated by d = 4 2; (d), (e) and (f) two lines with N =√ 34 impurities each separated by d = 6 2. Non-magnetic impurities are represented by filled yellow circles. Reˆ sults are obtained within the T-matrix formalism

trast,the impurity states shown in Fig.6.16(c) and (d) are “star-like” and extend almost radially outwards. Note that the“global” (i.e., larger length scale) spatial structure of the LDOS at §res = ±5 meV [Figs.6.16(c) and (d)] is identical, and thus particle-hole symmetric. This symmetry also exists for other impurity resonances, such as the ones at §res = ±11 meV. Only in the local structure of the impurity resonances can we identify differences in the spatial LDOS pattern

between particle-like and hole-like energies, such as a 45◦ rotation of the LDOS intensity, similar to the one observed near single impurities [28]. This dichotomy of global particle-hole symmetry and local particle-hole asymmetry is also exhibited by other nanostructure geometries. When the square is rotated by 45◦ such that its sides are parallel to the (100) direction [see Fig. 6.16(e)], specularly reflected electrons do not

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Fig. 6.16. (a)–(d) Spatial plot of the LDOS in the superconducting state for a square with N = 32 √ impurities and sides of length a = 8 2 parallel to the (110) and symmetry related directions. (e), (f) Spatial plot of the LDOS for a square with N = 40 impurities in (e) the normal state, and (f) the superconducting state. Results are obtained within ˆ the T-matrix formalism

experience a sign change in the superconducting pairing potential and hence no impurity resonances are formed. However, the square can still excite an eigenmode in the superconducting state that is also present in the normal state at energies |!| < 0 . Consider for example the normal state eigenmode at ! = −3 meV shown in Fig. 6.16(e). In the superconducting state, this eigenmode is located at ! = −1 meV [see Fig. 6.16(f)]. Note that this

eigenmode in the superconducting state does not arise from pairbreaking effects, such as for example, the impurity pairbreaking resonances shown in Figs. 6.15(a) and (b). It is therefore important to distinguish between impurity states arising from eigenmodes of the nanostructure and those that are pairbreaking resonances. In general, the spatial structure of impurity states arises from an interplay of eigenmodes and pairbreaking resonances,

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Fig. 6.17. Spatial plot of the LDOS at §res = 0 meV for two right isosceles triangles whose hypotenuse is aligned (a) along the (100) direction, and (b) along the (110) direction. Results are obtained ˆ within the T-matrix formalism

with eigenmodes (pairbreaking resonances) determining the spatial structure of the LDOS inside (outside) the nanostructure. The above results demonstrate that the resonances (and in particular the ZBCP-like state) induced by impurity nanostructures exhibit a strong dependence on the orientation of the nanostructure, which directly reflects the momentum dependence of the superconducting order parameter. Hence we conclude that nanostructures provide a valuable new tool for identifying the symmetry of unconventional superconductors. The above results suggest that it is already sufficient to construct a single nanostructure in order to identify the symmetry of the superconducting gap. Consider, for example, the right isosceles triangle shown in Fig.6.17.When the hypotenuse of the triangle is aligned along the (100) direction, and the two shorter legs are aligned along the (110) direction, a ZBCP-like resonance emerges only from the shorter sides, as shown in Fig. 6.17(a). After rotating the triangle by 45◦ , such that the hypotenuse is now aligned along the (110) direction,we still observe a ZBCP-like resonance, now emerging from the hypotenuse only, as shown in Fig. 6.17(b). Thus, it is possible to use a single nanostructure with appropriate geometry in order to identify the symmetry of the superconducting order parameter, and in particular, its nodal and antinodal directions. Another important quantum phenomenon arising from the interplay between the nanostructure’s geometry and the momentum dependence of the superconducting gap is the spatial screening of impu-

rity resonances in unconventional superconductors. This screening effect immediately follows from the observation that certain nanostructures, such as a square whose sides are parallel to the (100)-direction shown in Fig. 6.16(e), do not induce resonant impurity states. Consider, for example, the resonance state at §res = 0 meV shown in Fig. 6.18(a) that is induced by a single magnetic impurity. When the impurity is enclosed by a square whose sides are parallel to the (100)-direction, the single impurity resonance is completely confined to the interior of the square, as shown Fig. 6.18(b). Hence, the impurity resonance is screened by the nanostructure square. We find that even spatially more complex impurity resonances, such as ZBCP-like state shown in Fig. 6.18(c) can be screened when enclosed by a second square, as shown in Fig. 6.18(d). This new quantum phenomena of screening impurity states permits us to employ impurities or nanostructures to locally probe the electronic structure of an unconventional superconductor without inducing long-range changes in the superconductor due to its nodal structure. More complex nanostructures provide the possibility to create an abundance of novel spatial patterns of impurity resonances and, in particular, to form resonances that are confined to distinctively different parts of a nanostructure. For example, the two congruent squares shown in Fig. 6.19 induce impurity states that are either confined to the interior of the inner square [Fig. 6.19(a)], to the area between the two squares [Fig. 6.19(b)], or to the exterior of the outer square [Fig. 6.19(c)], where the latter corresponds to the ZBCP-like resonance of this nanostruc-

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Fig. 6.18. Spatial plot of the LDOS for (a) and (b) an unscreened and screened impurity resonance of a single magnetic impurity, respectively, and (c) and (d) an unscreened and screened impurity resonance of a nanostructure square, respectively. Results are obˆ tained within the T-matrix formalism

ture.Moreover,we also observe more complex spatial patterns of impurity states,such as the ones shown in Figs. 6.19(d), (e) and (f). These results exemplify the possibility of custom-designing the spatial structure of impurity resonances and thereby gaining further insight into the complex electronic structure of the cuprate superconductors.

6.6

Molecules on the Surface of dx 2−y 2 -wave Superconductors

Before investigating quantum interference phenomena in a molecular (dimer) nanostructure, we study the effects of a single molecule located on the surface of a dx2 −y 2 -wave superconductor, on the superconductor’s local electronic structure. As discussed earlier, the predominant effect of molecules that are located on the surface of a host systems is to provide a 2

localized dynamic defect in the form of a vibrational bosonic mode. Hence, the possibility that superconducting electrons can hop onto the electronic orbitals of the molecule, and thus “proximity-induce” superconducting correlations in the molecule, will not be considered here; an investigation of this effect, however, is the topic of currently ongoing work. To simplify the discussion we consider a particlehole symmetric dispersion2 with t  = = 0. In Fig. 6.20 we present the LDOS at the site of the molecule, r = (0, 0). Similar to a non-magnetic impurity, we find that the molecule induces a fermionic resonance state in its vicinity. In order to derive an equation for the resonance frequency, §res , we first calculate £loc (!) ≡ £(r, r, !) whose imaginary part for |§| ≥ !0 = 0 is given by   1 |§| − !0 |§| − !0 Im£loc (§) = − , (6.23) K  0 0

A band dispersion with strong particle-hole asymmetry, characteristic of the cuprate superconductors, yields qualitatively similar results.

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Fig. 6.19. Spatial plot of the LDOS for two congruent squares whose sides are aligned along the (110) direction (U = 2 eV). Results are obtained within the ˆ T-matrix formalism

where K is the complete elliptic integral of the first kind, while Im£loc ≡ 0 for |§| < !0 . Performing a Kramers–Kronig transform, we find for §/!0  1, Re£loc ≈ −A§/!0 where A = ∞ − 2 1 dx Im£loc (x)/x 2. In the limit §res  0 , the resulting equation for the resonance frequencies is −1   0 !0 40 A log §2res = . (6.24) g2 |§res |

We thus find that §res shifts to higher energies with increasing !0 , in agreement with our numerical results shown in Fig. 6.20(a). Moreover, for !0 → ∞ the LDOS approaches that of the unperturbed clean superconductor,as can be seen,for example,from the results for !0 = 200 meV. A qualitative difference to the effects of static impurities is that the logarithmic divergence in £loc at |§| = 0 + !0 leads to a dip in the LDOS, which is indicated by arrows in the inset

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Fig. 6.20. LDOS at r = 0 as a function of frequency for  = 0+ , g = 1.2 eV, and (a) at T = 0 for several values of !0 (inset: arrows indicate the “dip” in the DOS), and (b) for !0 = 10 meV and several T

of Fig. 6.20(a) (this dip was also found by Balatsky, Abanov and Zhu [84]). At non-zero temperature,a new scattering channel is opened since the ! = +!0 branch of the bosonic mode becomes populated. This yields significant changes in the LDOS. Specifically, if the resonance frequency at T = 0, satisfies |§res (T = 0)| < §−c with §−c = 0 − !0 we find that the resonance peak is shifted to lower energies with increasing temperature,as shown in Fig.6.20(b).In addition,the opening of a second scattering channel for T = 0 leads to a logarithmic divergence in £loc , and consequently a dip in the LDOS at |§| = §−c = 15 meV, as indicated by the arrows in Fig. 6.20(b). Consider next a molecular nanostructure consisting of two molecules located at r1 = (0, 0) and r2 = (1, 0) (case I) and r1 = (0, 0) and r2 = (2, 0) (case II). Note that for two molecules, the calculation of the LDOS involves not only the local Greens funcˆ i , rj ), with ri = rj , tion, Gˆ 0 (ri , rj ), and self-energy, £(r but also the non-local ones with ri = rj . As a result,

the LDOS is determined by the distance and orientation of the modes relative to the lattice, which is described by r = ri − rj = (nx , ny ). For the bandstructure considered here, superconducting correlations, as represented by the non-vanishing of the non-local anomalous Greens function, Fr , and selfenergy, ¥r (the off-diagonal elements of Gˆ 0 and £ˆ 0 , respectively), are important for the form of the LDOS if (nx + ny )mod2 = 1 which is satisfied for the molecules in case I, but not in case II. The LDOS for several values of the inter-molecular coupling J is shown in Fig. 6.21(a) and (b) for cases I and II, respectively. Since with increasing J, §2 increases while |gr(2) | decreases, the electronic scatter1,2 ing becomes weaker and the resonance peaks are shifted to higher energies. In case I, the peaks move close to ±0 as J increases from J = 0 to J = ∞, an effect that is independent of g and directly related to a non-zero Fr and ¥r . In contrast, in case II, the frequency shift is much smaller. For J = ∞, only the bosonic in-phase mode couples to the fermionic

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n+2 = 0 and several values of n+1 . For n+1 = 0, which corresponds to the equilibrium case at T = 0, we observe two peaks in the LDOS. With increasing n+1 , we find that each of the peaks is split,and a dip is induced in the LDOS at ±!1− with !1− = 0 − §1 . This new local method of manipulating the electronic structure of the host system can provide important insight into the complex behavior not only of the cuprate superconductors, but of strongly correlated electron systems in general.

6.7

Fig. 6.21. LDOS at R as a function of frequency for !0 = 0.60 and g = 150 at T = 0 and in equilibrium [(a) and (b)] and (c) for J = !0 /3 when the molecules are externally excited, leading to changes in the populations n+1,2

system. This coupling is much weaker in case I than in case II due to the non-vanishing superconducting correlations. The fact that the coupled energy levels of the two molecules can be externally excited, for example, by optical means [95], opens an intriguing possibility for manipulating the local electronic structure of the host material. In order to demonstrate this effect we assume that through excitations of the energy levels ±§1,2 we can arbitrarily change their population, − n± 1,2 .In Fig.6.21(c) we present the LDOS with n1,2 = 1,

Conclusions

Over the last 50 years, the investigation of impurity effects in superconductors has provided important insight into the nature of superconductivity. In the past, the main emphasis has been on the study of either disorder-averaged physical quantities, or of single impurity effects. However, recent experimental as well as theoretical studies suggest that novel quantum interference phenomena emerge when nanoscale impurity structures are placed on the surface of conventional or unconventional superconductors. While superconductors with s-wave symmetry are in general well understood, quantum interference in nanostructures leads quite unexpectedly to a series of novel and fundamental quantum effects. In particular, the interplay between particle-hole mixing in the superconducting state, the presence of a full gap in the LDOS and quantum interference leads to new types of quantum imaging and optical selection rules. Of particular importance for these phenomena is the fact that magnetic and non-magnetic impurities exert qualitatively different effects on the superconductor’s electronic structure, thus permitting the study of quantum interference separately from the creation of fermionic impurity states. One of our most important results pertains to the pairbreaking nature of a single magnetic impurity in a superconductor which was believed to be one of its most fundamental (and unalterable) properties. We showed, however, that when a magnetic impurity is located inside a (non-magnetic) nanostructure, destructive quantum interference can lead to a complete suppression of its pairbreaking ability. In other words,

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non-magnetic nanostructures can render magnetic impurities “harmless” for the superconductor. Moreover, we demonstrated that nanostructures can be used for the creation of quantum interference induced first order phase transitions. Since in these transitions, the spin polarization of the superconductor’s ground state changes, spin resolved scanning tunneling spectroscopy should be the primary tool for their investigation. Unconventional superconductors, on the other hand, give rise to novel types of quantum interference phenomena due to the momentum dependence of their order parameter. In particular, we showed that in superconducting host systems with dx2 −y 2 wave symmetry, the nature and existence of impurity resonances is determined by the interplay between the momentum dependence of the superconducting gap and the geometry and orientation of a nanostructure. We identified two important consequences of this interplay. First, by changing the geometry and orientation of the nanostructure, it is possible to identify the symmetry of the superconducting order parameter through the creation of a zero bias conductance peak. In particular, using a nanostructure in the form of a square as an example,we showed that the presence or absence of a ZBCP-like state depends on the relative orientation of the square with respect to the order parameter’s nodal directions. In other words, the nanostructure’s ZBCP is a phase sensitive probe of the order parameter’s symmetry. Second, while constructive quantum interference is responsible for the emergence of a ZBCP in nanostructures, destructive quantum interference leads to a complete suppression of impurity resonances around the nanostructure.To demonstrate this effect,we considered a square of impurities whose sides are aligned along the (100) direction. An important corollary of this effect is the possibility to screen impurity resonances in the superconducting state.In particular,we found that when a single magnetic impurity is placed inside the square, the impurity resonance induced by the single impurity is screened by the square, i.e., completely confined to its interior. Finally, we considered the effects of a new class of defects, namely molecules that possess an internal vibrational mode, on the local electronic struc-

ture of dx2 −y 2 -wave superconductors. Due to the scattering of the host system’s electrons off the vibrational mode, a fermionic resonance state is created in the vicinity of the molecule. While this effect is similar to that of conventional impurities, we identified a number of important qualitative differences, such as the emergence of dips in the superconductor’s LDOS with increasing temperature. These dips at frequencies ! < 0 arise from the opening of a second scattering channel due to thermal population. We also considered two molecules that are coupled by a “quantum mechanical spring” and demonstrated that their effect on the superconductor’s LDOS strongly depends on distance and orientation of the molecules; a dependence that arises from the spatial form of the superconducting correlations, as reflected in the anomalous Greens function, F(r). In a related study [85] (not discussed above), we recently showed that the LDOS can also provide insight into the nature of intermolecular interactions, about which very little is known at present. Moreover, we demonstrated that by externally exciting the energy levels of the molecular dimer, for example by optical means, it is possible to manipulate the superconductor’s local electronic structure. Hence, molecular nanostructures provide us for the first time with the ability to probe and potentially manipulate superconducting correlations in real space. How can one experimentally realize those quantum phenomena presented above that require a superconducting host with dx2 −y 2 -wave symmetry, i.e., a cuprate superconductor? One of the main experimental obstacles is the creation of nanostructures in close vicinity to the CuO2 planes. However, one of the candidate materials for the experimental realization of nanostructures in dx2 −y 2 -wave superconductors might be given by Ca2−x Nax CuO2 Cl2 in which (after cleaving) the topmost CuO2 plane is separated from the vacuum only by a single (Ca,Na)Cl plane [96]. Placing nanostructures on the surface of or inside the (Ca,Na)Cl layer might be sufficient to project localized scattering potentials into the CuO2 plane [97]. In contrast, for the extensively studied compound Bi2Sr2 CaCu2 O8+ı placing nanostructures close to the CuO2 layer might be somewhat more dif-

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ficult since a BiO plane and a SrO plane separate the topmost CuO2 layer from the vacuum. The results presented above demonstrate that the interaction of nanoscale impurity structures with superconducting host systems leads to a series of novel quantum phenomena. We expect these phenomena, though derived for the specific case of a dx2 −y 2 -wave superconductor, to exist in many other unconventional superconductors, since their existence only requires the vanishing of the local superconducting gap and the presence of line nodes. Moreover, the fact that these quantum effects as well as the superconductor’s local electronic structure can be manipulated by changing the geometry or orientation of a nanostructure or by exciting its internal degrees of freedom, provides us with fundamentally new possibilities to explore the nature and origin of superconductivity, and in particular, to identify the symmetry of the superconducting order parameter. These possibilities are of great importance for many unconventional superconductors in which the microscopic mechanism responsible for superconductivity is still unclear, such as the high-temperature, ferromagnetic, heavy-fermion or organic superconductors. The studies presented here suggest a number of different directions for future work. For example, up to now, nanostructures, and in particular quantum corrals, were employed for quantum imaging characteristic features (quantum candles) in the LDOS. However, some recent studies [98,99] have extended this idea and addressed the question whether superconducting correlations and order parameters in general can be quantum imaged. While quantum imaging of a spatially varying order parameter might be difficult to achieve [98], it was found that order parameter modes can be induced through their coupling to the LDOS [99]. Testing of these predictions is likely possible in the near future due to recent advances in the development of Josephson scanning tunneling spectroscopy [100]. Moreover, the above studies suggest that the emergence of novel quantum phenomena can also be expected for other host materials possessing broken-symmetry states with nontrivial electronic correlations, such as charge- and spin-density-wave systems,or phases with coexisting

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orders. Indeed, first studies of two-impurity quantum interference effects in the d-density wave phase [101] and triplet p-wave superconductors [102] have recently emerged. Another very interesting class of host materials include quantum critical systems, in particular those where competing phases near quantum critical points are present. Here, the ability of nanostructures to manipulate the local electronic structure can potentially identify the nature and microscopic origin of the competition. Moreover, in many strongly correlated materials, the question of which collective modes are responsible for the emergence of complex phases is at the center of intense scientific debates. For example, in the cuprate superconductors, it was argued that the unconventional behavior observed in the pseudogap region can be attributed to the presence of collective spin or charge modes [103]. The question thus naturally arises whether nanostructures can be used to identify the relevant collective modes, for example, by choosing a geometry which is commensurate with the modes’characteristic wave-vectors.Furthermore, we assumed above that the predominant effect of nanostructures is to provide localized scattering potentials.Whether superconducting electrons can hop onto the electronic orbitals of a nanostructure and thus “proximity-induce” superconducting correlations, is an open question and the topic of currently ongoing work. This new effect could potentially lead to interesting new pairing symmetries due to the projection of the host system’s pairing symmetry on the geometry of the nanostructure, and possible feedback effects. Finally, we recently showed [104] that quantum corrals provide a new venue for manipulating (on the atomic length scale) one of the most fundamental quantum phenomena in condensed matter physics,the Kondo effect.Specifically,it is possible,by using the spatial form of low energy eigenmodes of a quantum corral, to custom-design Kondo-screening by changing the location of a single magnetic impurity inside the corral. This idea can easily be generalized, for example, in order to study the possibility of manipulating the two-impurity Kondo problem inside a nanoscale impurity structure. In conclusion, research at the intersection of strongly correlated electron systems and nanoscale

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structures provides many possibilities for the emergence of novel quantum phenomena that will have important impact on a large number of fields, ranging from the study of quantum critical systems and unconventional phases to quantum information technology and spin electronics. Acknowledgements

rich, A. Kapitulnik, R. H. Nyberg, M. Randeria, K.-H. Rieder, E. Rossi, J. Schmalian, Q. Si, N.A. Stavropoulos,R.Wiesendanger,A.Yazdani,J.Yoon,and J.-X.Zhu for stimulating discussions. D. K. M. acknowledges financial support by the Alexander von Humboldt Foundation, the National Science Foundation under Grant No. DMR-0513415 and the U.S. Department of Energy under Award No. DE-FG02-05ER46225.

We would like to thank A. de Lozanne, A.V. Balatsky, K. Bennemann, J. C. Davis, L. Glazman,A. Hein-

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7

Nanostructured Superconductors

Guy Deutscher

School of Physics and Astronomy, Tel Aviv University, Israel

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.2 Nano-structured Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.3 The Granular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260 7.3.1 Grain Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.3.2 Granular to Amorphous Transformation and the Metal to Insulator Transition . . . . . . . . . . . . . . . 261 7.3.3 The Percolative Random Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.4 Normal State Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.4.1 Transport in Random Percolating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.4.2 Transport in Granular Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.4.3 Progressive Coulomb Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.5 Single Grain Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.5.1 Estimates of the Limiting Size of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.5.2 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.5.3 Experiments on Nano-size Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.6 Weakly Coupled Grains: Granular Insulators and Super-Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.6.1 Small Grain Case: Percolative Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.6.2 Large Grain Case: Superconducting Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.6.3 Pseudogap Regime in the Superconducting Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.6.4 The Superinsulator Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.7 Well Coupled Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 7.7.1 Granular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272 7.7.2 Random Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.8 Critical Temperature of Granular Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.8.1 Grain Size Dependence of the Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.8.2 Weak Screening Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

7.1

Introduction

We review the structure, transport and superconducting properties of compounds composed of small metallic grains connected in different ways. After a short description of the two main kinds of structures, normally granular,random and metal/insulator mix-

tures,an account is given of their normal state behavior near the metal to insulator transition. Superconductivity in single grains, weakly connected grains and well connected grains is then reviewed. Finally, some remarks are made on the Tc enhancement in granular superconductors.

260

7.2

G. Deutscher

Nano-structured Compounds

It has been known for a long time by metallurgists that eutectic systems composed of two immiscible phases (elements or compounds) can assume different structures depending on the procedure followed in their preparation such as cooling and annealing schedules. Granular matter falls into that category. For instance,when Al is evaporated in a poor vacuum system it condenses in the form of a film composed of small Al grains, weakly coupled together through thin amorphous Al2O3 barriers. This was discovered accidentally by Abeles et al. [1] in the early 1960s. Al-Al2O3 is a good eutectic system, because oxygen is essentially immiscible in Al, and Al2O3 is the only oxide of Al. This ensures a very sharp boundary between the metal and the oxide. In this way, Al grains as small as a couple of nanometers can be obtained. Another well studied eutectic system is Al-Ge, which can also assume a granular structure when the two elements are co-deposited onto a substrate held at room temperature, see Fig. 7.1. In that case also Al is crystalline and Ge amorphous [2]. But when such a film is annealed, or when it is deposited to start with at an elevated temperature, Ge is in crystalline form. The structure is then no longer granular. Rather, it is composed of Al and Ge small crystallites, randomly distributed in space. The structure is then of the percolative type [3]. Other eutectic systems having similar properties are Sn-SnOx , Sn-Ar, and Ni-SiO2. Abeles et al. discovered that granular Al films had many interesting properties. Their normal state resistivity was typically a few orders of magnitude higher than that of bulk Al. Their critical temperature could reach more than 3 K compared to 1.2 K for pure Al and the upper critical field several Tesla. These properties are structure sensitive. The same composite in its percolative form does not have an enhanced critical temperature. Its normal state properties are different, particularly in the vicinity of the metal to insulator transition at high insulator concentration. These observations raise some interesting questions.Why does the metal form in the granular structure? How should the “poor metal properties” be understood? Why is the critical temperature higher than

that of the bulk metal in the granular structure and not in the percolative one? Is the enhanced critical temperature a property of the small grain itself or is it a property of the coupled grains? Is it accidental that the maximum Tc enhancement occurs near the metal–to–insulator transition? This last property is the most intriguing one, because it resembles so much that of the cuprates whose highest Tc is also reached at an “optimum doping” very close to the metal–to–insulator transition. Is this similarity accidental? Is it a clue that may help understand the high Tc of the cuprates? Our answers to these questions are today still imperfect, but we certainly understand that granular matter is an even more interesting object of study than was realized at the time of its discovery.

7.3

The Granular Structure

Granular films can be obtained either by the reactive evaporation of the metal in the presence of a sufficient pressure of oxygen in the vacuum system as was first done un-intentionally by the RCA group or by the co-deposition of the metal and the insulator (for a review see for instance [3]). The metal and the insulator must be insoluble and must not form definite compounds. As already mentioned not all metal-insulator systems form granular structures and the type of structure that is obtained depends on the temperature of the substrate at the time of deposition. Granular structures may also transform into other structures upon annealing [5]. 7.3.1

Grain Size

An important observation is that in granular films the metal is crystallized and the insulator (or semiconductor) amorphous. This has been shown explicitly for Al-Al2O3 and for Al-Ge. Furthermore, in these two systems the thickness of the dielectric is of atomic size [2]. The size of the metal grains is a decreasing function of the dielectric concentration. At high dielectric concentration it is in the nanometer range. For dielectric concentrations roughly higher than 50 vol.%, the mixture turns amorphous, or at

7

Nanostructured Superconductors

261

Fig. 7.1. Phase diagram of the Al–Ge system as a function of the growth temperature T, and %Al (volume fraction). The crystalline phase has the random percolating structure. The amorphous phase is insulating. The granular phase can be either metallic or insulating

grow through diffusion of metal atoms in the amorphous as-deposited mixture. The dielectric atoms or molecules play a purely passive role in this process. They are expelled at the periphery of the growing metal grains. When a continuous dielectric layer is formed at the periphery, it acts as a barrier that slows down further grain growth. This continuous dielectric layer is formed at a grain size value that depends on the dielectric concentration in the composite. The higher it is the smaller is this grain size. This model explains the observed relationship between the dielectric volume fraction and the grain size as well as the uniformity of the latter. The prediction that the dielectric thickness is of molecular size, is also in good agreement with experiment [2].

Fig. 7.2. A typical histogram of grain size distribution n(d) of Al grains in the Al–Ge granular phase. The size d is the grain diameter in Ångstrom

least the grain size becomes smaller than the electron microscope resolution. Detailed studies of the grain size distribution have been performed by dark field electron microscopy. They have shown that this distribution is rather narrow, particularly at high dielectric concentration [6], see Fig. 7.2 for illustration. These observations have suggested the following model for the growth mechanism of granular films [2]. During deposition metal grains nucleate and

7.3.2

Granular to Amorphous Transformation and the Metal to Insulator Transition

The above correlations are is only valid at moderate insulator concentrations. Beyond a certain concentration the grain size becomes actually concentration independent. This limiting value depends on the temperature of deposition. It becomes smaller at lower temperatures suggesting that it is set by a diffusion mechanism. In this regime, as the dielectric concentration keeps increasing, the dielectric barriers between the metallic grains become thicker. Shortly thereafter the structure shifts to an amorphous mixture.

262

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The geometrical growth model assumes a priori that the granular structure is energetically more favorable than the amorphous one. A clue as to what drives the structural transformation may be the fact that the metal-to-insulator transition (discussed later in detail) appears first,but quite close to it.When electrons become localized in the grains,since the dielectric barriers become thicker, their kinetic energy increases. This increase is on the order of the energy level splitting, which is inversely proportional to the grain’s volume. For a grain of nanometer size, containing roughly a thousand electrons, this energy is of the order of a few tenths of an electron volt. Per grain this is a few times the thermal energy at room temperature. Depending on other factors such as the interface energy between the metal and the dielectric this may be sufficient to tip the balance in favor of the amorphous structure. Thus, the granular-toamorphous transition may in fact be electronically driven. 7.3.3

The Percolative Random Structure

If at the time of deposition both the metal and the dielectric crystallites can nucleate, one obtains a structure in which they are randomly dispersed. By this we mean that the probability to find, say, a dielectric crystal next to a metallic one, is solely dictated by the respective volume fractions, at least if they are of similar sizes. This structure is obtained if the mixture is deposited at a temperature that is high enough to allow the nucleation of dielectric crystals. It is well known that metals will crystallize even when deposited at cryogenic temperatures, but that dielectrics can easily form in an amorphous structure unless the deposition temperature is several hundred degrees Celsius above room temperature. For instance, a random Al-Ge mixture will be formed if deposited at a temperature higher than about 300 ◦ C. Even though metallic and dielectric crystallites are not located at well defined sites on a lattice, the obtained structure is essentially similar to a random percolative one. Metallic and insulating clusters are formed, whose structure is fractal on scales intermediate between the crystallites size and the percolation correlation length for each respective type of clus-

ter [3]. The percolation correlation length diverges at a certain critical volume fraction, whose value depends on the dimensionality. Below that volume fraction, only finite clusters are formed. Beyond it finite clusters are still present, but an infinite cluster is formed as well.In three dimensions there are distinct thresholds for the two constituents. The phase diagram comprises three regions: at low metal volume fraction only the insulator forms an infinite cluster; beyond the critical metal volume fraction, both the metal and the insulator form infinite clusters; at still higher metal volume fraction, only the metal forms an infinite cluster. In two dimensions there is only one threshold: either the metal or the insulator will form an infinite cluster.

7.4

Normal State Transport

In this section, we review the behavior of the electrical resistivity and of the Hall effect for both kinds of structure, the granular and the random one.We start with the latter which is better understood. 7.4.1

Transport in Random Percolating Structures

What is well understood in the framework of percolation theory is the case of random mixtures where the metal M and the insulator I are randomly distributed in space, as we have described above.The resistivity is given by  ∝ (p − pc )−t ,

(7.1)

where p is the metal volume fraction and pc the metal critical volume fraction at which an infinite metallic cluster is formed. The value of the exponent t depends on dimensionality, t3D = 2.0 and t2D = 1, 3. In two dimensions (2D) theory predicts and experiment verifies that the Hall constant RH does not diverge at the percolation threshold. In 3D, however, one gets RH ∝ (p − pc )−g . (7.2) The 3D experimental value of g = 0.38±0.05 [7] is in good agreement with the theoretical result g = 0.29±0.05 from numerical simulations [8]. Hence

7

percolation theory predicts and experiment verifies that in 3D one has RH ∝  0.15 .

(7.3)

As long as there exists an infinite metallic cluster (p > pc ) the temperature dependence of the resistivity of the composite has metallic character. It decreases as the temperature is lowered. The Hall constant is temperature independent. These properties are also well observed experimentally. They are non-trivial.For instance,close enough to the percolation threshold the room temperature resistivity will be very high typical of a semiconductor rather than of a metal. Yet, upon measuring its temperature dependence, it will appear that it has the metallic character [9]. This statement is correct except for some subtle weak localization effects that will appear at low temperatures on the weakest parts of the infinite cluster, if the crystallites are small enough. Transport in Granular Structures

The behavior experimentally observed in granular systems, where each metallic grain is surrounded by an insulating coating, is quite different from that of random systems. Contrary to the prediction of percolation theory verified in random systems the temperature dependence of the resistivity is activated beyond a certain value of the resistivity.Bandyopahyay et al.have found that in granular Al one finds RH ∝  0.5 [10] near the metal to insulator transition, while from percolation theory one would expect RH ∝  0.2 .In comparing their results with other theories, Bandyopahyay et al. noted that the scaling theory of localization predicts that RH should be temperature independent. Hence, in their samples where the conductivity is activated (RH /) should be temperature dependent contrary to their experiments. Theories based on interaction effects predict RH ∝  2 , also in disagreement with experiment. Hence, the transport properties of granular films are not well explained by either percolation localization (or interaction) theories. One essential difference between granular materials and random M/I mixtures consists in the existence in the former of a finite charging energy Ec

263

necessary for the transfer of an electron from grain to grain. This is due to the insulating coating surrounding each metal grain. The question is then whether the M-I transition is still driven by disorder effects, characterized in this case by a distribution of intergrain resistances R and for which a percolation description might apply as in the case of random mixtures, or whether the charging energy plays a dominant role. One would expect the first mechanism to apply for large grain sizes (small charging energy) and the second one for small grains. In the case of the granular films studied by Bandyopahyay et al. the grain size is only of a few nanometers Ec ≈ (e2/r). For the isolated grain it is of the order of 0.1 eV. One expects that electrostatic charging should play a dominant role near the M/I transition. We examine below the possible role of Coulomb blockade in transport phenomena in granular nano-structures. 7.4.3

7.4.2

Nanostructured Superconductors

Progressive Coulomb Blockade

A number of experiments support the idea of the Coulomb blockade [11] for electron tunnelling between small grains,even if this electrostatic effect can be washed out by quantum fluctuations in single tunnel junctions [12]. Thus, it is certainly not surprising that pure percolation models do not describe correctly the experimental situation of nano-structured granular films. On the other hand, since the samples of Bandyopay et al. have a finite conductivity in the normal state above the superconducting transition, one might be tempted to say that electrostatic effects are negligible.Why should’nt percolation effects then dominate? It has been proposed [13] that the behavior of RH near the M/I transition can in fact be successfully described in a semi-classical manner [14] by considering the effect of the charging energy in the form of a progressive Coulomb blockade in which more and more “free electrons” are frozen out from the conduction process. Following Abeles [15] we discuss the conduction process in granular compounds in terms of the intergrain resistance R and capacity C. We assume that electrostatic effects dominate over disorder, or in other words that R and C are uniform. As Abeles

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first pointed out, when the classical   relaxation time  = RC becomes shorter than e2 C,the link between two neighboring grains can in fact be considered as metallic. This occurs when R < eh2 independently of the capacitance value. Extending this two-grain criterion to the entire granular system, one obtains that it becomes macroscopically insulating beyond a critical resistivity c =( e2 )d, where d is the grain diameter.While this criterion is rather well verified empirically,it is of course not correct,since the existence of a Coulomb blockade depends on the connecting circuit itself, here the rest of the sample. However, we can modify Abeles argument so as to make it basically identical to that of Mott [16]. Consider a granular system and assume that it is metallic. We “isolate” one grain from the rest of the granular system to which it is coupled through R and C. If we charge this grain by the addition of one electron and if RC is larger than ( e2 )C, this grain is effectively isolated from the system and transfer of the electron requires a finite electrostatic energy. However,this grain is just like any other grain.Hence, all of them are isolated and the medium must be insulating,contrary to our initial assumption. Hence, if R > ( e2 ), the system cannot be metallic and it must be insulating. In fact c = ( e2 )d is identical, apart from a numerical factor to Mott’s minimum metallic conductivity criterion, with a carrier density d−3 . We can indeed recover it by using the conventional expression for the conductivity and assuming that each grain contributes only one free carrier. The effective carrier density is then n = (n0 /N), where no is the carrier density in the bulk metal and N is the number of free electrons in a grain and the conductivity is given by =

no e 2  . N m∗

(7.4)

Here, m∗ is the effective mass corresponding to the motion of the carriers in the lattice of grains. Denoting by l∗ , vF∗ , the mean free path and Fermi velocity corresponding to inter-grain motion, one has =

e 2 l∗ . m∗ vf∗ d 3

(7.5)

Ioffe and Regel were the first to point out that the minimum value of l∗ is the inter-atomic distance (the inter-grain distance) [17].Thus,the minimum metallic conductivity is given with dkf∗ ∼ 1and l∗ ∼ d. We recover the result    d. (7.6) c = e2 Just as the intergrain resistance value e2 allows the metallic transfer of one electron from a grain to a neighboring one, and the system behaves as if it had one free electron per grain, we now remark that the value Rp = p2e2 allows the simultaneous transfer of p electrons. In other words, a granular system with an inter-grain resistance Rp will behave as if it had p free electrons per grain. Its resistivity will be =

d , p2 e 2

(7.7)

and using the same approximation the Hall constant will be d3 RH = . (7.8) pe Hence, one has  2 1/2 e . (7.9) RH = d 5/2  Thus, the progressive Coulomb blockade model predicts that the Hall constant varies as the square root of the normal state resistivity at constant grain size. This is in agreement with the experiments of Bandyopay et al.. Keep in mind that near the M/I transition the grain size remains constant [6]. We conclude that the available data are consistent with a Mott description of the M/I transition in granular systems having a small grain size.This transition takes place by a progressive freezing out of the bulk conduction carriers, until just before the transition each grain contributes effectively only one conduction electron. Simultaneous transfer of more than one electron per grain requires a finite activation energy. This can be at the origin of the negative coefficient of temperature in such films,while there is some evidence that their conductivity is finite at zero temperature.. We note that in this borderline situation electronic screening is strong within each grain, but

7

weak between the grains. From that standpoint such grains become effectively zero-dimensional, which has some interesting consequences for superconductivity as will be discussed in more detail in a later section.

7.5

Single Grain Behavior

A fundamental question concerning superconductivity (as well as any other cooperative phenomenon) is to know the size below which superconductivity cannot persist. This question is all the more relevant for superconductivity in view of the fact that the surface boundary condition for the order parameter is a mild one due to the fact that the coherence length is much larger than interatomic distances. To a good approximation it is the gradient of the order parameter in the direction perpendicular to the surface that is zero, rather than the order parameter itself (on the scale of the coherence length). The boundary condition does not prevent the occurrence of superconductivity in grains smaller than the coherence length. An exception to this rule are high temperature and short coherence length superconductors. This we shall discuss briefly at the end of this chapter. What is then the limiting size of a superconducting grain? There are several ways in which we can try to answer this question. One of them is to calculate the thermodynamic fluctuations of the order parameter that result from the finite size and to determine at what size these fluctuations are of the same order of magnitude as the order parameter itself. The second one is to calculate at what size there will be only one Cooper pair per grain. A third one is to determine at what size the average separation between the single particle electronic levels is of the order of the gap.We shall briefly perform these estimates. A remarkable result is that they all give the same answer. 7.5.1

Estimates of the Limiting Size for Superconductivity

Assuming that the grain’s size is smaller than  the boundary condition for the order parameter, | ∇¦ |⊥ = 0, together with the requirement that

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265

the order parameter cannot vary substantially on a length scale smaller than ,imposes that the order parameter is constant within the grain. We can then use a simplified Landau–Ginzburg expression for the free energy density: Fs = Fn + a | ¦ | 2 +

b | ¦ |4 . 2

(7.10)

Here, a is a linear function of the temperature that changes sign at the critical temperature Tc , and b is a constant. Fluctuations of the order parameter consist of a deviation from its equilibrium value: ¦ = ¦0 + ı¦ .

(7.11)

Above Tc , ¦0 = 0, and the excess free energy resulting from fluctuations is simply: ıFs = a| ı¦ |2 § ,

(7.12)

where § is the grain’s volume.At a finite temperature T one has ıFs  kB T . (7.13) We are now in a position to calculate the quantity of interest: | ı¦ |2  kB T (7.14)   2 . a | ¦ 0 |2 § b

This result is easily understood: fluctuations of the order parameter are of the order of the order parameter itself when the thermal energy kB T is of the order of the condensation energy in the grain. If we move somewhat away from Tc (let us say, around Tc /2), T is still of the order of Tc , and the condensation energy is already of the order of its value at T = 0. Using the BCS expression for the condensation energy in the limit T = 0, and noting that the gap is of the order of kB Tc ,the same criterion gives 1 N(0)2 §   . 2

(7.15)

In other terms words one has N(0)§  1 .

(7.16)

Since N(0) is essentially the number of Cooper pairs per unit volume, this condition means that the number of Cooper pairs in the grain is of order unity.

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Finally, let us consider the last criterion. The average separation of the quasi-particles energy levels near the Fermi level is the Fermi energy EF , divided by the number of electrons n per unit volume times the volume of the grain. The last criterion then reads EF  . n§

(7.17)

In the free electron approximation it is N(0) = 32 EnF . This is the same condition as before. For a typical superconductor such as Al the limiting diameter for superconductivity comes out to be of the order of 100 Å. 7.5.2

Numerical Studies

The above estimates are only rough. Another factor to be taken into account to predict the properties of small grains is the odd or even number of electrons per grain.We can expect that when the number is odd and the size sufficiently small that the separation between the electronic levels is a substantial fraction of the gap. The unpaired electron will play the role of a pair breaker further weakening the superconducting ground state. In small grains with ı  , ab initio calculations of the effect of mesoscopic fluctuations on the heat capacity and the spin susceptibility have been recently performed [18]. These calculations take into account the odd or even number of electrons, and have been performed for an ensemble of monodispersed grains, where the single particle spectrum is statistically distributed and characterized by level repulsion. Such calculations are relevant for comparison with experiments where measurements are carried out on a very large number of grains. Results differ from those of calculations for equally spaced, non-interacting levels. This can give some spurious results. Results for the heat capacity predict for an ensemble of grains with even number of electrons an exponential decrease at low temperatures T  ı, and at temperatures T ∼ , an enhancement reminiscent of the heat capacity jump in bulk samples, see Fig. 7.3. For an ensemble of grains with odd number of electrons the low temperature behavior is the

Fig. 7.3. Temperature dependence of the heat capacity and of the susceptibility of superconducting grains.  is the superconducting gap, and ı the separation between the electronic levels in the normal state. When both energies are of the same order, a smeared peak in the heat capacity can still be observed at the transition temperature. At the same temperature, a dip in the susceptibility is predicted

same as that of normal grains with a T 2 behavior and no enhancement is predicted at T ∼ . In an experimental realisation, even and odd number of electrons will be mixed, so that the only effect related to superconducting pairing will be an enhancement at T ∼ . This enhancement is substantial (about 20%) for ı ∼ , but vanishingly small for ı ∼ 50. The spin susceptibility shows pairing effects for ensembles of grains with odd number of electrons. For even number of electrons the exponential decrease of the susceptibility at low temperatures is basically a normal state effect. For odd numbers the decrease below Tc (due to pairing) is followed by an upturn at lower temperatures due to the unpaired electron. This re-entrant behavior is a signature of pairing effects. It has been checked that in normal grains the susceptibility is continuously increasing as the temperature is lowered. 7.5.3

Experiments on Nano-size Grains

Experiments have not been performed so far on physically isolated nano-grains, but on grains effectively uncoupled to neighboring ones.This is realized for instance in the case of granular composites con-

7

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267

sisting of metal grains separated by dielectric barriers sufficiently thick to prevent inter-grain Josephson coupling. For instance, Al-Ge mixtures are like this. Another case consists of a metal grain embedded in a dielectric barrier. The first case is appropriate for heat capacity measurements and the second one for tunneling spectroscopy. Heat Capacity Measurements The heat capacity transition of Al-Ge thin film composites has been measured for various compositions, near the metal/insulator transition [19]. In this region, the Al grain size (diameter) is of about 10nm. Measurements were performed for films ranging from metallic to insulating. A transition, characterized by a broaden peak in the heat capacity, was observed even for samples that were macroscopically insulating. This is explicitly shown Fig. 7.4. For that grain size, the separation between the electronic levels is of the order of the gap of bulk Al, as we have calculated above. This result is in agreement with the calculations of Falci et al. [18] in the sense that it clearly shows that superconductivity is not quenched down to sizes at which ı ∼ . If anything,the experimentally observed peak in the heat capacity is rather larger than predicted. However, one has to be careful in view of the uncertainties on the exact grain size and gap values in the grain.These are probably larger than in the bulk in view of the enhanced critical temperature. Smaller grain sizes, down to a couple of nanometers, are achieved in the composite Al-Al2O3 . In that case, ı  . Heat capacity measurements on such films have shown that the transition disappears while the samples are still metallic, as shown Fig. 7.5. One thus reaches the conclusion that any sign of superconductivity in the heat capacity is quenched at a size between 10 nm and 2 nm, again in agreement with theory. Tunneling into Single Grains The spectroscopic properties of single grains have been studied by measuring the conductance of junctions where the current from one electrode to the

Fig. 7.4. Electrical resistance in ln scale (upper figure) and the heat capacity (lower figure) as a function of temperature for an Al–Ge sample in the granular insulating phase. The grain size distribution is similar to that shown in Fig. 7.2.Although the sample is macroscopically insulating, the heat capacity transition of the individual Al grains can be observed. The behavior of the heat capacity is similar to that predicted in [18], shown in Fig. 7.3

other passes through a grain, which is weakly connected to them [20]. This device is sensitive to both the single level splitting and the superconductng gap. The conductance shows a series of peaks marking the bias values at which the transfer of an electron from one electrode (lead) to the other. Each peak corresponds to tunnelling via an electronic level on the particle. The even (odd) number of electrons on the

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G. Deutscher

Fig. 7.5. Heat capacity transition temperatures Tc(e) of Al–Al2 O3 films at various levels of the normal state resistivity. As the resistivity is increased from about 0.5 × 10−3 §cm to 5 × 10−3 § cm, by increasing the oxide concentration, the transition goes from BCS like to very smeared. The grain diameter in these samples is about 40 Å, below the size at which an isolated grain can show a transition, according to [18] (ı/ is substantially larger than one). Notice the increased transition temperature compared to that of bulk Al (1.2 K), even in samples having a smeared transition. In this case, this increase cannot be attributed to the isolated grain, since for that size they are basically not superconducting

grain (in the zero bias state) can be identified by the presence (absence) of field splitting of these peaks. For the even case the bias where the first peak is observed decreases as a function of the applied field until a field is reached where it stays roughly constant. This decrease is associated with the decrease of the superconducting gap in the grain under the applied field. In this way, the value of the gap is determined. In the odd case the first peak corresponds to a state where an additional electron pairs with the single one, while in the second one there must be two unpaired electrons. The difference in energy must therefore be twice the gap and it must go down as the field is increased. This is observed.

Gap values can in this way be clearly identified as long as they are larger than ı. For the studied Al grains, this corresponds to sizes larger than about 10 nm. Gap values obtained are larger than the bulk Al value by about 50%, in agreement with the increase in Tc observed in granular Al films. These measurements confirm the conclusion drawn from the heat capacity data, that superconductivity persists certainly down to a size where ı ∼ . They also confirm that the increase in Tc quoted before is a property of single grains. It is not due to intergrain coupling.

7

7.6

Weakly Coupled Grains: Granular Insulators and Super-Insulators

We now consider the case of superconducting grains having some degree of coupling between them. By this we mean that the granular compound is macroscopically very close to being an insulator in the normal state. This is different from the case considered in a later section where we discuss critical currents and critical fields. There it is implicitly assumed that the compound is a metal in its normal state. Here we wish to review what is known for the more delicate situation where the coupling is not so weak that it can be ignored, but sufficiently weak that the compound is not really a metal. Some remarks concerning the normal state properties of such compounds have already been made. For the case of isolated grains, two energy scales had to be considered: the level spacing ı and the superconducting gap . Here additional scales come into play: the normal state tunnelling coupling energy Vt, the charging energy Ec and the intergrain Josephson coupling EJ . The degree of disorder in the compound has also some importance This includes the widths of the grain size and of the Josephson coupling distributions. We can distinguish between two cases, depending on whether EJ ≥ kB T, or EJ  kB T, It is clear that only in the first case can macroscopic coherence be achieved.

7.6.1

Nanostructured Superconductors

269

Small Grain Case: Percolative Behavior

An interesting case where disorder effects dominate is when the individual grains are smaller than the critical size defined above. Grains can then become superconducting only if they couple together.Assuming that the grains are all of the same size, but that the coupling between them (defined by the normal state intergrain resistance RN ) has a substantial distribution width, one can then show that a percolating structure appears below some critical temperature [21].As the temperature is lowered the condition EJ > kB T is met by an increasing fraction of junctions.When this fraction is sufficient an infinite cluster is formed. Under the simplifying assumption that only this infinite cluster contributes to the heat capacity the heat capacity transition can then be calculated.The only free parameter is the width of the RN ’s distribution.As shown Fig. 7.6, the result of the calculation is in good agreement with experiments [22]. 7.6.2

Large Grain Case: Superconducting Insulators

When the grains are larger than the critical size a relevant question to ask is whether the compound can be insulating in its normal state and still undergo a macroscopic transition to a coherent superconducting state. The tunnelling coupling energy Vt represents an effective band width for the granular compound. It is sufficient to make it a metal if it is larger than the electrostatic charging energy Ec and

Fig. 7.6. Fit to a percolation model of the heat capacity transition for one of the samples shown in Fig. 7.5

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G. Deutscher

vice versa. Calling t the intergrain tunneling matrix element, the coupling energy is given by Vt =

t2 , ı

(7.18)

where we have ignored numerical factors of order unity (such as the number of nearest neighbors) in view of the crudeness of our model for the metal/insulator transition. The compound is an insulator if t2 Ec > . (7.19) ı Hence, the question we are asking is whether the situation described by the conditions EJ > Ec >

t2 ı

(7.20)

can be realized. Note, this must be fulfilled if we want to have a superconducting insulator. The intergrain conductance can be calculated from the Thouless expression Vt , ı

(7.21)

  e eRN

(7.22)

g = gc

 2 where gc is the universal conductance e . Using this relation we can rewrite the Josephson expression EJ = as

t2 . (7.23) ı2 So we can have a superconducting insulator only if the gap is larger than the level spacing. This is our old large grain condition and not a very surprising result. The other condition that the Josephson coupling energy should be larger than the Coulomb energy is more interesting. For an isolated grain the e2 Coulomb energy is ( a ) where a is the grain’s radius. For a 10 nm grain this energy is of the order of 0.1 eV. This is several orders of magnitude larger than the Josephson coupling energy. Hence, the existence of a superconducting insulator at grain sizes typical of granular materials seems to be ruled out. Yet, early experiments by Shapira and Deutscher [23] had identified a narrow region near the M/I transition in Al-Ge films, where the superconducting insulating regime appeared to exist. The grain size was indeed of the order of 10 nm. The authors concluded that the Coulomb energy must in fact have been strongly reduced by a large effective dielectric constant eff near the Metal/Insulator transition. These early experiments have been recently extended down to lower temperatures [24]. The claim for the existence of a superconductor-insulator regime has been fully confirmed (Fig. 7.7). It is therefore established that the effective dielectric constant in the granular compound can be increased by several orders of magnitude larger (at least three) near the M/I transition. EJ = 

Fig. 7.7. Supra–insulator transition in an Al–Ge sample in the granular–insulating phase. At low temperatures, the resistance is higher when the grains are superconducting than when they are returned to the normal state by the application of a strong magnetic field [33]

7

It is well known that the M/I transition is one of the most difficult problems in solid state physics. Localization (a la Anderson),Coulomb interactions and disorder (percolation) may all play a role. A lingering question is whether this can be a (presumably second order) phase transition, and if such, what is the order parameter? So far the theory has not been able to give a full answer to this question. It could be that the true importance of the existence of the superconducting insulator near the M/I transition is that it implies a divergence of the dielectric constant. This would mean a second order phase transition. So far the experiments on Al/Ge granular compounds are the only ones that have clearly demonstrated this divergence. It could be that this is so, since the dielectric thickness in this compound is particularly uniform. More work needs to be done to establish firmly under what conditions the M/I transition can indeed be a phase transition. 7.6.3

Pseudogap Regime in the Superconducting Insulator

Nozi`eres and Pistolesi [25] have considered a model where a material has a gap Eg in the DOS in the normal state and at the same time can develop a superconducting order parameter below a temperature Tc . This model does not refer to the granular case in particular, although this could be one of its realizations. The parameters of the model are Eg and the critical temperature Tc0 that the material would have in the absence of the normal state gap. It assumes that the interaction parameter V of the BCS theory is not modified by Eg . The critical temperature, the quasi-particle energy gap and theorder parameter E are calculated as a function of 00g , where 00 is the BCS gap of the pristine material. The position of the Fermi energy is also taken into account.   E The main results are that as 00g is increased the critical temperature goes down, the single particle excitation energy goes up and the order parameter goes down up to a value where superconductivity collapses all together. One of the signatures of the pseudogap regime is that the strong coupling ratio (energy gap/critical temperature) becomes very

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large near the point of collapse of superconductivity. As is well known, this behavior has been one of the main recent findings in the high Tc cuprates [26]. One of the interpretations proposed for the large gap is that it represents the energy required to break up preformed pairs above Tc .However,it was shown that in the pseudogap regime the coherence energy scale follows the same doping dependence as Tc , rather than that of the pseudogap. The pseudogap in the cuprates may thus just be a manifestation of the fact that in the underdoped regime the cuprates are basically superconducting insulators. Transport measurements have indeed shown that their normal state conductivity increases at low temperatures [27]. Tunneling experiments on Al/Ge films in the superconducting insulator regime have given some evidence for a large strong coupling ratio [28]. 7.6.4

The Superinsulator Case

Out of the narrow concentration range where the superconducting insulator regime is observed one expects to enter into a regime where the grains are still superconducting (provided they are large enough), but the Josephson coupling is too weak to allow a macroscopically coherent state: Ec > EJ >

t2 . ı

(7.24)

Here, we have assumed that the decrease of the dielectric constant away from the M/I transition is the main reason for the inversion of the order of Ec and EJ . The main point is that Ec becomes quickly larger by orders of magnitude upon lowering the temperature. Hence, an increase of EJ will not help to recover the superconducting state. There is, however, still another interesting regime, where Ec is not much larger than the value of the superconducting gap in the grains: t2 U ≥  > EJ > . (7.25) ı In this regime, the intergrain resistance will be substantially increased by the known phenomenon of Giaever tunnelling between two superconductors which itself requires an energy of 2. As the temperature is lowered below the critical temperature of the

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G. Deutscher

Fig. 7.8. An Al–Ge sample still in the granular insulating phase having a lower resistance than that shown in Fig. 7.7 has a transition to a superconducting state near 1K

grains, the behavior will go from insulating (normal state exponential localization) to superinsulating. 1 Indeed, a change of slope in the lnR versus T − 2 representation has been clearly observed just outside the range of the superconducting insulator regime [23] at temperatures of the order of the critical temperature of the grains in Al/Ge films as shown Fig.7.8. This experiment is another proof that superconductivity persists in Josephson-uncoupled grains having a size of about 10 nm.

7.7

Well Coupled Grains

In this section we consider the case where electrostatic charging can be completely neglected. In other words the granular compound is a metal in its normal state. The key parameters are then the grain size and the inter-grain coupling [28]. In an homogeneous superconductor the critical current density jc and the upper critical field Hc2 are determined by the coherence length (T). The Ginzburg Landau (GL) theory predicts that: 4ens  jc = √ 3 3 m(T)

(7.26)

¥0 . 2 2(T)

(7.27)

and Hc2 =

In what follows we consider whether these relations are valid in a granular metal, taking care of course that the value of the coherence length correctly reflects the structure of the compound – granular or random. 7.7.1

Granular Case

The above relations remain valid for a granular superconductor as long as the grain size d is smaller than the coherence length. An expression for the coherence length valid for a granular superconductor has been given by Deutscher, Imry and Gunther (DIG) [29] on the basis of the free energy expression: F s = Fn + A

 i

| i |2 +

B | i |4 2

1  + C | i − j |2 , 2 i,j

i

(7.28)

where the sum in the last term of the rhs runs over neighboring grains.This expression assumes that the grain size is smaller than the value of the coherence length of the parent material so that inside each grain the order parameter  is constant. If there are spatial variations of the order parameter due to thermodynamical fluctuations, it will then vary discontinuously from grain to grain. This is expressed in the last term of the rhs. This term replaces the usual gradient

7

term of the GL theory. The value of the coefficient C is determined by the intergrain dielectric barrier. It is simply related to the strength Ec of the Josephson intergrain coupling: Ec = C |  |2 . The coherence length is given by  C . (T) = d|A|

(7.29)

(7.30)

This expression is valid as long as (T) > d. When this condition is fulfilled the granular superconductor is effectively three dimensional. The coefficients A and B are equal to the GL coefficients of the parent material. The explicit value of C can be calculated from the Josephson expression for the coupling energy near Tc :  |  |2 , (7.31) EJ = 2 8e RkB Tc where R is the intergrain normal state resistance. The expression for (T) allows to check whether it is indeed larger than d, which is the condition for its validity. Apart from coefficients of order unity we can write    (T)2 ı Tc  > 1, (7.32) = d2 Re2 kB Tc Tc − T where

EF (7.33) N is the distance between the electronic levels in the grain and N is the number of electrons per grain. The condition (T) > d is always valid near Tc . Far below Tc this is the case if the intergrain resis tance is smaller than the quantum resistance e2 , as long as the level splitting is of the order of kB Tc (or the gap). We have discussed in earlier sections the full importance of this condition. In the three dimensional regime, when the critical current density is reached, the phase changes by  over many grains. In the presence of vortices their core extends over many grains. The granularity of the material has the effect to turn the superconductor into a strong type II one.The grain size itself is not

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directly reflected in the superconducting properties. This was the case treated by the group of Abeles [4]. In the regime (T) < d the coherence length evidently looses its meaning, since inside each grain the order parameter is constant. The granular superconductor then becomes zero dimensional [6]. If the individual grains remain superconducting in the 0D regime, which will happen under conditions that we have discussed above, the critical current will be determined by the critical current of the individual junctions: jc = Ic d −2 ,

(7.34)

2e Ec . 

(7.35)

where Ic =

The critical field becomes equal to the critical field of the individual grains: ¥0 Hc (grain) = 2 d



T 1− Tc

 12

.

(7.36)

The transition to zero-dimensionality that occurs when (T) ≤ d is directly reflected in the temperature dependence of the critical field.While it is linear in the 3 dimensional high temperature regime where (T) > d, it becomes parabolic in the zero dimensional regime. The transition is thus marked by an upturn of the critical field [30], as seen Fig. 7.9.

ı=

Fig. 7.9. Upturn to the critical field of an Al–Ge sample in the granular metallic regime

274

7.7.2

G. Deutscher

Random Case

Here again the conventional expressions for the critical current density and of the upper critical field are valid as long as the coherence length is larger than the length scale that characterizes the inhomogeneity of the system. However now the inhomogeneity scale is not the grain size, but rather the percolation correlation length p : p = ˛d(p − pc )− ,

(7.37)

where ˛ is a numerical coefficient of order unity and  is the critical index that governs the divergence of p near the threshold pc .We are interested here in the case where p > pc . There is then an infinite cluster, which can be visualized as composed of “fat” blobs connected by thin weak links. The blobs consist of many interconnected threads, while the weak links are basically single chains of grains. More exactly, it has been shown that the number of parallel channels in a weak link does not diverge near pc . The physical meaning of p is that it is the distance that separates nearest weak links. At scales larger than p the infinite cluster can be considered as being homogeneous. At scales smaller than p , the actual structure of the blobs which is fractal will determine the properties of the system. The superconducting state probes the properties of the infinite cluster on the scale of the coherence length. Again, we have to consider two cases. (i) Homogeneous limit (T)  p : In this limit the percolating superconductor behaves as a homogeneous one. The critical current density is determined by the number of independent superconducting channels per unit cross section:  2 jc ∝ p−2 ∝ p − pc . (7.38) This dependence was observed experimentally for a series of Pb-Ge mixtures and offers directly a determination of the critical index of the correlation length. The coherence length and upper critical field are determined by the macroscopic coefficient of diffusion D which can be calculated from the Einstein relation for the macroscopic conductivity:  = 2e2N(0)D .

(7.39)

Here the density of states N(0) is given by the density of the infinite cluster which goes to zero at pc with exponent ˇ. Hence, D ∝ (p − pc )t−ˇ .

(7.40)

We can use the dirty limit expression to calculate the coherence length:  D Tc . (7.41) 8 kB T Tc − T Due to the divergence of (T) at Tc the homogeneous limit will always apply close to the transition temperature. The upper critical field will then vary linearly with temperature and is roughly proportional to the normal state resistivity (exponent t is substantially larger than the exponent ˇ (in 3D,t = 2 and ˇ = 0.4)).  2 (T) =

(ii) Inhomogeneous limit (T) < p : When (T) is smaller than the percolation correlation length, we are not allowed anymore to use the above expressions. This will happen when p is close to pc (D then goes to zero) and if we are not too close to Tc . The new and interesting property that then arises is that the coefficient of diffusion is now scale dependent. It is sensitive to the internal structure of the blobs as is the value of the upper critical field. At short length scales diffusion is not governed by the usual diffusion law x 2 = Dt ,

(7.42)

but rather by an anomalous diffusion law, which can be represented by an analogous equation, but where the diffusion coefficient is itself time dependent: x 2 = D(t)t .

(7.43)

At long time scales D goes back to its macroscopic, time independent value. At short length scales, namely at scales shorter than the percolation correlation length, it varies as a power law of t, in a way that is independent of the concentration. The idea here is that at these short length scales the diffusion process (or the conductance) is insensitive to the macroscopic structure of the cluster (whether it is finite or infinite).Now, if we apply a magnetic field we effectively probe the cluster on a scale  1 ¥0 2 (H) = . (7.44) 2H

7

If (H) > p , we are back to the usual case and the critical field is concentration dependent. It varies with the macroscopic conductivity in a way that is similar to that of a regular type II superconductor (there are some minor differences, due to the concentration dependence of the density of the infinite cluster). But if (H) < p , the cluster is probed on a scale that is concentration independent. Then the upper critical field,which is here the field that will destroy the“blobs”, is independent from the concentration, hence from the macroscopic conductivity.This saturation of the upper critical field in the vicinity of the percolation threshold, where the macroscopic resistivity diverges, has indeed been observed [31].

7.8

Critical Temperature of Granular Superconductors

It is a remarkable property of granular superconductors such as Al-Al2O3 and Al-Ge that their critical temperature increases as the grain size is decreased, reaching its maximum value near the metal to insulator transition. In this section we review the available data and make some remarks concerning the possible role of weak screening in the critical temperature enhancement. 7.8.1

Grain Size Dependence of the Critical Temperature

Enhancement of the critical temperature is observed in granular structures only. Random percolating mixtures of the same constituents,for instance Al and Ge, do not show any critical temperature enhancement. Critical temperatures increase as the dielectric volume fraction is increased up to near the metal to insulator transition. Critical temperatures are higher when the films are deposited unto substrates held at low temperatures. For instance, the maximum critical temperature observed in Al-Al2O3 is about 2 K when deposited at room temperature, and more than 3 K when deposited at liquid nitrogen temperature. As we have seen above the average grain size decreases when the dielectric volume fraction increases. The grain size is also decreased the deposition temperature is decreased. It has been shown

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that for a given eutectic system, Tc is in fact only a function of the grain size, itself determined by the dielectric volume fraction and the deposition temperature. More precisely, when plotted as a function of the surface to volume ratio  , the critical temperature values of Al-Al2O3 films fall on a single curve for samples deposited at room temperature or liquid nitrogen temperature. The results are consistent with a simple dependence of the BCS interaction parameter NV on  : NV = (NV )0 (1 + 0.3 ) .

(7.45)

Here we assume that the Debye temperature remains unchanged and that the interaction is enhanced over a surface layer having a thickness of about 3Å [27]. 7.8.2

Weak Screening Model

Many models, well reviewed by Abeles [4], have been proposed to explain the Tc enhancement. Amongst them, the idea that Tc increase is induced by phonon softening at the surface received considerable attention. Yet, detailed tunnelling measurements performed on granular Al failed to give quantitative evidence for this model [32]. In addition, it is not clear how it could explain why there there is no enhancement at all in the case of the percolating structure. Here, we consider a different approach, based on our description of the metal to insulator transition in the granular structure. When the Coulomb blockade becomes effective, reducing the number of electrons that can freely flow simultaneously from grain to grain, electrons at the grains surface become weakly screened while the screening in the interior of the grain remains strong. It is interesting to consider the effect of this weak surface screening on the interaction parameter. Weak screening evidently enhances all Coulomb interactions. For specificity let us consider the Jellium model of these interactions in a metal: V ="

!q2 4e2 4e2 + , ks2 + q2 ks2 + q2 ! 2 − !q2

(7.46)

where ks is the inverse screening length, q the momentum transfer and !q the frequency of a phonon of momentum q. The first term of the r.h.s. is the

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G. Deutscher

Coulomb repulsion, reduced by a factor " < 1, which takes into account its renormalisation due to retardation effects. In the limit of zero frequency the net interaction is: V =−

4e2 (1 − ) . ks2 + q2

(7.47)

This evidently gets enhanced by a reduced screening parameter. But this enhanced interaction does not lead to an enhanced Tc , on the contrary. Multiplying the interaction potential by the normal state density k2 of states, N = 4es 2 ,we obtain for a typical phonon wave vector q0 in the zero frequency limit k2 NV = 2 s 2 (1 − ) . ks + q0

(7.48)

Thus,the BCS parameter is in fact an increasing function of the screening parameter, as is well known. Although obtained here in the framework of a very crude model, we believe the conclusion to be quite general. Namely, that as a function of the screening parameter, the net interaction potential and the BCS parameter vary in opposite directions.This indicates a possible general way to obtain higher critical temperatures: one reduces screening under the condition that at the same time the density of states is kept unchanged. We can in principle do this, if reduced screening is obtained primarily through a reduced dimensionality, rather than through a reduced carrier density. In fact, in the 2D and 1D cases, it may even be possible to have an enhanced density of states, if the Fermi level is properly located (A15, van Hove). It would of course be risky to apply this very simple argument in a quantitative way to such complex materials as the cuprates or the organic superconductors. But it may not be unreasonable to apply this to the granular superconductors. We consider a grain of a weak coupling superconductor (in the BCS sense) that is large enough so that we can neglect special cluster size effects on its electronic structure. We assume that the grain is located in an insulating matrix.Then reduced screening takes place at the atomic surface layer of the grain, while the density of states remains unchanged to a first approximation. We write

−1 ks−1 = kso (1 + ˛ ) ,

(7.49)

where kso is the screening parameter of the bulk material and ˛ is a numerical coefficient of order unity whose value depends on the dielectric properties of the insulating matrix.The surface to volume ratio  is the surface of the grain divided by its volume, multiplied by a length of the order of the screening length, say roughly 1Å. Assuming   1,we can calculate the new value of the BCS parameter:   2k2 NV = NV0 1 + 2 so 2  , (7.50) kso + qo where Vo is the net interaction potential in the bulk material. Let us take as example Al. From the BCS expression for Tc we get NV0 = 0.18. The renormalized ∗ Coulomb  repulsion is = 0.1 From this we obtain  2 kso 2 +q2 ks0 0

= 0.3. Thus, NV = NV0 (1 + 0.6 ) .

(7.51)

In spite of the crudeness of our calculation, this result is in remarkable agreement with the data (7.30). No particular importance should of course be given to the value of the coefficient of  in view of the approximations made. The weak screening model can also explain why no critical temperature enhancement is seen in percolating random structures. Even very near the percolation threshold the majority of metal grains that constitute the infinite cluster are not at the surface of this cluster, but are well connected within “blobs”. Hence, the effective surface to volume ratio is too weak to have any effect on Tc . Note, this naive model is unlikely to be the last word on the intriguing Tc enhancement in granular materials. Electronic effects and other may play an interesting role. Acknowledgements This chapter summarizes many collaborations, some of them over long periods of time. Ben Abeles and Richard Cohen, then at RCA Laboratories, were the ones who attracted my attention to the challenges posed by the understanding of granular superconductivity. A long collaboration followed with the

7

neighboring Rutgers group, particularly with Peter Lindenfeld and the late William McLean, and their students and post docs, amongst them Tom Worthington and Tom Filler. At Tel Aviv University, Moshe Dayan, David Bergman, Henri Fenichel, Alexander Gerber, Meir Gershenson, Ilan Grave, Enrique Grunbaum, Y. Imry, Zvi Ovadyahu, Michael Rappaport, Yoash Shapira, O. Entin Wohlman all made important contributions.Inspiring discussions

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with the late Shlomo Alexander took place on a permanent basis. I take the opportunity to express my sincere gratitude to all of them. It is known that our work on granular Aluminium was one of the triggers that prompted Alex Muller to look for high temperature superconductivity in oxides. This may well have been the most important outcome of this work. This work was supported by the Oren Family Chair for Experimental Solid State Physics.

References 1. B. Abeles, R.W. Cohen and G.W. Cullen, Phys. Rev. Lett. 17, 632 (1966) 2. Y. Shapira and G. Deutscher, Thin Solid Films 87, 29 (1982) 3. G. Deutscher, A. Kapitulnik and M. Rappaport, In: Percolation, Structures and Processes, Eds. G. Deutscher, R. Zallen and J. Adler, Ann. Israel Phys. Soc. 5, 207 (1983) 4. B. Abeles, Adv. Phys. 24, 407 (1975); Applied Solid State Science 6, 1 (1976) 5. A. Kapitulnik, M. Rappaport and G. Deutscher, J. Phys. Lett. 42, L-541 (1981) 6. G. Deutscher, H. Fenichel, M. Gershenson, E. Grunbaum and Z. Ovadyahu, J. Low Temp. Phys. 10, 231 (1973) 7. U. Dai et al., Phys. Rev. B 36, 790 (1987) 8. D.J. Bergman et al., Phys. Rev. Lett. 50, 1512 (1983) 9. G. Deutscher, M. Rappaport and Z. Ovadyahu, Solid State Commun. 28, 247 (1978) 10. B. Bandyopadhyay et al., Phys. Rev. B 26, 3476 (1982) 11. A. Fulton and G.J. Dolan, Phys. Rev. Lett. 59, 109 (1987) 12. H. Devoret et al., Phys. Rev. Lett. 61, 1824 (1990) 13. G. Deutscher and M. Cyrot, unpublished 14. N.F. Mott and Davis, In: Electronic processes in non cristalline materials (Clarendon, Oxford, 1971) 15. B. Abeles, Phys. Rev. B 15, 2828 (1977) 16. N.F. Mott, Phil. Mag. 26, 1015 (1972) 17. J. Ioffe and A. Regel, Prog. Semiconduct. 4, 237 (1960) 18. G. Falci, A. Fubini and A. Mastellone, to be published 19. Y. Shapira, PhD Thesis, Tel Aviv University (1986) 20. D.C. Ralph, C.T. Black and M. Tinkham, Phys. Rev. Lett. 74, 3241 (1995) 21. G. Deutscher, O. Entin-Wohlman, S. Fishman and Y. Shapira, Phys. Rev. B 21, 5041 (1980) 22. R.L. Filler et al., Phys. Rev. B 21, 5031 (1980) 23. Y. Shapira and G. Deutscher, Phys. Rev. B 27, 4463 (1983) 24. P. Nozieres and F. Pistolesi, Eur. J. Phys. B 10, 649 (1999) 25. G. Deutscher, Nature 397, 410 (1999) 26. L. Nechustan, MSc Thesis, Tel Aviv University (2002) 27. G. Deutscher, M. Gershenson, E. Grunbaum and Y. Imry, J. Vac. Sci. Technol. 10, 674 (1973) 28. G. Deutscher, Phys. Lett. 35A, 28 (1971) 29. G. Deutscher, Y. Imry and L. Gunther, Phys. Rev. B 10, 4042 (1974) 30. G. Deutscher, O. Entin-Wohlman and Y. Shapira, Phys. Rev. B 22, 4264 (1980) 31. A. Gerber and G. Deutscher, Physica B 126, 477 (1984) 32. M. Dayan and G. Deutscher, Proceedings of Low Temperature Physics LT14, Vol. 2, 241 (North-Holland, Amsterdam, 1975) 33. A. Gerber et al., Phys. Rev. Lett. 78, 4277 (1997)

8

Proximity–Coupled Systems: Quasiclassical Theory of Superconductivity

Venkat Chandrasekhar Dept. of Physics and Astronomy, Northwestern University, Evanston, USA

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Transport Equations in the Diffusive Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 The Keldysh Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 The Quasiclassical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .288 Non-Equilibrium Green’s Functions for Superconducting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Quasiclassical Superconducting Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 The Dirty Limit: The Usadel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297 Parametrization of the Quasiclassical Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Applications of the Quasiclassical Equations to Proximity-Coupled Systems . . . . . . . . . . . . . . . . . . . . 304 8.9.1 Proximity-Coupled Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8.9.2 Superconductor-Metal Bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.9.3 The SNS Junction and Andreev Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .310 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.1

Introduction

Although the physics of normal metals (N) in close proximity to superconductors (S) has been studied extensively for more than thirty years, it is only in the past decade that experiments have been able to probe directly the region close to the NS interface at temperatures far below the transition temperature Tc of the superconductor. These experiments have been made possible by the availability of microlithography techniques that enable the fabrication of heterostructure devices with submicron scale features. This size scale is comparable to the relevant physical length scales of the problem, and consequently, a number of new effects have been observed. A variety of systems have been studied, the variation primarily being in the type of normal metal coupled to the superconductor. Canonical normal metals such as copper and gold, semiconductors, insulators, and

ferromagnets have been employed. Although a variety of theoretical techniques have been used to describe proximity-coupled systems, the quasiclassical theory of superconductivity [1–12] has proved to be a remarkably powerful tool in understanding the microscopic basis for the remarkable effects observed in these systems. A number of excellent recent articles [13–19] have explored the application of the quasiclassical theory of superconductivity to proximity coupled systems. In this review, a self-contained development of the quasiclassical theory is presented,starting from nonequilibrium Keldysh Green’s functions for normal metal systems. If the normal metal is clean, quasiparticles in the normal metal travel ballistically over long length scales. In the samples studied in the majority of recent experiments, however, the quasiparticles are scattered elastically within a short distance, so that the quasiparticle motion is diffusive. Here,

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we shall concentrate on this diffusive case, where the elastic scattering length  is the shortest relevant length scale in the problem. We shall also restrict ourselves to the case where the superconductor is of the canonical s-wave type, avoiding complications with non-spatially symmetric order parameters within the superconductor.

8.2

Transport Equations in the Diffusive Approximation

As an illustration of some of the issues that arise in dealing with non-equilibrium transport in mesoscopic diffusive systems, we consider first the classical Boltzmann equation in the diffusion approximation. Consider a one-dimensional diffusive normal metal wire of length L. We assume that all dimensions of the wire are larger than , the elastic mean free path of the conduction electrons. f (E, x),the distribution function of electrons at energy E and at a point x along the wire, obeys the diffusion equation D

d2 f (E, x) + C(f ) = 0 . dx 2

(8.1)

Here, D = (1/3)vF  is the three-dimensional electron diffusion constant, with vF being the Fermi velocity of the electrons. C(f ) is the collision integral, which takes into account inelastic scattering of the electrons,and itself depends on the distribution function f . If we consider a mesoscopic wire whose length L is much shorter than any inelastic scattering length Lin , this term can be set to zero. This diffusion equation has a simple solution in some specific geometries. Consider the case of a wire of length L sandwiched between two normal ‘reservoirs,’ which we shall call the left (at x = 0) and right (at x = L) reservoirs. The reservoirs are defined as having an equilibrium distribution function fL (E) and fR (E) respectively. Then the solution of the diffusion equation, Eq. (8.1), under the condition L  Lin (so that C(f ) = 0), and subject to the boundary conditions that f (E, x) = fL (E) at the left reservoir, and f (E, x) = fR (E) at the right reservoir, is given by [20]

x + fL (E) . f (E, x) = fR (E) − fL (E) L

(8.2)

The electrical current I through the wire in the diffusion approximation is given by  @f dE , (8.3) I = −eAD N(E) @x where A is the cross-sectional area of the wire. With f (E, x) given by Eq. (8.2), this can be written  eAD I =− N(E)[fR (E) − fL (E)]dE . (8.4) L In order to obtain a finite current, we must apply a voltage across the wire.Let us apply a voltage V to the left reservoir, keeping the right reservoir at ground (V = 0). This has the effect of shifting the energies of the electrons in the left reservoir by −eV , so that the electron distribution function there is given by fL (E) = f0 (E − eV ), where f0 (E) is the usual equilibrium Fermi distribution function f0 (E) =

1 eE/kB T

+1

(8.5)

and the energy E is measured from the Fermi energy EF . Figure 8.1 shows the distribution function f (E, x) in the wire. It has a step-function form, which varies linearly along the length of the wire. Evidence for such a non-equilibrium distribution has recently been seen experimentally in a series of beautiful experiments by the Saclay group [21].

Fig. 8.1. Nonequilibrium electron distribution function f in a normal wire with a potential V applied across it as a function of energy E and position x. The position x is normalized to the length L of the wire

8

In the limit of small voltages V , the difference of distribution functions in Eq. (8.4) can be expanded as fR (E) − fL (E) = f0 (E) − f0(E − eV ) ≈ f0(E)    @f0 − f0 (E) + eV − @E   @f0 (8.6) = −eV − @E so that e2 AD V I = L ≈





@f0 N(E) − @E

 dE

N(0)e2 AD V, L

(8.7)

where N(0) is the density of states at the Fermi energy. In performing the integral, we have assumed that we are at low enough temperatures so that the derivative of the Fermi function (−@f0 /@E) can be approximated by ı(E). The electrical conductance is then given in the Nernst-Einstein form G = N(0)e2 D(A/L) = 0 (A/L). The thermal current through the wire is given by an expression similar to Eq. (8.4): IT =

AD L

 EN(E)[fR (E) − fL (E)]dE .

(8.8)

The critical difference between the expression for the thermal current and the expression for the electrical current is the presence of an additional factor of E in the integrand in Eq. (8.8), which has important consequences in the calculation of the thermal properties. To obtain expressions for the thermal coefficients, let the temperature of the left reservoir be T and its voltage V , and let the temperature of the right reservoir be T + T and its corresponding voltage V = 0. The distribution function in the left electrode is then fL (E) = f0 (E − eV , T), and the distribution function in the right electrode is fR (E) = f0 (E, T + T). In the limit of small T, V , the difference in the distribution functions in Eq. (8.4) can be expanded as

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281

   @f0 E fR (E) − fL (E) = f0(E, T) + T − T @E    @f0 − f0(E, T) + eV − @E    @f0 E T − eV − . (8.9) = T @E Putting this expression into the equations for the electrical and thermal currents, Eqs. (8.4) and (8.8), one obtains the two transport equations     e2AD @f0 I= N(E) − dE V L @E     eAD @f0 − EN(E) − dE T , (8.10a) TL @E     @f0 eAD IT = − EN(E) − dE V L @E     AD @f0 + E2 N(E) − dE T . (8.10b) TL @E These equations are equivalent to the usual form for the transport equations for a metal: I = GV + T ,

(8.11a)

I = V + T ,

(8.11b)

T

with the linear-response thermoelectric coefficients defined by     e2 AD @f0 N(E) − dE , L @E     eAD @f0 =− EN(E) − dE , TL @E     eAD @f0  =− EN(E) − dE , L @E     AD @f0 = E2 N(E) − dE . TL @E

G=

(8.12a) (8.12b) (8.12c) (8.12d)

If we approximate the derivative of the Fermi function by a ı function at low temperatures,as we did for the electrical conductance G, we see that all other coefficients vanish. In order to obtain a finite value, we use the Sommerfeld expansion for the integrals [22]

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V. Chandrasekhar

  @f0 dE = (8.13) ¥ (E) − @E    2 (kB T)2 @2 ¥ (E) ¥ (0) + + ... 6 @E2 E=0

Using the fact that N  (0) = N(0)/2EF , we obtain the following expressions for the last three coefficients  2 kB2 T eAD N(0) , 6 L EF  2 (kB T)2 eAD N(0) =− , 6 L EF  2 kB2 T AD = N(0) . 3 L

=−

(8.14a) (8.14b) (8.14c)

Experimentally, the quantities often measured are the thermopower S and the thermal conductance GT . The thermopower is defined as the thermal voltage generated by a temperature differential T, under the condition that no electrical current flow through the wire (I = 0). Putting this condition into the first transport equation, Eq. (8.10a), we obtain S=

V   2 kB kB T =− = . T G 6 e EF

(8.15)

Note that since the expression for S contains a factor kB T/EF , the thermopower of a typical normal metal is very small. The thermal conductance GT is defined as the ratio of the thermal current I T to the temperature differential T, under the same condition of no electrical current flow (I = 0). From the equations above GT =

IT =  + S . T

(8.16)

For typical metals, the second term is much smaller than the first, and is usually ignored, so that GT ≈ . If we take the ratio of the electrical to the thermal conductance, we obtain GT  2 kB2 = T. G 3 e2

(8.17)

Consequently, one finds that the Wiedemann–Franz law holds, even though the scattering lengths for momentum relaxation () and energy relaxation (Lin )

are quite different. This is because the reservoirs are assumed to be perfect, so that any electron entering a reservoir immediately equilibrates with the other electrons in the reservoir. Before we go on to discuss normal metals in contact with superconductors, it is worthwhile to review some important assumptions in the calculations above.First,in our calculations,we have assumed that the diffusion coefficient D is a constant independent of the energy E and position x. When coherence effects are important, as in the case of the proximity effect, the diffusion coefficient becomes a function of both these parameters, D(E, x). The differential equation for the distribution function, Eq. (8.1) is then modified. D(E, x) itself in general is determined by the distribution function f (E, x), so one must solve a set of coupled differential equations to obtain a solution.This is difficult to do analytically in all but the simplest of cases, and more often must be done numerically. The remainder of this chapter will be devoted in great part to deriving the appropriate differential equations for the distribution functions and diffusion coefficient in the case of normal metals in contact with superconductors,using the quasiclassical equations for superconductivity. Second, apart from the electrical conductance G, the thermal coefficients derived above would all vanish if the density of states at the Fermi energy N(E), were assumed to be constant,so that it could be taken out of the integrals. The small variation in the density of states at the Fermi energy is responsible for finite (but small) values of the off-diagonal transport coefficients  and . For example, the thermopower S, which is non-zero only if there is an asymmetry between the properties of particles and holes near the Fermi energy, vanishes if N(E) is assumed constant. The small difference in N(E) above and below EF gives rise to the small but finite thermopowers of typical normal metals. The conventional quasiclassical theory of superconductivity assumes particlehole symmetry a priori in the definition of the quasiclassical Green’s functions, in that N(E) is assumed constant at EF . Consequently, thermoelectric effects cannot be calculated in the conventional quasiclassical approximation; an extension of the theory is required.

8

Finally, we have been discussing here currents and conductances, rather than current densities and conductivities. These are the more relevant quantities, since we will be discussing mesoscopic samples in which the measured properties are properties of the sample as a whole. This will be particularly important for the proximity effect, where long range phase coherence means that non-local effects are important.

8.3

Quasiclassical Theory of Superconductivity

The first Green’s function is just the one defined above, in this new notation. The second Green’s func˛˛ , except that it is tion is similar in definition to G12 ˜ defined with operator T instead of T, which orders the operators in reverse chronological order. The last two Green’s functions are defined without any timeordering operators. The four Green’s functions so defined are not linearly independent, but are related by linear equations of the form G˛˛ + Gˇˇ = G˛ˇ + Gˇ˛ .

The Keldysh Green’s Functions

The starting point for developing the quantum analog to the classical Boltzmann transport equation is the Keldysh diagrammatic technique. We shall begin our discussion of the Keldysh technique in the notation of Lifshitz and Pitaevskii [23]. As in the equilibrium case, we define a non-equilibrium Green’s function  : 9  G1 2 (X1 , X2 ) = −i n T ˆ1 (X1 ) ˆ+2 (X2 ) n . (8.18) Here X1 and X2 take into account the three spatial coordinates (denoted by r 1 and r 2 respectively), and the time coordinate (t1 and t2 ). The difference between the non-equilibrium Keldysh Green’s function above and the usual equilibrium Green’s function is that the average is taken over any quantum state |n > of the system, rather than just the ground state |0 >. The time ordering operator T has the effect G1 2 (X1 , X2 ) =  : 1 9  −i n  ˆ1 (X1 ) ˆ+2 (X2 ) n ,  : 9  +i n  ˆ+2 (X2 ) ˆ1 (X1 ) n ,

if t1 > t2 , if t2 > t1 ,

for fermion operators (the case of interest here). For simplicity, we label the spin indices and the coordinate indices by numbers. We shall define a set of Green’s functions that will be useful later: 9 : ˛˛ G12 = −i T ˆ1 ˆ2+ , (8.20a)   ˇˇ + G12 = i T˜ ˆ1 ˆ2 , (8.20b) : 9 ˛ˇ (8.20c) G12 = −i ˆ1 ˆ2+ , 9 + : ˇ˛ (8.20d) G12 = i ˆ2 ˆ1 .

(8.21)

The retarded and advanced Green’s functions GR and GA can be defined as in the equilibrium case 1 9 : −i ˆ1 ˆ2+ + ˆ2+ ˆ1 , if t1 > t2 , R G12 = (8.22) 0, if t2 > t1 and

1 A G12

=

0, 9 : i ˆ1 ˆ2+ + ˆ2+ ˆ1 ,

if t1 > t2 , if t2 > t1 .

(8.23)

and are related by  R ∗ A G12 = G21 .

(8.24)

GR and GA can be written in terms of the Keldysh Green’s functions defined earlier as (we set  = 1) GR = G˛˛ − G˛ˇ = Gˇ˛ − Gˇˇ , A

(8.19)

283

˛˛

ˇ˛

˛ˇ

ˇˇ

(8.25a)

.

(8.25b)

−1 (0)˛˛ G12 = ı(X1 − X2 ) , G01

(8.26)

G =G

−G

=G

−G

G˛˛ satisfies the equation of motion

where G0−1 is the differential operator (in the free electron approximation) G0−1 = i

∇2 @ + +

@t 2m

(8.27)

and the second subscript to G0−1 denotes that the differentials in Eq. (8.27) are with respect to coordinates corresponding to this subscript. The argument of the delta function Eq. (8.26) includes space, time (0)˛˛ and spin coordinates, and the notation G12 (with

284

V. Chandrasekhar

a superscript 0) signifies a Green’s function for an ideal gas. The ı function in Eq. (8.26) arises from the discontinuities in G˛˛ at t1 = t2 . GR and GA have similar discontinuities, and obey a similar equation. G(0)ˇˇ has a discontinuity of the opposite sign, and hence obeys the equation (0)ˇˇ

−1 G01 G12

= −ı(X1 − X2 ) .

(8.28)

G˛ˇ and Gˇ˛ have no discontinuities at t1 = t2 , and hence obey the equations −1 G01 G12

(0)ˇ˛

= 0,

(8.29a)

−1 (0)˛ˇ G12 G01

= 0.

(8.29b)

The diagram technique for Keldysh Green’s functions is similar to that for equilibrium Green’s functions, except that one needs to sum over the internal

Fig. 8.2. Diagrams corresponding to the first order corrections to the Keldysh Green’s function G˛˛ 12 in the presence of an external potential U , which is represented by a dashed line

indices ˛ and ˇ, with a corresponding increase in the number of diagrams. For example, Fig. 8.2 shows two diagrams corresponding to the first order correc˛˛ tions to G12 in the presence of an external potential, which is represented by a dashed line. In the same manner, Dyson’s equation for the Green’s function can be represented as shown in Fig. 8.3, where the ellipse represents the self-energy function £˛ˇ . In order to make the notation more compact and tractable, it is useful to introduce a matrix Green’s function and corresponding self-energy matrix:  ˇˇ ˛ˇ   ˇˇ ˛ˇ  G G £ £ ˇ ˇ G= , £= . (8.30) Gˇ˛ G˛˛ £ˇ˛ £˛˛ Dyson’s equation can then be written in matrix form as  0 0 ˇ ˇ + Gˇ 14 (8.31) £43 G32 d 4X3 d 4 X4 , Gˇ 12 = Gˇ 12 where the usual rules of matrix multiplication are used. As we saw earlier, the components of the Green’s function matrix are not independent, but are linearly related. One can therefore perform a transformation to set one of the matrix components to zero. There are many ways to do this; the one we shall use here is the one employed in most recent literature on the quasiclassical theory of superconductivity [11]. The

Fig. 8.3. Dyson’s equation in diagrammatic form for the Keldysh Green’s function G˛˛ 12 . The thick line represents the exact (0)˛˛ Green’s function G˛˛ , and the ellipse represents the 12 , the thin line represents the Green’s function for an ideal gas G12 self-energy £

8

matrices above are first rotated in Keldysh space usˇ where the  matriing the transformation Gˇ → 3 G, ces are identical in form to the Pauli spin matrices     10 01 0 1  = ,  = , 01 10     0 −i 1 0 2 3 ,  = , (8.32)  = i 0 0 −1 ˇ † , where the matrix and then transformed Gˇ → QGQ Q is given by 1 Q = √ ( 0 − i 2 ) . 2 The resulting matrices have the form  R K  R K £ £ G G ˇ ˇ , £= , G= 0 GA 0 £A

(8.33)

(8.35)

(8.36)

(8.37)

−1 on the Dyson equation (8.36) from the Operating G01 left, we obtain −1 ˇ G01 G = ı(X1 − X2 ) + £ˇ ⊗ Gˇ . 1

−1 ˇ 02 = ı(X1 − X2 ) + Gˇ ⊗ £ˇ . GG

(8.39)

Subtracting the two equations, we obtain ˇ = 0 , (8.40) (G0−1 ⊗ Gˇ − Gˇ ⊗ G0−1 ) − (£ˇ ⊗ Gˇ − Gˇ ⊗ £) −1 where we have suppressed the subscript “1” of G01 . This can be written as a commutator

ˇ − [£ˇ # G] ˇ = 0, [G0−1 # G] or

ˇ = [£ˇ # G] ˇ , [G0−1 # G]

(8.41)

The Mixed or Wigner Representation

Gˇ 0 obeys the equation of motion −1 ˇ 0 G01 G12 = ı(X1 − X2 ) .

The conjugate equation is1

(8.34)

£R , £A and £K are defined in a similar manner. In what follows, we shall use the “check” to denote the Keldysh matrices. In order to avoid writing integrals over the space and time coordinates, we introduce the binary operator ⊗, which has the effect of integrating over the free space and time coordinates and performing matrix multiplication when applied between two Keldysh matrices. Thus, the Dyson equation for the Keldysh Green’s function written above can be represented as Gˇ = Gˇ 0 + Gˇ 0 ⊗ £ˇ ⊗ Gˇ .

285

where the # operator defines the commutator of two operators [A # B] = A ⊗ B − B ⊗ A.

where the retarded and advanced Green’s functions GR and GA have been defined above, and the new Keldysh Green’s function GK is given by GK = G˛˛ + Gˇˇ = G˛ˇ + Gˇ˛ .

Quasiclassical Theory of Superconductivity

(8.38)

The Green’s function Gˇ 12 oscillates rapidly with the difference r 2 - r 1 , on the scale of the inverse Fermi wave vector kF−1 . In the physical systems to be discussed, we are interested in variations on much longer length scales. We therefore perform a transformation to center-of-mass coordinates, R, T, and difference coordinates, r, t, defined by the equations [24] r 1 = R − r/2 , t1 = T − t/2 ,

r 2 = R + r/2 , t2 = T + t/2 ,

(8.42a) (8.42b)

and then define a Fourier transform of Gˇ 12 with respect to the variables r and t:  ˇ ˇ G(R, T; r, t) = e−iEteip·r G(R, T; p, E) dp dE . (8.43) In Eq. (8.41) the part of the commutator involving G0−1 can be written as    @ @ −1 −1 G01 ⊗ Gˇ − Gˇ ⊗ G02 = i (8.44) + @t1 @t2  1 − (∇ 2 − ∇22 ) Gˇ , 2m 1

−1 From the right, we operate on the coordinate 2, hence we use the operator G−1 02 . G02 operated on the right is equivalent −1∗ to G02 operated on the left.

286

V. Chandrasekhar

using Eq. (8.27) in the free electron approximation. Transforming to the coordinates, R, T, r, t, this can be written   @ 1 i + ∇r · ∇R Gˇ . (8.45) @T m

where the Fourier transform of the Wigner distribution function h(R, T, p) is given by

From Eq. (8.35), the Keldysh component GK of the Green’s function can be written as the sum of the Green’s functions G˛ˇ and Gˇ˛ . Keeping in mind the definitions of these functions given in Eq. (8.20), we can define a non-equilibrium distribution function f (R, T, r), which is related to the function Gˇ˛ by 9 f (R, T, r) = ˆ + (R + r/2, T + t/2) : × ˆ (R − r/2, T − t/2 t=0

In equilibrium, h is given by

= −iGˇ˛ (R, T, r, t)t=0 .

(8.46)

Similarly, we have 9 : ˆ (R + r/2, T + t/2) ˆ + (R − r/2, T − t/2 t=0 = 1 − f (R, T, r) = iG˛ˇ (R, T, r, t)t=0 .

(8.47)

Subtracting the first equation from the second, we obtain 1 − 2f (R, T, r) = h(R, T, r)  = i G˛ˇ (R, T, r, t)t=0 + Gˇ˛ (R, T, r, t)t=0 = iGK (R, T, r, t)t=0 ,



(8.48)

where we have defined a new distribution function h(R, T, r). In terms of the mixed Fourier transform, Eq. (8.43), this can be written as h(R, T, r) = iGK (R, T, r, t)t=0  i eip·r GK (R, T; p, E) dp dE . (8.49) = 2 Taking the Keldysh component of Eq. (8.45) at t = 0, we obtain   @ 1 dE ip·r K i + ∇r · ∇R e G (R, T; p, E), dp @T m 2 (8.50) or, in terms of the Fourier components with respect to r, corresponding to momentum p, @h(R, T, p) p + · ∇R h(R, T, p) , @T m

(8.51)

h(R, T, p) =

i 2

 GK (R, T; p, E) dE .

h0( p ) = 1 − 2f0 ( p) = tanh( p /2kBT) .

(8.52)

(8.53)

Equation (8.51), which is the Keldysh component of the left hand side of Eq.(8.41),has the form of the one side of the classical Boltzmann equation for the distribution function. (Using the definition of the function h(R, T, p), this can also be written in a more conventional form in terms of f (R, T, p).) The right hand side of the Keldysh component of Eq. (8.41) must therefore correspond to the collision terms. The right hand side of the Keldysh component can be written as 2(£ˇ˛ G˛ˇ − £˛ˇ Gˇ˛ ). Taking the limit at t = 0, and writing in terms of the distribution function f (R, T, p) using Eq. (8.45) and Eq. (8.46), this Keldysh component of the right hand side of Eq. (8.41) can be written as [23]

−2i £ˇ˛ (R, T, p)(1 − f (R, T, p))  + £˛ˇ (R, T, p)f (R, T, p) .

(8.54)

The first term in Eq. (8.55) with the factor (1 − f (R, T, p)) has the usual form of a scattering-in term, corresponding to the gain of particles, while the second term has the form for a scattering-out term, corresponding to a loss of particles. Consequently, we see that the Keldysh component of the right-left subtracted Dyson equation gives the transport equation for the distribution function. From the diagonal components of the same equation, one can obtain ˇ More solutions for the other components of Gˇ and £. typically, the scattering terms on the right hand side of the Boltzmann equation make it difficult to solve, and some approximations must be employed. If the variation of the system with the center-of-mass coordinates T and R is small, then one can expand the Green’s functions and self-energies, which are functions of R, r, T, t about R and T in a Taylor’s series to first order in r and t. This is the gradient expansion

8

discussed by Kadanoff and Baym [24] and Larkin and Ovchinnikov [4], and we shall return to it at the end of this section. If instead of taking the difference of Eq. (8.38) and its conjugate equation, Eq. (8.39), we take the sum, we obtain the equation ˇ = 2ı(X1 − X2 ) + [£ˇ ⊕ G] ˇ , [G0−1 ⊕ G]

(8.55)

where the operator ⊕ defines the Keldysh anticommutator, in the same way as the operator # defines the Keldysh commutator. The left hand side of Eq. (8.55) can be written as    @ @ −1 −1 ˇ ˇ G01 ⊗ G + G ⊗ G02 = i (8.56) − @t1 @t2   1  2 2 ∇ + ∇2 Gˇ . − 2m 1 Transforming to the mixed representation, we obtain    1 1 2 @ 2 ˇ ∇ + ∇r G(R, T, r, t) . (8.57) 2i + @t m 4 R Now, the assumption we are making is that the variations of G on the scale of R are much slower than the variations on the scale of r. Hence, the terms in equation above involving derivatives with R contribute much less than the those with r, and can be neglected in this approximation. If we consider the equation for G0 , for which the terms involving £ on the right hand side of Eq. (8.55) are 0, we then obtain, after transforming to Fourier components [25] (E − p )G0 (R, T, p, E) = 1

(8.58)

or G0 (R, T, p, E) =

1 , (E − p )

(8.59)

where p2 − ∼ vF (p − pF ) . (8.60) 2m So far, we have assumed a free-electron model. If there is a slowly varying potential U (R, T), the equation above can be modified to 1 G0 (R, T, p, E) = . (8.61) (E − p − U (R, T)) p =

Quasiclassical Theory of Superconductivity

287

This equation has form of a Green’s function for a free particle, but in a slowly varying potential U (R, T). The operator G0−1 in the mixed representation can therefore be written as G0−1 = (E − p − U (R, T)) .

(8.62)

More accurately, one transforms Eq. (8.27) using the following equations @ 1 @ @ ± , = @t1,2 2 @T @t 1 ∇1,2 = ∇R ± ∇r , 2 1 2 ∇1,2 = ∇R2 ± ∇r · ∇R + ∇r2 . 4

(8.63a) (8.63b) (8.63c)

Again, assuming the functions in the mixed representation are slowly varying functions of R, we can ignore the second derivatives with respect to R, to obtain G0−1 =

@ ∇ 2 ∇r · ∇R 1 @ + + r + + . 2 @T @t 2m 2m

(8.64)

In most applications of interest here, we also ignore the slow T dependence.Adding the potential U (R, T), and writing in terms of the partial Fourier transform with respect to p and E, we obtain i G0−1 = E − p + v F · ∇R + − U (R, T) , 2

(8.65)

where we have replaced p/m by v F , since the important region of interest is at the Fermi surface. Note that we still keep account of the direction of the momentum. To conclude this section, we derive some expressions for the convolution of two operators in the mixed representation, the gradient expansion discussed above. Consider the convolution of two operators defined as an integral over the internal space and time coordinates   A ⊗ B = dr 3 dt3 A(r 1 , r 3 , t1 , t3 )B(r 3 , r 2 , t3 , t2 ) . (8.66)

288

V. Chandrasekhar

If we do a partial transform to p, R (but do not transform the time coordinates), A ⊗ B can be represented as  i A B A B A ⊗ B = dt3 e 2 (∇R ·∇p −∇p ·∇R ) (8.67) × A(R, p, t1 , t3)B(R, p, t3 , t2 ) , where the superscripts to the derivatives denote that they operate only those functions. If the derivatives are small, we need to take only the first order expansion of this expression   i A B A B ∇R · ∇p − ∇p · ∇R (A ⊗ B)(p, R) = dt3 1 + 2 × A(R, p, t1 , t3 )B(R, p, t3, t2 ) . (8.68) 

Similarly we get   i B ∇R · ∇pA − ∇pB · ∇RA (B ⊗ A)(p, R) = dt3 1 + 2 × B(R, p, t1 , t3 )A(R, p, t3, t2 ) . (8.69) 

If we were dealing with functions alone,then the multiplication of the two function A and B is commutative. When A and B are matrices, however, they do not in general commute, so we obtain (A # B)(p, R) = A ⊗ B − B ⊗ A  = dt3 [A, B] 

i dt3 {∇R A, ∇p B} + 2  (8.70) − {∇p A, ∇R B} , where [A, B] notation stands for the commutator of the two functions, and {A, B} stands for the anticommutator. A similar equation can be obtained if we transform the times to the mixed representation T, E (A # B)(T, E) = A ⊗ B − B ⊗ A  = dr 3 [A, B]  i dr 3 [{@T A, @E B} + 2 − {@E A, @T B}] . (8.71) When we transform both sets of variables, we obtain

(A # B)(p, R, T, E) = A ⊗ B − B ⊗ A = (8.72) i [A, B] + [({@T A, @E B} − {@E A, @T B}) 2  + {∇R A, ∇p B} − {∇p A, ∇R B} . In most cases, we are interested in stationary situations, where there is no T dependence. In this case, the equation above reduces to (A # B)(p, R, T, E) = A ⊗ B − B ⊗ A (8.73)

i {∇R A, ∇p B} = [A, B] + 2  − {∇p A, ∇R B} . These expressions will be useful in the calculations to follow.

8.4

The Quasiclassical Approximation

The non-equilibrium spectral function A is defined in the same way as in the equilibrium case [11,26] A=

i 1 (GR − GA ) = − Im(GR ) , 2 

(8.74)

where Im (GR) denotes the imaginary part of the retarded Green’s function. In the equilibrium case, A defines the spectrum of energy levels; for stationary quantum states, it has the form of a sum of ıfunctions at each state energy. In the quasiparticle approximation, these ı-functions are broadened, but the width  of each state, defining its lifetime, is still small compared to its energy. If  is large, the quasiparticle approximation breaks down,and one cannot obtain a kinetic equation for a distribution function by integrating over the energy E. However, for most perturbations of interest, the self-energies typically have a weak dependence on the magnitude of the momentum, this dependence being appreciable only near the Fermi energy. On the other hand, while the Dyson equation has a strong dependence on E and p , the subtracted Dyson equation, Eq. (8.41), from which we will obtain the equations of motion, has a very weak dependence on both E and p . In this case, it is possible to average over the particle energy to eliminate the dependence on the magnitude of the momentum, but keep the dependence on the direction of the momentum. Hence, one can think

8

of replacing the Green’s functions and self-energies by their values on the Fermi surface, multiplied by a ı-function in the form G(R, p, t1 , t2 ) → ı( p )g(R, pˆ , t1, t2 ) .

(8.75)

To this end, we define the so-called quasiclassical Green’s function  i ˆ g(R, p, t1 , t2 ) = d p G(R, p, t1 , t2 ) . (8.76)  Care must be taken in performing this integral, since the integrand falls off only as 1/ p for large p . To avoid this, one can introduce a cut-off in the integral, as done by Serene and Rainer [7], or following Eilenberger [1] use a special contour for integration. We would like to obtain an equation of motion for the quasiclassical Green’s functions. If we obtain an equation of motion by operating G0−1 in the form of Eq. (8.64) on Dyson’s equation, and then integrating over p , there are terms in the equation that will have large contributions.In order to eliminate these large terms, we start from the left-right subtracted equation of motion, Eq. (8.41), in which the troublesome terms are cancelled, and then integrate over p . The terms on the right hand side of Eq. (8.41) are of the form  ˇ 3 , t3 , x2, t2 ) . ˇ 1 , t1 , x3, t3 )G(x £ˇ ⊗ Gˇ = dt3 dx33 £(x (8.77) We first Fourier transform this term with respect to p, which takes care of the integral over x3. We then average the resulting equation with respect to p. The assumption here is that only Gˇ has a strong dependence on the momentum p, so the result of this averaging is a term of the form  ˇ dt3 £(R, (8.78) p, t1, t3 )ˇg (R, pˆ , t3 , t2 ) . Now, to complete the transformation, the self-energy ˇ which is a functional of the Green’s functions G, ˇ £, must become a functional only of the quasiclassiˇ G] ˇ → ˇ [ˇg ]. With this final cal Green’s functions gˇ , £[ change,Eq. (8.41) for the quasiclassical Green’s functions becomes [(g0−1 − ˇ ) ◦, gˇ ] = 0 .

(8.79)

Quasiclassical Theory of Superconductivity

289

The ◦ operator in the commutator [A ◦, B] involves an integral over the internal time coordinates in addition to the usual matrix multiplication for Keldysh matrices. If we transform the time coordinates as well,this integral can be removed,as shown at the end of the last section (Eq. 8.73). In this case, the commutator becomes a simple commutator (but involving matrix multiplication of the Keldysh matrices). It remains to express the physical quantities of interest in terms of the Green’s functions. The particle density is given by ˇ˛

(1) = −2iG11

(8.80)

and the current density in the absence of external fields is given by j(1) = −

e ˇ˛ (∇1 − ∇2 ) G12 |2=1 . m

(8.81)

From the definitions of GR , GA and GK in terms of G˛ˇ , Gˇ˛ and Gˇˇ , we can write the function Gˇ˛ as Gˇ˛ =

 1 K G + (GR − GA ) . 2

(8.82)

GR and GA depend on the equilibrium properties of the system, and so do not contribute to the current or nonequilibrium density. Consequently, these terms can be dropped in the expression for the particle density and current. Writing in terms of the quasiclassical Green’s functions,we can obtain expressions for the density and the current in the mixed representation.Consider the expression for charge density ı(R, T) = −ieGK (R, T, r = 0, t = 0) ,

(8.83)

where we use ı instead of  to emphasize that this does not include the equilibrium contributions. Expanding GK in terms of Fourier components in momentum space p 

 d3 p ip·r K e G (R, T, p, t) (2)3 r=0,t=0   d3 p K dE = −ie G (R, T, p, E) , 2 (2)3

ı(R, T) = −ie

(8.84)

290

V. Chandrasekhar

where we have used the fact that setting t = 0 in GK (t) is equivalent to integrating GK (E) over dE/2. The equivalent expression for the current density can also be written in terms of the quasiclassical Green’s function. From Eq. (8.81) and Eq. (8.82) we have in the mixed representation  e

(∇1 − ∇2 ) GK (R, T, r, p) r=0,t=0 . 2m (8.85) The integral over the momentum p can be rewritten as    d§p d3 p , (8.86) d → N 0 p 3 (2) 4 j(R, T) = −

where N0 is the density of states at the Fermi energy, and d§p is an element of solid angle in momentum space p. Writing ∇1 − ∇2 = 2∇r from Eq. (8.63b), we obtain    e d3 p ip·r K ∇r j(R, T) = − e G (R, T, p, t) . m (2)3 r=0,t=0 (8.87) Operating ∇r on the exponential within the integral gives ip. In the quasiclassical approximation, we assume that the major contribution comes from near the Fermi surface, so that p/m = v F , the Fermi velocity. With this approximation, we obtain   d3 p dE j(R, T) = −ie v F GK (R, T, p, E) . 2 (2)3 (8.88) If we use the definition of the distribution function h given by Eq. (8.48), we obtain  d3 p j(R, T) = −e v F h(R, T, p) , (8.89) (2)3 which is the expected classical form for the current. Recalling the definition of the quasiclassical Green’s function, Eq. (8.76), we obtain   d§p 1 j(R, T) = − eN0 dE v F g K (E, pˆ , R, T) , 2 4 (8.90) where we have used Eqs. (8.76) and (8.86). Note that writing these expressions in terms of the quasiclassical Green’s functions necessitates reversing the order of integration over E and p .

The transformation, Eq. (8.86), assumes that the density of states is constant at the Fermi energy, and hence assumes that particle-hole symmetry holds. Hence, within this approximation, we will not be able to obtain any results on physical phenomenon that depend on particle–hole asymmetry, in particular, thermoelectric effects. This should be contrasted with the derivation of the current in the diffusive limit that we performed in the introduction, where the energy dependence of the density of states was taken into account explicitly. So far,we have ignored the effects of external fields and potentials. In particular, the effect of a magnetic field is of interest. The magnetic field is introduced in the form of a vector potential A(R, T). Here, we shall consider only time independent fields. A(R) is introduced by making the change ∇R → ∇R − ieA(R) ≡ @R

(8.91)

in all equations involving the spatial derivative. For −1 is now written example, the operator G˜ 01 −1 G˜ 01 =i

@ @2 + r + − e . @t1 2m

(8.92)

In the equation above, we have also added a term e, corresponding to the presence of a scalar potential. In our formulation, we would like all observable quantities to be invariant under a gauge transformation of the vector potential A → A + ∇ (r) ,

(8.93)

which also transforms the potential  →−

@ . @t

(8.94)

Eq. (8.87) for the electrical current is then modified to 2e (8.95) [[∇r −ieA(R)] m   d 3p ip·r ˇ˛ e G (R, T, p, t) , × (2)3 r=0,t=0

j(R, T) =−

where we have written the current in terms of Gˇ˛ rather than GK . It follows that Eq. (8.88) can be written as

8

  d3 p dE j(R, T) = −i2e v F Gˇ˛ (R, T, p, E) 2 (2)3   dE d3 p 2ie2 A(R) Gˇ˛ (R, T, p, E) . (8.96) + m 2 (2)3

The second term in the equation above is called the diamagnetic term. It is cancelled by a contribution from the first term arising from energies far from the Fermi energy, which are not taken into account in the quasiclassical Green’s function as defined by Eq. (8.76) [11]. Note that such a cancellation does not occur in the case of a superconductor, and the second term gives rise to the supercurrent, which is proportional to the vector potential and the phase gradient according to Eq. (8.94). With this high energy contribution cancelled by the diamagnetic term, the equation above transforms into the expression Eq. (8.90) for the electrical current, written now in terms of the Keldysh component of the quasiclassical Green’s function. Such a cancellation does not occur in transforming ı in terms of the quasiclassical Green’s functions, and the contribution of the integrals in Eq. (8.84) must be explicitly calculated [11]. The result that was obtained by Eliashberg [3] can be written as  eN0 dE ı(R, T) = − (8.97) 2  d§p K g (E, pˆ , R, T)−2e2N0 (R, T) , × 4 where (R, T) is now the scalar electrochemical potential. This expression can also be obtained by invoking gauge invariance arguments. Since we must preserve charge neutrality in the system, ı = 0, so that (R, T) is given by   d§p K 1 (R, T) = − dE g (E, pˆ , R, T) . (8.98) 4e 4 For completeness, we also write expressions for the thermal current and the density of states. The expression for the thermal current in terms of the Keldysh Green’s function is given by  dE jth (R, T) = i (8.99) 2  d3 p × Ev F GK (R, T, p, E) , (2)3

Quasiclassical Theory of Superconductivity

291

and the corresponding form in terms of the quasiclassical Green’s functions  1 (8.100) jth (R, T) = N0 dE 2  d§p Ev F g K (E, pˆ , R, T) . × 4 As in the equilibrium case, the density of states is given directly by the spectral function A defined in Eq.(8.74),now expressed in the mixed representation N(E, R, T) = A(E, R, T) (8.101) 1 = − ImGR (E, R, T)   1 d3p R G (E, p, R, T) . = − Im  (2)3 Written in terms of the quasiclassical Green’s function, this becomes  d§p R N(E, R, T) = N0 Re g (R, E, T) , (8.102) 4 where the notations Re and Im stand for the real and imaginary components respectively.

8.5

Non-equilibrium Green’s Functions for Superconducting Systems

The formalism that we have developed for nonequilibrium Green’s functions for normal systems carries over into the superconducting case, except that the Green’s functions for the superconducting case are more complicated. In order to show how the Green’s functions are defined, we start by expressing the Green’s functions in the superconducting case in terms of field operators in the Nambu–Gor’kov formalism. In this formalism, the field operators can be written as two-component column matrices ¦ˆ 1 =



ˆ1↑ + ˆ1↓

 ,

(8.103)

where the up and down arrows refer to the spin indices, and the numbers now refer only to the time

292

V. Chandrasekhar

  ˛˛ G˜ 12 (8.109a) = −i 3 T ¦ˆ 1 ¦ˆ 2+ 9 % : 9 : + T ˆ1↑ ˆ2↑ T ˆ1↑ ˆ2↓ : , = −i 3 9 + + : 9 + T ˆ1↓ ˆ2↑ T ˆ1↓ ˆ2↓

and space coordinates. The Hermitian adjoint of this operator can be written as a two-component row matrix  +  ˆ1↓ . ¦ˆ 1+ = ˆ1↑ (8.104)

  ˇˇ G˜ 12 = −i 3 T˜ ¦ˆ 1 ¦ˆ 2+  ⎛ + T˜ ˆ1↑ ˆ2↑  = −i 3 ⎝ + + T˜ ˆ1↓ ˆ2↑

The multiplication of two such operators is defined as the tensorial product of the two matrices. For example, %  + ˆ1↑ ˆ2↓ ˆ1↑ ˆ2↑ + . (8.105) ¦ˆ 1¦ˆ 2 = + + + ˆ1↓ ˆ2↑ ˆ1↓ ˆ2↓ It is natural then to define the Green’s functions in terms of these products.For example,the natural definition of the Green’s function corresponding to G˛˛ , Eq. (8.20a), would be   ˛˛ G12 (8.106) = −i T ¦ˆ 1¦ˆ 2+ % 9 :9 : + T ˆ1↑ ˆ2↓ T ˆ1↑ ˆ2↑ : . = −i 9 + + : 9 + T ˆ1↓ ˆ2↑ T ˆ1↓ ˆ2↓ However, we would like to keep the same form for the equations of motion and Dyson’s equation as for the normal case, except, of course, the quantities will be matrices in Nambu–Gor’kov space. To see if the definition above fits this requirement, let us operate i@/@t1 on the Green’s function for an ideal gas, as defined above. With the definition h01 = −

∇12 − , 2m

(8.107)

we have the matrix Equation : 9 :% 9 + T ˆ1↑ ˆ2↓ T ˆ1↑ ˆ2↑ @ (0)˛˛ i G12 = h01 : 9 + +: 9 + @t1 ˆ2↑ − T ˆ1↓ ˆ2↓ − T ˆ1↓  % ı(X1 − X2 ) 0 + . 0 ı(X1 − X2 ) (8.108) This does not have the required form of the analogous equation for the normal case, Eq. (8.26). To remedy this, we define our Green’s functions with an extra factor of  3 . The definitions corresponding to Eq. (8.20) are then

 

T˜ ˆ1↑ ˆ2↓ + T˜ ˆ1↓ ˆ2↓

⎞

(8.109b)

⎠ ,

  ˛ˇ (8.109c) G˜ 12 = −i 3 ¦ˆ 1¦ˆ 2+ 9 % : 9 : + ˆ1↑ ˆ2↑ ˆ1↑ ˆ2↓ : , = −i 3 9 + + : 9 + ˆ1↓ ˆ2↑ ˆ1↓ ˆ2↓ and

  ˇ˛ (8.109d) G˜ 12 = i 3 ¦ˆ 2+ ¦ˆ 1 9 % : 9 : + ˆ2↑ ˆ1↑ ˆ2↓ ˆ1↑ : . = i 3 9 + + : 9 + ˆ2↑ ˆ1↓ ˆ2↓ ˆ1↓

Here, the “tilde” over the Green’s functions denotes that they are matrices in Nambu–Gor’kov space. The operator corresponding to Eq. (8.26) is then defined as @2 @ −1 G˜ 01 (8.110) = i 3 + 1 + . @t1 2m Here it is understood that any ‘scalar’ quantities in the equation above and in what follows are multiplied by the identity matrix  0 . With these modifications, all the equations derived for the Keldysh Green’s functions for the normal case can be carried over directly to the superconducting case. In particular, the advanced and retarded Green’s functions G˜ R and G˜ A are defined in terms of the Green’s functions in Eq. (8.109) in the same manner as before. The Keldysh matrices corresponding to Eq. (8.34) are then  R K  R K G˜ G˜ £˜ £˜ ˆ Gˆ = , £ = , (8.111) A ˜ 0 G 0 £˜ A Since each element of these matrices are themselves 2x2 matrices, the resulting Keldysh matrices for superconductors are 4x4 “supermatrices” and we shall

8

denote them by a “hat” symbol (ˆ). The equation of motion equivalent to Eq. (8.41) is ˆ = [£ˆ # G] ˆ . [Gˆ 0−1 # G]

(8.112)

Before we go any further, we need to specify the self˜ At the temperatures of interest in the exenergy £. periments on proximity systems, the important selfenergy terms are due to electron-phonon scattering (£˜ e−p ), and electron impurity scattering (£˜ imp ). For conventional superconductors, the elastic component (£˜ Re−p +£˜ Ae−p ) of the electron-phonon contribution is the one that leads to the coupling between superconducting electrons [8] (the other contributions of electron–phonon scattering will be ignored, under the assumption that we are at low enough temperatures so that inelastic electron-phonon scattering can be ignored). Perhaps a simpler way of dealing with this component of the electron-phonon interaction is to start directly with the Gor’kov equations of motion [26] for the Green’s function Eq. (8.109b), which we write in the form (ignoring impurity scattering for the moment) ⎛

⎞ @ ∇12  ⎜ i @t + 2m +

⎟ ˛˛ ⎜ 1 ⎟ G˜ = ı(X1 − X2 ) , 2 ⎝ ⎠ 12 @ ∇ −∗ −i + 1 +

@t1 2m

(8.113)

with the pair potential  defined as2 ˛˛  =  limit < T ˆ1↑ ˆ2↓ >= i limit [G˜ 12 ]12 2→1+

2→1+

(8.114) in terms of the upper left component of the Green’s ˛˛ function G˜ 12 , which is frequently called the anomalous Green’s function (or pair amplitude), and denoted by F. Here  is the coupling constant. For a uniform bulk superconductor  is real. For a normal metal,  vanishes, and hence, although the pair amplitude in a normal metal may be finite, the pair potential vanishes. Eq. (8.114) defines a self-consistent equation for the pair potential ; it is defined in terms of the anomalous Green’s function F, which in turn is determined by an equation that depends on . Note 2

Quasiclassical Theory of Superconductivity

293

˛˛ ˛˛ that unlike [G˜ 12 ]11 , [G˜ 12 ]12 is continuous at t2 = t1 , ˛ˇ so that  can also be written in terms of [G˜ 12 ]12 or ˇ˛ [G˜ 12 ]12 at t2 = t1 . Still ignoring impurity scattering, the equation of motion Eq. (8.112) can then be written in compact form as    ∇12 3 @ ˆ ˆ iˆ + +  # G = 0. (8.115) + @t1 2m

Here ˆ represents a 4 × 4 matrix   ˜ ˆ =  0 , 0 ˜ where ˜ =



0  −∗ 0

(8.116)

 ,

with  defined by Eq. (8.114), and  3   0 3 . ˆ = 0 3

(8.117)

(8.118)

Before we move on to making the quasiclassical approximation, it is useful to obtain expressions for measurable quantities in terms of the Green’s functions defined in this section.Following Eq.(8.49),one can define a distribution function by averaging the Keldysh component of the Green’s function defined in Eq. (8.111)  ˜h(R, T, p) = i dE G˜ K (R, T; p, E) . (8.119) 2 However, since G˜ K is a 2 × 2 matrix, h˜ is also a 2x2 matrix, so that its interpretation as a simple distribution function (in the flavor of Eq. (8.49) for the equilibrium case, for example) is not immediately clear. A more physical interpretation can be obtained by diagonalizing the Gor’kov equations given in Eq. (8.113) [12]. The matrix on the left hand side of this equation can be diagonalized by a unitary transformation; this transformation, of course, is just the Bogoliubov–Valatin transformation, which results in a diagonal “energy” matrix with eigenvalues Ep2 = p2 + 2 .

(8.120)

 defined this way differs from the conventional definition by a factor of i. Note that the signs associated with  in the matrix above are different from the conventional definition of Gor’kov’s equation, because of the  3 factor in the definition of the Green’s functions.

294

V. Chandrasekhar

The same transformation also diagonalizes the equilibrium distribution matrix h˜ 0, which now has the form   0 ˜h0  1 − 2f0↑ (Ep ) . (8.121) 0 2f0↓(Ep ) − 1 The top-left component is for electron-like excitations, and the bottom-right component for holelike excitations. This suggests that for our combined Nambu–Gor’kov–Keldysh Green’s functions, the equations for the electric current and thermal current should be modified to  ie dE j(R, T) = − (8.122) 2 2 

 d3 p × v F Tr  3 G˜ K (R, T, p, E) , (2)3 and i 2  ×

jth (R, T) = −



dE (8.123) 2

 d3 p Ev F Tr G˜ K (R, T, p, E) . 3 (2)

Taking the trace of the matrix takes the contributions of both electrons and holes. Note that the expression for electrical current includes a factor of  3 in the argument of the trace, while the thermal current does not.

8.6

Quasiclassical Superconducting Green’s Functions

In principle, the properties of the superconducting system may be calculated starting from Gor’kov’s equations. In practice, however, such calculations are difficult in all but the simplest cases. The problem is that Gor’kov’s equations contain information at length scales much finer than those of interest. The way around this is to make the quasiclassical approximation as we did for the case of the normal Green’s functions, which we proceed to do below. Taking into account impurity scattering, the equation of motion for the superconducting Green’s function can now be written as

   @12 3 @ ˆ ˆ ˆ iˆ + +  − £imp # G = 0 , + @t1 2m (8.124) where the impurity self-energy £ˆ imp includes contributions from both spin-flip and spin-independent elastic scattering. The elastic contribution to the self-energy can be written as  2 d 3 p  ˆ  ) . (8.125) £ˆ 0 (p) = Ni v(p − p ) G(p 3 (2) Here v is the impurity potential, and Ni the number of impurities per unit volume. We assume that v(p) is independent of the magnitude of p, so that  2 d§p  ˆ  ) . (8.126) v(pˆ · pˆ  ) G(p £ˆ 0 (p) = Ni N0 d p 4 Defining the elastic scattering rate 1/ in the Born approximation by  2 d§p  1 = 2Ni N0 v(pˆ · pˆ  ) , (8.127)  4 we can write £ˆ 0 =

1 2



ˆ d p G(p) .

(8.128)

ˆ Here G(p) is a function of the magnitude of p alone. Similarly, for the contribution from spin-flip scattering, one obtains  1 ˆ 3 . £ˆ sf = d p ˆ3 G(p)ˆ (8.129) 2sf From Eq. (8.111), one has    @2 @ ˆ £ˆ 0 − £ˆ sf # Gˆ = 0 . (8.130) iˆ 3 + 1 + + − @t1 2m Converting the left hand side of this equation to relative coordinates using Eq. (8.63) (neglecting any derivatives with respect to the center-of-mass time T), and using only the first term of Eq. (8.74) to lowest order, we obtain ˆ G]+iv ˆ ˆ = 0. ˆ £ˆ 0 + £ˆ sf , G] [ˆ 3 E+ , F · @R G−[

(8.131)

Note that there are no integrations over space or time coordinates in the equation above, but only matrix

8

multiplications. Written in terms of the quasiclassical Green’s functions, this becomes ˆ gˆ ] + iv F · @R gˆ − [ˆ 0 + ˆ sf , gˆ ] = 0 , (8.132) [ˆ 3 E + , where 1 gˆs , 2

(8.133)

1 3 3 ˆ gˆs ˆ . 2sf

(8.134)

ˆ 0 = −i and ˆ sf = −i

The subscript to the quasiclassical Green’s functions denotes that they are averaged over all angles of p. Equation (8.132) is Eilenberger’s equation [1] and is the starting point for many calculations on superconducting systems. To complete the transformation to quasiclassical Green’s functions, we express the equations for physically observable quantities in terms of the quasiclassical Green’s functions. From Eq. (8.114), the gap parameter  can be expressed as ˛ˇ  = i[G˜ 11+ ]12 =

i K [G˜ 11+ ]12 2

(8.135)

using the definition Eq. (8.35) of the Keldysh Green’s function. In the mixed representation, this can be written as   i dE d3 p = [GK (R, T, p, E)]12 2 2 (2)3   d§p K  dE [g (R, t, pˆ , E)]12 = N0 4 4   dE (8.136) = N0 8  > d§p = 1 × Tr ( − i 2 )g K (R, t, pˆ , E) . 4 From Eqs. (8.123) and (8.124), the electrical current and thermal currents are given by  eN0 dE (8.137) 4 

 d§p × v F Tr  3 g˜ K (R, T, p, E) 4

j(R, T) = −

Quasiclassical Theory of Superconductivity

and

295

 N0 dE (8.138) 4 

 d§p × Ev F Tr g˜ K (R, T, p, E) . 4

jth (R, T) = −

One can also define an equation that gives the amount of charge associated with quasiparticle excitations.In a superconductor in equilibrium,the number of particles and holes are equal, so the quasiparticle charge should vanish. If there is an imbalance in the population of electrons and holes, one can obtain a charge imbalance [2, 27], usually denoted by Q∗ , which is given by  

 dE d3 p ie Q∗ = − Tr G˜ K (R, T, p, E) 3 2 2 (2)    d§p K eN0 dE Tr g˜ (R, T, p, E) =− 4 4 (8.139) Q∗ is essentially the first term on the right hand side of Eq. (8.98). Invoking charge neutrality, this then would result in a electrochemical potential given Eq. (8.98) 1 ∗ (R, T) = − Q . (8.140) N0 e The Normalization Condition and the Distribution Function In obtaining the equation of motion of the Green’s functions by subtracting the Dyson equation from its conjugate, some information was lost regarding the norm of the quasiclassical Green’s function g. ˆ The norm of gˆ can be obtained by the normalization condition for the quasiclassical Green’s function obtained by Eilenberger [1] and Larkin, and Ovchinnikov [4] gˆ gˆ = ˆ0 , (8.141) where (4 × 4) matrix multiplication is implied. This normalization condition can be obtained by explicit calculation for a bulk system in equilibrium. Furthermore, it can be shown that the normalization condition is consistent with Eilenberger’s equation Eq. (8.132).

296

V. Chandrasekhar

Eq. (8.141) is equivalent to the three (2 × 2) matrix equations g˜ R g˜ R =  0 ,

(8.142a)

g˜ A g˜ A =  0 ,

(8.142b)

and g˜ R g˜ K + g˜ K g˜ A = 0 .

(8.142c)

As in the case of the E-integrated Green’s functions, we would like to obtain an equation of motion for a distribution function, equivalent to the quantum Boltzmann equation derived earlier. We introduce such a distributionfunction h˜ by the ansatz [11] g˜ K = g˜ R h˜ − h˜ g˜ A .

(8.143)

This form of the quasiclassical Keldysh Green’s function satisfies the normalization condition Eq. (8.142c), as can be verified by direct substitution g˜ R (˜g R h˜ − h˜ g˜ A ) + (˜g R h˜ − h˜ g˜ A )˜g A = 0

(8.144)

using Eqs. (8.142a) and(8.142b). Note that the func˜ h˜  , tion h˜ is not uniquely defined; if h˜ is replaced by h+ where h˜  = g˜ R x˜ + x˜ g˜ A (8.145) and x˜ is an arbitrary matrix function, the right hand side of Eq. (8.143) is unchanged [7]. This arbitrariness allows us some flexibility in choosing the dis˜ At low frequencies, for examtribution function h. ple, following Schmid and Sch¨on [2], and Larkin and Ovchinnikov1 [5], we may choose h˜ to be diagonal in particle–hole space h˜ = hL  0 + hT  3 .

(8.146)

The subscripts refer to the longitudinal (hL ) and transverse (hT ),a terminology introduced by Schmid and Sch¨on to refer to changes that are associated with the magnitude (hL ) or phase (hT ) of the complex order parameter. In equilibrium, hL(E) = 1 −2f0(E) and hT (E) = 0. Another possible choice is one introduced by Shelankov [28] h˜ = h1  0 + g˜ R h2 . (8.147)

This representation has the advantage that the equation for the distribution function reduces to a form very similar to the Boltzmann equation when the quasiparticle approximation is valid. In what follows, we shall use the representation of Schmid and Sch¨on. The representation of the Keldysh Green’s function in terms of a distribution function allows us to obtain an equation of motion for the distribution function from the equation of motion for the Green’s function.From Eilenberger’s equation (8.132),we obtain three equations of motion for the three components of the quasiclassical Green’s function ˜ g˜ R ] + iv F · @R g˜ R − [˜ R , g˜ R ] = 0 , (8.148a) [E 3 + , ˜ g˜ A ] + iv F · @A g˜ A − [˜ A , g˜ A ] = 0 , (8.148b) [E 3 + , ˜ g˜ K ] + iv F · @R g˜ K − ˜ R g˜ K − ˜ K g˜ A [E 3 + , + g˜ R ˜ K + g˜ K ˜ A = 0 ,

(8.148c)

where ˜ = ˜ 0 + ˜ sf . Substituting g˜ K from Eq. (8.143) into Eq. (8.148c), we obtain ˜ g˜ Rh˜ − h˜ g˜ A ] + iv F · @R (˜g R h˜ − h˜ g˜ A ) [E 3 + , − ˜ R (˜g R h˜ − h˜ g˜ A ) − ˜ K g˜ A + g˜ R ˜ K (8.149) + (˜g R h˜ − h˜ g˜ A )˜ A = 0 . Subtracting from the equation above Eq. (8.148a) multiplied by h˜ on the right, and adding Eq. (8.148b) multiplied by h˜ on the left, we obtain ˜ − B[h]˜ ˜ gA = 0 , g˜ R B[h]

(8.150)

where ˜ + ˜ K ˜ = [E 3 + , ˜ h] B[h] − (˜ R h˜ − h˜ ˜ A ) + iv F · @R h˜ . (8.151) Equation(8.150) is the required equation of motion ˜ for the distribution function h. As an example, we shall calculate the Green’s functions for the equilibrium case for a bulk superconductor, in the limit where  = 0. Equation(8.148a) in this limit has the form ˜ g˜ R ] = 0 . [E 3 + ,

(8.152)

8

First, if we represent the retarded Green’s function by %  g11 g12 R . (8.153) g˜ = g21 g22 The normalization condition Eq. (8.142a) and Eq. (8.152) together imply that g11 = −g22 , 2 + g21g12 = 1 , g11 ∗ g21 = −g12 

(8.154a) (8.154b) (8.154c)

and

E . (8.154d)  Solving these equations, we obtain ⎛ ⎞ E    ⎜ E2 − ||2 E2 − ||2 ⎟ ⎟. g˜ R = ⎜ (8.155) ⎝ ⎠ ∗ E − − E2 − ||2 E2 − ||2 g11 = g12

In taking the square roots in the equation above, a factor of i will appear if E2 < ||2 . This is the expected solution from Gor’kov’s equations. Note that g11 is just the normalized BCS density of states for E2 > ||2 , so that g11 represents a generalized density of states. From the equations above, it is clear that g21 = ∗ −g12 . To obtain g˜ A , we can use the relation Eq. (8.24), which in terms of Nambu-Gor’kov matrices reads G˜ A = − 3 G˜ R+  3 , g˜ A = − 3 g˜ R+  3 ,

or (8.156)

where the G+ , etc. denotes the Hermitian conjugate.

Quasiclassical Theory of Superconductivity

297

of the momentum is still its value at the Fermi surface, pF . With this in mind, let us consider a system with strong impurity scattering, which we define as a system in which the impurity scattering rate 1/, as defined by Eq.(8.127),is much larger than any energy in the problem (E, ). In this case, we expand the quasiclassical Green’s function to first order in momentum (essentially an expansion in spherical harmonics) gˆ = gˆs + pˆ gˆp ,

(8.157)

where pˆ denotes a vector of unit magnitude in the direction of p, and gˆs and gˆp are independent of the direction of p, and the subscripts stand for s-wave and p-wave components of gˆ . The assumption is that gˆp  gˆs . The self-energy is expanded in a similar fashion ˆ = ˆ 0 + ˆ sf = ˆ s + pˆ ˆ p .

(8.158)

We would like to calculate the components of ˆ in terms of the components of gˆ , for which we shall need the definition of ˆ in terms of gˆ . Ignoring spinflip scattering for the moment, this relation is given by Eq. (8.126), which can be rewritten in terms of the quasiclassical Green’s functions in the form  d§p ˆ 0 (p) = −iNi N0 |v(pˆ · pˆ )|2 gˆ (p ) (8.159) 4  d§p |v(pˆ · pˆ )|2 (ˆgs + pˆ  gˆp ) , = −iNi N0 4 using the expansion Eq. (8.157) for the Green’s function above.In order to do this,we take the dot product of pˆ with both sides of the equation above pˆ · ˆ = pˆ ˆ s + (pˆ · pˆ )ˆ p = pˆ ˆ s + ˆ p .

(8.160)

We perform a similar operation on Eq. (8.160)

8.7

The Dirty Limit: The Usadel Equation

In systems where elastic impurity scattering is strong, the motion of quasiparticles is not ballistic, but diffusive. In this case, the effect of the strong impurity scattering is to randomize the momentum of the quasiparticles, so that it makes sense to average the properties of system over the directions of the momentum, keeping in mind that the magnitude

pˆ · ˆ 0(p) = −iNi N0 (8.161)  d§p |v(pˆ · pˆ )|2 (pˆ gˆs + (pˆ · pˆ  )ˆgp ) . × 4 Since both pˆ and gˆs are independent of the integration over d§p , the first term under the integral in the equation above can be written as  d§p i −i pˆ gˆs Ni N0 |v(pˆ · pˆ )|2 = − pˆ gˆs (8.162) 4 2

298

V. Chandrasekhar

using the definition Eq. (8.127) of 1/. If we consider non-spin-flip scattering alone, then ˆ can be obtained by equating like terms in Eqs. (8.160) and (8.162). If we include spin-flip scattering, one can write i i 3 3 ˆ s = − gˆs − − ˆ gˆs ˆ , (8.163) 2 2sf as expected from Eqs. (8.133) and (8.134). For the p-wave component, we consider only the contribution from non-spin-flip impurity scattering, under the assumption that it is much stronger than the spinflip scattering. The second term under the integral in Eq. (8.162) can be written as  d§p −iNi N0 |v(pˆ · pˆ )|2 (pˆ · pˆ  )ˆgp = (8.164) 4  d§p |v(pˆ · pˆ  )|2 [1 − (1 − pˆ · pˆ  )] , −iNi N0 gˆp 4 where gˆp can be taken out of the integral since it is independent of the direction of p. The first term in the square bracket can be seen to be related to the elastic scattering rate 1/. The remaining terms can be written in terms of the transport time, defined by  d§p 1 |v(pˆ · pˆ  )|2 (1 − pˆ · pˆ  ) , (8.165) = 2Ni N0 tr 4 that is well known in the transport theory of metals. With this definition, ˆ p can be written as 1 i 1 ˆ p = − ( − )ˆgp . 2  tr

(8.166)

By putting Eq. (8.157) into the normalization condition for the quasiclassical Green’s function, and neglecting terms quadratic in gˆp , we also obtain the two equations gˆs gˆs = 1 (8.167a) and gˆs gˆp + gˆp gˆs = 0 .

(8.167b)

We now proceed to expand the Eilenberger equation, Eq. (8.132), in terms of the s and p-wave expansions of gˆ and ˆ . Replacing v F by vF pˆ , we obtain ˆ gˆs ] + pˆ [ˆ E − , ˆ gˆp ] [ˆ E − , (8.168) + ivF pˆ · @R (ˆgs + pˆ gˆp ) − [ˆ s , gˆs ] − pˆ [ˆ s , gˆp ] + [ˆ p , gˆ s ] + pˆ · pˆ [ˆ p , gˆp ] = 0 . 3

3

The last term is second order in the small quantity gˆp , and can be neglected. Collecting the terms that are even in pˆ , we obtain ˆ gˆs ] + ivF (pˆ · pˆ )@R gˆp = 0 . [ˆ 3 E + ,

(8.169)

Averaging this equation over all directions of pˆ gives ˆ gˆs ] + i [ˆ 3 E + ,

vF @R gˆp = 0 . 3

(8.170)

Ignoring spin-dependent scattering for the moment, the terms that are odd in pˆ can be written ˆ gˆp ] + ivF @R gˆs − [ˆ 3 E + ,

i [ˆgp , gˆs ] = 0 , (8.171) 2tr

where we have used Eqs. (8.163) and (8.166) to write ˆ s and ˆ p in terms of gˆs and gˆp . If elastic scattering is strong, the first term in the equation above can be neglected compared to the third, so we obtain ivF @R gˆs +

i gˆs gˆp = 0 , tr

(8.172)

where we have used Eq. (8.167b) to simplify the second term. Multiplying this equation by gˆs on the left and using Eq. (8.167a) we obtain vF gˆs @R gˆs = −

1 gˆp , tr

(8.173)

or writing gˆp in terms of gˆs we get gˆp = vF tr gˆs @R gˆs = −ˆgs @R gˆs .

(8.174)

Here,we have introduced the elastic scattering length  = vF tr . Putting this into Eq. (8.171) we obtain ˆ gˆs ] − i [ˆ 3 E + ,

vF  @R gˆs @R gˆs = 0 . 3

(8.175)

Writing this in terms of the diffusion coefficient D = (1/3)vF , and reintroducing the spin-flip scattering term, we get [ˆ 3 E + ˆ − ˆ sf , gˆs ] − iD@R gˆs @R gˆs = 0 .

(8.176)

8

This is the equation first derived by Usadel [29] and forms the starting point for most discussions of dirty superconducting systems. In the remainder of our development we shall neglect the spin-flip scattering term. Writing gˆs as a matrix  R K g˜ g˜ gˆs = s sA , (8.177) 0 g˜s

Quasiclassical Theory of Superconductivity

299

The equation for jth is obtained in a similar way jth (R, T) =

N0 D 4 ×

(8.182)

 dE E Tr g˜sR @R g˜sK + g˜sK @R g˜sA .

(8.178b)

The kinetic equation, Eq. (8.178c), can be recast in terms of a differential equation for a distribution ˜ the hopes of separating the equilibrium function h,in properties of the system, represented by g˜sR and g˜sA , from the non-equilibrium properties of the system, ˜ To this end, as before, we substitute represented by h. Eq.(8.143) for g˜ sK in Eq.(8.178c),which then becomes

 ˜ g˜sK ] = iD@R (˜gsR @R g˜sK ) + (˜gsK @R g˜sA ) . [ 3 E + , (8.178c)

˜ − [ 3 E + , ˜ g˜sR h] ˜ h˜ g˜sA ] = [ 3 E + ,   iD@R g˜sR (@R g˜sR )h˜ − h˜ g˜sA (@R g˜sA )   + @R h˜ − g˜sR @R h˜ g˜sA . (8.183)

we can write the matrix equation,Eq.(8.176),as three separate equations ˜ g˜sR ] = iD@R (˜gsR @R g˜sR ) , [ 3 E + , ˜ g˜sA ] = iD@R (˜gsA @R g˜sA ) , [ 3 E + ,

(8.178a)

and

As before, the first two equations come from the diagonal components of the Usadel equation, and describe the equilibrium properties of the system,while the third equation comes from the off-diagonal or Keldysh component,and represents the kinetic equation for the distribution function. These equations are supplemented by the equations for measurable quantities corresponding to Eqs. (8.138), (8.139) and (8.138).Expanding g˜ K in Eq.(8.138 using Eqs.(8.157) and (8.138), we have  eN0 dE (8.179) j(R, T) = − 4     d§p × vF pˆ Tr  3 g˜sK + pˆ g˜pK 4 

 eN0 D dETr  3 (˜gs @R g˜s )K , = 4 where the angular average over the p-wave component gives a factor of (1/3), and we have replaced (1/3)vF  by the diffusion coefficient D. Now (˜gs @R g˜s )K = (˜gsR @R g˜sK ) + (˜gsK @R g˜sA )

(8.180)

so that Eq. (8.180) above can be rewritten as j(R, T) =

eN0 D 4  ×

(8.181)

  dETr  3 g˜sR @R g˜sK + g˜sK @R g˜sA .

Taking the trace of both sides of the equation above gives 0 = iD@R Tr @R h˜ + g˜sR (@R g˜sR )h˜

˜ g A , (8.184) − h˜ g˜sA (@R g˜sA ) − g˜sR (@R h)˜ s using Eqs. (8.178a) and (8.178b). In the linear regime, we can use the diagonal representation of h˜ given by Eq. (8.146) to obtain 

D@R (@R hL )Tr 1 − g˜sR g˜sA   + hT Tr  3 g˜sR (@R g˜sR ) − g˜sA (@R g˜sA )

 − (@R hT )Tr g˜sR  3 g˜sA = 0 . (8.185) Here, we have used the fact that > = > = Tr @R (˜gsR g˜sR) = Tr @R (˜gsR )˜gsR + g˜sR @R (˜gsR ) = > = 2Tr g˜sR (@R (˜gsR ) = 0 . (8.186) Taking the trace after multiplying both sides of Eq. (8.183) by  3 gives =

> (8.187) D@R @R hT Tr 1 − g˜sR  3 g˜sA  3 = 3 R > R A A + hL Tr  g˜s (@R g˜s ) − g˜s (@R g˜s ) > = − @R hL Tr g˜sR g˜sA  3 =  = > = > ˜ − 2hT Tr (˜ ˜ gsR + g˜sA ) . i hL Tr  3 [˜gsR − g˜sA , ]

300

V. Chandrasekhar

Equations (8.185) and (8.188) form a set of coupled differential equations for the distribution functions hL and hT . Let us define the quantities Q=

> 1 = 3 R Tr  g˜s (@R g˜sR ) − g˜sA (@R g˜sA ) 4

(8.188)

and Mij =

> 1 = 0 Tr ıij  − g˜sR  i g˜sA  j . 4

(8.189)

Then Eqs. (8.185) and (8.188) can be written in the form @R [M00 (@R hL ) + QhT + M30 @R hT )] = 0, (8.190a) i @R [M33 (@R hT ) + QhL + M03 @R hL )] = 4D  = 3 R > A ˜ × hL Tr  [˜gs − g˜s , ] = > ˜ gsR + g˜sA ) . − 2hT Tr (˜ (8.190b) These equations are in the form of diffusion equations for the distribution function, more general forms of the diffusion equation discussed for the normal case in the introduction. As we shall see, the quantity Q is related to the spectral supercurrent in the system, DMij are now the energy and position dependent diffusion coefficients, and Eq. (8.190) are essentially continuity equations for the spectral thermal and electric current. In the normal limit, g R =  3 , g A = − 3 , and ˜ = 0, so that M00 = M33 = 1, and Q = M03 = M30 = 0. Equations (8.190) then reduce to Eq. (8.1), as expected. Noting that the term in square brackets in Eq. (8.178c) is the same as the term in parenthesis in Eq. (8.182) and the term in square brackets in Eq. (8.183), the electric current can be written as j(R, T) = eN0 D (8.191)  × dE (M33 (@R hT ) + QhL + M03 @R hL ) . The first term corresponds to quasiparticle (or dissipative) current, and the second term to the supercurrent. The third term, which is proportional to the derivative of hL , is associated with an imbalance between particles and holes. The thermal current can be written in a similar way

jth (R, T) = N0 D (8.192)  dE E[M00 (@R hL ) + QhT + M30 @R hT ] . For the charge-imbalance Q∗ , we note that only the s-wave part of the Keldysh Green’s function in the square brackets in Eq. (8.139) survives after angular averaging. Writing gsK in the form given by Eq. (8.143), with h˜ given by Eq. (8.146), we have  > = eN0 ∗ Q =− dE hL Tr gsR − gsA 4  = > eN0 dE hT Tr  3 (gsR − gsA ) − 4  (8.193) = −eN0 dE hT N(E) , since gsR and gsA are traceless. Here we have defined the normalized superconducting density of states by > 1 = N(E) = Tr  3 (gsR − gsA ) , (8.194) 4 which reduces to the conventional BCS density of states in the equilibrium case. We can also recast the kinetic equations Eq.(8.190) in a slightly different form sometimes used in the literature. To do this, we subtract Eq. (8.178b) from Eq. (8.178a), multiply by  3 , and take the trace. The result is = > = ˜ = −iD@R (Tr  3 [˜gsR @R g˜sR Tr  3 [gsR − gsA , ] > − g˜sA @R g˜sA ] ) . (8.195) Using the definition of Q, we have > i = 3 R ˜ . Tr  [gs − gsA , ] (8.196) @R Q = 4D If we choose a gauge in which  is real, the right hand side of the equation vanishes, so that @R Q = 0. Clearly, @R Q = 0 also for a normal metal (even a proximity-coupled normal metal), where  vanishes. In either case, the spectral supercurrent Q is conserved. The right hand side of the equation above multiplied by hL is the same as the second term of Eq. (8.190b). Subtracting Eq. (8.195) multiplied by hL from Eq. (8.190b), we obtain @R [M33 (@R hT ) + M03 (@R hL )] + Q(@R hL ) = = > i  ˜ gsR + g˜sA ) . − 2hT Tr (˜ (8.197) 4D

8

For real , when @R Q = 0, Eq. (8.190a) can be written in a similar manner @R [M00 (@R hL ) + M30 (@R hT )] + Q(@R hT ) = 0 . (8.198) Finally, we can also write the kinetic equations in a form similar to Eq. (8.150) ˜ − B[h]˜ ˜ gA = 0 , g˜sR B[h] s

(8.199)

by performing similar manipulations on Eq. (8.178) as were performed on Eq. (8.148). For the diffusive ˜ is a functional of h˜ now given by case, B[h]  ˜ − iD (@R g˜ R )(@R h) ˜ ˜ = [ 3 E + , ˜ h] B[h] s  1 R 2˜ ˜ g A (8.200) g˜s (@R h) − (@R2 h)˜ + s 2  ˜ R g˜ A ) . − (@R h)(@ s Boundary Conditions for the Quasiclassical Equations of Motion Equation (8.178) or their derivatives form a set of coupled differential equations for the quasiclassical Green’s functions and the distribution function. In order to obtain a solution, however, we need to specify the boundary conditions for the Green’s functions and the distribution function. To this end, we define the concept of a reservoir, where the Green’s functions and distribution function have well-defined values. For a normal reservoir, the retarded and advanced quasiclassical Green’s functions are given by gNR 0 =  3 ;

gNA0 = − 3 ,

(8.201)

and for the superconducting case, by Eq. (8.155), which we reproduce here ⎛



E





⎞

⎜ E2 − ||2 E2 − ||2 ⎟ ⎟, g˜ R = ⎜ ⎝ ⎠ ∗ E − − E2 − ||2 E2 − ||2 A is given by Eq. (8.156). with gS0

(8.202)

Quasiclassical Theory of Superconductivity

301

The equilibrium distribution function h˜ is given by Eq. (8.121) in both normal and superconducting reservoirs (since we are dealing only with excitations), where, as we noted earlier the (1, 1) component of the matrix applies to particle-like excitations, and the (2, 2) component of the matrix applies to hole-like excitations. From this point of view, the Fermi functions in Eq. (8.121) are given in terms of the usual equilibrium Fermi function Eq. (8.5) by f0↑ (E) = f0 (E) and f0↓ (E) = f0 (−E). (Since we are dealing with the static limit in all that follows, we again use the symbol T to refer to the temperature.) Looking ahead to where we might have a finite voltage V on a reservoir, the equilibrium form of h˜ can then be written as ⎛  ⎞  E + eV 0 ⎜tanh 2k T ⎟ B ⎟ . (8.203)  h˜ 0 = ⎜ ⎝ E − eV ⎠ 0 tanh 2kB T If we write h˜ 0 in the form of Eq. (8.146) the equilibrium values of hL and hT can then be expressed as      1 E + eV E − eV tanh ± tanh . hL,T = 2 2kB T 2kB T (8.204) If a finite voltage is put on the superconductor,we will obtain a time evolution of the phase in accordance with the Josephson relations.Since we have restricted ourselves here to the static case, we must assume that the voltage on the superconducting reservoir V = 0. In this case,hT = 0 for the superconducting reservoir. For a system with perfect interfaces between the superconducting and normal parts, the boundary conditions defined by the equations above are sufficient to solve the differential equations, the implicit assumption being that the Green’s functions and the distribution functions are continuous across any interface.When the transparency of the interface is less than unity, this is no longer true. Zaitsev [30] derived the boundary conditions for the Green’s functions at an interface of arbitrary transparency. Kupriyanov and Lukichev [31] simplified these equations for the diffusive case in the limit of small barrier transparency. Consider then an interface at x = 0 between

302

V. Chandrasekhar

two metals, say one a normal metal in the half-plane x < 0 (labeled by the index ‘1’) and one a superconductor in the half-plane x > 0, (labeled by the index ‘2’), although it also could be an interface between two different normal metals or superconductors. The boundary conditions of Kupriyanov and Lukichev are then vF1 D1 gˆs1 (@x gˆs1 ) = vF2 D2 gˆs2 (@x gˆs2 ) , 1 gˆs1 @x gˆs1 = [ˆgs1 , gˆs2 ] . 2r

(8.205a) (8.205b)

Here @x denotes a derivative in the positive x direction, and r = Rb /RN is a parameter that is nominally the ratio of the barrier resistance Rb to the normal wire resistance per unit length RN /L, but is inversely proportional to the transmission t of the interface. The first of the equations is clearly related to the conservation of current across the interface. The right hand side of the second equation has been shown to be the first term in an expansion of terms of the transmission coefficient t [14], and hence, it is only valid for low t. The diagonal part of Eq. (8.205b) gives the boundary condition for the Green’s functions R R g˜s1 @x g˜s1 =

 1 R R R R g˜s1 g˜s2 − g˜s2 g˜s1 . 2r

(8.206)

The off-diagonal part of Eq. (8.205b) gives the boundary condition for the distribution function  1 R K R K K A K A R K K A g˜s1 g˜s2 + g˜s1 g˜s1 @x g˜s1 + g˜s1 @x g˜s1 = g˜s2 − g˜s2 g˜s1 − g˜s2 g˜s1 . 2r

(8.207) If we put in Eq. (8.143) for g˜sK , with Eq. (8.146) for ˜ and then take as before the trace of the resulting h, equation, and the trace of the equation multiplied by  3 , we will obtain boundary conditions for hT and hL . Noting that the left hand side of Eq. (8.207) is simply the term in square brackets on the right hand side of Eq. (8.178c), we obtain the two equations 2r [M00 (@R hL ) + QhT + M30 @R hT )] = ˛1 , (8.208a)

where

= ˛1 = Tr g˜1R (˜g2R h˜ 2 − h˜ 2 g˜2A ) + (˜g1R h˜ 1 − h˜ 1g˜1A )˜g2A > − g˜ R (˜g R h˜ 1 − h˜ 1g˜ A ) − (˜g R h˜ 2 − h˜ 2 g˜ A )˜g A , (8.208c) 2

1

1

2

2r [M33 (@R hT ) + QhL + M03 @R hL )] = ˛2 , (8.208b)

1

and

= ˛2 = Tr ( 3 g˜1R (˜g2R h˜ 2 − h˜ 2 g˜2A )+(˜g1R h˜ 1 − h˜ 1 g˜1A )˜g2A > − g˜ R (˜g R h˜ 1 − h˜ 1 g˜ A )−(˜g R h˜ 2 − h˜ 2 g˜ A )˜g A ) . (8.208d) 2

1

1

2

2

1

We note that although the boundary conditions of Kupriyanov and Lukichev are valid for large r, they also give the correct boundary conditions (that of continuity of the Green’s functions and distribution functions) in the limit of r → 0. For arbitrary transmission of a barrier with n channels, the boundary condition can be represented as [19] e2  [ˆgs1 , gˆs2 ] , 2Tn  n 4+Tn (ˆgs1 gˆs2 + gˆs2 gˆs1 −2) (8.209) where is a constant factor, and Tn is the transmission of the nth channel. It can be seen that for a single channel with small transmission T, this equation reduces to the boundary condition of Kupriyanov and Lukichev. gˆ s1 @x gˆs1 =

8.8

Parametrization of the Quasiclassical Green’s Function

The normalization Eq. (8.167a) permits a parametrization of the quasiclassical Green’s functions that is very convenient for calculations. Equation (8.167a) is a matrix equation that is equivalent to the three equations (8.142) for the s-wave component of the Green’s function. To take into account the macroscopic phase of the superconductor, we note that a gauge transformation that transforms the vector and scalar potentials according to Eqs. (8.93) and (8.94) transforms the field operators according to the equations ˆ → ˆ ei , ˆ → ˆ e +

and

2

+ −i

(8.210a) .

(8.210b)

The Nambu–Gor’kov Green’s functions defined in Eq.(8.109) are transformed accordingly.For example,

8 ˇ˛

two components of G˜ 12 would transform according to ˇ˛ ˇ˛ [G˜ 12 ]11 → [G˜ 12 ]11e−i( (r 1)− (r 2 )) , ˇ˛ ˇ˛ [G˜ ]12 → [G˜ ]12ei( (r 1 )+ (r 2 )) . 12

12

(8.211a) (8.211b)

Making the transformation (8.42a) to mixed coordinates, and taking the limit as r → 0, we see that the off-diagonal components of the Nambu–Gor’kov Green’s functions are multiplied by a phase factor ei (R) or e−i (R) , while the diagonal components remain unchanged. Consequently,  also transforms as  → ei (R) . Keeping this in mind, we can express g˜sR as   cosh  sinh ei R g˜s = , − sinh e−i − cosh 

(8.212)

(8.213)

where  and are complex functions of the energy E and position R. This form satisfies the normalization condition gˆsR gˆsR =  0 . Note, one can also express gˆsR equivalently in terms of sin and cos. For completeness, we also give the expression for g˜sA :  ∗ − cosh  ∗ − sinh  ∗ ei A ∗ . (8.214) g˜s = sinh  ∗ e−i cosh  ∗ We now put this into the Usadel equation for g˜ R , Eq. (8.178a). Keeping in mind that the matrix for ˜ involves additional factors of ei and e−i due to the gauge transformation, the diagonal (1,1) component of this matrix equation is D sinh2  @R2 + D sinh 2 @R @R  − 2iIm() sinh  = 0 ,

(8.215a)

and the off-diagonal component (1,2) is D D@R2  − sinh 2 (@R )2 + 2Ei sinh  2 − 2iRe() cosh  = 0 , (8.215b) where we have used Eq. (8.215a) to simplify Eq. (8.215b). Defining a current js (E, R) by the equation js (E, R) = sinh2 (E, R)@R (E, R) ,

(8.216)

Quasiclassical Theory of Superconductivity

303

we can rewrite Eq. (8.215a) as D@R js (E, R) − 2iIm() sinh  = 0 .

(8.217)

js (E, R) is proportional to sinh2 , which is proportional to the square of the pair amplitude, and it is also proportional to the gauge-invariant gradient of the phase. Consequently, it is similar in form to the conventional definition of the supercurrent, and is called the spectral supercurrent. Indeed, Eq. (8.217) is simply another way of writing Eq. (8.196), and Q and js are related by Q(E, R) = −Im(js (E, R)) .

(8.218)

As we noted before, @R Q = 0 if  is purely real, and from Eq. (8.217), it can be seen that @R js (E, R) = 0 also if  is real. Equations(8.215) form a set of coupled equations that can be solved in principle for (E, R) and (E, R). In the case of a negligible spectral supercurrent, the equations decouple, and one needs to solve only Eq. (8.215b). In the limit of a bulk superconductor with a uniform real order parameter and no phase gradient, we recover the bulk value of the Green’s function, Eq. (8.155). The differential equations must be supplemented by boundary conditions. From Eq. (8.155), in a superconducting reservoir, we have E cosh S0 =  , (8.219) 2 E − ||2 so that the value of  in the superconducting reservoir is given by ⎧  1 || + E ⎪ ⎪ if E < || , ⎨−i + ln 2 2 || − E S0 = (8.220) 1 E + || ⎪ ⎪ ⎩ ln if E > || . 2 E − || The value of in a superconducting reservoir is just the macroscopic phase of the superconductor. In a normal reservoir,  = 0. The value of in a normal reservoir is meaningless, of course, and any choice that results in no phase gradient is valid. In terms of the  parametrization, the boundary conditions of Kupriyanov and Lukichev can be expressed as r sinh 1 (@R 1 ) = sinh 2 sin( 2 − 1 ),

(8.221a)

304

V. Chandrasekhar

and

 r @R 1 + i sinh 1 cosh 1 (@R 1 ) = cosh 1 sinh 2 e

(8.221b)

where performing the angular average in Eq. (8.137) gives the s-components of the Green’s functions. Putting in h˜ in the form of Eq. (8.146), we obtain

− sinh 1 cosh 2 .

i( 2 − 1 )

Note, that for r = 0, the boundary conditions reduce to 1 = 2 and 1 = 2 . In the absence of a supercurrent, (@R 1 ) = 0, so that the equations above simplify to 1 = 2 , (8.222a) and r(@R 1 ) = sinh(2 − 1 ) .

(8.222b)

Finally, we can write expressions for physical quantities in terms of  and . These quantities can be written in terms of the Mij 1

1 + cosh  cosh  ∗ 2  − sinh  sinh  ∗ cosh(2Im( )) , (8.223a) 1

1 + cosh  cosh  ∗ = 2  + sinh  sinh  ∗ cosh(2Im( )) , (8.223b) i = − sinh  sinh  ∗ sinh(2Im( )), (8.223c) 2

 = N0

 4

M03 and

M30 =

i sinh  sinh  ∗ sinh(2Im( )) . 2

 = N0

 4



∗ dE hL (sinh ei + sinh  ∗ ei ) (8.227) ∗  − hT (sinh ei − sinh  ∗ ei ) .

As an example, consider the case of a bulk superconductor, where = 0 and hT = 0. We then have  = N0

 2

 dE hL Re(sinh ) ∞

= N0  0

(8.228)

   , dE tanh(E/2kB T)Re √ E2 −2

which is the usual self-consistent equation for the gap.

(8.223d)

If is real, then the M03 = M30 = 0, and M00 and M33 simplify to (8.224a) M00 = cos2 (Im()) and M33 = cosh2 (Re()) .

dE [hL (˜gsR − g˜sA )+hT (˜gsR  3 −  3 g˜sA )]12 .

(8.226) With g˜sR and g˜sA given by Eqs. (8.213) and (8.214), we obtain

M00 =

M33



(8.224b)

These relations will be used in the next section when we discuss applying the Usadel equation to derive the transport properties of some simple device geometries. To conclude this section, we shall write an expression for the gap in terms . Replacing Eq. (8.143) for g˜ K into Eq. (8.137) for the gap, we have    = N0 dE [˜gsR h˜ − h˜ g˜sA ]12 , (8.225) 4

8.9

Applications of the Quasiclassical Equations to Proximity-Coupled Systems

We shall conclude our discussion of the quasiclassical theory by applying the equations that we have derived to some simple devices incorporating normal metals in close proximity with superconductors. Since the equations of motion in the diffusive limit are in general nonlinear, solving them usually requires numerical techniques, except in the limit of large resistances between the superconductor and the normal metal, where the Usadel equation can be linearized. We shall restrict ourselves to onedimensional examples; these are the ones discussed most in the literature.

8

8.9.1

Proximity-Coupled Wire

M00 (@R hL ) = K1 ,

(8.229a)

M33 (@R hT ) = K2 ,

(8.229b)

and where K1 and K2 are constants of integration. On integrating these equations from x = 0 to x = L, we obtain L hL (x = L) − hL (x = 0) = K1 0

L hT (x = L) − hT (x = 0) = K2 0

1 dx , M00

(8.230a)

1 dx . M33

(8.230b)

To calculate the conductance of the normal-metal wire in the linear approximation, we apply a small voltage V on the normal-metal reservoir, keeping the superconducting reservoir at V = 0. If we consider the second equation, hT (x = 0) = 0, and expanding hT (x = L) in a Taylor’s expansion to first order, we obtain from Eq. (8.230b) ⎡ K2 =

2kB T cosh2



E 2kB T

⎣

L 0

⎤−1 1 dx ⎦ . M33 (E, x) (8.231)

3

305

The electric current in the linear response regime can then be obtained from Eq. (8.192)

We start with the simplest possible device, a onedimensional normal-metal wire of length L connected on one end to a superconducting reservoir and at the other end to a normal metal reservoir. For definitiveness, let us take the superconducting reservoir to be at x = 0 and the normal-metal reservoir at x = L, and let us consider first the case of a perfect SN interface, so that the interface barrier resistance parameter r = 0. In this geometry, there can be no supercurrent, so that Q (or alternatively, js ) is zero. Furthermore, we can take the phase to be zero in the superconducting reservoir without loss of generality, and we note again that  = 0 in the normal metal. Under these conditions, only M00 and M33 are non-zero in Eq. (8.190), which now read

eV

Quasiclassical Theory of Superconductivity

j=

N0 e2VD (8.232) 2kB T ⎤−1 ⎡ L   1 1 ⎣ × dE dx ⎦ . M33 (E, x) cosh2(E/2kB T) 0

There are two differences between this equation and the equivalent equation for a normal metal in the classical regime (Eq. (8.10a)) that we derived earlier. First, the equation above involves an integral over energy and position. One can define a energy and position dependent electrical diffusion coefficient D3 (E, x) = DM33 (E, x)

(8.233)

instead of the constant diffusion coefficient D in Eq. (8.10a). Second, the current in Eq. (8.233) does not involve temperature differentials. Indeed, if one assumes that the superconducting reservoir is at a temperature T, and the normal reservoir at a temperature T + T, and expand hT (x = L) to first order in T, the terms involving T cancel, so that there is no term proportional to T. As a consequence, thermoelectric phenomena cannot be described using the quasiclassical equations, and an extension to the theory is required to take them into account.3 From Eq. (8.233), one can define a spectral or energy dependent conductance of the wire ⎡ L ⎤−1  1 dx ⎦ , G(E) = GN ⎣ (8.234) M33 (E, x) 0

where GN is the normal state conductance of the wire. The total conductance is then  G(E) . (8.235) G = dE 2kB T cosh2 (E/2kB T) Figure 8.4 shows the results of a numerical calculation of G(E)/GN as a function of the normalized energy E/Ec . The normalization factor Ec = D/L2 is called the correlation energy or Thouless energy (from the theory of disordered metals, where it also occurs) [32], and is dependent on the length L of

See, for example, F. Wilhelm, PhD Thesis, Universit¨at Karlsruhe, 2000.

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V. Chandrasekhar

Fig. 8.4. Spectral conductance G(E) of a one-dimensional wire of length L as a function of energy E, normalized to the correlation energy Ec . The barrier interface parameter r=0, and the gap is set to be  = 1000Ec

Fig. 8.5. Temperature dependent resistance R(T) normalized to the normal state resistance RN , as a function of the temperature T normalized to Ec , for different values of the interface resistance parameter r. The gap is set to be  = 32Ec

the wire. At high energies, G(E) approaches its normal state value, as expected.As the energy is lowered, the conductance increases, as might be expected in a proximity-coupled normal metal. However, instead of continually increasing as the energy is reduced, it reaches a maximum of around 1.15 GN at an energy of E  5Ec , and then decreases, reaching its normal state value at E = 0. This non-monotonic behavior of the conductance is called reentrance, and has been observed in experiments by a number of groups [33].It should be emphasized that the relevant energy scale where the maximum in conductance is observed is set not by the gap  of the superconductor, but by Ec , which itself depends inversely on the square of the length of the sample L. Hence in very long or macroscopic samples, the energy (and correspondingly, the temperature) at which the minimum would occur is far below the experimentally accessible range, and one regains the monotonic behavior expected from the simple Ginzburg–Landau theory of de Gennes [34]. The temperature dependent conductance G(T) can be obtained from G(E) using Eq. (8.235); the result of this calculation, plotted in terms of the normalized resistance R(T)/Rn, is shown in Fig. 8.5. In obtaining this plot, we have used a value of  = 32Ec , corresponding to a weak-coupling transition temperature of Tc = 1.764/kB , typical parameters for Al films. We have also assumed that the gap is temperature dependent. Like G(E), R(T) is also nonmonotonic, with a minimum at some intermediate temperature.We would expect the minimum in R(T) to be around T  5Ec /kB, based on the behavior of G(E). However, the temperature dependence of the superconducting gap modifies this behavior, so that the minimum in resistance occurs at a somewhat higher temperature when the interface between the normal metal and the superconductor is perfect. Figure 8.5 shows additional curves corresponding to progressively increasing values of the interface barrier parameter r. Increasing the resistance of the NS interface decreases the leakage of superconducting correlations from the superconductor into the normal metal, and consequently results in a smaller increase in the conductance of the proximity-coupled normal metal. In addition, the temperature Tmin at

8

Quasiclassical Theory of Superconductivity

307

ignored the effects of electron decoherence on the proximity correction. Phase coherence is essential to observing the proximity effect; if the phase coherence length L is less than the length of the sample L, a finite spatial cutoff of the proximity effect is introduced. Phenomenologically, this can be taken into account by saying that the length of the wire is now L instead of L. Since L is typically of the order of a few microns even at low temperatures, this sets the dimensions of the samples that are required to observe this mesoscopic proximity effect. While L sets the upper cutoff for observing the proximity effect, a second relevant length scale for the problem can be obtained by considering the length at which Ec is equal to kB T. This length is  Fig. 8.6. Density of states N(E, X) normalized to the normal state density of states N0 for a one-dimensional wire of length L, as a function of E/Ec and position x along the wire. The superconducting reservoir is at x = 1.0, and the normal reservoir is at x = 0. The density of states is suppressed at low energies near the superconducting reservoir

which the minimum in resistance occurs is also shifted down. From Eq. (8.194), the normalized density of states N(E) can be expressed in terms of  as N(E) = cosh (Re()) cos (Im()) .

(8.236)

Figure 8.6 shows the density of states as a function of energy and position along the wire of length L. There is a proximity-induced decrease of N(E) near the superconducting reservoir. In fact, at the NS interface, there is a divergence in N(E) at the gap energy, and it goes to zero at E = 0, just as one would expect for a superconductor. However, unlike a superconductor, it is not strictly zero for E < , but still has a finite amplitude. As one moves away from the NS interface into the proximity-coupled normal wire, both the amplitude of this effective gap and the divergence are smoothly reduced,so that at the normal reservoir, one recovers the normal density of states. This position dependent variation of the density of states has been observed in experiments [35]. In our analysis above of the proximity effect in a normal metal coupled to a superconductor, we have

LT =

D , kB T

(8.237)

where LT is called variously the thermal diffusion length or the Thouless length, again from the theory of disordered metals, where it also occurs [32]. (We put in here explicitly the factor of .) In fact, here it is simply the diffusive form of the superconducting coherence length in the normal metal, familiar from the de Gennes/Ginzburg–Landau theory of the proximity effect [34], which in the clean limit is given by N =

vF . kB T

(8.238)

At low temperatures, when LT is longer than L, the superconducting correlations induced in the normal metal extend throughout its length. At higher temperatures, they are restricted to a region of length LT near the superconductor. LT is also on the order of a few microns in typical metallic samples in accessible temperature regimes, so it also sets a limit on the dimensions of the samples in which one can see a proximity effect. To calculate the thermal conductance of the wire, we proceed from Eq. (8.230a). We now consider a small temperature differential T applied across the wire. Expanding hL in a first-order Taylor’s expansion, as we did for hT , we obtain

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⎡ K1 = −

ET 2kB T 2 cosh2



E 2kB T

⎣

L 0

⎤−1 1 dx ⎦ . M00 (E, x)

(8.239) We then obtain from Eq. (8.193)  E2 N0 DT jth = − dE 2kB T 2 cosh2 (E/2kB T) ⎤−1 ⎡ L  1 dx ⎦ . (8.240) ×⎣ M00 (E, x) 0

As with the electrical conductance, we can define a thermal diffusion coefficient D0 (E, x) = DM00 (E, x) ,

(8.241)

and a spectral thermal conductance ⎡ L  Gth (E) = GthN ⎣ 0

⎤−1 1 dx ⎦ M00 (E, x)

,

(8.242)

where GthN is related to the normal state electrical conductance by Eq. (8.17) GthN = GN

 2 kB2 T. 3 e2

(8.243)

Finally, the thermal conductance itself is given by an integral over energy  E2 Gth (E) 3 1 Gth = 2 dE . (8.244) 3  2(kB T) cosh2 (E/2kB T) Of course, as noted by Andreev [36], the thermal conductance of a normal metal wire sandwiched between a normal-metal reservoir on one end and a superconducting reservoir on the other end must vanish, since the superconductor acts as a thermal insulator, so that no thermal current can flow through the device as a whole.However,one may consider a normal-metal wire with the superconducting reservoir connected off to one side, so that it does not block the flow of thermal current through the proximity-coupled wire. One may then consider the thermal conductance of the normal metal wire itself. Figure 8.7 shows the thermal conductance of

Fig. 8.7. Thermal conductance Gth of a normal wire connected to a superconducting reservoir on one end, and a normal metal reservoir on the other, as a function of the normalized temperature T, for a number of different values of the interface barrier parameter r. The gap is set to be  = 32Ec , corresponding to a transition temperature of T = 18.14Ec

this geometry, as a function of temperature, for different transmissivities of the interface barrier. The thermal conductance shows a monotonic decrease as T is lowered below Tc , although there are no distinct features at any particular temperature, unlike for the electrical conductance. In a superconductor, the exponential decrease in the thermal conductivity is associated with the opening of the gap in the quasiparticle density of states, since it is the quasiparticles that carry the thermal current.Noting the decrease in the density of states in the proximity-coupled normal metal wire, shown in Fig. 8.6, it is not surprising that this system will also show a decrease in the thermal conductance. The thermal conductance of the wire is strongly dependent on the transmission of the NS interface, characterized by the parameter r, and approaches the normal state thermal conductance as r increases. The thermal conductance of Andreev interferometers has recently been measured [37], and calcula-

8

tions of the thermal conductance of this more complicated geometry have also been performed [38]. We note here again that, in our current approximation, a small voltage drop across the S/n–wire/N device will not result in a contribution to the thermal current through the system, since any terms proportional to voltage in the expansion of hL will cancel. This is the converse of the case for the electrical current, where a small temperature drop did not contribute to the electrical current, emphasizing again that the conventional quasiclassical approximation cannot take into account thermoelectric effects.

8.9.2

Instead of a normal reservoir on one side of the wire, if we consider a wire of length L connected only on one end to a superconducting reservoir (with the other end open), then a true gap in the density of states opens up in the proximity-coupled normal metal. The magnitude of the gap is related to Ec ; hence, one can consider the proximity-coupled normal-metal in this case as a superconductor with a gap of Ec . If the superconductor is not a reservoir, but a thin layer itself, then one will suppress superconductivity in the superconducting layer due to the proximity of the normal metal, an inverse proximity effect. The suppression of superconductivity is expected to reduce the gap in the superconductor. It is an interesting exercise to calculate the transition temperature of the bilayer in the quasiclassical approximation. We shall loosely follow here the treatment given by Martinis et al. [39] and Gu´eron [40]. Let the thickness of the superconductor be tS , and the thickness of the normal metal tN . We take the origin, x = 0, at the NS interface; the superconductor extends from x = −tS to x = 0, and the normal metal from x = 0 to x = tN . Near the transition, the order parameter in the superconductor is small, so we may the small  limit of Eq. (8.215b). Since the phase is not important in this problem, we take = 0. The resulting equation is (8.245)

309

where we have assumed that the gauge is chosen so that  is real. Let us assume that  at x = 0 in the superconductor is 0S , and that variations of  about this mean value are small.Under these conditions,we can also assume that the gap  in the superconductor is uniform. We can then expand S to second order in x S = 0S + ax + bx 2 . (8.246) Now, at x = −tS , (and also at x = tN ), we have a vacuum interface, where @x  = 0. Hence a = 2btS , and from the differential equation (8.245) taken at x = 0, b = (i/D)( − E0S ), so that S = 0S +

Superconductor-Metal Bilayer

D@x2  + 2Ei − 2i = 0 ,

Quasiclassical Theory of Superconductivity

i ( − E0S )(2tS x + x 2) . D

(8.247)

i E0N (2tN x − x 2) . D

(8.248)

Similarly N = 0N +

From the boundary condition Eq. (8.222b), we have the two equations 2irtS ( − E0S ) = 0N − 0S , D 2irtN E0N = 0N − 0S . D

(8.249a) (8.249b)

Solving this pair of equations for 0S , we have 0S =

 D2 tS (tS + tN ) + 4r 2 tN2 tS2 − 2irDtn2 . (8.250) E D2 (ts + tN )2 + 4r 2tS2 tN2

Putting this into Eq. (8.229) for the gap, with T = Tc , and with the approximation that sinh()  , we obtain  dE  1 = N0 tanh(E/2kB Tc ) (8.251) 2 E   tN (tS + tN ) . × 1− (tS + tN )2 + (4r 2/D2)tS2 tN2 The first term in the square brackets gives the bare transition temperature Tc0 of the superconducting film, and the second term corresponds to the corrections associated with the inverse proximity effect.For a perfect interface, with r = 0, the suppression of Tc

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V. Chandrasekhar

is directly proportional to the normal fraction of the bilayer,and Tc → 0 as tN increases.For a highly resistive barrier (r → ∞), the second term in the square brackets goes to zero, so that there is little effect of the normal metal film on Tc of the superconducting film, as expected. 8.9.3

The SNS Junction and Andreev Interferometers

As our final example of the application of the quasiclassical equations of superconductivity, we consider the case of a dirty SNS junction. The model we consider is a normal metal wire of length L sandwiched between two superconducting reservoirs. As is well known, the application of a phase difference between the two superconducting reservoirs will result in the flow of a supercurrent through the normal wire. The phase difference can be applied, for example, by connecting one of the superconducting reservoirs to the other, thereby forming a loop with two different arms, one superconducting and one normal. This configuration is commonly called an Andreev interferometer. The phase between the two superconducting reservoirs can be varied by applying an Aharonov–Bohm type magnetic flux through the area of the loop; in this respect, we put together all contributions in the gauge-invariant phase . Due to the single-valued nature of the wave functions, a phase factor of 2¥ /¥0 is picked up in going around the loop, where ¥ is the magnetic flux threading the Andreev interferometer, and ¥0 = h/2e is the superconducting flux quantum. In a superconductor, the supercurrent IS that is generated is directly proportional to the phase gradient; if IS is small compared to the critical current Ic , the phase dropped across the superconductor will be small. Since Ic of the superconducting part of the Andreev interferometer is so much greater than the critical current of the proximity coupled normal-metal wire, most of the phase change will occur across the length L of the normal metal wire. This fact allows us to map the Andreev interferometer that is coupled with an Aharonov–Bohm flux ¥ to a SNS system with a phase difference  = 2¥ /¥0 across it. In terms of our model, we consider the superconductors to be reservoirs; this means apply-

ing a boundary condition for the gauge-invariant phase at the superconducting reservoirs. For our purposes, we apply this boundary condition antisymmetrically, with a phase L = −¥ /¥0 at the superconducting reservoir at x = 0, and R = ¥ /¥0 at the superconducting reservoir at x = L. We then must solve Eq. (8.215) in the normal-metal wire for  and , with  = 0, subject to the boundary condition for noted above, and the boundary condition  = S0 (where S0 is given by Eq. (8.220)) at both x = 0 and x = L. In general, both  and are complex functions of x and E, and the solution of Eq. (8.215) must be done numerically. Following Yip [41], we consider first the supercurrent Q, given by Eq. (8.218). Some insight into the contributions to js can be gained by looking again at the case of a bulk superconductor. From Eq. (8.202), the major contribution to js comes from energies near the gap. For a long proximity wire with Ec < , however, the major contribution comes from energies of order Ec . Figure 8.8(a) shows a plot of Q as a function of energy for various values of the phase difference  between the two superconducting reservoirs, for the case of a perfectly transparent interface, and  = 32Ec . Of course, for zero phase difference, the supercurrent vanishes. As  is increased from zero, there is a peak in Q(E) at E  6Ec . This peak moves down in energy as  increases. At larger values of E, Q becomes negative. For shorter wires, this region of negative Q is less prominent. The total supercurrent is given by the second term in Eq. (8.192)  Js = eN0 D dE Q(E)hL (E) , (8.252) and therefore depends also on the distribution of quasiparticles. Any change in this distribution will affect the supercurrent.For example,the distribution can be changed by increasing the temperature, which has the result of decreasing the supercurrent. As we demonstrated in the introduction,a non-equilibrium quasiparticle distribution can also be generated by injecting a normal current into the proximity wire in the SNS geometry by attaching two additional leads to the center of the normal wire, forming a normal cross, with two of the wires attached to superconducting reservoirs, and the two other wires attached

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Quasiclassical Theory of Superconductivity

311

Fig. 8.8. (a) Spectral supercurrent Q in a normal wire between two superconducting reservoirs, as a function of the normalized energy E/Ec , for different values of the phase difference  between the superconducting reservoirs. (b) Geometry of an Andreev interferometer, essentially a cross with one arm connected to superconducting reservoirs, and the other arm connected to normal reservoirs

to normal reservoirs, as shown in Fig. 8.8(b). The current in the SNS junction is then a function of the current injected between the normal reservoirs, and the supercurrent can even change sign depending on the magnitude of the injected normal current [41,42]. This effect has been observed in recent experiments [43]. Due to long-range phase coherence, the Green’s function in the arms of the cross attached to the normal reservoirs will also depend on the phase difference between the two superconducting reservoirs in the structure shown in Fig. 8.8(b). Consequently, the electrical and thermal conductance measured between the two normal reservoirs will also be a periodic function of the phase difference between the two superconducting reservoirs. Experimentally, both the electrical conductance [44–46] and the thermal conductance [37] of such Andreev interferometers have been found to oscillate periodically with an applied external flux, with a fundamental period of 0 = h/2e. Periodic oscillations are also observed in the thermopower of Andreev interferometers [47], although these thermopower oscillations cannot be

described within the framework of the current quasiclassical theory.

8.10

Summary

The quasiclassical theory of superconductivity has proved to be a powerful tool for the quantitative description of long-range phase coherent phenomena in diffusive proximity coupled systems. As we have shown, the linear electrical and thermal conductance of complicated devices incorporating normal and superconducting elements can be calculated in principle,although the solutions frequently involve numerical techniques. Extension to the nonlinear regime, with finite voltages across the normal reservoirs, is also conceptually straightforward, although numerically challenging. Application of finite voltages to the superconducting elements is trickier, as it involves time dependent evolution of the phase,and is only beginning to be examined theoretically. Finally, the quasiclassical theory for diffusive systems, in its present form, does

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not deal at all with thermoelectric phenomena. Extensions to incorporate thermoelectric effects in the theoretical framework have been attempted [48], but still require further work to be complete.

Acknowledgements It is a pleasure to acknowledge illuminating discussions with Wolfgang Belzig, John Ketterson and Zhigang Jiang. This work was supported by the US National Science Foundation through grant number DMR-0201530.

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32. B.L. Altshuler, P.A. Lee and R.A. Webb, Mesoscopic Phenomena in Solids (North-Holland, Amsterdam, 1991) 33. See, for example, P. Charlat, H. Courtois, Ph. Gandit, D. Mailly, A.F. Volkov and B. Pannetier, Phys. Rev. Lett. 77, 4950 (1996) 34. P.G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, Redwood City, 1966) 35. S. Gu´eron, H. Pothier, N.O. Birge, D. Esteve, and M.H. Devoret, Phys. Rev. Lett. 77, 3025 (1996) 36. A.F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964), Sov. Phys. JETP 19, 1228 (1964) 37. Z. Jiang and V. Chandrasekhar, Phys. Rev. B 72, 020502(R) (2005) 38. Z. Jiang and V. Chandrasekhar, Phys. Rev. Lett. 94, 147002 (2005) 39. J.M. Martinis, G.C. Hilton, K.D. Irwin and D.A. Wollman, Nucl. Instrum. Methods Phys. Res. A 444, 23 (2000) 40. S. Gu´eron, PhD thesis, Universit´e Paris 6 (1997) 41. S.-K. Yip, Phys. Rev. B 58, 5803 (1998) 42. F.K. Wilhelm, G. Sch¨on and A.D. Zaikin, Phys. Rev. Lett. 81, 1682 (1998) 43. J.J.A. Baselmans, A.F. Morpugo, B.J. van Wees and T.M Klapwijk, Nature 397, 43 (1999) 44. H. Pothier, S. Gu´eron, D. Esteve, and M.H. Devoret, Phys. Rev. Lett. 73, 2488 (1994) 45. V.T. Petrashov, V.N. Antonov, P. Delsing, and T. Claeson, Phys. Rev. Lett. 74, 5268 (1995) 46. S.G. den Hartog, B.J. van Wees, T.M. Klapwijk, Yu.V. Nazarov and G. Borghs, Phys. Rev. B 56, 13738 (1997) 47. J. Eom, C-J Chien and V. Chandrasekhar, Phys. Rev. Lett. 81, 437 (1998) 48. F. Wilhelm, PhD thesis, Universit¨at Karlsruhe (2000)

9

Principles of Josephson-Junction-Based Quantum Computation

S. E. Shafranjuk Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA J. B. Ketterson Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 9.2 Josephson-Junction-Based Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.2.1 Junction Parameters and Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.2.2 The Basic Josephson Qubit categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .320 9.2.3 Phase Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .320 9.2.4 Charge Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 9.2.5 Flux Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 9.3 Single Qubit Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 9.3.1 Transitions Within Two-Level Systems; the Rabi Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 9.3.2 Manipulation, Readout and Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 9.3.3 Phase Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332 9.3.4 Charge Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 9.4 Quantum Oscillations in Two Coupled Charge Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345 9.4.1 The Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 9.4.2 The Two-Qubit Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 9.4.3 Two Qubit Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 9.5 SISIS Two-Qubit Gate with Intrinsic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 9.5.1 Proximity Coupling in a Multilayered Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 9.5.2 Inter-Qubit Coupling Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.5.3 Control of the Inter-qubit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 9.5.4 Additional Control Using Tangential Supercurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 9.5.5 Leakage and Fidelity of the Two-Qubit Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Appendix. Elementary Quantum Logic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 A.1 Boolean Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 A.2 Required Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 A.3 Universal Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 A.4 Quantum Sum and Carry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 A.5 Shor’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 A.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .365 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

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Introduction

One of several approaches that have been proposed for implementing quantum information processing is to utilize mesoscopic, artificially fabricated, solid-state structures which,based on the underlying physics, can be designed to behave as single “quantum particles.” The primary reason for a solid state approach is that it offers the likelihood of scalability by exploiting fabrication strategies made possible by the semiconductor industry; scalability is essential if quantum information processing is to become a practical reality. Aside from practical applications, the study of such mesoscopic structures is of interest in its own right. Quantum information processing is typically based on an assembly of quantum bit devices, socalled qubits [1], a term the community has adopted. Such devices involve externally controlled transitions between two quantum states, |0 and |1, corresponding to two different eigenenergies, E0 and E1 . There are now many proposals for qubits involving various two-level systems. However such elementary quantum logic devices must satisfy strict requirements if they are to be used as quantum logic elements in realistic information processing circuits. A practical circuit utilizing qubits must permit: (i) controlled manipulations of the quantum state of each qubit without disturbing adjacent elements and (ii) controlled inter-qubit coupling; also required are (iii) a limited influence of the external environment and (iv) sufficiently long dephasing and decoherence times [2,3]. If these requirements cannot be fulfilled, reliable qubit circuits cannot be realized, which currently eliminates many qubit proposals. Quantum computation (QC) employs and requires a programmable unitary evolution of the individual qubits [4]. Because the proposed systems exploit quantum parallelism and quantum algorithms, they are capable of efficiently solving certain classes of problems, which are intractable on classical computers. A striking example is the factorizing of large numbers [5], which is far more efficient on a quantum computer than on a conventional one. Along with the development of the theory of quantum information, there has been a parallel interest in

finding physical systems where quantum computation can be implemented. Toward this end, one must first identify a suitable two-level system, which is sufficiently decoupled from any source of decoherence [6], and for which the coupling among like elements can be controlled, thereby allowing the realization of single- and two-qubit operations. In principle one can then carry out any computational task if requirements (i)–(iv) are fulfilled [2,3,7] For the implementation of quantum algorithms various physical systems have been suggested. Some of these proposals involve: ions in traps, [8] QED cavities, [9] and NMR-based approaches [10]. To achieve large-scale integrability and flexibility in the design, approaches involving micro- or even nanotechnology are being examined including: smallcapacitance Josephson junctions [11–15], coupled quantum dots [16, 17], neutral atoms in optical lattices [18], and phosphorus dopants in silicon crystals1 [19]. Most of the solid-state based efforts concentrate on superconducting qubits, specifically Josephson junction based qubits [3], and we will review recent junction-based experiments and their theoretical interpretation. The experimental evidence for the superposition of charge states in Josephson junctions [20,21] and recent achievements in controlling the coherent evolution of quantum states in a “Cooper pair box” [22] make superconducting circuits very promising candidates to implement solid state quantum computing. Figure 9.1 shows a schematic representation of a Josephson junction. Superconducting qubits exploit Cooper pair tunneling between the superfluid condensates in adjacent superconducting electrodes. The Josephson interaction across a dielectric barrier has an intrinsic non-linear origin, which plays an important role in the design of the qubit devices. Josephson junction based qubits, depending on the external configuration, exploit the order-parameter phase, magnetic flux, or electric charge, as shown schematically in Fig. 9.2. Combinations of these configurations are also exploited. This chapter is organized as follows.In Sect.9.2 we briefly describe the three Josephson junction based qubit devices, the phase qubit, the charge qubit, and the flux qubit, along with experiments on some spe-

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Fig. 9.1. A schematic representation of a Josephson junction and the equivalent circuit

Fig. 9.2. The three major Josephsonjunction-based qubit implementations

cific implementations. We also discuss so-called hybrid qubits that combine aspects of both charge and flux qubits. Section 9.3 starts with a review of the quantum mechanics of two-level systems, which underlies all qubit operations. This is followed by a discussion of the effects of decoherence arising from coupling to external degrees of freedom. It is crucial to maintain the coherence of the two basic qubit states during a quantum computation [2]. The decoherence time, dec , of the individual qubits should be larger, by a factor of order 104 , than a time top , required to carry out a single operation; one can then carry our arbitrarily long quantum computations by exploiting various error-correction techniques. In solid state implementations, due to the complexity of the environment, there are many degrees of freedom that can couple to the qubit states and cause decoherence. Compared with the photonic and atomic qubit strategies, the superconducting persistent current qubit is subject to more severe decoherence. Dominant sources of decoherence include both “internal” effects, such as dissipation, e.g., from quasiparticle re-

sistance, and dephasing from qubit interactions with the external environment. These mechanisms depend strongly on the geometry, and on fluctuations of the environment (e.g., nuclear spin fluctuations in aluminum), on background charge noise, and on noise in the control currents. It is also possible to couple to an environmental degree of freedom, without a dissipative mechanism, that will still lead to decoherence. We will outline the general formalism that can be used to deal with the dephasing caused by thermal fluctuations, including quasiparticle dissipation, charge oscillations, nuclear spin relaxation, etc. However we will not, for the most part, try to analyze such mechanisms in detail. Section 9.3 then goes on to discuss some experiments on single qubit operations that have been carried out on phase and charge qubits. The challenge in performing accurate qubit operations lies in effectively isolating the two energy levels from the rest of Hilbert space. In other words, how does one operate as quickly, and with as little error as possible, on the qubit subspace, while simultaneously isolating the remaining Hilbert space. This is especially

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important when the coherence time of the system is short.As an example, a Josephson phase qubit can be described by three energy levels |0, |1, and |2, with energies E0 , E1 , and E2 . The qubit space is formed by |0 and |1, and hence we wish to operate only within this subspace. Clearly, the higher-order state can be avoided when working in the |0,|1 sub-space provided the energies differ sufficiently and the excitation pulse is long enough. However, because one wants to maximize the number of logic operations within a fixed coherence time, there is a need to mix the |0 and |1 states as quickly as possible without affecting other states. Operations on charge qubits can be carried out using the gate voltage alone. However much better control is achieved using a hybrid qubit where the charging and Josephson energies can be individually controlled. Extensive experiments with this device demonstrating controlled decoupling from the external environment are also described. Section 9.4 describes a two-qubit gate involving two capacitively coupled hybrid qubits and some associated experiments.Section 9.5 discusses a recently proposed and promising two-qubit gate where the qubit coupling is engineered into a single device and which is based on a multilayer SISIS junction. Finally Sect. 9.6 discusses our conclusions where we examine the outlook for Josephson-junctionbased quantum computing. Although it is not the focus of this article, we briefly outline the elementary quantum logic operations that are necessary for performing quantum computations in Appendix A. Errors induced by the gate operations themselves must be considered if fault-tolerant quantum computation is to be achieved. The most obvious example is fluctuations in the control parameters of the gate, which act as random noise and thus degrade the unitarity of the time evolution of the computational degrees of freedom. In addition, the actual gate operations can change the qubit coupling to the external environment (even if the coupling is negligible during storage periods) thereby enhancing decoherence. Most sources of error can be analyzed by properly modeling the qubit-environment coupling. However, there are errors, which are not due to (or cannot be

described in terms of) the action of an external environment. An (intrinsic) source of error in gate operations, [23] which is common to several of the proposed solid-state implementations,is so-called quantum leakage. It occurs when the computational space is a subspace of a larger Hilbert space. The effects of such states have been investigated for ion trap quantum computers where estimates have been obtained of the number of operations before decay processes induce dissipative transitions outside the computational space [8].A procedure for estimating the leakage for a phase qubit will be given in Sect. 9.3.2.

Fig. 9.3. Schematic view of a qubit with leakage (according to [23]). The two low-energy states constitute the computational Hilbert space. The system, however, evolves under the action of some unitary operator and can leak out to the higher excited states. In the case of Josephson junction qubits, leakage is due to the Josephson tunneling to higher charge states. In the case of two-qubit operations, the computational space is spanned by the states |00, |01, |10, |11 and the coupling with the higher charge states is due both to EJ and to inter-qubit coupling EI . The two lowenergy states constitute the computational Hilbert space. However the system can leak out to the higher states. If the energy difference between the low-lying and the excited states is large compared to the other energy scales of the system, the probability to leak out is small

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Principles of Josephson-Junction-Based Quantum Computation

Non-computational states affect the gate dynamics even in the absence of dissipative processes. Such an analysis applies to the situation illustrated schematically in Fig. 9.3; the quantum states |0 and |1 (with the level splitting ∼ 2Ech , Ech being the charging energy) are affected by the higher excited states |n and |n + 1, which have the same order of level splitting ∼ Ech due to Josephson tunneling (associated with an energy EJ ). In two-qubit gates the states also leak due to the inter-qubit coupling which energy is EI .

9.2

319

Fig. 9.4. Sketch of the SIS tunneling Josephson junction and its equivalent circuit

Josephson-Junction-Based Qubit Devices

In this section we briefly review some approaches to realizing Josephson-junction-based qubits. The flexibility of the Josephson-based circuits allows different implementations, some of which are now being developed by various laboratories. 9.2.1

Junction Parameters and Energetics

Figure 9.4 shows, schematically, a Josephson tunnel junction together with its equivalent circuit. The junction is parameterized by a critical current, I0, a capacitance, C, and a (in general nonlinear) shunt resistance, R, which here we assume to be large. The superconducting layers are characterized by LandauGinzburg order parameters i = Ai ei'i with amplitudes, A1 and A2 , and phases '1 and '2 of the respective superfluid condensates. The Josephson coupling energy between the layers depends on the phase difference ' = '1 − '2

Fig. 9.5. Potential energy (in units of maximal Josephson energy EJ ) of a tunnel junction biased by a supercurrent, I

and plays the role of a “potential energy”. We will show shortly that in the presence of an external bias supercurrent this potential energy has the form   ¥0   U ' = I0 sin ' − ' I ; 2

(9.1)

i.e., it has the shape of a“tilted washboard”, as shown in Fig. 9.5. This energy profile is exploited in the large area Josephson junction qubits. Figure 9.6 shows the

Fig. 9.6. The interior of the ellipses shows the tilting of the “washboard potential” at different positions on the voltage–current characteristic of the Josephson junction. The parameters of the junction (Lukens group, SUNY) are shown in the inset

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current-voltage characteristic (the I–V curve) of a voltage biased tunnel junction (the parameters being indicated in the inset); also shown is the property that the “washboard” tilt depends on the position on the I-V curve. When the junction is embedded in a circuit having an external inductance Ls , the behavior depends on its size relative to a characteristic inductance LJ ≡ ¥0 /2I0 (where ¥0 = h/2e is the flux quantum). One then has two limiting cases: Ls  LJ , in which the induced flux in the loop is unimportant, and Ls  LJ where it plays a key role. Circuits of the first type (Ls  LJ ) are usually based on aluminum junctions while circuits of the second type (for which Ls  LJ ) are usually based on niobium. When Ls  LJ the properties are determined by the relation between the maximum Josephson coupling energy, EJ = I0 ¥0 /2, and the elementary Coulomb energy, e2 /2C. 9.2.2

The Basic Josephson Qubit Categories

We now briefly review some Josephson-junctionbased qubits. The flexibility of the Josephson-based circuits allows different implementations, some of which are now being developed by various laboratories. Depending on the variable manipulated, and the junction energetics, we identify the following qubit categories: (i) the phase qubit with EJ  Ec (by a factor 103 ), where the current density j is the variable; (ii) the charge qubit with EJ < Ec , where the charge Q is the variable; and (iii) the flux qubit with EJ  Ec (by a factor 102–103 ), where the magnetic flux ¥ is the variable. We now give a brief description of each of these three cases. 9.2.3

Phase Qubits

The total current across the junction in Fig. 9.4b is given by I = V /R + I0 sin ' + C V˙ .

(9.2)

Using the Josephson relation, V = (/2e)'˙ , and assuming the current I is constant, which allows us to write I = @ (I' )/@' , we can rewrite Eq. (9.2) as

 C

¥0 2

2

  1 ¥0 2 '¨ + '˙ R 2   ¥0 @ ¥0 −I0 cos ' − I ' = 0, + @' 2 2

(9.3)

where ¥0 = h/2e is the superconducting flux quantum. The first term in Eq. (9.3) can be associated with a “kinetic energy” K which takes the various forms  2 Q2 1 2 ¥0 K= = CV = '˙ 2 . (9.4) 2C 2 2 The potential energy of the junction itself is given by   d' I0 ¥ 0 sin ' dt U = IJ V dt = 2 dt (9.5) I0 ¥ 0 cos ' ; =− 2 from this expression it follows that we can interpret Eqs. (9.2) and (9.3) in terms of the motion of a classical particle in a “tilted washboard” potential of the form ¥0 ¥0 U˜ = −I0 cos ' − I '; (9.6) 2 2 this system is shown in Fig. 9.7. Neglecting damping (R → ∞), the potential energies given by Eqs. (9.5) and (9.6) together with the above kinetic energy, K = q2 /2C, yield the Hamiltonian under which the classical system evolves in time H(q, ' ) = K + U˜ Q 2 I0 ¥ 0 I¥0 − cos ' − ' , (9.7) 2C 2 2 where the charge q plays the role of a momentum, C is a mass, and ' is a coordinate. We now make the transition to quantum mechanics by writing =

ˆ = H

1 2 I0 ¥ 0 I¥0 cos ' − ', Qˆ − 2C 2 2

(9.8)

where Qˆ = (2e/i)@/@' and we have the commuta  ˆ ' = 2ei. Quantum mechanical betion relation Q, havior can be observed for large area junctions for which I0 ¥0 /2 = EJ  EC = e2/2C and when the bias current I is somewhat smaller than the critical current I0 . In this regime the potential U˜ (' )

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top. The escape rates, n and n+1 , from levels n and n+1 to an adjacent well differ significantly from each other; a typical ratio is, n/n+1 ∼ 102. Microwave bias currents induce transitions between levels at a frequency !mn = Emn / = (Em − En ) /, where En is the energy of state |n. In the cubic approximation the two lowest transitions have the frequencies [24]   5 !p !10 ≈ !p 1 − (9.9) 36 U 

and !21 ≈ !p Fig. 9.7. The motion of a classical particle in a “tilted washboard” potential. When the tilt is small (the lower profile), the particle oscillates inside the local “washboard” minimum. However at steeper inclines, the particle escapes to an adjacent minimum or, if the damping is small, rolls steadily down the “washboard” (corresponding to as finite voltage state)

can be expanded (about the displaced minimum resulting from the constant current I) and accurately approximated by a cubic polynomial involving a  √ barrier height U (I) = 2 2I0 ¥0 /3 [1 − I/I0 ]3/2 and a quadratic curvature at the bottom of the well that gives a classical oscillation frequency !p (I) = 21/4 (2I0 /¥0C)1/2 [1 − I/I0]1/4 . The commutation relation leads to quantized energy levels in this cubic potential, which are shown schematically in Fig. 9.8. When operating the junction as a qubit, one adjusts the tilt to achieve two states lying deeper in the potential well with a third level positioned near the

Fig. 9.8. Some quantized states in the quantum well created by the Josephson energy profile

10 !p 1− 36 U

 .

(9.10)

These two frequencies must differ if we are to access the two-state system in a controllable way. The ratio U /!p parameterizes the anharmonicity of the cubic potential with regard to the qubit states, and gives an estimate of the number of states in the well. The result of the quantization is to create states inside a local minimum of the washboard, as shown in Fig. 9.8. (Due to tunneling these states are more correctly viewed as resonances with widths n .) The dependence of !10 , !21 , and !32 on the anharmonicity ratio, U /!p , is given in Fig. 9.9. The challenge in performing accurate qubit operations lies in successfully isolating the two lowest energy levels from the rest of the state space; clearly

Fig. 9.9. The dependence of the transition frequencies !10 , !21 and !32 between levels on the anharmonicity ratio U /!p

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one wants to operate quickly in the qubit subspace (to minimize the effects of decoherence, see Sect. 9.3.2) with as little coupling to unintended states as possible. This is especially important when the coherence times of the system are short. A Josephson phase qubit involves three energy levels |0, |1, and |2, with energies E0 , E1 , and E2 . The qubit space is formed by |0 and |1, while the third level is used as a readout (by exploiting its high tunneling rate). The |0 ↔ |2 transition can be minimized by exciting the |0 ↔ |1 transition (having a frequency !10 ) with a sufficiently long pulse. However, because one wants to maximize the number of logic operations within a fixed coherence time, there is a need to excite the |0 ↔ |1 transition as quickly as possible without populating other states. The state of the qubit is determined by a combination of a dc bias current, Idc , and a time-varying microwave bias current, I w (t), at a frequency ! = !10 I (t) = Idc + I (t) = Idc − I w (t) cos (!t + ) . (9.11) 9.2.4

Charge Qubits

The Single Junction Charge Qubit We now consider the case EC >EJ (by a few orders of magnitude); this leads us to a different class of devices, the so-called charge qubits. The simplest example of such a device is shown schematically in Fig. 9.10. It involves two superconducting strips, S1 and S2 one of which slightly overlaps the other, but which are separated by an oxide barrier so as to form a Josephson junction; this junction is characterized by a capacitance CJ and critical current I0 where the latter is associated with a Josephson energy, EJ . One of these strips (designated S2 in the figure) is called the island or Cooper charge box; it has a selfcapacitance CI and it is coupled to an adjacent third strip, called the gate, through a capacitance CG to which a voltage VG is applied. It is assumed that there is no Josephson coupling between the gate and the island and that the gate can be biased relative to the island by a voltage VG . A typical island capacitance, CI , is ≤ 10−15 F, while the gate capacitance is typically smaller.

Fig. 9.10. The basic circuit of a single junction charge qubit

It is assumed the device is configured such that the superconducting energy gap, , is the largest energy in the system and hence at low temperatures (  kB T) quasiparticle tunneling can be neglected. Under this circumstance coherent tunneling of Cooper pairs is the only channel between the island and the superconducting electrode and the charge of the island is restricted to 2Ne where N, the number of excess Cooper pairs, is an integer. The electrostatic energy associated with devices of this type is somewhat subtle. We will not discuss this problem here but rather refer the reader to an analysis by Tinkham who obtains the form [25] ECoulomb = 4EC (N − NG )2 + const.,

(9.12)

where the constant can be ignored, EC ≡ e2 /2CI is the single electron charging energy of the island, and NG is the number of Cooper pairs induced electrostatically by the gate on the island; NG can be written in terms of the gate voltage as NG = CG VG /2e. When there are no excess Cooper pairs on the island (N = 0) the energy increases quadratically with the gate voltage, as with any capacitor; this behavior is shown by the black parabola in Fig. 9.11. The energy in the presence of a single excess Cooper pair on the island is shown by the red parabola. Note ECoulomb vanishes for NG = 1 or, equivalently, at a gate voltage VG = 2e/CG. On the other hand, at the point NG = 1/2, where the red and black parabolas in Fig. 9.11 cross each other, two many-body superconducting ground states of the island would be

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Fig. 9.11. The energy diagram of a charge qubit

degenerate in the absence of the Josephson coupling (which we have been assuming small compared with the Coulomb energy); i.e. the states with that N = 0 and N = 1 would have the same energy. Hence VG can be used as a control parameter to perform quantum superpositions of these many-body superconducting charge states.1 As with the phase qubit discussed above, the dynamics of the device are again governed by a classical Hamiltonian that is the sum of an electrostatic kinetic energy, given by Eq. (9.12), and the Josephson potential energy. To transition to quantum mechanics, in a representation where the Josephson phase is diagonal, the electrostatic energy is interpreted as a kinetic energy operator, K = 4EC (Nˆ − NG )2 ; here Nˆ = −i@/@' is the Cooper pair number operator, which is conjugate to the phase variable, ' , of the superconducting order parameter of the island. From Eq (9.6) (with I = 0), the potential energy is written U = EJ cos ' , where we again defined a Josephson coupling energy, EJ =I0 ¥0/2. The Hamiltonian of the system is then H = K + U = 4EC (Nˆ − NG )2 + EJ cos ' ; 1

(9.13)

this Hamiltonian   is quantized via the commutation ˆ ' = i. relation N, The Double Junction Charge Qubit The single junction device described above has the disadvantage that the only parameter that can conveniently be tuned is the gate voltage. We will show in Sect. 9.3.3 that, in spite of this limitation, it is still possible to perform arbitrary single qubit operations; however the procedures required turn out to be somewhat awkward and for this reason it is desirable to have a second, independent, parameter. As shown schematically in Fig. 9.12, this can be accomplished if we couple the island to a second Josephson junction that is connected to the first junction by a superconducting link such that a closed loop is formed; it is now assumed that a flux ¥x is induced in this loop (by a second loop driven by an external current) such that a supercurrent passes through the two junctions (in addition to the island); this alters the Josephson critical current and with it the Josephson coupling energy and provides an independently

Since the charging energy depends on the square of the gate potential, it is periodic in the Cooper pair number, and undergoes a splitting where NG = (2N + 1)/2, the E vs. NG curves have the appearance of the E vs. k curves encountered in the theory of nearly free electrons in a one-dimensional crystal. In particular the point NG = 1/2 corresponds to an electron at the first Brillouin zone point.

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adjustable parameter. The device is variously called a hybrid device or Bloch transistor. Assuming identical junctions one obtains the Hamiltonian as  2 Hˆ = EC Nˆ − NG + EJ (cos '1 + cos '2 ) (9.14)  2 = EC Nˆ − NG + 2EJ cos 'x cos ' ; here 'x = '1 + '2 = ¥x /¥0 and arises from the induced d.c. current flowing through the two junctions while ' = '1 −'2 where '1 and '2 are the phase shifts across the first and second junction respectively. The second term now acts as an effective Josephson coupling energy the strength of which can be tuned through the current I. Writing 2EJeff = 2EJ cos (/2) our Hamiltonian becomes  2 (9.15) Hˆ = EC Nˆ − NG + 2EJeff cos ' with NG = NG (VG ) Fig. 9.12. A double junction qubit biased by a flux transformer

and EJeff = EJeff (¥x ) .

Fig. 9.13. Single-Cooper-pair box with a probe junction. Left: Circuit diagram of the device. The C’s represent the capacitance of each element and the V’s are voltages that can be applied to each electrode. Right: Micrograph of the sample; light areas are electrodes. The electrodes were fabricated by electron-beam lithography and shadow evaporation of Al on a SiN x insulating layer (400 nm thick) above a gold ground plane (100 nm thick) on the oxidized Si substrate. The“box”electrode is a 700 ×50 ×15 nm Al strip containing ∼ 108 conduction electrons. The reservoir electrode was evaporated after a slight oxidation of the surface of the box so that the overlapping area becomes two parallel low-resistance junctions (∼ 10 k§ in total) with Josephson energy EJ which can be tuned by a magnetic flux ¥ penetrating through the loop. Before the evaporation of the probe electrode the box is further oxidized to create a highly resistive probe junction (R b < 30 M§). Two gate electrodes (d.c. and pulse) are capacitively coupled to the box electrode. The sample was placed in a shielded copper case at the base temperature (T < 30 mK; kB T < 3 ‹eV) of a dilution refrigerator. The single-electron charging energy of the box electrode EC = e2 /2CI was 117 ± 3 meV, where CI is the total capacitance of the box electrode. The superconducting gap energy  was 230 ± 10 ‹eV

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Principles of Josephson-Junction-Based Quantum Computation

The Double Junction Charge Qubit with a Readout Junction By including a third or “probe” junction, readout can be facilitated [26]. Such a device is shown in Fig. 9.13. The right side is an electron micrograph of the device itself while the left side shows a schematic diagram. As with the simplified double junction device, it involves a small superconducting island on which N excess Cooper-pairs (relative to some neutral reference state) can reside. The island is again electrostatically coupled through a capacitance CG to a control gate G, that biases the charge on the island; however an additional capacitance, Cp , is also included to allow control pulses to be inserted. As noted, the inclusion of a third junction allows read out; this

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is accomplished by biasing the device far from the degeneracy point causing the superposition state to collapse. Figure 9.14 shows a similar device that the original authors [27] refer to as a quantronium circuit. Activating the bias current Ib , drives the qubit away from its optimal working point and is again used to readout the quantum state.The circuit was fabricated by depositing aluminum through a suspended mask that was in turn patterned by e-beam lithography. It consists of a superconducting loop interrupted by two small Josephson tunnel junctions, each with capacitance CJ (having a low series capacitance) and Josephson energy EJ , a superconducting island with capacitance CI , and by a large Josephson junction EJ0 with energy EJ0 ∼ = 20EJ . The island is again charge-

Fig. 9.14. Top: scanning electron micrograph of the circuit. Bottom: schematic diagram showing the tuning, preparation and readout blocks

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biased by a voltage source VG through a gate capacitance CG ; it is flux-biased by a loop that generates clockwise or counter-clock wise currents through the junctions. Experimental studies have been performed by several groups with aluminum tunnel junctions with dimensions below 100 nm [4,28]. The superposition of charge states in circuits in the charging regime has been demonstrated [29–31] and is in quantitative agreement with theory [32,33]. The Heisenberg uncertainty principle has been demonstrated when EJ ≈ EC . [34] When EJ > EC topological excitations involving vortices exist (which we do not discuss) and quantum mechanical interference of these quantities has been observed [34]. Unfortunately circuits of the first type in the charging regime are sensitive to fluctuating “off-set charges” that are present in the substrate [35,36]. These random offset charges make the design of a controllable array of quantum circuits difficult and introduce a strong source of decoherence. The Charge-State Basis In a bulk superconductor one usually characterizes the macroscopic quantum state by the GinzburgLandau phase and regards the Cooper pair number as a fluctuating quantity. However when the island charging energy significantly exceeds the Josephson coupling energy, Ec  EJ ,states |N numbered by the excess number of Cooper pairs N on the island form a good basis. In this basis the Hamiltonian (9.13) is written  2  Hˆ = 4EC Nˆ − NG |N N| (9.16) N

 1 − EJ [|N N + 1| + |N + 1 N|] . 2

When N differs significantly from NG the energy levels are dominated by the charging part of the Hamiltonian. However, when NG is approximately a halfinteger, the charging energies of two adjacent states that differ by a Cooper pair are close to each other (e.g., at VG =Vdeg ≡ e/CG ), and the Josephson tunneling strongly mixes them (see Fig. 9.11). If we focus

on voltages near such a degeneracy point, only two charge states (say N = 0 and N = 1) play a role; all other charge states have a much higher energy and can be ignored. In this case the Cooper box Hamiltonian (9.16) reduces to a two-state quantum system (a qubit) with a Hamiltonian that in spin-1/2 notation can be written as 1 1 ˆ = − Bz ˆ z − Bx ˆ x . (9.17) H 2 2 The charge states N = 0 and N = 1 associated with the diagonal operator ˆ z are given by     1 0 |↑ = ; |↓ = (9.18) 0 1 respectively and the effective magnetic field Bz corresponds to the charging energy, which is controlled by the gate voltage and is given by Bz = ıEch ≡ 4Ec (1 − 2NG ) .

(9.19)

On the other hand, Bx ≡ EJ , associated with the off diagonal operator ˆ x , couples states differing by one Cooper pair. We now rewrite the Hamiltonian (9.17) as   Hˆ = −E () cos  ˆz + sin  ˆx /2 , (9.20) where the mixing angle  ≡ tan−1 (Bx /Bz )

(9.21)

determines the direction of the effective magnetic field in the x–z plane. The eigenvalues of (9.20) are given by (9.22) E () = B2x + B2z = EJ / sin  and the eigenstates, |0 and |1, by   (9.23a) |0 = cos |↑ + sin |↓ , 2 2   (9.23b) |1 = − sin |↑ + cos |↓ . 2 2 The point where the two charge states are degenerate corresponds to Bz = 0, or equivalently  = /2; here E = EJ . We can now rewrite the Hamiltonian in diagonal form as 1 Hˆ = − E () ˆz , 2

(9.24)

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Principles of Josephson-Junction-Based Quantum Computation

where we have introduce a second set of Pauli matrices, , ˆ that operate in the |0, |1 basis, while retaining the ˆ operators for the charge-state basis, |↑, |↓. The Hamiltonian (9.24) is similar to the ideal single-qubit model. Ideally the bias energy (the effective magnetic field in the z direction) and the tunneling amplitude (the field in the x direction) are controllable, a property of the hybrid devices discussed above. As noted above, it turns out that we can perform all qubit operations using only a single parameter,the bias energy (through the gate voltage); this situation will be considered in Sect. 9.3.3. Operations that include tuning the tunneling amplitude, which fixes the Josephson energy, will be considered in Sect. 9.3.3. 9.2.5

Flux Qubits

In circuits of the second type identified in Sect. 9.2.1 (Ls  LJ ), the quantum variables can be related to the flux in the loops and their time derivatives. The circuits involve a superconducting ring that is interrupted by one or more Josephson junctions in which a persistent current flows with its associated magnetic flux. The flux-based qubits emerge from the following ideas. We recall that flux is quantized within a superconducting loop; i.e.,

If we now insert a Josephson junction in the loop (see Fig.9.15) the phase difference across the junction will be related to the flux ¥ within the loop,which in units of the flux quantum, ¥0=h/2e, is written  '  , ¥ = ¥0 n − 2

where   Q2 ¥ ; ; EJ = −E0J cos 2 2CJ ¥0 (9.27) 1 2 (¥ − ¥x )2 LI = ; 2 2L EC =

here L is the self-inductance of the loop and CJ is the capacitance of the junction. In quantizing (9.26) we interpret the charge as an operator Q = −i@/@¥ that is canonically conjugate to the flux ¥ . If the self-inductance is large, such that the pa rameter ˇL = E0J / ¥02/4 2 L is larger than 1, and the externally applied flux ¥x is close to ¥0 /2, the second and third terms in the Hamiltonian (9.26),     2 1 U = U0 ' − 'x − ˇL cos ' , (9.28) 2 form a double-well potential (see Fig. 9.16); here ' = 2¥ /¥0,'x ≡ 2(¥x /¥0),and U0 = ¥02/(4 2 L). At low temperatures only the lowest states in the two wells contribute.

Fig. 9.15. A Josephson junction shunted by an external inductor

(9.25)

where n is an integer number and ' = 2(¥ /¥0). When the system is biased by an externally applied flux ¥x , the Hamiltonian (which includes the Josephson, charging, and magnetic contributions to the energy) is written as H = EC + EJ +

1 2 LI , 2

(9.26)

327

Fig. 9.16. A typical two-level system

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Analogous to Eq. (9.17) for the charge qubit, the reduced Hamiltonian of this effective two-state system can again be written in the form Hˆ = −Bz ˆ z /2 − Bx ˆ x /2 . The diagonal term Bz plays the role of a bias; when ˇL − 1  1 we can expand the cosine and the asymmetry of the double-well potential can then be represented by a field       Bz 'x = 4 6 ˇL − 1 EJ 'x − 1/2 . (9.29) Bz can be tuned through the applied flux ¥x via an applied current Ix . The off-diagonal term Bx describes the tunneling amplitude between the wells,which depends on the height of the barrier and thus on the Josephson energy EJ ; this energy can be controlled if the junction is replaced by a dc SQUID with the flux ¥˜ x introduced as another control variable [8]. With these two external control parameters the elementary single-bit operations, i.e., z and x rotations, can be performed, which are equivalent to the manipulations described in the previous section for charge qubits. Flux qubit operations can be performed either by abrupt switching of the external fluxes ¥x and ¥˜ x for a finite time, or with the use of r.f. fields and resonant pulses. Such a devices constitutes the r.f. SQUID used in the experiments [37–40]. To permit coherent manipulations, the parameter ˇL in Eq. (9.29) should be chosen larger than unity (so that two minima with well-defined levels appear), but not so large that the resulting barrier height between the minima, U ,overly suppresses the tunneling; these energetics are shown in Fig. 9.16. The r.f. SQUID described above, which mimics an asymmetric quantum well (shown in Fig. 9.17), was discussed in the mid 1980’s as a realization of a two-state quantum system. Some features of macroscopic quantum behavior were demonstrated, such as macroscopic quantum tunneling of the flux, resonant tunneling, and level quantization [41–44]. However, only very recently has the level repulsion near a degeneracy point been demonstrated [45,46]. For the r.f. SQUID, thermal activation of macroscopic quantum states [47] has been observed as well

Fig. 9.17. The asymmetric quantum well; biasing is achieved by a flux ¥x . The level structure is probed using macroscopic resonant tunneling

Fig. 9.18. The SQUID level splitting versus the magnetic flux ¥x in the double-well potential U . The two states have circulating currents of opposite sign

as macroscopic quantum tunneling between states shown schematically in Fig. 9.18 [48]. Caldeira and Leggett [49] proposed these systems in the mid 1980s as test objects to study various quantum-mechanical effects, including macroscopic quantum tunneling of the phase (or flux) as well as resonant tunneling. Both the effects have been observed in several experiments [50–53]. An advantage of such devices is that the two persistent current states can be externally distinguished, since they involve circulating currents of opposite sign (see Fig. 9.18). This leads to alternative qubit design that exploits circuits of the first type (with aluminum), but which utilizes states associated with circulating currents having opposite sign (as in circuits of the second type). These circulating current

9

Principles of Josephson-Junction-Based Quantum Computation

Fig. 9.19. The energy level positions versus the tilt of the double well potential in a SQUID-type device. The avoided level crossing is indicated by the dashed line

states typically create a magnetic flux ∼ 10−3 ¥0 socalled “persistent current (PC) states”. These states obey all of the five requirements for a quantum bit: (1) they can readily be prepared in the ground state (at a sufficiently low temperature); (2) they can be precisely manipulated with magnetic fields; (3) two qubits can be coupled inductively, and that coupling can be switched on and off; (4) the flux associated with the PC states can be measured using a SQUIDtype detector; and (5), the states can be made insensitive to background charges and effectively decoupled from their electrostatic environment (in contrast with charge quantum states in Josephson circuits); the magnetic coupling to the environment can also be effectively suppressed. Another important quantum effect has been reported recently: The groups at Stony Brook [54] and Delft [55] have experimentally observed the avoided level crossing due to coherent tunneling of the flux in a double-well potential. In principle, all other manipulations discussed in the previous section should be possible with Josephson flux devices as well. The group at Stony Brook probed, spectroscopically, the superposition of excited states in different wells. The experimental results are shown in Fig. 9.19 for an r.f. SQUID with a self-inductance L = 240 pH and ˇL = 2.33. In the experiment a substantial separation of the minima of the double-well potential (of order ¥0 ) and a high inter-well barrier made the tunnel coupling between the lowest states in the wells neg-

329

ligible. However, both wells contain a set of higher localized levels (under suitable conditions one state in each well) with relative energies that are also controlled by ¥x and ¥˜ x . Because they were closer to the top of the barrier, these states mixed more strongly and formed eigenstates, which were superpositions of localized flux states from different wells. External microwave radiation was then used to pump the system from a well-localized lowest state in one well to one of these eigenstates. The energy spectrum of these levels was studied for different biases ¥x , ¥˜ x , and the properties of the model associated with Eq. (9.28) were confirmed. In particular, the level splitting at the degeneracy point indicated a superposition of distinct quantum states. They differed in a macroscopic way: the authors estimated that the parameters associated with the two superimposed flux states differed by a flux ¥0 /4, a current of 2–3 mA, and a magnetic moment of 1010 B.

9.3

Qubit Dynamics

In this section we begin by discussing the manipulation of single qubit devices by an external time dependent field. In realistic mesoscopic systems, there are usually several discrete energy levels (e.g. the three levels utilized in a phase qubit). However a qubit utilizes only two of these levels and therefore care must be taken to isolate these levels from the remaining levels. Assuming for the moment that this is the case, we first review the quantum mechanics associated with transitions within a two-level system, which is fundamental to the operation of all qubits. 9.3.1

Transitions Within Two-Level Systems; the Rabi Frequency

Assume we are given a system with discrete levels En and states ¥ (q, t) (where q denotes all spatial coordinates) that is perturbed by an external timedependent field, Vˆ (t). We seek an approximate solution to the time-dependent Schr¨odinger equation, i

  @¦ ˆ 0 + Vˆ (t) ¦ , = H @t

(9.30)

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by expanding in terms of the stationary states, ¦k0 , of the unperturbed system in the form ¦ =



ak (t)¦k0 ,

(9.31)

k

where, as indicated, the coefficients ak (t) are timedependent. Substituting (9.31) in (9.30), and keeping in mind that @¦ 0 i k = Hˆ 0 ¦k0 , (9.32) @t one gets i

 k

¦k0

@ak  = ak Vˆ (t) ¦k0 . @t

(9.33)

k

After multiplying both sides of (9.33) by ¦m0∗ and integrating over configuration space one finds i

@am  = Vˆ mk ak , @t

(9.34)

k

where  Vmk (t) =

¦m0∗ Vˆ ¦k0 dq = Vmk ei!mkt , !mk

0 Em − Ek0 . 

=

(9.35)

In a two-level qubit one normally uses a weak periodic perturbation of the form ˆ −i!t + Fˆ †ei!t Vˆ = Fe

(9.36)

which yields Vkn (t) = Vkn ei!kn t ∗ i(!kn +!)t = Fkn ei(!kn −!)t + Fnk e .

(9.37)

The external field frequency is selected to satisfy 0 − En0 =  (! + ") , Em

(9.38)

where " is a small detuning frequency. Near resonance, one of the coefficients in the expansion (9.31) becomes large since the essential contributioncomes

from terms for which !mn − ! is small. One then obtains the following system of equations @am = Fmn ei(!mn−!)t an = Fmn ei"t an (9.39) @t @an ∗ −i"t i = Fmn e am . (9.40) @t Substituting an ei"t = bn (9.41) i

we obtain the pair of equations   ∗ am (9.42) i˙am = Fmn bn , i b˙ n − i"bn a˙ m = Fmn which give 1 b¨ n − i" b˙ n + 2 |Fmn |2 bn = 0 . (9.43)  The two independent solutions of this equation are chosen as ˛1 (9.44) an = Aei˛1 t , am = −A ∗ ei˛2 t , Fmn ˛2 an = Be−i˛2 t , am = −B ∗ ei˛1 t , (9.45) Fmn where A and B are integration constants (being determined from initial conditions) and " " ˛1 = − + § , ˛2 = + § , 2 2 7 (9.46) Fmn " 2  2 §= +  , = , 4  where  is the so-called Rabi frequency, which determines the rate at which the system cycles between the states. Hence the perturbation transforms the wave functions of a quantum mechanical two-level system as ¦m0 , ¦n0 → an ¦n0 + am ¦m0 ; (9.47) i.e., it mixes the initial unperturbed states in a deterministic manner. A two-level system can be used to implement a quantum bit provided it can be manipulated, projectively readout, and coupled to similar devices in a controlled way. But such systems, in practice, always couple to the external environment at some level, and that introduces a loss of coherence on a time scale characterized by a coherence time. The number of single-qubit operations that can be conducted during a coherence time is then a measure of the

9

Principles of Josephson-Junction-Based Quantum Computation

qubit performance. The coherence time is limited by interactions with the other degrees of freedom in the environment, including the manipulation and readout systems. Achieving full control and long coherence times in a qubit are somewhat incompatible goals; strong coupling is desirable when preparing a state; in addition, readout requires strong coupling to another physical system which is measured at the macroscopic level. On the other hand, a high degree of isolation is required as the prepared state evolves. Although most qubit devices have their own unique features, their coupling to the environment can often be described in a generic way, which we now discuss. 9.3.2

Manipulation, Readout and Decoherence

We start by considering the effect on a qubit circuit ˆ c = X ˆ that couof a term in the Hamiltonian H ples a qubit variable Xˆ to an external variable. The coupling entangles the qubit with the environment, thereby introducing decoherence. The readout gives ˆ the expectation   value  of X, and  the measured signal ˆ ˆ is X01 = 1 X  1 − 0 X  0 . This signal is directly related to variations of the qubit transition energy !01 (or frequency !01 ) with the average value of the external variable  treated as a control parameter: @(!01 ) X01 = . (9.48) @ When the coupling is weak, the decoherence is minimal and does not modify the qubit states; the qubit will then be projected onto an eigenstate on a time scale determined by a characteristic coherence time [56–59]. Decoherence can be described in terms of two processes: relaxation processes, in which an energy !01 is exchanged between the qubit and its environment, and dephasing processes, in which the relative phase between the two qubit states grows randomly with time. The latter, which formally represents an entanglement between the qubit and its environment, is characterized as arising from a modulation of the 2

3

331

qubit transition frequency caused by the fluctuations of the control variable, measured by .2 The random phase-shift accumulated between the two qubit states in a time t is then given by t ı' = 0

@!01     ı t dt . @

(9.49)

To characterize ı (t  ) one introduces a fluctuation spectrum, S (!). This spectrum is usually modeled as being constant below a characteristic cut-off frequency !C , but9 for times: longer than !C−1 , the coherence factor, exp(iı' ) , decays exponentially; the characteristic time is given by [60,61] T' =   @!01 2 @

2 S (! = 0)

.

(9.50)

In general, the spectral density S (0) is the sum of contributions arising from many sources: 1) thermal fluctuations of the environment (which are proportional to temperature), 2) non-equilibrium excessnoise arising from uncontrolled variables (e.g. substrate charge fluctuations in charge qubits), and 3) the readout system (e.g. noise arriving from input/output leads).3 On comparing (9.48) and (9.50) it is clear that the dephasing time T' and the measured signal X01 are closely interrelated.In addition,coupling the qubit to a readout system can introduce extra noise over that required by the measurement process itself, which in many cases cannot be avoided. However by working at a stationary point where @!01/@ = 0, long dephasing times can still be achieved since this will suppress the coupling to all noise sources to first order. At such a point we have X01 = 0; hence  must be shifted prior to readout. Below we will discuss a readout strategy for a qubit circuit based on the Cooper pair box (where quantum coherence has already been demonstrated).In this so-called quantronium device, activating the readout automatically drives the qubit away from this optimal working point [62].

This semi-classical approximation (which breaks down in the zero-temperature limit) is applicable to Josephson junction based qubits studied to date. A special treatment is required when the spectral density diverges at low frequency (so called 1/f noise).

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9.3.3

Phase Qubit Dynamics

The Hamiltonian The basic ideas underlying the Josephson phase qubit were discussed in Sect. 9.2.3. Here we discuss their dynamics in more detail. If we restrict ourselves to three levels,the Hamiltonian of a biased Josephson junction is written as 

H=

I¥0 1 2 I0 ¥ 0 Q − cos ' − ', 2C 2 2

(9.51)

where I is the bias supercurrent, I0 is the critical current, C is the junction capacitance, and ¥0 is flux quantum. The operators Q and ' correspond to the charge and phase difference on the junction. The above Hamiltonian can be approximated by [63] ⎛ ⎞ E0 0 0 H = ⎝ 0 E1 0 ⎠ 0 0 E2 ⎡ 9   : 9   : 9   :⎤ 0 '  0 0 '  1 0 '  2 ⎢ 9 ¥0 ⎢   : 9   : 9   : ⎥ + I 1 ' 0 1 ' 1 1 ' 2 ⎥. 2 ⎣ 9   : 9   : 9   : ⎦ 2 '  0 2 '  1 2 '  2 (9.52) where I = −I w (t) cos(!t + ), ! is the frequency of the r.f. current, and  is the phase shift. Here E0 , E1 and E2 are the energies 9  of :the three lowest levels. The matrix elements m '  n are calculated as follows. We assume the presence of a dc current such that we can approximate the potential as a cubic polynomial. This Hamiltonian can be diagonalized numerically in terms of the energies 9 of the: three lowest states and the matrix elements m '  n computed. Calculating 9   : m '  n in this manner, one obtains ⎛

⎞ E0 g (t) ei(!t+) 0 ⎜ ⎟ √ −i(!t+) E1 2g (t) ei(!t+) ⎟ H ≈⎜ ⎝g (t) e ⎠+Hnr √ −i(!t+) 0 2g (t) e E2

(9.53) where the basis states are |0, |1, and |2 from left to right, ! is the frequency of the applied √ time-varying r.f. current, and g (t) = 1.014I w (t) /2!01C/2 is an “envelope function” related to the time-varying current, I w (t). The numerical coefficient 1.014 has

been obtained from the bias supercurrent magnitude. A shaped pulse is generated using the envelope function, g(t). The remaining contributions to the Hamiltonian, Hnr , involve additional diagonal and non-diagonal elements, all of which are sufficiently far from the resonances associated with !10 and !12 that they can be neglected. One can then perform operations between the |0 and |1 states while minimizing those between |1 and |2. The Rotating Frame To calculate the effect of the envelope function it is convenient to move into a doubly rotating frame, defined by the unitary operator ⎛ ⎞ 10 0 ⎜ i!t ⎟ ⎟. ˆ V(t) =⎜ (9.54) ⎝0 e 0 ⎠ 0 0 e2i!t  : Let | = Vˆ  be a state in the rotating frame gen : erated by Vˆ where  is a state in the laboratory frame. The equation of motion for | can then be derived from the Schr¨odinger equation, and has the Schr¨odinger-like form i

@ | = H˜ | , @t

(9.55)

˜ is the rotating frame Hamiltonian given by where H H˜ = V †HV − iV † (@V /@t) and is given by ⎞ ⎛ 0 g (t) ei 0 ⎟ ⎜ √ −i 2g (t) ei ⎟ H˜ = ⎜ ⎠ , (9.56) ⎝ g (t) e E1 − ! √ 0 2g (t) e−i E2 − 2! where we have set E0 = 0. We will limit ourselves here to on-resonance excitation involving the |0 ↔ |1 transition. Using E1 = !10 , !=!10 , and defining the energy difference between the two transitions as E2 − 2E1 = ı! , Eq. (9.56) becomes ⎛ ⎜

0

g (t) ei

−i H =⎜ ⎝ g (t) e 0

0

⎞ 0 √ ⎟ 2g (t) ei ⎟ ⎠.

√ 2g (t) e−i ı!

(9.57)

This rotating frame Hamiltonian can be used to directly calculate the effect of flat topped (termed hard)

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Principles of Josephson-Junction-Based Quantum Computation

pulses.“Shaped”pulses can be treated by discretizing an assumed form for g(t) as a series of steps,gj ,acting for times,t.During the jth slice,Uj = exp(−iH˜ j t/) and by multiplying a succession of these forms we obtain the total evolution as A A U= Uj = exp (9.58) j

j

⎡

⎛

0

gj ei

⎢ it ⎜ ⎜ gj e−i 0 × ⎣−  ⎝ √ 0

⎞⎤ 0 ⎟⎥ √ 2gj ei ⎟ ⎠⎦ .

2gj ei ı!

From U we can calculate the“leakage”out of the twolevel qubit manifold. From Eq. (9.58) we see that the effects of ı! scale as tı! /2; an appropriate time unit is therefore pw = tı! /2, and the results given below are plotted in terms of this parameter. Shaped and Composite Pulses Shaped pulses are widely used in NMR quantum computing since they can significantly enhance the selective excitation of a qubit compared with hard pulses. In contrast to NMR, where each qubit is represented by a spin-1/2 particle, the two transitions in our phase qubit share the same potential well, leading to a more complex dynamics. Although it is not immediately obvious that pulse shaping can be useful, detailed calculations [63] (see below) show that some benefit is possible. Here we will limit ourselves to a 180◦ rotation (a  pulse in NMR jargon) which involves the |0 ↔ |1 and |1 ↔ |0 transitions, since it is usually most difficult to selectively carry out this rotation. Calculations have been performed for several pulse shapes including Gaussian and the so-called Hermite forms. The r.f. envelope, g(t), for the Gaussian is given by     ggauss (t) = a/tg exp −t 2 /2tg2 ; (9.59) here tg is the characteristic pulse width and a is the pulse amplitude. This form can be “truncated” by setting g(t) = 0 for |t| > ˛tg , where ˛ is a cutoff parameter (usually 3 to 5); the total pulse width tpw is then 2˛tg . For 180◦ pulses a ≈ 1.25 (for typical values of ˛ The Hermite shape is defined as a Gaussian

multiplied by a second order polynomial  2     ghrm (t) = 1 − ˇ t/˛tg a/tg   × exp −t 2 /2tg2 ,

333

(9.60)

where ˇ determines how strongly the Gaussian pulse is modulated and the remaining parameters are the same as above. For ˇ = 4 and a 180◦ rotation angle, a ≈ 2.2 for ˛ = 3 and a ≈ 1.67 for ˛ = 3. The performance of various pulses has been calculated using these parameters [63]. An error measure can be defined as " = 1 − |U (3, 3)|2 where U (3, 3) denotes the (3,3) element of the resulting unitary transform; " is equivalent to the probability of the system being in the state |2 after the application of the pulse, when starting from an arbitrary superposition of the form a |0 + b |1. Ideally, the error measure vanishes (" = 0) if operation is confined to the |0, |1 subspace. The measure " serves as a lower bound. Of course, the rotation may deviate slightly from an ideal rotation even when " = 0. One also has to account for the so-called transient Bloch-Siegert shifts,similar to those seen in NMR; however these effects can be corrected using a method similar to the one described in [64]. One can calculate " numerically using Eq. (9.58), however to gain an intuitive understanding of the utility of a given envelope function one can use a simple bandwidth argument. Given that the response is approximately linear for small rotation angles, one can Fourier transform a given envelope function; this procedure is extensively used in NMR, even when the rotation angles are large. For an untruncated Gaussian pulse, the relative power at a frequency ı! from !10 is given by  

 "(pw ) ≈ exp −ı! tg2 = exp −(pw/ ˛)2 . (9.61) This form is plotted in Fig. 9.20 along with that for a truncated Gaussian pulse and the exact calculation using (9.58). For small pulse widths, the exact calculation and the one based on Fourier analysis of the untruncated shape are similar, but the exact calculation flattens out for pw > 4. From this example it is clear that Fourier analysis gives a rough estimate of the error, especially when the pulses are truncated. Fourier transforming tells us immediately that hard

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Fig. 9.20. Numerical results for the error " as a function of normalized pulse width pw = tı! /2 via Fourier analysis using the untruncated and truncated Gaussian shape, compared with the exact calculation

pulses are inferior relative to Gaussian or Hermite shapes. In order to quantify the performance accurately however, one must calculate the error numerically using Eq. (9.58). One can also use a “composite pulse” consisting of a sequence of individual pulses that is designed to reduce certain types of errors, but possibly at the expense of a longer total duration. The individual pulses are typically square wave pulses (as they are the easiest to generate via r.f. switches), but could be shaped pulses as well. Such pulses have also been utilized in NMR. The unitary evolution U of a composite pulse is also calculated via Eq. (9.58), where Uj are just the evolutions associated with the individual pulses in the sequence. Decoherence Studies in Phase Qubits; Junction Resonances Thus far we have only considered the ideal case where the frequency connecting the two lowest levels of the three-level system lies far from those connecting the third, but no sources of decoherence are present. However, in a real Josephson phase qubit, the quantum state can tunnel through the barrier of the cubic well, and this process acts as a source of decoherence. Tunneling has the following consequences. First we

know that the tunneling rate depends exponentially on the barrier height and width, and hence we expect the upper level to be most susceptible to tunneling. When the tunneling rate out of this level is high, it acts as a significant source of decoherence of the |0, |1 manifold if the |2 state becomes partially populated. Therefore it is important to keep the amplitude of the |2 state as small as possible during single qubit rotations. As discussed above, shaped and composite pulses can improve the fidelity of single-qubitoperations, but estimates of the associated rates show that tunneling can still be the dominant source of decoherence; for this reason operating a Josephson phase qubit with three level is problematic. The decoherence can be reduced by using a four level scheme (see Fig. 9.21b). For typical junction parameters (e.g. !10 /2 ≈ 10 GHz,U /!p ≈ 4,and ı! = 0.04!10 ), the associated pulse widths tpw are a few to tens of nanoseconds. The electronics needed to achieve accurate circuit timings, and simultaneously allow the application of shaped and composite pulses at these time scales, is non trivial. Possibly some of the techniques used in NMR spectroscopy can be applied to superconductor-based qubit systems. We now discuss some experiments that were designed to specifically address microscopic decoherence mechanisms. One such mechanism, studied by the authors of [65], is coupling to unwanted electromagnetic modes. These workers found that decoherence arising from coupling to spurious junction resonances strongly affects the amplitude of the oscillations. They proposed a model (to be discussed below) in which these resonances arise from fluctuations in the tunnel barrier, and related them to previous measurements of the I–V characteristics and critical-current 1/f noise. Such resonances appear to be a major source of decoherence and must therefore be minimized if a successful Josephson quantum computer is to be constructed. Figure 9.21a shows the circuit used in the experiments. The junction, which is based on an aluminum technology, was placed in a superconducting loop with an inductance L (to minimize quasiparticle generation and self-heating when the qubit state is probed); by coupling flux through a transformer with a mutual inductance M, the junction

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Fig. 9.21. (a) Circuit diagram for the Josephson junction qubit studied by the authors of [65]. Junction current bias I is set by I¥ and microwave source I w . Parameters are I0 = 11.659 A, C = 1.2pF, L = 168pH, and L/M = 81. (b) Potential energy diagram of qubit, showing a case where there are four levels, |0, |1, |2, and |4, in the cubic well to the left. Measurement of |1 state is performed by driving the 1 → 3 transition, tunneling to right well, followed by relaxation of state to bottom of right well. Post-measurement classical states “0” and “1” differ in flux by ¥0 , which is readily measured by  : the readout SQUID. (c) Schematic description of states in a symmetric well. Tunneling between states produces ground g and excited |e states separated in energy by !r . (d) Energy-level diagram for coupled qubit and resonant states for  : !10 ≈ !r ; coupling strength between states 1g and |0e is given by Hˆ int

can be biased close to its critical current I0. Unlike the simple washboard potential discussed in Sect. 9.3.2, the circuit is configured such that there are two asymmetric wells. With this arrangement the states in the left well (which can be modeled with a cubic potential) can tunnel into the right well where (on decaying) they change the flux through the loop by ∼ ¥0=h/2e, which can be read-out with a pulsed critical-current measurement using an independent SQUID detector. By applying a microwave current, I w (!), in the presence of a d.c. current-bias pulse, the various transition frequencies can be probed.Figure 9.22a shows the results for the 0 → 1 transition frequency, !10 , vs. bias current, I0 ; the resonance is inferred from an increase in the tunneling rate. As expected, the frequency falls as the bias current increases (and the potential becomes shallower). The currents used are such that the left well supports approximately 3–4 energy levels. Note that a number of small spurious resonances (at the dotted vertical lines in Fig. 9.22) are observed that are characteristic of energy-level repulsion in a coupled twostate system. The splitting of these extra resonances is distributed in size; the largest ones are ∼ 25 MHz, and there is approximately 1 spurious resonance per ∼ 60 MHz.

Rabi oscillations (see Sect. 9.3.1) were observed between the |0 and |1 states by applying a microwave pulse at the 0 → 1 transition frequency, !10 , for a time tr , which is followed by measuring the occupation probability of state |1 using a second microwave pulse resonant at the 1 → 3 transition frequency, !31 . Figure 9.23a–c shows the occupation probability vs. pulse width for three values of the microwave power; the microwave frequencies utilized correspond to currents I0 following from the spectroscopic studies shown in Fig. 9.22a. The decay of the oscillations is approximately exponential and yields a coherence time of 41 ns. The Rabi frequency vs. microwave amplitude shows the expected linear behavior, as can be seen in Fig. 9.23d. Note that strong Rabi oscillations occur only between the spurious resonances (dashed lines) in Fig. 9.22b, as is apparent from the coloring; i.e., the spurious resonances strongly disrupt the Rabi oscillations. Figure 9.24 shows the time dependence of the decay of the Rabi oscillations for various bias currents ((a) through (f) in Fig. 9.22) in the neighborhood of the spurious resonances.Near a spurious resonance non-exponential oscillatory decays are observed which appear to involve beating, although the overall decay time may remain constant; clearly some

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Fig. 9.22. (a) Measured probability of state “1” versus microwave excitation frequency !/2 and bias current I for a fixed microwave power. Data indicate the !10 transition frequency. Dotted vertical lines are centered at spurious resonances. (b) Measured occupation probability of the |1 state vs. the Rabi-pulse time tr and bias current I. In panel (b), a color change from dark blue to red corresponds to a probability change of 0.4. Color modulation in a time tr (vertical direction) indicates Rabi oscillations.

kind of mode mixing is occurring. The maximum amplitude of the Rabi oscillations is approximately 30%, which is significantly less than unity. The spurious resonances appear to be distributed randomly,both in size and frequency,and a few small resonances are likely to be near an arbitrary bias point. Even for the major resonances, the presence of several small resonances can still degrade the amplitude, thereby reducing the overall amplitude. Spurious resonances have been found in other experiments [68, 69], although the Rabi oscillation data is usually reported only for the maximum response. Coherence times and Rabi amplitudes for several experiments are collected in Table 9.1 [65, 69]; all the experiments show some reduction in amplitude.

Table 9.1. Coherence times and Rabi amplitudes observed in various experiments. All data show a coherence amplitude smaller than unity. Materials for the junction electrodes and barrier are also listed.Values in parentheses are estimated from published data Reference Junction Coh. Time Coh. Amp.(%) [69] Al/AlOx /Al 150 ns 50 [65] Al/AlOx /Al 1 ‹s (30) [66] NbAl/AlOx /Nb 20 ns (15) [67] NbN/AlN/NbN 4.9 ‹s (1) [68] Al/AlOx /Al 41 ns 30

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Fig. 9.23. (a)–(c) The measured occupation probability of state |1 as a function of the Rabi pulse width tr for three values of microwave power, taken at a bias of I = 11.609 ‹A (see Fig. 9.22). The microwave power for the responses shown in (a), (b), and (c) is 0.1, 0.33, and 1.1 mW, respectively. (d) The Rabi oscillation frequency ( in Eq. 9.40) versus microwave amplitude; the theory predicts a linear dependence

Fig. 9.24. Measured occupation probability of state |1 versus time duration of the Rabi pulse tr for current biases a–f as noted by arrows in Fig. 9.22. Data a–e is offset for clarity. Note that when the bias is changed, the coherence is degraded mainly as a loss in amplitude, not by a decrease in coherence time

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Time-Dependent Decoherence The NIST group [65] also observes that the frequencies and magnitudes of the spurious resonances occasionally change in time, not only when thermally cycled to room temperature, but even when the device is cold. The latter strongly implies that at least some resonances are microscopic in origin, as opposed to modes arising from leads to the device or other sources. These authors constructed a model, similar to one describing 1/f fluctuations in the junction critical current [66,67],involving two-level states within the barrier having large tunneling matrix elements corresponding to a microwave frequency [67]. Consider two states in the barrier having configurations A and B that produce critical currents I0A and I0B ; the interaction Hamiltonian between the resonance and the critical current is then ¥0 cos 'ˆ ⊗ |¦A  ¦A | Hint = −I0A 2 (9.62) ¥0 cos 'ˆ ⊗ |¦B ¦B| , − I0B 2 where 'ˆ is an operator corresponding to the phase difference across the junction, and ¦A,B are wave functions for the two configurations. Assuming a symmetric potential with energy eigenstates separated  :by !r , the ground √ and excited states will be g = ( |¦A  + |¦B ) / 2 and |e = √ ( |¦A  − |¦B) / 2. Using matrix elements for cos 'ˆ appropriate for a phase qubit [70],and including only the dominant resonant terms arising from this interaction Hamiltonian, one has  I0  ˜ int = H (9.63) 2 2!10 C  :   9  × |0 1| ⊗ |e g  + |1 0| ⊗ g e| ,

where I0 = I0A − I0B . Figure 9.25 shows an energy level diagram for the case where !r ≈ !10 . The coupling of the two intermediate energy levels through H˜ int produces repulsion in the energy eigenstates. From the of the level repul magnitude  sions at resonance 2 H˜ int  / ≈ 25 MHz, one obtains I0 ≈ 65 × 10−6I0 , which is consistent with parameters obtained from the current-voltage characteristics.

Fig. 9.25. The energy level diagram in the Josephson junction potential versus the phase difference

A mesoscopic theory [71] involving a nonuniform tunnel barrier has been invoked to account for the Josephson and quasiparticle currents. The model assumes the current is carried by an array of independent tunneling channels. The associated transmission coefficients i can be obtained from steps in the quasiparticle current of magnitude 2/i that arises from n-th order multiple Andreev reflections at voltages 2/n [72]. If it is assumed that the current is carried by only Nch channels, then the measured current-voltage characteristics [73] imply a characteristic = 4 × 10−3 for a critical-current density of ∼ 40 A/cm2 . For the 32 ‹m2 junction with a normal-state resistance RN = 29 § used in the experiments, one obtains Nch = h/2e2 RN  = 1.3×105,implying a channel density of 4 × 103/‹m2 = 1/(16 nm)2 , which in turn implies that Nch (I0 /I0) ≈ 8 channels are switched on and off between the two junction states A and B.Since typical junctions have a distribution of channels and the spectroscopy measurement is dominated by the largest resonances, the magnitude of I0 likely involves single channels with i ≈ 2 × 10−2. As noted above the model of these microwave resonances is similar to that describing 1/f criticalcurrent noise at audio frequencies. Measurements on submicrometer Josephson junctions have shown discrete changes in the critical current which are associated with fluctuating links [66,67,74].A recent experiment on an 0.08 ‹m2 aluminum junction showed a change in critical current of I0 ≈ 10−4 I0 associated with a single fluctuator [67].Assuming this link turns a channel on and off, the data imply an areal density

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of fluctuating links Nch/‹m2 ∼ 1.25×105,which is 30 times greater than the channel density 4 × 103 /‹m2 found in the above microwave resonance data. In addition, the density of frequency fluctuators can also be estimated from previous experiments.Various experiments suggest there is approximately one resonance per decade in frequency for junctions with an area of 0.1 ‹m2 [66, 67]; this turns out to be twice estimated for the microwave experiment described above. Given they involved phenomena with characteristic frequencies differing by orders of magnitude, the magnitude and density with frequency of the microwave and 1/f noise measurements are in rather good agreement; this strongly suggesting that they arise from the same microscopic behavior in the tunnel barriers. A compilation of 1/f noise data indicates that junctions made from oxides of Al, Nb, and Pb-In have similar magnitudes of 1/f noise [67]. Clearly alternatives to thermal or plasma oxidized junctions should be investigated. In conclusion,the spurious microwave resonances in Josephson junction qubits significantly degrade the amplitude of Rabi oscillations, and represent a major challenge. These resonances can be understood as arising from two-level fluctuating links within the tunnel barrier, which couple to the qubits states through the critical current. Clearly, better qubits will require better junctions. 9.3.4

Charge Qubits

We now discuss single qubit operations involving charge qubits. As noted earlier all such operations can be carried out using the gate voltage alone, and we treat this approach first. Operations Using the Gate Voltage Only The Hamiltonian describing charge qubits was discussed in Sect. 9.2.4 [see Eq. (9.17)]. We start by assuming the qubit is in an“idle state”in which the gate voltage VG is set at a value sufficiently far to the left of the degeneracy point such that the eigenstates |0 and |1, Eq. (2.18), are close to the pure |↑ and |↓ 4

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states, Eq. (9.18), respectively. Switching the system suddenly to the degeneracy point for a time t and back produces a rotation in spin space, ⎛ ˛ ˛⎞ cos i sin  ˛  ⎜ 2 2⎟ U1−bit (˛) = exp i x = ⎝ ˛ ˛⎠, 2 i sin cos 2 2 (9.64) where the angle ˛ = EJ t. One can then produce a superposition state with any chosen weights simply by choosing the proper time interval t [62]. On the other hand when the system is biased slightly to one side or the other, this produces a small shift in energies E0 and E1 with the result that these two states precess at slightly different rates which, acting for a time t, results in a rotation by an angle ˇ about the x-axis; i.e., we can mimic a field Bx . Hence a combination of the two parameters, VG and t, effectively give control of Bz and Bx . It turns out that unitary rotations by Bx and Bz are sufficient to perform all manipulations of a single qubit. Implementing a sequence of no more than three such elementary rotations one achieves any unitary transformation of a qubit’s state.4 The former example, with control of Bz only, yields an approximate spin flip for the situation in which the idle point is far from degeneracy and EC  EJ . But a spin flip in the logical basis can also be performed exactly. One switches from the idle point (corresponding to Q = . . . , −2, 0, 2, . . ., see Fig. 9.6) to the point where the effective magnetic field is orthogonal to the idle one (i.e., Q = . . . , −1, 1, . . .),  = idle + /2. This changes the Hamiltonian from Hˆ = −E (idle ) ˆz /2 to Hˆ = −E (idle + /2) ˆx /2. This is achieved by increasing NG by EJ /(4EC sin 2idle ). When idle  1 as above,the operating point lies near the degeneracy point,  = /2. Another way of manipulating the qubit is to use resonant pulses, i.e., r.f. pulses with frequency close to the qubit level spacing. We do not describe this technique here as it is well known from NMR methods. Up to this point we have been concerned with the time evolution during elementary rotations involv-

This is the analogue of rigid body rotations where three successive rotations parameterized by the three Euler angles, but involving only two axes, allow an arbitrary orientation of the body.

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ing a single qubit. In practice, however, the quantum state must be controlled for some time, for instance, while other qubits are manipulated. Even when in an idle state,  = idle ,the energies of the two eigenstates differ and hence their phases evolve relative to each other, which leads to coherent oscillations, typical of a quantum system in a superposition of eigenstates. One must therefore keep track of this time dependence with high precision, starting from the time t0 when the manipulations are begun. In principle the independent phase factors can be removed from the eigenstates if all the calculations are performed in the interaction representation, with the zero-order Hamiltonian being the one at the idle point. However, one must then incorporate an additional time dependence in the Hamiltonian during operations, which is introduced by the transformation to the interaction representation. The choice of the qubit logical basis is by no means unique. As follows from the preceding discussion, one can perform x and z rotations in the charge basis, |↑ and |↓, which allows any unitary operation. On the other hand, since one can perform any unitary transformation, other logical basis can be used as well. The Hamiltonian at the idle point is diagonal in the eigenbasis Eq. (9.23a,b) while the controllable part of the Hamiltonian, the charging energy, favors the charge basis, Eq. (9.18). The preparation procedure (thermal relaxation at the idle point) is more easily described in the eigenbasis, while coupling to the meter is diagonal in the charge basis. So the choice of the logical states remains a matter of convention. Operations Combining SQUID Flux and Gate Voltage The above discussion has shown that logic operations with a charge qubit can be performed using the gate voltage alone. However it is clearly better to have a second parameter available. The Josephson coupling energy is such a parameter since it can be tuned using the hybrid design; here two junctions in a loop configuration replace the single Josephson junction [61] as shown earlier schematically in Fig. 9.12, and experimental realized with the devices shown in Figs. 9.13 and 9.14. This so called “dc

SQUID” can be biased by an external flux ¥x , which is coupled into the system through an inductive loop. If the self-inductance of the SQUID loop is low, the potential energy term in the Hamiltonian takes the form [see Eq. (9.14)]   U = 2EJ0 cos 'x cos ' , (9.65) where 'x = ¥x /¥0 and ¥0 = hc/2e denotes the flux quantum. Here we assume that the two junctions are identical with the same EJ0 . The effective junction capacitance is the sum of individual capacitances of two junctions; in symmetric cases CJ = 2CJ0 . When the parameters are chosen such that only two charge states play a role, one again arrives at the Hamiltonian (9.14) but where now the effective Josephson coupling, Bx = EJ (¥x ) = 2EJ0 cos 'x ,

(9.66)

is tunable.Varying the external flux ¥x by amounts of order ¥0 changes the coupling between 2EJ0 and zero. The SQUID-controlled qubit is thus described by the ideal single-bit Hamiltonian, with the field components Bz (t) = ıEch [Vg (t)] and Bx (t) = EJ [¥x (t)] controlled independently by the gate voltage and the flux. If we fix conditions such that VG = Vdeg and 0 ¥x = ¥0 /2, the Hamiltonian is zero, Hqb = 0, and the state does not evolve in time. Hence there is no need to control the total time t0 from the beginning of the manipulations. If we change the voltage or the current, the modified Hamiltonian generates rotations around the z or x axis respectively, which generate the elementary one-bit operations. Typical time intervals for single-qubit logic gates are determined by the corresponding energy scales, which are of order /EJ , /ıEch for x and z rotations, respectively. If only one of the fields, Bz (t) or Bx (t), is applied, the time integrals of their envelopes determine the result of the operation; these envelopes can then be chosen to optimize the speed and simplicity of the manipulations. The introduction of the SQUID not only permits simpler and more accurate single-bit manipulations, but also allows control of the two-bit couplings. The time evolution of the hybrid qubit is controlled by i) applying microwave pulses (t) with

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Fig. 9.26. Relaxation time at the degeneracy point

frequency  = 01 to the gate,and ii) by applying biascurrent pulses with a small amplitude (compared to the critical current).Resonant modulation of the gate voltage induces Rabi precession (see Sect. 9.3.1) between the two-qubit states; the bias-current can be used to shift the qubit transition frequency. Starting from |0, any superposition |¦  = a |0 + b |1 can be prepared. The relaxation following a  pulse for the device shown in Fig. 9.14 is shown in Fig. 9.26. For readout,one may implement a strategy similar to the Stern and Gerlach experiment: the information about the quantum state is transferred onto another variable, the phase difference ' (' and 'x are the phase across the large junction and across the series combination of the two junctions in the loop correspondingly, see Fig. 9.14) and the two states are discriminated via the supercurrent flowing in the loop. For this purpose,a trapezoidal readout NG pulse Ib (t) with a peak value slightly below the critical current, I0 = EJ0 / 9 0,: is applied to the circuit. When starting from 'x = 0 ('x is the phase difference across the large junction as shown in Fig. 9.14) the phases ' and 'x grow during the current pulse, and consequently a state-dependent supercurrent develops in the loop. This current adds to the bias-current in the large junction, and by precisely adjusting the amplitude and duration of the Ib (t) pulse, the large junction switches during the pulse to a finite voltage state with a large probability p1 for state |1 and with a small probability p0 for state |0. The effi-

ciency of this projective measurement is expected to exceed = p1 − p0 for optimum readout conditions. The readout part of the circuit has been tested [75] by measuring the switching probability p as a function of the pulse height Ip for a current pulse duration of r = 100 ns, at thermal equilibrium. The discrimination efficiency was then estimated using the calculated difference |0 and |1 states. The value obtained,  = 0.6, was lower than the expected one, possibly due to noise coming from the large bandwidth current-biasing line. Hybrid Qubit Manipulations The switching of the large junction to the voltage state was detected by measuring the voltage across it with an amplifier at room temperature. By repeating the experiment (∼ 104 times), the switching probability can be measured, which gives the weights of the two states. One performs spectroscopic measurements by applying a weak continuous microwave radiation to the gate, which is suppressed just prior to the readout current pulse. The variations of the switching probability with the microwave frequency display a resonance; Figure 9.27 shows the center frequency as a function of the control parameters. Figure 9.28 shows the measured transition frequency (symbols) as a function of reduced gate charge NG for a reduced flux ' = 0 (right panel) and as a function of ' at NG = 1/2 (left panel), at 15 mK. The

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Fig. 9.27. Calculated transition frequency 01 as a function of the control parameters ng and ' = ı/2pi

experimental measurement results of the narrowest lineshape, obtained at the saddle point, are shown in Fig. 9.29 along with a Lorentzian fit. The linewidth in vicinity of the optimal point is presented in Fig. 9.30. The optimal working point, the linewidth was found to be minimal with a 0.8 MHz FWHM, corresponding to a Q of 2 × 104 . These spectroscopic data allow one to determine the relevant circuit parameters.

Preparation of a Coherent Superposition

Fig. 9.28. Measured center transition frequency (symbols) as a function of reduced gate charge ng for reduced flux ' = 0 (right panel) and as a function of ' at ng = 1/2 (left panel), at 15 mK. Spectroscopy is performed by measuring the switching probability p (105 events) when a continuous microwave radiation of variable frequency is applied to the gate before readout. The continuous line shows best fits used to determine circuit parameters. Inset: narrowest lineshape, obtained at the saddle point (Lorentzian fit with a FWHM of 0.8 MHz)

Close to the optimal point, controlled rotations of the spin can be performed using large amplitude microwave pulses at the transition frequency. The switching probability of a sinusoidal pulse as a function of pulse duration is shown in Fig. 9.31, and is in agreement with the expected behavior for Rabi oscillations (see Sect. 9.3.1). The linear dependence of the Rabi frequency on the microwave amplitude was used to calibrate the rotation angle. Note that the amplitude of the Rabi oscillations is smaller than the estimated efficiency. This may arise from relaxation of the level population during the measurement it-

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Fig. 9.29. Narrowest lineshape, obtained at the saddle point (Lorentzian fit with a FWHM of 0.8 MHz)

Fig. 9.30. Linewidth close to the optimal point

self. Better control is needed for achieving a perfect “single-shot” readout. Measurement of the Coherence Time The coherence time during free evolution can be obtained using a two-pulse sequence with a delay during which the qubit evolves freely; the sequence is shown in Fig. 9.32. After the first pulse, the spin direction rotates to a new position, which corresponds to a mixed state. The final state depends on the phase of the second pulse. If the pulse is inverted (shifted

by  as shown in the upper panel), the spin returns to the initial position. However if the pulse is the same, the spin flips. Figure 9.33 shows the switching probability for a given detuning of the microwave frequency  after a two-pulse sequence as a function of the pulse delay t, at 15 mK. Note it displays decaying oscillations of frequency  , which correspond to “beating” between the spin precession and the external microwave field; this is equivalent to the Ramsey fringe experiment.The envelope of the oscillations yields the coherence time, which for the case shown corresponds to 8000 free precession cycles.

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Fig. 9.31. Left: Rabi oscillations of the switching probability p measured just after a resonant microwave pulse of duration t. Data taken at 15 mK for nominal amplitudes (dots, from top to bottom). Solid lines are sinusoidal fits used to determine the Rabi frequency. Right: test of the linear dependence of the Rabi frequency with the microwave amplitude

Fig. 9.32. Measuring the coherence lifetime

The measured coherence time is shorter than the relaxation time T1 = 1.8 ‹s deduced from the exponential decay of the switching probability when the readout is delayed after a single pulse. Decoherence of the quantum state is thus dominated by dephasing and not by relaxation from the excited state to the ground state.

Full Qubit State Manipulation When working at the optimal point, the two-pulse sequence can be used to probe the phase shift that is induced between both qubit states by a small adiabatic change of the bias current applied during a pulse. The switching probability vs rotation angle ˛ induced by a 100 ns ramped bias supercurrent pulse

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Adding and Removing Decoherence

Fig. 9.33. Dots: switching probability after a two-pulse sequence as a function of the pulse delay t, at 15 mK. The total acquisition time was 5 s. Continuous line: fit by an exponentially damped sinusoid with frequency 20.6 MHz, equal to the detuning frequency  , and decay time constant T¥ = 0.5 ‹s

Fig. 9.34. Switching probability following a two-pulse sequence as a function of the rotation angle ˛ induced by a bias-current pulse with variable amplitude. The 100 ns bias current pulse is applied between the two microwave pulses

applied between the two microwave pulses is shown in Fig. 9.34. Combining bias-current pulses and microwave pulses, which correspond to two independent rotation angles, one can produce any unitary evolution of the qubit.

When departing from the optimal point, the coherence time T' decreases rapidly. By inserting a  pulse between the two /2 pulses, NMR-like echo experiments can be performed to probe the spectral density of the noise sources responsible for dephasing (so-called homogeneous line broadening). In this sequence, the random phases accumulated during the two free evolution periods (i.e., between the  pulses), with durations t1 and t2 , will compensate when t1 = t2 ; provided the transition frequency does not change on this time-scale. Echoes can be observed at times for which Ramsey fringes (free induction decay) are completely washed out. This indicates that in this situation decoherence is essentially due to charge fluctuations at frequencies lower than 1 MHz. In the opposite limit, no echo was seen in experiments attempting to probe the phase noise, suggesting the phase noise extends over a wide range of frequencies. Controlling the quantum evolution of an individual qubit is a necessary first step towards functional quantum circuits. It is however still necessary to improve the coherence time by a factor ≈ 100,to achieve high fidelity readout, and to implement controlled qubit interactions. Coupling multiple quantronium circuits can in principle be achieved using on-chip capacitors and/or ultra small junctions. Coupling schemes have been proposed for other Josephson qubits as well [56, 57, 76]. Thus in principle, quantum gates could be implemented and probed by measuring quantum correlations induced in multi-qubit entangled states.It seems that no fundamental obstacle blocks the realization of an elementary quantum processor based on Josephson junctions. Some experiments on a two-qubit gate will be discussed next in Sect. 9.4.

9.4

Quantum Oscillations in Two Coupled Charge Qubits

9.4.1

The circuit

In this section we discuss experiments in which a coherent coupling between two different qubits has

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9.4.2

The Two-Qubit Hamiltonian

The Hamiltonian of the system in the two-qubit charge basis |00, |10, |01 and |11 is given by: ⎞ ⎛ E00 − 12 EJ1 − 12 EJ2 0 ⎟ ⎜ 1 ⎜ − EJ1 E10 0 − 12 EJ2 ⎟ ⎟ ⎜ 2 H=⎜ 1 ⎟ ; (9.67) 1 ⎟ ⎜ − EJ2 0 E − E 01 2 J1 ⎠ ⎝ 2 0 − 12 EJ2 − 12 EJ1 E11 here the total electrostatic energy of the system is given by En1 − n2 = Ec1 (NG1 − N1 )2 + Ec2 (NG2 − N2 )2 + Em (NG1 − N1 )(NG2 − N2 ) ,

Fig. 9.35. Two-qubit gate with capacitive coupling

been observed [77]. A Cooper-pair box implementation was used for the individual qubits in which two charge states, say |0 and |1, differing by one Cooper pair are coherently mixed by Josephson coupling [78–80]. As discussed in Sect. 9.3.4, quantum state manipulation in such a system can be done using a non-adiabatic pulse technique and read-out can be performed by a properly biased probe electrode. However the experiments have been carried one step further toward implementing quantum logic gates by integrating two charge qubits which are coupled electrostatically by an on-chip capacitor, Cc ,as shown schematically in Fig. 9.35. The qubits have a SQUID geometry to allow control of the Josephson coupling to their reservoir. Both qubits have a common pulsed gate (not shown in the figure) but separate dc gates, probes and reservoirs.The pulsed gate has nominally equal coupling to each box (see also Table 9.2).

(9.68)

(N1 , N2 ) = 0, 1 is the number of excess Cooper pairs in the first and the second box, EJ1,2 are the Josephson coupling energies of the two boxes and 2 the reservoir, Ec1,2 = 4e2 C2,1/2(C1 C2 − Cm ) are the effective Cooper-pair charging energies, C1,2 are the sum of all capacitances connected to the corresponding island 1(2) (including the mutual coupling capacitance Cm , i.e., C1,2 = CL,R + CG1,2 + Cm ), and NG1,2 = (CG1,2 VG1,2 + Cp Vp )/2e are the normalized charges induced on the corresponding qubit by the d.c. and pulse gate electrodes. The detailed derivation of the Coulomb part of the Hamiltonian (9.67) is given in [79]. The coupling energy Em depends not only on Cm , but also on the total capacitance of the 2 boxes: Em = 4e2 Cm /(C1 C2 − Cm ); for spin effects see Table 9.2. A schematic of the classical electrostatic components is shown in Fig. 9.36.Applying the gate voltages VG1 and VG2 controls diagonal elements of the Hamiltonian given in Eq.(9.70).The circuit was designed to

Table 9.2. Quantum versus classical spin–spin correlations for two qubits Qubit 1 Qubit 2 x y z

x

y

z

−1 0.5 (< 0.33) 0.5 (< 0.33) 0.5 (< 0.33) −1 0.5 (< 0.33) 0.5 (< 0.33) 0.5 (< 0.33) −1

Fig. 9.36. The electric circuit of the classic double-dot system

9

Principles of Josephson-Junction-Based Quantum Computation

have EJ1,2 , Em < Ec1,2; this ensures a coherent superposition of the four charge states, |00, |10, |01, and |11 in the vicinity of NG1 = NG2 = 0.5 (see also Table 9.2), while keeping other charge states separated by large energy gaps, thereby justifying a four-level description of the system. In the absence of Josephson coupling,the ground-state charging diagram (N1 , N2 ) shown in Fig. 9.37a consists of hexagonal cells5 whose boundaries separate two neighboring charge states with degenerate electrostatic energies.According to [81] the precise shape of the cells depends on ratios of the inter-dot capacitances. The points R and L in Fig. 9.37a correspond to a degeneracy between the states |00, |10 and the states |00, |01 differing by one Cooper pair in the first and the second Cooper-pair box, respectively. If we choose the dc gate charges ng1 and ng2 far from the boundaries but within the (0,0) cell, then, because of the large electrostatic energies, we can assume that the system remains in the state |00. Since a pulse applied to the gate couples equally to both qubits, the state of the system moves along a line tilted by 45◦ , as indicated by arrows in Fig. 9.37a. The charging energy dominates except for the small Josephson coupling which manifests itself on the boundaries where charge states become superposed. If the system is driven non-adiabatically to the points R or L, it behaves like a single qubit and oscillates between the degenerate states with a frequency !12 = EJ12 /. By applying various pulses and measuring the oscillations of the probe currents I1 and I2 , the Josephson energies of each qubit can be determined. The accuracy of the measured EJ1,2 is very high, since the electrostatic coupling through Cm has a minimal effect on !1,2 near the points R and L. At the ‘co-resonance’ point X (corresponding to NG1 = NG2 = 0.5), the system is doubly degenerate, i.e., E00 = E11 , E10 = E01 , and the dynamics of the quantum evolution become more complex and reflect the coupling between the qubits. The crosssection of the energy bands through the point X is shown in Fig. 9.37b. Exactly at the co-resonance, all four-charge states are mixed and the state of the system can be expressed in general as 5

347

Fig. 9.37. Pulse operation of quantum dot device: (a) tilting of the two quantum dot states, (b) energy bands

 :  (t) = c1 |00 + c2 |10 + c3 |01 + c4 |11 , (9.69) where |ci | (i = 1, 2, 3, 4) are the time-dependent probability  amplitudes obeying a normalization condition 4i=1 |ci |2 = 1. Using the Hamiltonian (9.67) and initial conditions, one can calculate the probabilities |ci |2 of each charge state. However, in the read-out scheme, one measures a probe current I12 proportional to the probability p12 (1) of each qubit having a Cooper pair on it, regardless of the state of the other qubit; that is, I1 ∝ p1 (1) ≡ |c2 |2 + |c4 |2 and I2 ∝ p2(1) ≡ |c3|2 + |c4 |2 . Assuming that the initial state at t = 0 is |00, one obtains the time evolution of these probabilities for an ideal rectangular pulse shape of length t as p1,2 (1) =

  1 2 − 1 − 1,2 cos {(§ + ") t} 4 

  − 1 + 1,2 cos {(§ − ") t} ,

(9.70)

In the spirit of our earlier analogy between the charging energy profiles and nearly free electron motion in one dimension, for the coupled two qubit system the cell boundaries make up the two dimensional Brillouoin zone.

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2 2 2 where 1,2 = (EJ2,1 − EJ1,2 + Em /4)/(42§2 ), § = ((EJ1 + EJ2 )2 + (Em /2)2)1/2/2, and " = ((EJ1 − EJ2 )2 + (Em /2)2)1/2/2.Unlike the single qubit case,there are two frequencies present, § + " and § − ", both dependent on EJ1 , EJ2 and Em . One can identify these frequencies with the energy gaps in Fig. 9.37b. Note that in the uncoupled situation,which corresponds to Em = 0, each qubit oscillates with its own frequency !12 . However, the above consideration is valid only in the ideal case when the pulse has zero rise/fall time, and the time evolution occurs exactly at the co-resonance point.

9.4.3

Two Qubit Experiments

The idea of the experiment is shown schematically in Fig. 9.37b. Starting from the |00 state (shown as a black dot), a gate pulse (represented by the solid arrow) brings the system to the co-resonance point. At this point a microwave pulse is applied for a time t, which produces a superposed state of the form (9.69). as indicated by gray circles. After the pulse terminates,the system evolves in the superposed state until it decays (indicated by the grey arrows) to the ground state by emitting quasiparticles into the probe junctions biased at Vb1,2 ≈ 600 ‹eV. To accumulate a signal, a series of 3 × 105 pulses was applied to the

Fig. 9.38. Quantum oscillations in two coupled qubits. The upper two traces depict single qubit like oscillations, corresponding to the points R and L in Fig. 9.37a; the lower figures involve states near the co-resonance point X (in the “corner”), where strong four-state superpositions occur

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Principles of Josephson-Junction-Based Quantum Computation

pulsed gate. The repetition time of 64 ns between the pulses is long enough (compared with the quasiparticle relaxation time of 10 ns) to let the system decay through a Josephson-quasiparticle, and give rise to a probe current proportional to p1,2. The estimated amplitude of the applied pulses is Vp ≈ 30 mV. The results obtained in this way are presented in Fig.9.38.First,by controlling NG1 or NG2 ,individually, the system can be brought to the points R or L; one then performs single qubit measurements by exciting autonomous oscillations in one of the qubits (upper two traces in Fig. 9.38).The spectra of the oscillations can be fitted to a cosine function with an exponential decay time of about 2.5 ns. The spectra of the oscillations (right panels of Fig. 9.38) obtained by Fourier transforming contain one pronounced component at 13.4 GHz for the first qubit and at 9.1 GHz for the second qubit. These values are identified with EJ1 and EJ2 . From the experiments (see, for example, [82]) one concludes that the values obtained are close to those expected for the device fabrication parameters (i.e., overlap area and oxidation conditions). By controlling both NG1 and NG2 , the system can be driven to the vicinity of the co-resonance point, X, and the induced quantum oscillations traced using the same technique. The oscillation patterns, shown in the lower two traces in Fig. 9.38 (with the Fourier transforms given to the right), are then more complex; two clear frequencies now appear in the spectrum. For the above parameters (EJ1 = 13.4 GHz and EJ2 = 9.1 GHz) measured in the single qubit experiments, and a value for Em = 15.7 GHz estimated from independent measurements of dc current– voltage/gate-voltage characteristics, the two peaks in the spectrum are close to the expected frequencies § + " and § − " predicted from Eq. (9.70); these frequencies are indicated by the arrows and dotted lines in lower right of Fig. 9.38. The decay time (0.6 ns) of the coupled oscillations near X is shorter than that observed for the independent oscillations near R or L, as is expected because an extra decoherence channel appears for each qubit after coupling it to its neighbor. The amplitudes of the spectral peaks do not agree that well with those predicted by Eq. (9.70). This is attributed to the non-ideal pulse shape (finite rise/fall time 35 ps), and the fact that a small

349

shift of NG1 and VG2 off the co-resonance drastically changes the oscillation pattern. Also, even far from the co-resonance, a small contribution to the initial state is still observed which arises from charge states other than |00 distorting the oscillations. Numerical simulations have been performed of the oscillation pattern, taking into account a realistic pulse shape and an initial condition corresponding to a non-pure |00 state, but assuming the system is exactly at co-resonance. The resulting fits are shown in Fig. 9.38 as solid lines. A slightly different value of Em = 14.5 GHz, close to the value estimated above from the d.c. measurements, gives better agreement with the experimental data. Finally, the dependence of the oscillation frequencies on EJ1 was controlled by a weak magnetic field (up to 20 G). The results are shown in Fig. 9.39. The plot contains the data from both qubits represented by open triangles (first qubit) and open circles (second qubit). Without coupling (Em = 0), the single peaks in each qubit would follow the dashed lines with an intersection at EJ1 = EJ2 . The coupling introduced modifies this dependence by creating a gap, and shifting the frequencies to higher and lower values; the spacing between the two branches is equal to Em /2h when EJ1 = EJ2 . A comparison of the ob-

Fig. 9.39. EJ1 -dependence of the spectrum components obtained by Fourier transform of the oscillations at the coresonance

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S. E. Shafranjuk and J. B. Ketterson

Fig. 9.40. Possible sources of decoherence

served dependence with the prediction of Eq. (9.68) (given by solid lines) gives excellent agreement. The observed quantum coherent dynamics of coupled qubits in the vicinity of the co-resonance, in particular, the double-frequency structure of the probability oscillations in both qubits, structure of the probability oscillations in both qubits, and frequency ’repulsion’ at EJ1 ≈ EJ2 (see Fig. 9.38b) indicates that the two qubits become entangled during the course of coupled oscillations, although a direct measurement of the degree of entanglement was not possible. Simple calculations [83] based on the standard expression for the entanglement of the pure states show that,with an ideal pulse shape and the |00 initial condition, the wave function given in Eq. (9.69) evolves through a maximally entangled state in the case of equal Josephson energies.The numerical simulations confirm that the amount of entanglement does not decrease significantly when realistic experimental conditions are taken into account.The relatively large observed oscillation amplitude (about 50% of the expected value) also suggests the existence of entangled states even in multi-pulse averaged experiments. For decoherence effects see Fig. 9.40.

9.5

SISIS Two-Qubit Gate with Intrinsic Coupling

In majority of realistic systems under consideration, there are usually several discrete energy levels. However a qubit utilizes only two of these levels and therefore care must be taken to isolate these levels from the remaining Hilbert space. We considered a multi-terminal double barrier SISIS junction (S and I denote a superconductor and an insulating barrier respectively) is suggested as a two-qubit gate with tunable intrinsic coupling. The presentation in this section is based on [84]. Two quantum wells are formed in vicinities of the left and right SIS subjunctions. Thisgives two individual qubits, which are intrinsically coupled via the middle S layer due to phase coherence. The inter-qubit coupling J is tuned by two bias supercurrents I1 and I2 across each of the SIS subjunctions independently. Additional coupling is accomplished by transport supercurrents Iltr along adjacent S layers. Using a microscopic model we compute major qubit characteristics and study sources of the intrinsic decoherence. One computes the entanglement of the two qubit states, leakage and fidelity characteristics versus J, and discuss the readout process. Each qubit is a two-state quantum

9

Principles of Josephson-Junction-Based Quantum Computation

system, which behaves like a spin-1/2 particle and can be entangled with other qubits. In recent experiments [85,86] a quantum-coherent dynamics of two Josephson qubits coupled through a capacitance was studied. A tunable interaction between different types of Josephson qubits was examined also theoretically (see [76,87] and the references therein). The qubit gate [88, 89] consisted of two current-biased Josephson junctions coupled via a capacitance,which allowed performing of arbitrary two-qubit quantum logic operations (see Fig. 9.40). The method of capacitive coupling has, however, certain disadvantages. In particular, additional circuit elements and wiring serve as potential sources of disturbance in the system: the electric charges, accumulated on the capacitance, disturb the quantum states and cause errors during quantum logic operations. An alternative way to introduce a tunable coupling between different quantum subsystems is exploiting of a bias-tuned intrinsic coupling taking place in multilayered multiterminal structures [90–92]. Though general properties of the intrinsic entanglement in a solid state device were considered earlier (see, e.g., [93] and references therein), its implementation to the two-qubit gates is not well known yet. We payed our attention to a simple two-qubit gate based upon intrinsic properties of a double-barrier multi-terminal SISIS junction (S is the superconductor, I is the insulating barrier) with a proximity-type coupling between the left and right SIS subjunctions (see Fig.9.4) across their common S layer.Elementary quantum logic operations on qubits [75, 94–98] are associated with controlled manipulations involving two states 0 and 1 of the same qubit, the superposition of which forms a mixed state | ¦ . Quantum : computing presumes :also superposition | ¥1,2 of the states | ˛1  and | ˇ2 of two different qubits, 1 and 2. An ideal two-qubit Hamiltonian in spin-1/2 notations takes the form [3] ˆq = H



[" (k) ˆ zk +  (k) ˆ xk + ˛ˆ k ]

k=1, 2

+

 nm

    Jnm iˆ n1 ⊗ iˆm2 ,

(9.71)

351

where " (k) and  (k) are the energy level spacing and the inter-level tunneling matrix element in the k-th qubit, ˆ n1 and ˆ m2 , are Pauli matrices associated with the first and second qubits; Jnm is the inter-qubit coupling energy, {n, m} = {x, y, z}. Each of the qubits is independently controlled by fields ˛ˆ k = ˛l exp(ˆ yk k (t)), where ˛k and k are control field amplitudes and phases, and k = 1, 2. For controlled manipulations of the qubit the coefficients of the Hamiltonian are modified by adiabatic change of the Josephson supercurrents. The adiabaticity is required to eliminate transitions between different two-qubit gate states. The parameters of Eq. (9.71) depend also on particular design of the qubit gate. Each of the qubits in the two-qubit gate described by Eq. (9.71) should behave individually. One assumes that r.f. control pulses address both qubits with no disturbance of other circuit elements. The two qubits labeled as 1 and 2 are built using the tilted washboard Josephson energy potentials U1('1 ) and U2 ('2 ), where '1 and '2 are the phase differences across the left (1) and right (2) SIS sub-junctions. The sets of quantized energy levels QL positioned at "n(l) (l is the qubit index and n is the level quantum number) are formed in the washboard quantum wells U1 and U2. The tilting of U1 and U2 is controlled by the bias supercurrents I1 and I2 as shown in the Fig. 9.41. We will see that directions and magnitudes of I1 and I2 not only affect the inter-level spacing " l = "1l − "0l (l = 1, 2) in both the qubits but actually determine the strength of the inter-qubit interaction Jnm . In this way one accomplishes arbitrary singleand two-qubit quantum logic operations with applying appropriate r.f. and d.c. bias currents to the SIS sub-junctions. 9.5.1

Proximity Coupling in a Multilayered Junction

The coupling between the left and right SIS subjunctions of the SISIS junction (see Figs. 9.41, 9.42) is conveniently described in terms of Andreev reflection. Here we are interested in a non-local process when an incoming electron and the reflected hole belong to different electrodes spatially separated by a distance ∼  ,where  being the superconducting coherence length (see, i.e., [91] and references therein).

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Fig. 9.41. (a) The quantized levels (QL) formed inside the quantum wells U1 and U2 controlled by bias supercurrent I1 and I2 across the SIS subjunctions of a multiterminal SISIS junction. The phase differences '1 and '2 if transport supercurrents Il flow along the S electrodes. (b) Schematic illustration of the current biasing of the SISIS junction

a singlet state of a Cooper pair spread between two different leads . The NA was studied experimentally in [99] using a double-barrier three-terminal NISIN junction. Similar double-barrier three-terminal SINIS devices were recently examined also in [92]. The quantized oscillation spectrum of a highly transparent SISIS junction was studied in [91]. The coherent interaction through the double-barrier junction is noticeable even if the barrier transparency is relatively low (i.e., D ≈ 10−5, as in the NISIN structures used in [99]). Then, quantum coherence (QC) established across the whole SISIS junction yields a finite coupling between the right and the left SIS subjunctions via their common middle S layer. The QC coupling energy J12 between the two subjunctions 1 and 2 (see Fig. 9.42) is defined as = > J12 ('2 ) = max W − W (0) , (9.72) '1

where   W ' =

'

ISSISIS ('  )d' 

(9.73)

is the energy of the whole SISIS junction. The energy of two single SIS junctions connected in series is W

Fig. 9.42. (a) The two subjunctions. (b) Two SIS junctions coupled capacitively. The dashed line with a cross indicates that the coupling is not phase coherent (see Fig. 9.40). (c) The current-phase relationship in the two-barrier multiterminal junction

Though the two electrodes are separated, they couple via the superfluid condensate. Following [91,99] we quote this process as a non-local Andreev reflection (NA). The NA process is actually equivalent to injecting of two spin-entangled electrons, which form

(0)

  ' =2

'

ISSISIS ('  )d'  .

(9.74)

The two energies W and W (0) are not equal to each other because the current across the middle electrode of SISIS junction is evidently a phase-coherent supercurrent, while the electric current between two single SIS junctions connected in a sequence [as shown by the dash line with a cross in Fig. 9.41c] is not phase-coherent. This difference between W and W (0) coming from the phase coherence in SISIS serves as a source of our inter-subjunction coupling. The QC coupling between the left and right SIS subjunctions may be tuned by attaching a third terminal to their common S layer.Then the phase differences '1 and '2 across the left and right barriers of the SISIS junction are controlled independently from each other and the quantum states of each subjunction are addressed individually. Though a microscopic calculation of J may be performed in various ways, here we implement a quasiclassical Green

9

Principles of Josephson-Junction-Based Quantum Computation

function method with special boundary conditions at the interface barriers [90,100–102]. The approach allows a direct microscopic delineation of the interjunction coupling and is applicable to junctions with arbitrary interface transparency and pureness of the electrodes. The QC coupling is tuned by the bias supercurrents I1 and I2 . The bias is applied to each of the SIS subjunctions individually, as shown in Fig. 9.41. If I1 = I2 , then the corresponding phase differences '1 and '2 are not equal to each other (i.e., '1 = '2 ), while the net supercurrent Is '1 , '2 , x inside the middle S layer of the SISIS junction depends also on the coordinate x in the direction perpendicular to interfaces. The approach [90, 102–104] gives a tractable microscopic description of the bias-controlled QC effect in multilayered superconducting junctions. The basic elementary process responsible for the coherent coupling in the system is the Andreev reflection. In this approach, a moving hole creates a new electron with a reversed trajectory of motion. Most important is that the multiple processes of the electronhole conversion keep the packet on a classical trajectory in the r-p space. The particles may switch to another classical trajectory at the knots, where the scattering occurs with a certain probability described by special boundary conditions. The quasiclassical approximation had proven to be effective when studying the phase coherent transport in multilayered superconducting structures [75,95,99]. The   non-uniform supercurrent Is '1 , '2 , x is obtained from the one-point quasiclassical Green function gˆ as   epF Is '1 , '2 , x = 0 2m 1   (9.75) = > × d" dTr Imgˆ (", , '1 , '2 , x) ,

where spinors

±

353

(x) are

¯ = −i

T

  ˆ y = v −u ,

(9.77)

and T means transposing. The electron and hole envelope wavefunctions u and v are defined on classical trajectories. The trajectories are coupled to each other at the knots,associated with the interface barriers. The non-uniform supercurrent Eq. (9.75) is used to control the coupling between the two SIS subjunctions as will be discussed in the next section. 9.5.2

Inter-Qubit Coupling Energy

The inter-qubit coupling energy J12 is computed using Eqs. (9.75), (9.76); see Fig. 9.43. The microscopic equations (9.75), (9.76) describe the properties of junctions with arbitrary transparency.They allow accounting for the tunneling across the interface barriers I, for the elastic electron scattering on atomic impurities, and for the inelastic scattering on phonons in the junction’s electrodes. As a first step toward the computing J12 we solve the quasiclassical Andreev equation ˆ ¦ = E¦ H (9.78) for ¦+ (x) and ¦− (x). The equation is completed by special boundary conditions [96–99] at the interface barrier positions x = xL(R) , and at x = ±∞ for the geometry of Fig. 9.41. The solution serves as a plug into Eq. (9.76) for the quasiclassical retarded Green function gˆ (x) . Substituting gˆ (x) into  Eq. (9.75)  one computes the total supercurrent Is '1 , '2 , x . The

−1

where  = cos # , # being the electron incidence angle, pF is the Fermi momentum, 0 is the normal electron density of states at the Fermi level. The Green function gˆ in Eq. (9.75) is expressed as  −1 gˆ (x) = ¯− (x) + (x)   × + (x) ¯− (x) + − (x) ¯+ (x) (9.76)

Fig. 9.43. The QC coupling energy of the left SIS subjunction. The inset shows the maximum coupling strength Jmax versus the thickness of the middle layer for“clean”(curve 1) and “dirty” limits (curve 2)

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S. E. Shafranjuk and J. B. Ketterson

local current-phase relationships obtained for Is inside the middle layer of the SISIS junction  are shown in Fig. 9.42c where Is is denoted as IsL '1 for fixed   '2 = /4 at the left barrier x = xL , as Is(0)  '1 at the middle of the junction x = 0, and as IsR '1 at the right barrier x = xR.Here we assume that '1, 2 = '1, 2 . Then the local supercurrent Is (x) depends upon two phase differences '1 and '2 , therefore, one may define a critical supercurrent in respect just to one variable (e.g.,'1 ),and consider its dependence versus another variable (i.e., '2 ). This behavior is interpreted as the QC coupling between the two SIS subjunctions of a symmetric SISIS junction characterized by an energy J('1 , '2 ). The coupling energy of the left SIS subjunction, which depends also on the phase difference across the right SIS subjunction '2 is obtained from Eq. (9.72). In Eq. (9.72)   ¥0   1   1 W '1 , '2 = Ic − W1 '1 , '2 − W2 '1 , '2 , 2 2 2 (9.79) is the SISIS “washboard” Josephson energy. In Eq. (9.79), Ic is the absolute (i.e., in respect to both '1 and '2 ) critical current of the SISIS junction while the Josephson energy of the coupled SIS subjunction 1(2) is defined as 

W1(2) '1 , '2



'1(2) ¥0 IS ('1 , '2 , xL )d'1(2) . = 2

(9.80)

−∞

Equation (9.72) completed by Eqs. (9.75), (9.76), (9.79), and (9.80) constitutes the QC coupling J12 ('2 ) of the left SIS subjunction to the right SIS subjunction versus '1 and '2 , tuned by corresponding bias supercurrents I1 and I2 (assuming here that '1 = '1 and '2 = '2 ). We emphasize that the coupling J12 comes entirely from the phase coherence between the two SIS subjunctions. The computed dependence J12 ('2 ) at x = xL is plotted in Fig. 9.43, from which one can see that the sign and magnitude of J12 ('2 ) is controlled by the phase difference '2 across the counterpart SIS subjunction. The QC coupling magnitude J(d) = max '2 {J12 ('2 )} (d is the middle layer thickness in units of the BCS coherence length) is plotted in inset to Fig. 9.43, which shows how J depends on the

thickness and purity of the middle layer characterized by the electron elastic mean free path li : if it is very thick (d  ), the mutual interaction of subjunctions J vanishes. When the electron motion inside the middle layer is ballistic (li ≥ ), the SIS subjunctions interact with each other though the middle layer is relatively thick (curve 1 in Fig. 9.43 for which li = 5). If, however, the middle  layer is impure, the coupling range shortens: ∗ ∼ = li , which is confirmed by our numerical calculations of J (see curve 2 in Fig. 9.43 computed for li = 0.3). This circumstance can be utilized to optimize the coupling in two-qubit gates. 9.5.3

Control of the Inter-qubit Coupling

In the setup shown in Fig. 9.41, the two SIS subjunctions are used as two coherently coupled qubits. : The two-qubit idle state | ¥1,2 is realized when the strength J12 of the inter-qubit coupling vanishes at some value of '2 . The manipulations with quantum states and the inter-qubit coupling are controlled by applying bias voltages and supercurrents between the S electrodes.An additional independent control is furnished when '1 = '1 and '2 = '2 (see Fig. 9.41) as will be discussed in next section. In this section we analyze basic two qubit gate parameters (i.e., the level splitting " and the inter-qubit coupling strength J) semi-qualitatively. We give a simple illustration to our description using approximate formulas. Properties of each individual SIS qubit either 1 or 2 are conveniently described as a motion of a particle with the mass C1(2) in the “tilted washboard” potential ¥0 I1(2) '1(2) . (9.81) 2 The motion inside the quantum well leads to quantized states as sketched in the upper part of Fig. 9.41. The Josephson energy W1(2) of an SIS subjunction 1(2) entering Eq. (9.81) is given by Eq. (9.80). The subjunction 1(2) is coupled to its counterpart subjunction 2(1). The quantum eigenstates and eigenvalues of the system in the potential U1(2) are obtained numerically for given geometry of the double barrier junction. The input parameters for numeric computations include the capacitances C1,2 of the left (right) U1(2) = W1(2) −

9

Principles of Josephson-Junction-Based Quantum Computation

SIS subjunctions, the S layer thickness ds , the barrier transparency D, the junction area A, the energy gap , the elastic electron-impurity scattering rate

i , and the control supercurrents I1,2. The inter-level spacing " (l) = "1(l) − "2(l) is tuned by altering Il . Typically, one sets the JE profiles to exploit just two levels in each quantum well, while the third level (used for the readout of the quantum state) is positioned just below the top of the well (see the upper part of Fig. 9.41), which is achieved in large area junctions when Ic ¥0/2  e2/2C and Ic − I1, 2  Ic . In this approximation, the SISIS junction behaves like a non-relativistic two-body system.Then the whole behavior of the SISIS gate is reduced to a trivial motion of the center of mass. The relative one-body motion is executed by a particle of mass C = C1 C2 /(C1 + C2 ) (where C1 and C2 are the capacitances of the left and right SIS subjunctions). The motion is described by a “coordinate”  = '1 − '2 about a fixed center under the action of an “elastic force” F = −@U ()/@. Here we are interested in a relative motion of the reduced “mass” C in the two-well model potential U () shown in Fig. 9.44. Simple analytical formulas are obtained when the two-well U () is further approximated by a function pieced together from two qubic parabolas. Each of the qubic parabolas has a quadratic curvature at the bottom,which gives a classical oscillation frequency    Il 1/4 1/4 2Ic !l = 2 1− (9.82) ¥0 C Ic controlled by the supercurrent Il . The two characteristic barriers in the model potential U () have the height   √ Ic ¥ 0 Il 3/2 Ul = 2 2 1− , (9.83) 3 Ic where l is related to the left (l = 1) and right (l = 2) wells. The two wells are separated by a hump, which in a symmetric case has the height U h ∼ = C!p2 02/2, where 0 is the distance between the two wells (see Fig. 9.44). Although the applicability of such an approximation is limited, it serves as a good illustration when modeling qubit switches. Generally speaking,

355

Fig. 9.44. A two-qubit gate potential with two wells separated by a hump, which is controlled by the bias supercurrents I1 and I2 . The quantized levels "n (l) are formed in each of the wells, while the interlevel splitting "n is controlled by the hump height

the energy spacing " (l) = "1(l) − "0(l) and the tunneling matrix element , which enter Hamiltonian (9.71), depend on the reduced phase difference  in quite a complicated way.A finite inter-qubit coupling J = 0 splits each level additionally, i.e., "n(l) → "n(l)± (n = 0 . . . 3 and l = 1, 2 ), so the relevant splitting magnitude is ı"n(l) = "n(l)+ − "n(l)− . In the limit of a weak inter-qubit coupling,for a symmetric two-qubit gate,within our two-body motion model one approximately finds [101,102]:   E0 = P− + " +  /3 , √ (9.84)   E1, 2 = E± = −P− /2 + " +  /3 ∓ i 3P+ /2 and ı"n(2) = 2!p ( /)1/2 exp (− ) ,

(9.85)

where  = C!p 02 /, and the quantized level energies of a non-interacting qubit are 

 "1 = "0 + !p 1 − (5/36) !p /U (9.86) and 

 "2 = "1 + !p 1 − (11/36) !p /U .

(9.87)

In the above formulas we omitted the qubit index l for brevity. 9.5.4

Additional Control by Transport Supercurrents : The two-qubit quantum state | ¥1,2 is controlled by two bias supercurrents I1 and I2 as shown in Fig. 9.41.

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By changing the magnitude and direction of both I1 and I2 one shifts the quantized level positions "n(l) and their splitting ı"n(l) . However the two parameters I1 and I2 are generally not enough for a full control of the two-qubit gate. In addition to the manipulating with each individual qubit one must tune the inter-qubit coupling as well. This requires three independent control parameters at least.An additional independent control over the two-qubit gate is accomplished with applying transport supercurrents Iktr (k = 1 . . . 3) along the S electrodes as shown in Fig. 9.45a (see also Fig. 9.41). If the magnitudes of ıI1tr = I1tr − I2tr and ıI2tr = I2tr − I3tr are finite, the “washboard” Josephson energy U is modified, which : in turn changes the two-qubit state | ¥1,2 . When the SISIS junction in the lateral z-direction is sufficiently

long, i.e., L ≥ J (where J = c 2 / 8e(2L + dB )jc is the Josephson penetration depth, L is the London penetration depth, dB is the thickness of the insulating barrier, jc is the Josephson critical supercurrent density), the distribution of Josephson supercurrent js (z) inside each of the SIS subjunctions becomes non-uniform [103] and depends also on the lateral coordinate z. Such an inhomogeneous distribution of the supercurrent density js across each of the SIS subjunctions causes a finite phase change ı'l along

the z-direction providing that ı'1 = '1 − '1 = 0 and ı'2 = '2 − '2 = 0 (see also Fig. 9.41). This situation resembles the penetration of magnetic flux into a long Josephson junction [103]. In our case ıIltr plays a role similar to the y-component of the external d.c. magnetic field Hy , which was the case in [94,105].Accordingto [102,106], the js (z) profile depends on the ratio L/J : the supercurrent density is highly inhomogeneous for large L/J when a magnetic flux enters the subjunction. An actual js (z) dependence versus ıIltr is obtained from the sineGordon equation: 1 @2 'l = sin 'l , @z2 J

(9.88)

completed by the boundary conditions     @'l /@z |z=0 = 0 and @'l /@z |z=L = ıIltr /ens = , where ns is the superfluid charge carrier concentration [94]. In the BCS approximation one gets ns = n2 T

  k

 2 T 2 (2k + 1)2 + 2

−3/2

,

(9.89)

n is the normal charge carrier concentration, T is the temperature,  is the energy gap. The solution 'l (z) of Eq. (9.88) is given in elliptic functions [90, 107]. In Fig. 9.45b we show a typ-

Fig. 9.45. (a) An additional control of the two-qubit states with transport supercurrents I flowing in the adjacent S layers. (b) A nonuniform distribution of the supercurrent js (z) versus the lateral coordinate z (in units of 10J /L) along adjacent S layers. (c) The Josephson energy potential

9

Principles of Josephson-Junction-Based Quantum Computation

ical distribution of the supercurrent js (z) /jc along the left subjunction with L/J = 10 and ıI1tr = 0. In Fig. 9.45c we plot the Josephson energy profile U1 ('1 , ı'1) in the left quantum well at fixed I2 for a long junction with L/J = 10.One can see that the Ul profile depends on ı'l (which actually is ∝ ıIltr ).The tr transport supercurrents I1,2 renormalize the height of the characteristic barriers U1,2 () which acquire the dependence on  = ıIltr /ens . From Eq. (9.88) at z  J and   Ic /ens one finds     U1, 2 ,  = 0   , (9.90) U1, 2 ,  = Z ,    where U1, 2 ,  = 0 = U1, 2 () and the renormalizing function is 1   =1+ Z , 



  z2 1 −  2  1+ , +   2 2J

(9.91)

and we denoted  = U () /U0 , U0 = U () |=0 ,  = @ /@. For a non-interacting junction  = 1 − cos , while for the subjunctions of the SISIS junction the expression for  is more complicated and is obtained numerically for given S layer thickness dS , barrier transparency D,junction area A,energy gap ,elastic electron-impurity scattering time i , and control sutr percurrents I1,2 .When finite transport supercurrents tr I1, 2 = 0 are applied, the levels "1,2 are shifted versus . The characteristic barriers U1,2 [see Eq. (9.90)] entering  Eqs.  (9.86), (9.87) for "1,2 are renormalized by Z ,  as follows from Eqs. (9.89), (9.91). In this way,Eqs.(9.86),(9.87) in conjunction with Eqs.(9.89), (9.91) constitute the energy level dispersion versus tr the control transport supercurrents I1,2 . Our solu: tion demonstrates that the two-qubit states | ¥1,2 are tr readily controlled by the transport supercurrents I1,2 in addition to the bias supercurrents I1,2 . 9.5.5

Leakage and Fidelity of the Two-Qubit Gate

The intrinsic inter-qubit coupling in the SISIS gate is executed via weak Josephson supercurrents, which switching frequency is relatively low (! ∼ 106–109 s−1 ). The supercurrents produce no noise in the junction [103], thus the coupling is “quiet”. Since there is

357

no ac electric field in the SISIS junction, the disturbance to the surrounding circuit elements is negligible. Therefore, the current and charge noises are typically very low in the SISIS setup. However, the supercurrent is generated by an external circuit and the qubit will be exposed to noises of the circuit. Additionally, dissipation in real SISIS gates is possible due to a tunneling of a quantum state through the characteristic barriers U1, 2 () (see Fig. 9.44). Since the tunneling rate  depends on the barrier height and width exponentially, the populated upper level "2 may in principle contribute to the dissipation, especially if one biases the system such that the tunneling rate 2 out of |2 is 2 = ("2 + "0 − 2"1 )/2 . The relevant tunneling probability is  (2) Pt = p2 2 dt .

(9.92)

(9.93)

Although the Pt(2) magnitude may be significant, the tunneling probabilities Pt(0) and Pt(1) out of states |0 and |1 are about 103 and 106 times lower as compared to Pt(2) . The dissipation W2 due to a transient population [105] p2 of state |2 is estimated as: " p2 2 dt/ .

W2 =

(9.94)

2

The last formula means that W2 is roughly proportional to p2 and to the time of measurement tm . The dissipation is eliminated by keeping the population p2 of "2 as small as possible, and tm as short as possible during the quantum logic operations [105]. More essential in SISIS qubit gate is an intrinsic source of errors. Such errors come up when the interaction is turned on (J = 0) and are caused by the quantum leakage. The leakage is taking place if the Hilbert space of the real gate is larger than the qubit’s computational subspace. A different time evolution in the real space and in the computational subspace causes an error in the gate operation.An ideal unitary gate operation UI (t) is obtained when turning on the inter-qubit coupling for a time t0 . By choosing t0 one may accomplish an arbitrary gate operation [3].

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The output of an operation is related to the input state via the map ¢UR(t)¢, where UR (t) is the unitary operator which acts on the full Hilbert space, ¢ is the projector to the computational subspace ˆ = (|00| + |11|)H(|00| ˆ acting as ¢H¢ + |11|), where |00| = (1ˆ − z )/2, |11| = (1ˆ + z )/2, |01| = (x −iy )/2,and |10| = (x +iy )/2.The error of an operation is minimized by setting ¢UR(t)¢ as close as possible to U1 (t0 ). In general an optimal operation requires t = t0 as all the system eigenvalues are modified by states outside the computational subspace. The fidelity and leakage probabilities are defined as

¢UR(t)¢ = =

 n

 n

e−ien t ¢|¥n¥n |¢ e−ien t |00| + |11|•|¥n × ¥n |(|00| + |11|)

(we set here  = 1). For simple three-state analyses one uses that ⎛ ⎞ ⎛ ⎞ 0 1 (9.98) |0 = ⎝ 0 ⎠ ; |1 = ⎝ 1 ⎠ . 0 0

The above Eqs. (9.95)–(9.98) allow modeling of the basic two qubit characteristics and simulation of the two qubit gate dynamics with qubit parameters computed in the former sections.According to Eq. (9.98), basic parameters include the qubit energy level positions "0,1,2, the inter-level tunneling matrix element  and the inter-qubit coupling energy Jnm . Major two-qubit gate characteristics, which are determined by the inter-qubit coupling and readout, are studied within the 3 × 3 state analysis. In that case one works with the 6-dimensional Hilbert space of the real gate, which includes 3 states | 0n , | 1n , and | 2n for each of the qubits n = 1, 2 affected by the interaction J. The computation is done in a straightforward way, using computer algebra and numeric methods. In this way one begins with microscopic computation of all the two-qubit parameters for given SISIS geometry, electrode purity and temperature. The next stage involves a study of the inter-qubit interaction J and coupling to external meters. Finally one computes the major dissipative and dynamic properties of the two-qubit gates, i.e., the leakage and fidelity described by the Hamiltonian (9.71). An illustrative insight into the two-qubit dynamics and dissipative characteristics is given by analytical formulas derived in the former sections. Analytical solutions are available for a three-state single qubit gate and a 2 × 2-state two-qubit gate. One may for instance find an analytical expression for a resonant contribution to the leakage of a three-state system. The three-state system is described by the Hamiltonian ⎞ ⎞ ⎛ ⎛ −" √0 E0 0 0 H 3s = ⎝ ı √" 2 ⎠ → ⎝ 0 E1 0 ⎠ , (9.99) 0 0 E2 0 2ı " + 

and that the eigenfunctions | ¥n  of the Josephson energy potential of a non-interactive SIS junction at I1,2 = 0 are expressed in terms of Mathieu functions. So, the evolution in the computational subspace for a time interval t is:

where the lowest level is positioned at −", ı, and stand for the matrix elements of inter-level tunneling with the external field factors included. The arrow in Eq. (9.99) indicates the diagonalization of H 3s . The eigenvectors of (9.99) are:

F = 1 − min UI (z0 ) − ¢UR(t)¢ /2

(9.95)

     †  . UR (t)¢UR (t)

(9.96)

{t}

and L (t) = 1 − min

The norm . . . of an arbitrary operator K is defined as   K = Sup K|  = Sup

9   K † K 

:

(9.97)

= > over the vectors | :  |  = 1 of the computational subspace. This definition implies that K = √ M , where M is the largest eigenvalue of K † K. The evolution operator is  UR (t) = e−ien t |¥n ¥n | n

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Principles of Josephson-Junction-Based Quantum Computation

⎛ ⎞ Ÿ0 1 |¥0  = √ ⎝ ıC 0 ⎠ , N √2ı 2 ⎛ ⎞ Ÿ1 1 |¥1  = √ ⎝ ıC 1 ⎠ , N √2ı 2 ⎛ ⎞ Ÿ2 1 |¥2  = √ ⎝ ıC 2 ⎠ , N √2ı 2

(9.100)

where   C0 = P− − 2 " +  /3 ,   √   C1, 2 = C± = −P− ∓ i 3P+ /2 − 2 " +  /3 ,   P± = B20 ∓ 4" 2 + 2" + 2 / (3B0 ) , B0 = (3"2 + 3 + 3B1 − 8" 3 − 6" 2 )1/3 , B1 = i(12" 4 2 + 3" 2 4 + 12" 33 )1/2 . The above Eqs. (9.99), (9.100) allow for a direct computing of the resonant contribution to the leakage at the frequency of external field ! = ("1 − "0 ) / from which one obtains 

L (t) t = 1 −

 Ÿ04 + Ÿ14 + Ÿ24 + Ÿ02Ÿ12 + ¡ 4 , N4

359

(9.101)

  where ¡ 4 =   Ÿ0 Ÿ1 (Ÿ0 C1 + Ÿ1 C0 ) and the normalizing factor is N 4 = Ÿ04 + Ÿ14 + Ÿ24 + 2Ÿ02 Ÿ12 + 2Ÿ12Ÿ22 + 2Ÿ02Ÿ22 . (9.102)   Here  Ÿ0, 1, 2 = Ÿ E0, 1, 2 and the auxiliary function Ÿ  is      Ÿ  =  −" " − − . (9.103)

Fig. 9.46. Fidelity of the SISIS two-qubit gate for the three levels and different tunneling matrix elements  = 0.01, 0.6, and 0.9 for curves 1, 2, and 3, correspondingly. Dark spots in the left inset show regions of maximum entanglement. The right inset shows the related leakage probability curves versus time

One can see that the fidelity of the SISIS gate shows anomalies when the inter-qubit coupling is sufficiently large, i.e., when it exceeds the splitting energy, J/" > 1. If the coupling is small, the qubit gate fidelity is close to 1. The leakage probability versus the evolution time t oscillates with a large period which depends on the qubit parameters and the inter-qubit coupling, though it is small for t ≤ 103 . Optimal working regions are associated with avoided level crossing [108,109] where one achieves the maximum entanglement between the quantum states of the two SIS subjunctions. The entanglement is computed using Schmidt decomposition [110], which represents any entangled state by a biorthogonal expression with positive real coefficients   : ¦ (1, 2) = (9.105) ci |˛i  ⊗ ˇi , i=1

The eigenenergies of the Hamiltonian (9.99) are obtained as   " + , E0 = P− + 3   (9.104) √ P+ " + P− + ∓i 3 . E1,2 = E± = − 2 3 2

where |˛i  and |ˇi  are orthonormal states of subjunctions 1 and 2 and ci are real and positive. The entanglement of a partially entangled pure state [110] is  E=− ci2 log2 ci2 , (9.106)

The corresponding analytical expression for the fidelity is much more cumbersome. Typical calculation results of the fidelity and leakage probability versus time t are shown in Fig. 9.46.

Since the inter-level spacing ", tunneling amplitude , and the inter-qubit coupling J depend on the bias current Il , the entanglement is computed selfconsistently. Typical results for entanglement are

i

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shown in the contour plot (see the left inset in Fig. 9.46), where we assumed the junction transparency D = 10−4 , d = 3, and li / = 5. Dark spots in the {'1 , '2 }-plane correspond to the avoided level crossing (working) regions with the maximum interqubit entanglement. The same QC coupling mechanism may be used for a controlled readout of the two-qubit state. Since the measurement is executed with weak supercurrents, there is no direct energy dissipation in the multibarrier Josephson junction. Instead the dissipation occurs rather indirectly via the tunneling through the characteristic barriers, as was mentioned at the beginning of this section. Quantum measurement of the qubit states is furnished in a six-barrier multi-terminal SISI-SISIS-ISIS junction, where the two additional SISI subjunctions act as quantum meters. The SISI- and -ISIS sections measure the left and right side SIS subjunctions of the SISIS junction via the QC coupling. During the quantum measurement event the quantum states of SISIand -ISIS sections are entangled with the adjacent SIS subjunction for a short time tm  tdc (tdc is the decoherence time). To insure a sufficient sensitivity of the measurement, one biases the measured SIS subjunction to get the upper level "2 be located at the top of the characteristic barrier U1,2. The readout : of the two-qubit states | ¥1,2 is conducted by applyac ing short pulses of ac supercurrent I1,2 (t) to the SIS subjunctions. The "2 level becomes populated with probability p2 (t). The system then tunnels out from the populated state | 2 with the tunneling rate "2 . To evaluate the energy dissipation W2, we establish the time dependence of the upper level population p2 (t). This is done by introducing of a fourth fictional auxiliary state | 3n (see,e.g.,[111] and references therein). The tunneling from the state | 21,2 to the auxiliary state | 3n is described by the amplitude damping on fictitious qubits [112] within the 4 × 4-state analysis. The full space now includes 4 states for each of the qubits affected by the interaction J, which gives in total the 8-dimensional Hilbert space of the real two-qubit gate. Computation of the qubit characteristics and simulation of the qubit work both are conducted numerically. Initially one finds the parameters of the two-qubitHamiltonian (9.99) as described

in the former sections. Then the tunneling dynamics is modeled by using the operator-sum representation, where an initial density matrix ˆi is mapped to the final density matrix ˆf as  † , (9.107) ˆf = Eˆ m ˆi Eˆ m m

where Eˆ m are the Kraus operators [112]. The numeric analysis is performed by slicing the time dependence Eˆ m (t) in many discrete steps as described in [110]. The input parameters include the junction parameters listed above, completed with the a r.f. pulse amplitude and duration. We obtain the explicit form of the density matrix ˆi before the tunneling event. Then ˆi is mapped to ˆf in accordance with Eq. (9.107). Finally, we determine the time dependence p2 (t), which gives the dissipation W2 for each of the meters and for given parameters of the sixbarrier setup. In general, the dissipation introduced by a meter depends on the measurement time tm and sensitivity of the meter. The sensitivity actually determines a minimal threshold magnitude of the energy dissipation W2 , which must be sufficient for a firm readout. For a two-qubit gate with parameters used in Fig. 9.46 and J = 0.2, a most effective readout is achieved when ln(tdc /tm ) ∼ = 10−2 , = 10 and I0ac /Ic ∼ ac where tdc is the decoherence time and I0 is the amplitude of the r.f. supercurrent pulse used to excite "2 during the readout. We conclude that the intrinsic coupling assures a low decoherence and dephasing of the two-qubit gate. If the measurement time tm is sufficiently short while the tunneling current from the upper state through the characteristic barrier is small, the decoherence and dephasing are caused by the leakage only. The leakage comes from the induced changes in the Hilbert space of the real gate as considered above. This makes the intrinsic coupling mechanism to be attractive for using in two-qubit gates. The suggested model quantitatively describes general properties of the two-qubit gate with an externally controlled coupling. Since the coupling is intrinsic and coherent, it allows eliminating of additional circuit elements and wirings. The QC coupling is tuned by the bias Josephson supercurrents which typically are much smaller than currents circulating

9

Principles of Josephson-Junction-Based Quantum Computation

in flux qubits.No charge accumulates on the coupling element in contrast to the case of the two-qubit gates with a capacitive coupling. In this way noises and external sources of decoherence in the system under study can be minimized.Nevertheless one should pay attention to remaining intrinsic sources of decoherence, which persist during the two-qubit quantum logic operations.

9.6

Conclusions

(ii)

361

thermalization is assured at a working temperature of 50 mK. The superconducting gap should be slightly higher,  > EC . We can therefore conclude that aluminum is a suitable material. If we further choose EJ = 100 mK, the time scale 1 of one-qubit operations will be op = /EJ ∼ −10 10 s. The impedance of the gate-voltage circuit is typically R ∼ 50§ that of the transmission line that drives it. The associated voltage fluctuations then limit the dephasing time to values of order ' ∼ 10−4 s, thereby allowing coherent manipulations of a single qubit. To assure sufficiently fast two-bit operations one can choose CL ∼ CJ . Two-bit operations are then about 102 times slower than the one-bit operations with an associated reduction the maximum number. The quantum measurement process introduces additional constraints on the parameters, which can be met in realistic devices as demonstrated by the following example for typical parameters: For the junction and gate capacitances we choose CT = 1.5 × 10−17 F, CgSET = 0.5 × 10−17 F respectively and, for the coupling capacitance between the SET and qubit, Cint = 0.5 × 10−17 F. Finally, a quantum measurement is needed to read out the quantum information, either at the final stage or during the computation, e.g., for the purposes of error correction.

We have discussed how Josephson-junction devices with suitable parameters can be manipulated in a (iii) quantum-coherent fashion. They are clearly promising physical realizations (i.e. they can be the hardware) of future devices to be used for quantumstate engineering. We have discussed various modes of operation associated with different designs (in (iv) the charge- and the flux-dominated regimes), constraints on the parameters,various dephasing effects, and the quantum-mechanical measurement process. In order to demonstrate that the constraints on the circuit parameters can be met by available technologies,we now summarize them and suggest a suitable set. (v) (i) Necessary conditions for a Josephson charge qubit are  > EC  EJ , kB T. The superconducting energy gap  must be large enough to suppress quasiparticle tunneling. The temperature also has to be low relative to the charg- To conclude,fabrication and controlled coherent maing energy to assure initial thermalization, i.e., nipulations of Josephson-junction qubits are possikB T  EC , and to reduce dephasing effects. A ble using present-day technologies. In these systems sufficient choice is kB T ∼ EJ /2, since further fundamental properties of macroscopic quantumcooling does not reduce the dephasing (relax- mechanical systems can be explored. Experiments ation) rate in a qualitative way.6 As an explicit on elementary systems have now been performed example we choose some reasonable parameters successfully. More elaborate designs as well as furand estimate the corresponding time scales: For ther progress in materials and patterning technoljunctions with a capacitance CJ = 10−15 F, corre- ogy will provide longer coherence times and allow sponding to (in temperature units) EC ∼ 1 K,and sequences of coherent manipulations as well as scala (smaller) gate capacitance Cg = 0.5×10−17F (to ing to larger numbers of qubits. The application of reduce the coupling to the environment), initial Josephson-junction systems as elements of a quan6

It will, however, affect relaxation far from the degeneracy point, i.e., for h = 0, or if we switch off the Hamiltonian, Hctrl = 0. However, during manipulations EJ is the typical energy difference and sets the time scale for both the manipulation times and the dephasing.

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tum computer, i.e., with a very large number of manipulations and large number of qubits, will remain a challenge for some time. On the other hand, many aspects of quantum information processing can initially be tested on simple circuits, as described here. Additional fallout can be expected with further developments in quantum-state engineering. We have also described how a dissipative quantum system when coupled to a qubit can serve as a quantum measuring device in an accessible range of parameters. Explicitly we described a single-electron transistor coupled capacitively to a charge qubit, and the process of measurement has been discussed using the time evolution of the reduced density matrix of the coupled system. It is found that the dephasing time is shorter than the measurement time, and the mixing time, i.e., the time scale over which the transitions induced by the measurement occur, was estimated. Similar scenarios were discussed for flux qubits measured by a SQUID coupled to it inductively,by switching the interqubit interactions on and off.In terms of the physics,the two types of operation allow the creation of arbitrary superpositions and nontrivial coupled states, such as entangled states, respectively.

Appendix. Elementary Quantum Logic Operations This chapter has focused primarily on individual Josephson-based qubit strategies as well as a twoqubit gate. In this Appendix we will briefly outline some of the logical aspects associated with quantum computing. Those wishing to pursue this aspect in depth are referred to the text by Nielsen and Chuang [112]. We start by sketching the elementary quantum logic operations. and the possibility of performing them with macroscopic Josephson qubits. We then focus on the feasibility of implementing Shor’s quantum factoring algorithm using such devices. This will also gives us an opportunity to “design” a twodimensional Josephson qubit array, capable of performing useful quantum computations. The discussion closely follows that given in [113].

A.1 Boolean Logic Boolean logic operates with two distinguishable states – False or True, or simply 0 or 1 – and serves as the basis for most contemporary computers. Now, according to general quantum mechanical principles,if there are two distinguishable quantum states, in addition to the system being in either the “0” or “1” state, it can also exist in various coherent superpositions of these states. Because the smallest piece of information in a two-state quantum system has this new fundamental property we can call it a quantum bit or qubit. In systems with more than one qubit one may introduce quantum logical gate operations, which are the building blocks of a quantum computer. Because of quantum mechanical superpositions of qubits, a quantum computer has many more degrees of freedom and with it new computational capabilities not available with classical computers using conventional Boolean operations. After the discovery of the quantum factorization algorithm by Shor in 1994, it was proved that quantum computers could perform certain important tasks that cannot be completed by classical computers within a realistic period of time. The Shor algorithm is exponentially faster than any known classical algorithm [114]. For this reason, quantum computation theory stimulated efforts to realize practical quantum computers. The Josephson-based qubits are among the more promising devices proposed for performing quantum logic operations. In particular, they can be easily initialized, precisely manipulated, and individually addressed by conventional techniques. However, as noted of the beginning in Sect. 9.1, macroscopic quantum systems tend to suffer from the problem of decoherence,which degrades and ultimately destroys quantum coherent superpositions by irreversible interactions with the environment. As we have discussed in earlier sections,superconducting tunnel junction circuits have been designed, fabricated, and characterized that have a high quality factor characterizing the quantum coherence (see, e.g., [115–117]). Such experiments now demonstrate that decoherence need not necessarily be an obstacle in building a quantum computer from Josephson circuit elements [117].

9

Principles of Josephson-Junction-Based Quantum Computation

If one defines the quantum computing process as a set of quantum logic operations, such operations can then be defined in terms of a unitary evolution of a quantum network that takes a specified initial state into some final state [4]. This network will be a computing device consisting of an array quantum logic gates, and each gate is a unitary operation on one or more qubits. Quantum computations are then accomplished by assembling quantum logic circuits from many quantum logic gates. Since a unitary transformation is reversible, quantum gates need to be reversible and cannot be assembled from their classical counterparts. We will start by describing the elementary quantum logic operations, so-called “universal quantum gates”. Following this some simulation work is presented in Sect.A.3.In Sect.A.4 we present the strategy to perform Shor’s factoring algorithm using arrays of superconducting Josephson qubits. A.2 Required Properties Figure 9.47 depicts a device consisting of a loop with three junctions in series that encloses a magnetic flux ¥ which is controlled by an external magnet. When the enclosed magnetic flux is close to half a superconducting flux quantum ¥0 (= h/2e, where h is Planck’s constant), the loop may have multiple stable persistent current states, and the system behaves as a particle in a double-well potential. The two wells are coupled via quantum tunneling through the barrier between the wells, and the loop is a macroscopic quantum two level system. The system can be engineered to have two stable states designed as 0 and 1, involving oppositely circulating persistent currents [118]. The device is operated using resonant microwave modulation of the enclosed magnetic flux applied through a superconducting control line; the control line is located on top of the qubit, separated by a thin insulating layer. Measurements can be made using superconducting magnetometers (superconducting quantum interference devices or SQUIDs). Two or more qubits can be coupled by means of the flux that the circulating persistent current generates [119]. An alternate circuit proposed in [120] consists of a Cooper pair box, a

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Fig. 9.47. A three-junction Josephson qubit

superconducting island coupled to two small Josephson junctions in a superconducting loop which includes a third, much larger, Josephson junction (see Sect. 9.5.2). Arbitrary quantum states of the circuit are manipulated by applying microwave pulses and read out by a special readout circuit. A.3 Universal Quantum Gates As in the case of classical computers, certain sets of quantum gates are universal in the sense that any quantum computation can be performed by appropriately connecting them. We now examine a minimal set of universal quantum gates: i) the single qubit rotation and ii) the quantum controlled-NOT gate, or CNOT.An arbitrary single qubit rotation can be written as e−iˆ t = cos t − iˆ sin t for some Pauli matrix combination ˆ = aˆx +bˆ y +c ˆ z where a2 +b2 +c 2 = 1 and we used that, e.g.,    it  0 eit e 0 iz t eix t = . (9.108) = ,e 0 e−it eit 0 By applying a magnetic pulse for certain duration, a 3-Josephson junction qubit can be “rotated” (see Sect. 9.4.3). Figure 9.48 shows, schematically, that the unitary transformation U rotates the qubit a (which is described by a 2 × 2 matrix).We use here the notations of [114]. The red arrow across the circle (which denotes the rotation itself) indicates the “direction” of rotation. Figure 9.49 depicts a controlled NOT which is a two-qubit quantum logic gate that flips the value of the second qubit if the value of the first qubit is 1; i.e., |00 → |00, |01 → |10, |10 → |11, |11 → |01. The system of two coupled qubits is now described by the 4 × 4 matrix. The rotation affects the lower right 2 × 2 submatrix. By controlling

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Fig. 9.48. Single qubit rotation

Fig. 9.50. Quantum sum Fig. 9.49. Controlled NOT (CNOT)

the coupling of two qubits, a so-called controlled not gate can be performed as indicated in Fig. 9.49. Given a  pulse (i.e., by shifting the qubit phase by ), a controlled NOT gate is constructed from the controlled rotation gate. A controlled NOT can be combined with single qubit rotations to perform arbitrary quantum logic operations. A.4 Quantum Sum and Carry The addition of two quantum registers |a and |b can be written as |a, b → |a, a + b. Since the input (a, b) can be reconstructed from the output (a, a + b) (i.e., there is no loss of information in the computation), the calculation can be carried out reversibly. The sum operation can be implemented with two CNOT gates (Fig. 9.50, where time goes from left to right). However the carry operation has to be obtained in a more complicated way [31] (Fig. 9.50). A sequence of RF pulses with different frequencies and durations are injected into the quantum network during operation. During a pulse, the involved qubits switch their states.

A.5 Shor’s Algorithm Although there is still a long way to go before quantum computers come into practice, it is believed that they can solve problems that are intractable with present classical computers, for example, the factorization of large numbers into prime numbers. The best algorithm available for factoring an integer n using classical computation is exponential in time, which rapidly becomes impractical with increasing n. Shor discovered a quantum algorithm for factoring, which is polynomial in n, along with some polynomial (in log n) processing time on a classical computer [4]. To factor an odd number n, Shor’s algorithm first finds the least integer r such that, x r ≡ 1 (modn); i.e. the period of f (A) = x A mod n for A from 0 to n − 1, where x is random with x < n and gcd (x, n) = 1. Then it finds factors of n by calculating gcd(x r/2 − 1, n) and gcd(x r/2 + 1, n) if r is even and x r/2 = ±1 mod n; otherwise, it repeats the algorithm. Here, gcd(a, b) is the greatest common divisor of a and b and it can be effectively computed with Euclid’s algorithm on a classical computer. To per-

Fig. 9.51. The quantum carry operation as performed with RF pulses

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Principles of Josephson-Junction-Based Quantum Computation

form Shor’s factoring algorithm, we start with two quantum registers, one of which is prepared to be superpositions |a and the other is in |0 state. We compute x a (modn) in the second register and keep the first register in the |a state, i.e., performing the modular exponentiation  : Ux,n |a, 0 → a, x a (modn) , (9.109) which can be built from a reversible network of quantum sum and carry operations [9].We next perform a quantum Fourier transformation on the first register to find the period r. The discrete Fourier transform is a unitary transformation and can be implemented by a network of quantum CNOT and qubit rotation gates [4].Finally,we calculate the factors of nwith Euclid’s algorithm by classical computing.We see that in the realization of Shor’s factoring algorithm,classical computation is indispensable; i.e., the experimental realization of a quantum network can be greatly simplified by incorporating some classical components.

365

A.6 Conclusions We have presented strategies for performing some elementary quantum logic operations using macroscopic Josephson qubits, supported by some simulations. The feasibility of building a quantum system to perform Shor’s factoring algorithm was explored by implementing the algorithm on a Josephson qubit network.As is discussed by authors of [121] it should eventually be possible to build a two-dimensional Josephson qubit array,possibly accompanied by classical computing components, capable of performing useful quantum computations, e.g. Shor’s factoring algorithm. But the practical realization depends heavily on advances involving the individual devices.

Acknowledgements We thank C. Bennemann for technical assistance preparing the text.

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A. I. Larkin University of Minnesota Lab. of Physics Minneapolis, USA A. A. Varlamov COHERENTIA-INFM, CNR, Rome, Italy

To our friend Lev Aslamazov, in memoriam

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 10.2 Ginzburg–Landau Formalism: Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 10.2.1 Fluctuation Contribution to Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 10.2.2 Ginzburg–Levanyuk Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 10.2.3 Scaling and Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.2.4 Fluctuation Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 10.3 Fluctuations Below the Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .390 10.3.1 Effect of Fluctuations on Superfluid Density and Critical Temperature . . . . . . . . . . . . . . . . . . . . 390 10.3.2 Phase Fluctuations in 2D Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 10.3.3 Phase-Slip Events in 1D Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 10.3.4 Fluctuations of the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 10.4 Ginzburg–Landau Theory of Fluctuations in Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 10.4.1 Time-Dependent GL Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 10.4.2 Paraconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 10.4.3 General Expression for Paraconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399 10.4.4 Fluctuation Conductivity of Layered Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .401 10.4.5 Magnetic Field Angular Dependence of Paraconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 10.5 Fluctuations Near Superconductor–Insulator Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 10.5.1 Quantum Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 10.5.2 3D Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 10.5.3 2D Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 10.6 Microscopic Derivation of the Time-Dependent Ginzburg–Landau Equation . . . . . . . . . . . . . . . . . . . .410 10.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 10.6.2 The Cooper Channel of Electron–Electron Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 10.6.3 Superconductor with Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 10.7 Microscopic Theory of Fluctuation Conductivity of Layered Superconductors . . . . . . . . . . . . . . . . . . .416 10.7.1 Qualitative Discussion of Different Fluctuation Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .416 10.7.2 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 10.7.3 The Aslamazov–Larkin Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 10.7.4 Contributions from Fluctuations of the Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 10.7.5 The Maki–Thompson Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 10.7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 10.8 Manifestation of Fluctuations in Various Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 10.8.1 The Effects of Fluctuations on Magnetoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

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10.8.2 Fluctuations Far from Tc or in Strong Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 10.8.3 The Effect of Fluctuations on the Hall Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 10.8.4 Fluctuations in the Ultra-Clean Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 10.8.5 The Effect of Fluctuations on the One-Electron Density of States and on Tunneling Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 10.8.6 The Nonlinear Fluctuation Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 10.8.7 The Effect of Fluctuation on the Optical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 10.8.8 Thermoelectric Power above the Superconducting Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.8.9 The Effect of Fluctuations on NMR Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .448 10.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

10.1

Introduction

A major success of low temperature physics was achieved with the introduction by Landau of the notion of quasiparticles. According to his hypothesis, the properties of many-body interacting systems at low temperatures are determined by the spectrum of some low energy, long living excitations (quasiparticles). Another milestone of many-body theory is the Mean Field Approximation (MFA), which permitted achieving considerable progress in the theory of phase transitions. Phenomena which cannot be described by the quasiparticle method or by MFA are usually called fluctuations. The BCS theory of superconductivity is a bright example of the use of both the quasiparticle description and MFA. The success of the BCS theory for traditional superconductors was determined by the fact that fluctuations give small corrections with respect to the MFA results. During the first half of the century after the discovery of superconductivity the problem of fluctuation smearing of the superconducting transition was not even considered. In bulk samples of traditional superconductors the critical temperature Tc sharply divides the superconducting and the normal phases. It is worth mentioning that such behavior of the physical characteristics of superconductors is in per-

fect agreement both with the Ginzburg–Landau (GL) phenomenological theory (1950) [1] and the BCS microscopic theory of superconductivity (1957) [2]. The characteristics of high temperature and organic superconductors, low dimensional and amorphous superconducting systems studied today, strongly differ from those of the traditional superconductors discussed in textbooks. The transitions turn out to be much more smeared out. The appearance of superconducting fluctuations above the critical temperature leads to precursor effects of the superconducting phase occurring while the system is still in the normal phase, sometimes far from Tc . The conductivity, the heat capacity, the diamagnetic susceptibility, the sound attenuation, etc. may increase considerably in the vicinity of the transition temperature. The first numerical estimation of the fluctuation contribution to the heat capacity of a superconductor in the vicinity of Tc was done by Ginzburg in 1960 [3]. In that paper he showed that superconducting fluctuations increase the heat capacity even above Tc . In this way fluctuations change the temperature dependence of the specific heat in the vicinity of critical temperature,where,in accordance with the phenomenological GL theory of second order phase transitions, a jump should take place. The range of

10

temperatures where the fluctuation correction to the heat capacity of a bulk clean conventional superconductor is relevant was estimated by Ginzburg1 to be  4 ıT Tc Gi = ∼ ∼ 10−12 ÷ 10−14 , (10.1) Tc EF where EF is the Fermi energy. The correction occurs in a temperature range ıT many orders of magnitude smaller than that accessible in real experiments. In the 1950s and 1960s the formulation of the microscopic theory of superconductivity, the theory of type-II superconductors and the search for high-Tc superconductivity attracted the attention of researchers to dirty systems, superconducting films and filaments. In 1968, in the papers of L.G. Aslamazov and A.I. Larkin [5], K. Maki [6] and a little later in the paper of R.S. Thompson [7] the fundaments of the microscopic theory of fluctuations in the normal phase of a superconductor in the vicinity of the critical temperature were formulated. This microscopic approach confirmed Ginzburg’s evaluation [3] for the width of the fluctuation region in a bulk clean superconductor. Moreover, it was found that the fluctuation effects increase drastically in thin dirty superconducting films and whiskers.In the cited papers it was demonstrated that fluctuations affect not only the thermodynamical properties of superconductor but also its dynamics. Simultaneously the fluctuation smearing of the resistive transition in bismuth amorphous films was experimentally found by Glover [8], and it was perfectly fitted by the microscopic theory. In the BCS theory [2] only the Cooper pairs forming a Bose-condensate are considered. Fluctuation theory deals with the Cooper pairs out of the condensate. In some phenomena these fluctuation Cooper pairs behave similarly to quasiparticles but with one important difference. While for the well defined quasiparticle the energy has to be much larger than its inverse lifetime, for the fluctuation Cooper pairs the “binding energy” E0 turns out to be of the same 1

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371

order. The Cooper pair life time GL is determined by its decay into two free electrons. Evidently at the transition temperature the Cooper pairs start to condense and GL = ∞. So it is natural to suppose from dimensional analysis that GL ∼ /kB (T − Tc ). The microscopic theory confirms this hypothesis and gives the exact coefficient: GL =

 . 8kB (T − Tc )

(10.2)

Another important difference of the fluctuation Cooper pairs from quasiparticles lies in their large size (T). This size is determined by the distance on which the electrons forming the fluctuation Cooper pair move away during the lifetime of the pair GL . In the case of an impure superconductor the electron motion is diffusive with the diffusion coefficient D ∼ vF2 √( is the electron scattering time2 ), and √ d (T) = DGL ∼ vF GL . In the case of a clean superconductor, where kB T  , impurity scattering no longer affects the electron correlations. In this case the time of electron ballistic motion turns out to be less than the electron-impurity scattering time  and is determined by the uncertainty principle: bal ∼ /kB T. Then this time has to be used in this case for determination of the effective size instead of :  c (T) ∼ vF GL /kB T. In both cases the coherence length grows with the approach to the critical temperature as (T − Tc )−1/2 , and we will write it in the unique way ( = c,d ):  (T) = √ ,

=

T − Tc . Tc

(10.3)

The microscopic theory in the case of an isotropic Fermi surface gives for  the precise expression:  1  v2  2 2 (D) ( + ) (10.4) =− F D 2 4kB T  1   1 − ( )− ( ) , 2 4kB T 2

The expression for the width of the strong fluctuation region in terms of the Landau phenomenological theory of phase transitions was obtained by Levanyuk [4]. So in the modern theory of phase transitions the relative temperature width of fluctuation regions is called the Ginzburg–Levanyuk parameter Gi(D) , where D is the effective space dimensionality. Strictly speaking, in most of the following results  should be understood as the electron transport scattering time tr . Nevertheless, as it is well known, in the case of isotropic scattering these values coincide, so for sake of simplicity we will use hereafter the symbol .

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where (x) is the digamma function and D = 3, 2, 1 is the space dimensionality. In the clean (c) and dirty (d) limits: 7 7 vF 3 3 c = 0.133 = 0.740 , (10.5) kB Tc D D  7  vF l 3 3 d = 0.36 = 0.85 0 l . (10.6) kB Tc D D Here l = vF  is the electron mean free path and 0 = vF /(0) is the conventional BCS definition of the coherence length of a clean superconductor at zero temperature. One can see that (10.5) and (10.6) coincide with the above estimations.3 Finally it is necessary to recognize that fluctuation Cooper pairs can really be treated as classical objects, but these objects instead of Boltzmann particles appear as classical fields in the sense of Rayleigh–Jeans. This means that in the general Bose–Einstein distribution function only small energies E(p) are involved and the exponent can be expanded: 

n(p) = exp

1 E(p) kB T



= −1

kB T . E(p)

(10.7)

This is why the more appropriate tool to study fluctuation phenomena is not the Boltzmann transport equation, but the GL equation for classical fields. Nevertheless at the qualitative level the treatment of fluctuation Cooper pairs as particles with density  dD p 4 n(D) = n(p) (2) D often turns out to be useful. s Below will be demonstrated both in the framework of the phenomenological Ginzburg–Landau theory and the microscopic BCS theory that in the vicinity of the transition one has 3

4

E(p) = ˛kB (T − Tc ) + =

p2 2m∗

(10.8)

 1 2 2  / (T) + p2 . 2m∗

Far from the transition temperature the dependence n(p) turns out to be more sophisticated than (10.7), nevertheless one can always write it in the form   m∗ kB T 2 (T)p n(p) = . (10.9)  (T) f 2  In classical field theory the notions of the particle distribution function n(p) (proportional to E −1 (p) in our case) and the Cooper pair mass m∗ are poorly determined. At the same time the characteristic value of the Copper pair center of mass momentum can be defined and it turns out to be of the order of p0 ∼ /(T). So for the combination m∗ E(p0 ) one can write m∗ E(p0) ∼ p20 ∼ 2 / 2(T). In fact the particles’ density enters into many physical values ∗ in the combination n(D) s /m . As the consequence of the above observation it can be expressed in terms of the coherence length: n(D) kB T s = ∗ ∗ m m E(p0 )



p0 

D ∼

kB T 2−D  (T) , (10.10) 2

where pD0 estimates the result of momentum integration. For example, we can evaluate the fluctuation Cooper pairs’ contribution to conductivity by using the Drude formula  =

2 n(D) kB T s e  ⇒ 2 d D−3  2−D (T)(2e)2GL ∗ m  D/2−2 ×( ) ∼ .

(10.11)

Let us stress some small numerical difference between our expression (10.4) and the usual definition of the coherence length. We are dealing with the near critical temperature, so the definition (10.4) is natural and permits us to avoid many numerical coefficients in further calculations. The cited coherence length 0 = vF / (0) = 0.18vF /kB Tc , as is evident, was introduced for zero temperature and an isotropic 3D superconductor. It is convenient to determine the coherence length also from the formula for the upper critical field: Hc2 (T) = A(T)¥0 /2  2 (T). A(Tc ) = 1, while its value at T = 0 depends on the impurities concentration. For the dirty case the appropriate value was found by K. Maki [9] Ad (0) = 0.69, for the clean case by L.Gor’kov [10] A2D c (0) = 0.59, A3D c (0) = 0.72. This particle density is defined in the (D)–dimensional space. This means that it determines the normal volume density of pairs in the 3D case, the density per square unit in the 2D case and the number of pairs per unit length in 1D. The real three dimensional density n can also be defined: n = d D−3 ns(D) , where d is the thickness of the film or wire.

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Analogously a qualitative understanding of the increase in the diamagnetic susceptibility above the critical temperature may be obtained from the well known Langevin expression for the atomic susceptibility:5

10.2

Ginzburg–Landau Formalism: Thermodynamics

10.2.1

Fluctuation Contribution to Heat Capacity

9 2: 4e2 kB T e2 n(D) s R ⇒ − 2 2 d D−3  4−D (T) ∼ − D/2−2 . 2 ∗ c m c  (10.12) Besides these examples of the direct influence of fluctuations on superconducting properties, indirect manifestations by means of quantum interference in the pairing process and of renormalization of the density of one-electron states in the normal phase of the superconductor take place. These effects are much more sophisticated and have a purely quantum nature, and in contrast to the direct Cooper pair contributions they require microscopic consideration. This is why in developing phenomenological methods through the first five sections of this review, we will deal with the direct fluctuation pair contributions only. The sixth section is devoted to the microscopic justification of the time-dependent Ginzburg– Landau equation. The description of the microscopic theory of fluctuations, including indirect fluctuation effects, and a discussion of their manifestations in various physical properties of superconductors will be given in Sects. 10.7 and 10.8. The first seven sections are written in detail, so they can serve as a textbook. On the contrary, in the last (eighth) section a wide panorama of fluctuation effects in different physical properties of superconductors is presented. Thus this section has more of a handbook character, and the intermediate calculations are often omitted. Finally we would like to mention that the number of articles devoted to superconducting fluctuations published in the last 33 years is of the order of tens of thousands, so our bibliography list does not pretend to be either complete or to establish rigorous priorities.

The complete description of the thermodynamic properties of a system can be done through the exact calculation of the partition function:6 %2 1  B H . (10.13) Z = Tr exp − T

=−

5

6 7

The GL Functional

As discussed in the Introduction, in the vicinity of the superconducting transition, fluctuation Cooper pairs of a bosonic nature appear in the system side by side with the fermionic electron states.As has already been mentioned, they can be described by means of classical bosonic fields ¦ (r), which can be treated as “Cooper pair wave functions”. So the calculation of the trace in (10.13) can be separated into a summation over the “fast” electron degrees of freedom and a further functional integration carried out over all possible configurations of the “slow” Cooper pairs wave functions:  Z = D2 ¦ (r)Z[¦ (r)] , (10.14) where



F [¦ (r)] Z[¦ (r)] = exp − T

 (10.15)

is the system partition function in a fixed bosonic field ¦ (r), already summed over the electronic degrees of freedom. Here it is assumed that the classical field-dependent part of the Hamiltonian can be chosen in the spirit of the GL approach:7  F[¦ (r)] = FN + dV (10.16)   b 1 2 4 2 |∇¦ (r)| . × a|¦ (r)| + |¦ (r)| + 2 4m

This formula is valid for the dimensionalities D = 2, 3, when the fluctuation Cooper pair has the possibility to “rotate” in the applied magnetic field and the average square of the rotation radius is R2  ∼  2 (T).“Size” effects, important for low dimensional samples, will be discussed later on. Hereafter  = kB = c = 1. For simplicity the magnetic field is assumed to be zero.

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Let us discuss the coefficients of this functional. In accordance with the Landau hypothesis, the coefficient a goes to zero at the transition point and depends linearly on T − Tc . Then a = ˛Tc ; all the coefficients ˛,b and m are supposed to be positive and temperature-independent. Concerning the magnitude of the coefficients it is necessary to make the following comment. One of these coefficients can always be chosen arbitrarily: this option is related to the arbitrariness of the Cooper pair wave function normalization. Nevertheless, the product of two is fixed by dimensional analysis: ma ∼  −2 (T). Another combination of the coefficients, independent of the wave function normalization and temperature, is ˛ 2/b. One can see that it has the dimensionality of the density of states. Since these coefficients were obtained by a summation over the electronic degrees of freedom, the only reasonable candidate for this value is the electron density of states for one spin at the Fermi level  . The microscopic theory gives the precise coefficients for the above relations: 8 2 4m˛Tc =  −2 ; , (10.17) ˛ 2/b = 7(3) where (x) is the Riemann zeta function, (3) = 1.202. One can notice that the arbitrariness in the normalization of the order parameter amplitude leads to unambiguity in the choice of the Cooper mass introduced in (10.16) as 2m. Indeed, this value enters in (10.17) in the product with ˛ so one of these parameters has to be fixed. In the case of a clean D-dimensional superconductor it is natural to suppose that the Copper pair mass is equal to two free electron masses, which results in ˛(D) =

2D 2 Tc . 7(3) EF

(10.18)

As the first step in the Landau theory of phase transitions ¦ is supposed to be independent of position. This assumption in the limit of a sufficiently large volume V of the system permits a calculation of the functional integral in (10.14) by the method of steepest descent. Its saddle point determines the equilibrium value of the order parameter 1 −˛T c /b, < 0 2 | = |¦ . (10.19) 0, > 0

Choosing ˛ in accordance with (10.18) one finds that this value coincides with the superfluid density ns of the microscopic theory [2]. The fluctuation part of the free energy related to the transition is determined by the minimum of the functional (10.16): ⎧ ˛ 2Tc2 2 ⎨ FN − V, < 0  . F = (F [¦ ])min = F [¦ ] = 2b ⎩ FN , > 0 (10.20) From the second derivative of (10.20) one can find an expression for the jump of the specific heat capacity at the phase transition point:     Tc @SN Tc @SS C = CS − CN = − = V @T V @T  2  1 ˛2 8 2 @ F =− Tc =  Tc . (10.21) = 2 VTc @ b 7(3) Let us mention that the jump of the heat capacity was obtained because the system volume was taken to infinity first, and after this the reduced temperature was set equal to zero. Zero Dimensionality: The Exact Solution In a system of finite volume fluctuations smear out the jump in heat capacity. For a small superconducting sample with the characteristic size √ d  (T) the space-independent mode ¦0 = ¦ V defines the main contribution to the free energy:   2 Z(0) = d ¦0Z[¦0 ] =  d|¦0 |2 % b ( a|¦0 |2 + 2V |¦0|4 ) (10.22) × exp − T 7  3 VT = exp(x 2)(1 − erf (x))|x=a√ V . 2bT 2b 

By evaluating the second derivative of this exact result [11] one can find the temperature dependence of the superconducting granular heat capacity (see Fig. 10.1). It is evident that the smearing of the jump

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375

effective dimensionality on the basis of the following observation. The volume of the specimen may be divided into regions of size (T), which are weakly correlated with each other. Then the whole free energy can be estimated as the free energy of one such zerodimensional specimen (10.23), multiplied by their number n(D) = V  −D (T): F(D) = −TV  −D (T) ln

Fig. 10.1. Temperature dependence of the heat capacity of superconducting grains in the region of the critical temperature

takes place in the region of temperatures in the vicinity of transition where x ∼ 1, i.e.     7(3) 03 1 Tc0 cr = Gi(0) = , ≈ 13.3 √ 2 EF V  Tc V where we have supposed that the granule is clean; Tc0 and 0 are the critical temperature and the zero temperature coherence length (see footnote 3 in the Introduction) of the appropriate bulk material. From this formula one can see that even for a granule with the size d ∼ 0 the smearing of the transition still is very narrow. Far above the critical region,where Gi(0)   1, one can use the asymptotic expression for the erf (x) function and find  F(0) = −T ln Z(0) = −T ln . (10.23) ˛ Calculation of the second derivative gives an expression for the fluctuation part of the heat capacity in this region: ıC(0) = 1/V 2 . (10.24) The experimental study of the heat capacity of small Sn particles in the vicinity of transition was done in [12]. Arbitrary Dimensionality: Case T  Tc It is possible to estimate the fluctuation contribution to the heat capacity for a specimen of an arbitrary

 . ˛

(10.25)

This formula gives the correct temperature dependence for the free energy for even dimensionalities. A more accurate treatment removes the ln dependence from it in the case of odd dimensions. Let us begin with the calculation of the fluctuation contribution to the heat capacity in the normal phase of a superconductor. We restrict ourselves to the region of temperatures beyond the immediate vicinity of transition, where this correction is still small. In this region one can omit the fourth order term in ¦ (r) with respect to the quadratic one and write the GL functional, expanding the order parameter in a Fourier series:   k2 F[¦k ] = FN + |¦k |2 a+ (10.26) 4m k   = FN + ˛Tc +  2k 2 |¦k |2 . k



Here, ¦k = √1V ¦ (r)e−ikr dV and the summation is carried out over the vectors of the reciprocal space. Now we see that the free energy functional appears as a sum of energies of the independent modes k. The functional integral for the partition function (10.15) can be separated to a product of Gaussian type integrals over these modes:   A k2 Z= d 2 ¦k exp −˛( + )|¦k |2 . 4m˛Tc k (10.27) Carrying out these integrals, one gets the fluctuation contribution to the free energy:   . F( > 0) = −T ln Z = −T ln  k2 ˛ + 4m˛T k c (10.28)

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The appropriate correction to the specific heat capacity of a superconductor at temperatures above the critical temperature may thus be calculated. We are interested in the most singular term in −1, so the differentiation over the temperature can be again replaced by that over :  2  1  1 1 @ F ıC+ = − =  2 . 2 VT c @ V k2 k + 4m˛T c (10.29) The result of the summation over k strongly depends on the linear sizes of the sample, i.e. on its effective dimensionality. As is clear from (10.29), the scale with which one has to compare these sizes is 1 determined by the value (4m˛Tc )− 2 , which, as was already mentioned above, coincides with the effective size of Cooper pair (T). Thus, if all dimensions of the sample considerably exceed (T), one can integrate over (2)−3 Lx Ly Lz dkx dky dkz instead of summing over nx , ny , nz . In the case of arbitrary dimensionality the fluctuation correction to the heat capacity turns out to be  VD dD k 1 ıC+ =  2 V (2)D k2 + 4m˛T c D

VD (4m˛Tc ) 2 = #D , D V 2− 2

(10.30)

where VD = V , S, L, 1 for D = 3, 2, 1, 0. For the coefficients #D it is convenient to write an expression that is valid for an arbitrary dimensionality D, including fractional ones. For a space of fractional dimensionality we just mention that the momentum integration in spherical out according to  coordinates is carried  the rule: dD k/ (2)D = D kD−1 dk, where D

In the case of small particles with characteristic sizes d  ( ) the appropriate fluctuation contribution to the free energy and the specific heat capacity coincides with the asymptotics of the exact results (10.23) and (10.24). From the formula given above it is easy to see that the role of fluctuations increases when the effective dimensionality of the sample or the electron mean free path decrease. Arbitrary Dimensionality: Case T < Tc The general expressions (10.14) and (10.16) allow one to find the fluctuation contribution to heat capacity below Tc . For this purpose let us restrict ourselves to the region of temperatures not very close to Tc from below, where fluctuations are sufficiently weak. In this case the order parameter can be written as the  (see (10.19)) and fluctuasum of the equilibrium ¦ tion (r) parts:  + (r) . ¦ (r) = ¦

(10.33)

Keeping in (10.16) the terms up to the second order , one can find in (r) and up to the fourth order in ¦     2 + b/2¦ 4 A a¦ ] = exp − Z[¦ d Re k d Im k T k    1 k2 2  +a+ 3b¦ Re 2 k × exp − T 4m    k2 2 2  Im k . (10.34) + b¦ + a + 4m Carrying out the integral over the real and imaginary parts of the order parameter one can find an expression for the fluctuation part of the free energy: 1 Tc T ln F=− 2+a + 2 3b ¦ k

k2 4m

+ ln

2

Tc 2 + a + b¦

k2 4m

.

(10.31)

(10.35)

and  (x) is a gamma-function. The coefficient in (10.30) can also be expressed in terms of the gammafunction:  (2 − D/2) #D = , (10.32) 2D  D/2 yielding #1 = 1/4, #2 = 1/4 and #3 = 1/8.

Let us discuss this result. It is valid both above and below Tc . The two terms in it correspond to the contributions of the modulus and phase fluctuations of  ≡ 0 and these the order parameter. Above Tc ¦ contributions are equal: phase and modulus fluctua represent just two equivations in the absence of ¦ lent degrees of freedom of the scalar complex order

D =

2D  D/2  (D/2

+ 1)

10

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377

Table 10.1. Values for the Ginzburg–Levanyuk parameter Gi(L) Gi(3)  80

Tc EF

1.6  3 pF l

4 , (c) 

Tc EF

 , (d)

Gi(2)   Tc , (c) EF 0.27 , (d) pF l 1.3 , (d) film p2F ld

Gi(1) 0.5, (c)  −2/3 1.3 p2F S (Tc )−1/3 , (d) wire

Hence, in the framework of the proposed theory we found that the heat capacity of the superconductor tends to infinity at the transition temperature. Strictly speaking, the restrictions of the above approach do not permit us to seriously discuss this divergence at the critical point itself. The calculations are, in principle, valid only in that region of temperatures where the fluctuation correction is small. In the next section we will discuss the quantitative criteria for the applicability of this perturbation theory. The Ginzburg–Levanyuk Criterion

The fluctuation corrections to the heat capacity obtained above allow us to answer quantitatively the 8



 −2/3 2.3 p2F S , (c) whisker

parameter. Below Tc , the symmetry of the system decreases (see (10.19)). The order parameter modulus fluctuations remain of the same diffusive type as above Tc , while the character of the phase fluctuations, in accordance with the Goldstone theorem, changes dramatically. Substitution of (10.19) in (10.35) results in the disappearance of the temperature dependence of the phase fluctuation contribution and, calculating the second derivative, one sees that only the fluctuations of the order parameter modulus contribute to the heat capacity.As a result the heat capacity, calculated below Tc , turns out to be proportional to that found above: D ıC− = 2 2 −1 ıC+ .

10.2.2

Gi(0)

 7(3) 1 Tc0 √ ≈ 13.3 2 EF  Tc V

03 V

question: where are the limits of applicability of the GL theory? This theory is valid not too near to the transition temperature, where the fluctuation correction is still small in comparison with the heat capacity jump. Let us define as the Ginzburg–Levanyuk number Gi(D) [3,4] the value of the reduced temperature at which the fluctuation correction (10.30) equals the value of C see (10.21):8 2   4−D D 1 VD D 2 −1 2 #D b(4m) Tc Gi(D) = . (10.36) ˛ V Substituting into this formula the microscopic values of the GL theory parameters (10.17) one can find 

Gi(D)

7(3)#D = 8 2



VD V



1 D Tc  D

2  4−D

.

(10.37)

−1 1−D −1 Since D Tc ∼ D vF /c ∼ pD−1 c one can F c ∼ a convert this formula to the form    2  7(3)#D VD c aD−1 4−D Gi(D) ∼ , 8 2 V D

where a is the interatomic distance. It is worth mentioning that in bulk conventional superconductors, due to the large value of the coherence length (c ∼ 10−6 ÷ 10−4 ) cm, which drastically exceeds the interatomic distance (a ∼ 10−8 cm), the fluctuation correction to the heat capacity is extremely small. However, the fluctuation effect increases for small effective sample dimensionality and small electron

One can see that some arbitrariness occurs in this definition. For instance, the Gi number could be defined as the reduced temperature at which the AL correction to conductivity is equal to the normal value of conductivity (as was done in [13,14]). Such a definition results in the change of the numerical factor in the Gi number: Gi(2, ) = 1.44Gi(2,h.c.).

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mean free path. For instance, the fluctuation heat capacity of a superconducting granular system is readily accessible for experimental study. Using the microscopic expression for the coherence length (10.4), the Ginzburg-Levanyuk number (10.37) can be evaluated for different cases of clean (c) and dirty (d) superconductors of various dimensionalities and geometries (film, wire, whisker and granule are supposed to have 3D electronic spectrum): One can see that for the 3D clean case the result coincides with the original Ginzburg evaluation and demonstrates the negligibility of the superconducting fluctuation effects in clean bulk materials. 10.2.3

Scaling and Renormalization Group

In the above study of the fluctuation contribution to heat capacity we have restricted ourselves to the temperature range out of the direct vicinity of the critical temperature: | |  Gi(D) . As we have seen the fluctuations in this region turn out to be weak and neglecting their interaction was justified. In this section we will discuss the fluctuations in the immediate vicinity of the critical temperature (| |  Gi(D) ) where this interaction turns out to be of great importance. We will start with the scaling hypothesis, i.e. with the belief that in the immediate vicinity of the transition the only relevant length scale is (T). The temperature dependencies of all other physical quantities can be expressed through (T). This means, for instance, that the formula for the fluctuation part of the free energy (10.25) with the logarithm omitted is still valid in the region of critical fluctuations9 F(D) ∼ − −D ( ) ,

(10.38)

the coherence length is a power function of the reduced temperature: ( ) ∼ − . The corresponding formula for the fluctuation heat capacity can be rewritten as ıC ∼ − 9 10

@2 F ∼ D −2 . @ 2

(10.39)

As was demonstrated in the Introduction, the GL functional approach, where the temperature dependence of (T) is determined only by the diffusion of the electrons forming Cooper pairs, ( ) ∼ −1/2 and ıC ∼ −1/2. These results are valid for the GL region ( | |  Gi) only, where the interaction between fluctuations can be neglected. In the immediate vicinity of the transition (the so-called critical region), where | |  Gi, the interaction of fluctuations becomes essential. Here fluctuation Cooper pairs affect the coherence length themselves, changing the temperature dependencies of ( ) and ıC( ). In order to find the heat capacity temperature dependence in the critical region one would have to calculate the functional integral with the fourth order term,accounting for the fluctuation interaction, as was done for the 0D case. For the 3D case up to now it is only known how to calculate a Gaussian type functional integral. This was done above when, for the GL region, we omitted the fourth order term in the free energy functional (10.16). The first evident step in order to include in consideration of the critical region would be to develop a perturbation series in b.10 Any term in this series has the form of a Gaussian integral and can be represented by a diagram,where the :solid lines correspond 9  to the correlators ¦ (r)¦ ∗ (r ) . The “interactions” b are represented by the points where four correlator lines intersect (see Fig. 10.2). This series can be written as C∼

 n=0

 cn

Gi(D)

 4−D 2 n

.

For  Gi it is enough to keep only the first two terms to reproduce the perturbational result obtained above. For  Gi all terms have to be summed. It turns out that the coefficients cn can be calculated for the space dimensionality D → 4 only. In this case the complex diagrams in Fig. 10.2 (like the diagram similar to an envelope) are small by the parameter " = 4 − D and in order to calculate cn it

The logarithm in (10.25) is essential for the case D = 2. This case will be discussed later. Let us mention that this series is an asymptotic one, i.e. it does not converge even for small b. One can easily see this for small negative b, when the integral for the partition function evidently diverges. This is also confirmed by the exact 0D solution (10.23).

10

Fluctuation Phenomena in Superconductors

379

Fig. 10.2. Examples of diagrams for the fluctuation contribution to b

is sufficient to sum the relatively simple “parquet” type diagrammatic series. Such a summation results in the substitution of the “bare” vertex b by some effective interaction  b, which diminishes and tends to zero when the temperature approaches the transition point. Such a method was originally worked out in quantum field theory [15–17]. For the problem of a phase transition such a summation was first accomplished in [18]. Instead of a direct summation of the diagrams it is more convenient and physically obvious to use the method of the renormalization group. In the case of quantum field theory this method was known long ago [19,20]. For phase transition theory it was proposed in [18, 21], but the most simple and evident formulation was presented by Wilson [22]. The idea of the renormalization group method consists in separating the functional integration over “fast” ( |k|  ) and “slow” ( |k|<  ) fluctuation modes. If the cut off   is large enough the fast mode contribution is small and the integration over them is Gaussian. After the first integration over fast modes the functional obtained depends on the slow ones only. They can, in turn, be divided into slow (|k| <  1 ) and fast ( 1 < |k| <  ), and the procedure can be repeated. Moving step by step ahead in this way one can calculate the complete partition function. As an example of the first step of renormalization, the partition function calculation below Tc can be recalled. There we separated the order parameter into  (“slow” mode) and the the space-independent part¦

fluctuation part (r) (“fast” mode), which was believed to be small in magnitude. Being in the GL region it was enough to average over the fast variables just once,while in the critical region the renormalization procedure requires subsequent approximations. The cornerstone of the method consists in the fact that in the critical region at any subsequent step the free energy functional has the same form.For D close to 4 this form coincides with the initial free energy GL functional but with the coefficients a  and b  depending on  . We will perform these calculations by the method of mathematical induction. Let us suppose that after the (n − 1)-st step the free energy functional has the form:    n−1 ] = FN , n−1 + dV a n−1 |¦  n−1 |2 F [¦  b n−1 1 4 2   |¦ n−1 | + |∇ ¦ n−1 | . + 2 4m

(10.40)

 n−1 in the form ¦  n−1 = ¦  n +  n and Writing ¦ choosing  n close enough to  n−1 it is possible to make  n so small that one can restrict the functional to the quadratic terms in  n only and perform the Gaussian integration in complete analogy with (10.34). The important property of the spaces with dimensionalities close to 4 is the possibility to  n . In choose  n   n−1 and still to have  n  ¦  n can be taken as an independent coorthis case ¦ dinate, and one can use the result directly following from (10.34):

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A. I. Larkin and A.A. Varlamov

 n ] = FN , n−1 + F [¦



  n |2 dV a n |¦

 b n 1  n |4 +  n |2 |¦ |∇ ¦ 2 4m 1 Tc T  ln − k2  2 (3b n |¦ n |2 + a n + 4m )  n |k| n−1 2 Tc + ln (.10.41) 2  b n |¦ n |2 + a n + k +

4m

 n Expanding the last term in (10.41) in a series in ¦  one can get for F [¦ n ] the same expression (10.40) with the substitution of  n−1 →  n . From (10.41) it follows that Fn, n = Fn, n−1 − T

   n |k| n−1

b n = b n−1 − 5T

  n |k| n−1

k2 4m

a n +

 n |k| n−1

a n = a n−1 + 2T

Tc

ln 

b n  a n +

k2 4m

b2 n



a n +

k2 4m

,

 , (10.42) 2 .

 2 4m

⎧ @a( ) b( ) D−1 ⎪ ⎪  = −2T D  ⎪ ⎪ ⎪  2 @  ⎪ a( ) + 4m ⎨ ⎪ @b( ) b2 ( ) D−1 ⎪ ⎪ = 5T

⎪ D  2 ⎪ ⎪  2 ⎩ @  a( ) + 4m 11

,

 2 /4m  Tc0Gi(D)

(10.45)

in order to move away from the first approximation. Let us recall that in the framework of the Landau theory of phase transitions the coefficient a(Tc0 ) = 0 at the transition point and this can be considered as the MFA definition of the critical temperature Tc0 . The same statement for the function a = a( ) in the framework of the renormalization group method can be written as the a(Tc0,   ∼  −1) = 0. With the decrease of   the effect of critical fluctuations is increasingly taken into account and the renormalized value of the critical temperature decreases, being defined by the equation: a(Tc ( ),  ) = 0. Finally, after the application of the complete renormalization procedure, one can define the real critical temperature Tc , shifted down with respect to Tc0 due to the effect of fluctuations, from the equation: a(Tc ,   = 0) = 0 .

Passing to a continuous variable  n →   one can rewrite these recursion equations as the set of differential equations: @F( ) Tc = −T D  D−1 ln  @  a( ) +

These renormalization group equations are evidently valid for small enough   only, where the transition from discrete to continuous variables is justified.This means at least

(10.43)

(10.46)

It is easy to find this shift in the first approximation. Indeed, let us integrate the first equation of (10.44) over   in limits [0,  −1 ]. The main contribution to the integral will be determined by the region where  2 a( )  4m . Being far from the critical point one can assume that the coefficient b = const. and then −1 a(  )

1/ da = −8mT D b  D−3 d  .

a( ) = ˛ıTc = −1

.

a(0)

(10.44)

0

For the 3D case this gives the shift of the critical temperature ıTc due to fluctuations11

In the same way we can also analyze the shift of the critical temperature in the 2D case and obtain: ıTc(2) 1 = −2Gi(2) ln . Tc 4Gi(2)

(10.47)

As we will show below both 3D and 2D results for ıTc coincide with those obtained by the analysis of the effect of fluctuations on superconducting density in the perturbation approach.

10

b 2mb ıTc(3) 1 =− ∼− (10.48) Tc ˛ 2Tc ˛ 2  3 7(3) 8 =− = − Gi(3) . 16 3  Tc  3  Let us come back to study of properties of the system of equations (10.44). One can find its partial solution at T = Tc in the form: 4−D  2 , (10.49) 4m[5 + (4 − D)] 5 4−D  4−D . b(Tc ,  ) = 2 16m T D [5 + (4 − D)]2 a(Tc ,  ) =

These power solutions are correct in the domain of validity of the system (10.44) itself, i.e. for small enough   defined by the condition (10.45). Nevertheless, in a space of dimensionality close to 4 (D = 4 − ", "  1) it is possible to extend their validity up to the GL region and to observe their crossover to the GL results: a(Tc ) = 0; b(Tc ) = b0 = const. Indeed, in this case, due to proportionality of a( ) to " → 0, one can omit it in the denominators of the system (10.44) and to write down the solution for b( ) in the form 80m2 T D ( D−4 −  4−D ) . (4 − D) (10.50) We have chosen the constant of integration as b−1 0 − 80m2 4−D T

 in order to match the renormalizaD (4−D) tion group and GL solutions at the value of   =  max ∼  −1. Now let us pass to study of the function a(T,  ) for the same interesting case of space dimensionality D → 4 for temperatures slightly different (but still close enough) from Tc , where one can write b−1 (Tc ,  ) = b−1 0 +

a(T,  ) = a(Tc ,  ) + ˛(Tc ,  )Tc . The first term on the right hand side is determined by (10.49). In order to determine ˛(Tc ,  ) let us expand the first equation in (10.44) in terms of @˛(Tc ,  ) b(Tc ,  ) D−1 = 2T D  2 ˛(Tc ,  ) . @   2 a(Tc ,  ) + 4m (10.51)

Fluctuation Phenomena in Superconductors

381

For  2 /4m  ˛(Tc ,  )Tc

(10.52)

we can again use the solution (10.50) for b(Tc ,  ) and omit a(Tc ,  ) in the denominator of (10.51). The constant of integration, appearing in the process of solution of (10.51) is chosen in accordance with the condition that for   =  max ∼  −1 we match ˛(Tc ,  ) = ˛(Tc0,  −1) = ˛0 with the GL theory:  −2/5 80m2 b0 ˛(Tc ,  ) = ˛0 1 + T D ( D−4 −  4−D ) . (4 − D) (10.53) The condition (10.52) can be written as     −1 (T), where (T) is the generalized coherence length, determined by the equation:    −2 (T) = 4m˛ Tc ,  −1 (T) Tc . Such a definition is valid at any temperature. For example, far enough from  the critical point, in the GL region, ˛ Tc ,  −1 (T) = ˛0 and one reproduces the result (10.3). Vice versa, in the critical region the main contribution on the right hand side of the (10.53) results from the second term containing  D−4 so, putting  −1 (T) =  , one can rewrite the selfconsistent equation for (T) and get (T) = (4m)(1−D)/2

4−D √ − , 20b0T D Tc ˛0

(10.54)

where 2 = [1 − (4 − D)/5]−1. As has already been mentioned, strictly speaking this result was carried out for " = 4 − D  1, so it is confident up to the first in " expansion only:  = 1/2 + "/10. Nevertheless, extending it to " = 1 (D = 3) one can obtain 3 = 3/5. Let us pass to the calculation of the critical exponent of the heat capacity in the immediate vicinity of the transition. For this purpose one can calculate the second derivative of equation (10.43) with respect to : @C( )  D−1 ˛ 2 ( ) = T 2 D  2 . @   2 ˛( )T c + 4m

(10.55)

The heat capacity renormalized by fluctuations has the value C(  = 0), which is the result of integration over all fluctuation degrees of freedom. Carrying out

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A. I. Larkin and A.A. Varlamov

the integration of (10.55) over all     −1 one can divide the domain of integration on the right hand side in two:     −1 (T) and  −1 (T)      −1 . In the calculation of the integral over the region     −1 (T)  2 holds, and the functhe inequality ˛( )T c  4m tion ˛( ) can be omitted in the denominator. In the numerator of (10.55) one can use for ˛( ) the solution (10.53). In the region     −1 (T) one has to use the partial solution (10.49) for ˛( ) and can find that the contribution of this domain has the same singularity as that from the region     −1 (T), but with a coefficient proportional to (4−D)2 = " 2 ,hence negligible in our approximation. The result is  

 15 4 (4 − D) 4/5 2 2 C(  = 0) = ˛0 (4mT) D 5 b0 5 4−D  5 (T) . × 4−D Substituting the expression for (T) one can finally find   15 m2 T 2 12/5 2 C = 2 ˛0 5 D 4 −˛ , (10.56) b0 (4 − D) confirming the validity of the scaling hypothesis and the relation (10.39).The critical exponent in (10.56) is ˛=

(4 − D) ≈ "/10 . 10[1 − (4 − D)/5]

One can see that, generally speaking, the critical exponents  and ˛ appear in the form of series in powers of ". More cumbersome calculations permit finding the next approximations for them in " = 4 − D. Nevertheless it is worth mentioning that even the first approximation, giving 3 = 3/5 and ˛3 = 1/10 for " = 1, is already weakly affected by the following steps of the expansion in powers of " [23]. One can notice that the exercise performed in this section has more of an academic than a practical character. Indeed, the results obtained turn out to be applicable to the analysis of the critical region of a 3D superconductor only if Gi is so small that the theoretical predictions are hardly experimentally observable. Nevertheless, we demonstrated the RG method, which helps one to see the complete picture of the fluctuations manifestation in the vicinity of the critical temperature.

10.2.4

Fluctuation Diamagnetism

Qualitative Preliminaries In this section we discuss the effect of fluctuations on the magnetization and the susceptibility of a superconductor above the transition temperature. Being the precursor effect for the Meissner diamagnetism, the fluctuation induced magnetic susceptibility has to be a small correction with respect to the diamagnetism of a superconductor, but it can be comparable to or even exceed the value of the normal metal diamagnetic or paramagnetic susceptibility and can be easily measured experimentally.As was already mentioned in the Introduction the temperature dependence of the fluctuation induced diamagnetic susceptibility can be qualitatively analyzed on the basis of the Langevin formula, but some precautions in the case of low dimensional samples have to be taken. As regards the 3D case we would like to mention here that expression (10.12), presented in terms of (T), has a wider region of applicability than the GL one. Namely, the scaling arguments are also valid for diamagnetic susceptibility and one can write the general relation 1 1,  Gi 2 −1/2  1/2− (3) ∼ −e T(T) ∼ − P ,  Gi , Gi (10.57) which is also valid in the region of critical fluctuations in the immediate vicinity of the transition temperature. Here, in order to define the scale of fluctuation effects, we have introduced the Pauli paramagnetic susceptibility P = e2vF /4 2. Moreover, the Langevin formula permits us to extend the estimation of the fluctuation diamagnetic effect to the other side beyond the GL region: to high temperatures T  Tc . The coherence length far from the transition becomes a slow function of temperature. In a clean superconductor, far from Tc , (T) ∼ vF /T, so one can write (3c) (T  Tc ) ∼ −e2T(T) ∼ − P

(10.58)

and see that the fluctuation diamagnetism turns out to be of the order of the Pauli paramagnetism even

10

far from the transition.More precise microscopic calculations of (3) (T  Tc ) lead to the appearance of ln2 (T/Tc ) in the denominator of (10.58). In the 2D case, (10.12) is applicable for the estimation of (2) in the case when the magnetic field is applied perpendicular to the plane, permitting 2D rotations of fluctuation Cooper pairs in it: n 2 EF R  ∼ e2 T 2(T) ∼ − P . m T − Tc (10.59) This result is valid for a wide range of temperatures and can exceed the Pauli paramagnetism by factor EF even far from the critical point (we consider the T clean case here). For a thin film (d  (T)) perpendicular to the magnetic field the fluctuation Cooper pairs behave like effective 2D rotators,and (10.12) can still be used, though one has to take into account that the susceptibility in this case is calculated per unit square of the film. So for the realistic case (from the experimental point of view) of the dirty film, one just has to use in (10.59) the expression (10.6) for the coherence length:   e2T 2 Tc l (2d) ∼  (T) ∼ − P . (10.60) d d T − Tc (2c) (T) ∼ e2

Now let us discuss the important case of a layered superconductor (for example, a high temperature superconductor). It is usually supposed that the electrons move freely in conducting planes separated by a distance s. Their motion in the perpendicular direction has a tunneling character, with effective energy J. The related velocity and coherence length can be estimated as vz = @E(p)/@p⊥ ∼ J/p⊥ ∼ sJ and z,(c) ∼ sJ/T for the clean case. In the dirty case the anisotropy can be taken√into account√ in the spirit of (10.6), yielding z,(d) ∼ D⊥ /T ∼ sJ /T. We start from the case of a weak magnetic field applied perpendicular to layers. The effective area of a rotating fluctuation pair is x ( )y ( ). The density of Cooper pairs in the conducting layers (10.10) has to be modified for the anisotropic case. Its isotropic 3D value is proportional to 1/( ), which now has to  be read as ∼ 1/ x ( )y ( ). The anisotropy of the 12

Fluctuation Phenomena in Superconductors

383

electron motion leads to a concentration of fluctuation Cooper pairs in the conducting layers,and hence to  an effective increase of the Cooper pairs density of x ( )y ( )/z ( ) times its isotropic value. This increase is saturated when z ( ) reaches the interlayer distance s,so finally the anisotropy factor appears in the form x ( )y ( )/ max{s, z ( )}, and the square root in its numerator is removed in the Langevin formula (10.12), which is rewritten for this case as (layer,⊥) ( , H → 0) ∼ −e2 T

x ( )y ( ) . (10.61) max{s, z ( )}

The existence of a crossover between the 2D and 3D temperature regimes in this formula is evident: as the temperature tends to Tc the diamagnetic susceptibility √ temperature dependence changes from 1/ to 1/ . This happens when the reduced temperature   reaches its crossover value cr = r z ( cr ) ∼ s . The anisotropy parameter

r=

4z2 (0) s2

⎧  ⎪ , T  1 J ⎨ 4 = T ⎪ ⎩ 7(3) , T  1 8 2 T 2

(10.62)

plays an important role in the theory of layered superconductors.12 It is interesting to note that this intrinsic crossover, related to the spectrum anisotropy, has an opposite character to the geometric crossover that occurs in thick enough films when (T) reaches d. In the √ latter case the characteristic 3D 1/ -dependence takes place far enough from Tc (where (T)  d), is changed to the 2D 1/ law (see (10.60)) in the immediate vicinity of transition (where (T)  d) [24]. It is worth mentioning that in a strongly anisotropic layered superconductor the fluctuationinduced susceptibility may considerably exceed the normal metal diamagentic and paramagnetic effects even relatively far from Tc [25,26]. Let us consider a magnetic field applied along the layers. First it is necessary to mention that the fluctuation diamagnetic effect disappears in the limit J ∼ z → 0.Indeed,for the formation of a circulating

We use here a definition of r following from microscopic theory (see Sect. 10.6).

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A. I. Larkin and A.A. Varlamov

current it is necessary to tunnel twice, so

Zero-Dimensional Diamagnetic Susceptibility

z2 (layer,) ∼ −e2 T max{s, z (T)}   sJ J/T √ . √ ∼ − P vF max{ , J/T} In the general case of an anisotropic superconductor, choosing the z-axis along the direction of magnetic field H, the following extrapolation of the results obtained may be written: ∼ − e2 T ×

(10.63) x2( )y2( )

max{a, x ( )} max{b, y ( )} max{s, z ( )}

.

This general formula is useful for the analysis of the fluctuation diamagnetism of anisotropic superconductors or samples of some specific shape: granular, quasi-1D, quasi-2D, and3D. It is also applicable to the case of a thin film (d  z ( )) placed perpendicular to the magnetic field: it is enough to replace z ( ) by d in (10.64). Nevertheless, (10.64) cannot be applied to the cases of thin films in parallel fields, wires and granules.In those cases the Langevin formula 9 : (10.12) can still be used with the replacement of R2 → d 2 :  (D) ∼ − P

T vF

  2−D d D−1 ∼ D/2−1 .

The magnetic field dependence of the fluctuation part of free energy in these cases is reduced only to account for the quadratic shift of the critical temperature versus magnetic field. For 3D systems or in the case of a film in a perpendicular magnetic field the critical temperature depends on H linearly, while the magnetic field de∗ pendent part of the free energy for H  Hc2 (− ) (the ∗ line Hc2(− ) is mirror-symmetric to the Hc2( ) with respect to y-axis passing through T = Tc ) is proportional to H 2 . This is why the magnetic susceptibility is determined by (10.64) for weak enough magnetic fields H  ¥0 /[x ( )y ( )] = Hc2( )  Hc2(0) only. In the vicinity of Tc these fields are small enough. 13

For quantitative analysis of the fluctuation diamagnetism we start by writing down the GL functional for the free energy (see (10.16)) in the presence of the magnetic field   b F [¦ (r)] = FN + dV a|¦ (r)|2 + |¦ (r)|4 2  2 H·B 1 B 2 | (−i∇−2eA) ¦ (r)| + − , + 4m 8 4 (10.64) where A is vector potential. As long as fluctuation effects are comparatively small, the average magnetic field in the metal B may be assumed to be equal to the external field H. Thus we omit the last two terms in (10.64) (see later on). The fluctuation contribution to the diamagnetic susceptibility in the simplest case of a “zerodimensional” superconductor (spherical superconducting granule of diameter d  ( )) was considered by V. Shmidt [11]. In this case the order parameter does not depend on the space variables and the free energy can be calculated exactly for all temperatures including the critical region in the same way as was done for the case of the heat capacity in the absence of a magnetic field. Formally the effect of a magnetic field in this case is reduced to the renormalization of the coefficient a, or, in other words, to the suppression of the critical temperature. This is why one can use the same formula (10.23) for the partition function with the critical temperature Tc shifted by magnetic field as:13   4 2  2 2 Tc (H) = Tc (0) 1 − A  . ¥02

(10.65)

Here ¥0 = e is the magnetic flux quantum and . . . means the averaging over the sample volume. Such a trivial dependence of the properties of 0D samples on magnetic field immediately allows one to understand its effect on the heat capacity of a granular sample. Indeed, with the growth of the field the

Let us stress the difference between the H 2 shift of the critical temperature for a zero-dimensional granule and the linear shift in the case of bulk material.

10

temperature dependence of the heat capacity presented in Fig. 10.1 just moves in the direction of lower temperatures. In the GL region Gi(0)  one can write the asymptotic expression (10.23) for the free energy: F(0) ( , H) = −T ln

 ˛ +



.

4 2  2 A2  ¥02

In the case of a spherical particle the relation A2  = 1 ( 12 H × r)2  = 40 H 2 d 2 can be used in full analogy with the calculation of the moment of inertia of a solid sphere.In this way an expression for the 0D fluctuation magnetization valid for all fields H  Hc2(0) can be found: M(0) ( , H) = −

1 @F(0) ( , H) V @H

= −T 

6 2 5¥02 d

+

22 2 2 H d 10¥02

Fluctuation Phenomena in Superconductors

one has to choose the gauge of the vector-potential 9 : 2 2 A = 12 H × r yielding A2 (wire,) = H32d (the calculation of this average is analogous to that of the moment of inertia of a solid sphere). For a wire in a perpendicular field, or a film in a parallel field, the gauge has to be chosen in the form A =(0, Hx, 0) (to avoid the appearance of currents perpendicular to 9 : 2 2 surface). One can find A2 (wire,⊥) = H16d for a wire 9 : 2 2 and A2 (film,) = H12d for a film. Calculating the second derivative of (10.28) with the appropriate magnetic field dependencies of the critical temperature one can find the following expressions for the diamagnetic susceptibility:

(D) ( ) = −2 H.

(10.66)

One can see that the fluctuation magnetization turns out to be negative and linear up to some crossover field, which can be called the temperature dependent ¥0 upper critical field of the granule Hc2(0) ( ) ∼ d( ) = √  H (0) at which it reaches a minimum.At higher d c2 fields Hc2(0) ( )  H  Hc2(0) the fluctuation magnetization of the 0D granule decreases as 1/H. In the weak field region H  Hc2(0) ( ) the diamagnetic susceptibility is:   6T02 1  1 2 (0) ( , H) = − ≈ − · 10 , P d 5¥02d which coincides with our previous estimate in its temperature dependence but the numerical factor found is very large. Let us underline that the temperature dependence of the 0D fluctuation diamagnetic susceptibility turns out to be less singular than the 0D heat capacity correction: −1 instead of −2 . The expression for the fluctuation part of free energy (10.28) is also applicable to the cases of a wire or a film placed in a parallel field; as was already mentioned above all its dependence on magnetic field is manifested by the shift of the critical temperature (10.65). In the case of the wire in a parallel field

385

T vF

(10.67)

⎧ 1 ⎪ wire in parallel field √ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2 wire in perpendicular field × P √ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d 1 ⎪ ⎩ ln , film in parallel field. 3

GL Treatment of Fluctuation Magnetization Let us analyze quantitatively, on the basis of the GL functional, the temperature and field dependencies of the fluctuation magnetization. We will carry on the discussion for a layered superconductor. As has already been mentioned this system has great practical importance because of its direct applicability to high temperature superconductors,where the fluctuation effects are very noticeable. Moreover, the general results obtained will allow us to analyze 3D and 2D situations as limiting cases. The effects of a magnetic field are more pronounced for perpendicular orientation, so let us consider this case first. The generalization of the GL functional for a layered superconductor (Lawrence–Doniach (LD) functional [27]) in a perpendicular magnetic field can be written as

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A. I. Larkin and A.A. Varlamov

FLD [¦ ] =

 l

 b d r a |¦l |2 + |¦l |4 2 2

(10.68)

 2 1  + ∇ − 2ieA |¦l  + J |¦l+1 − ¦l |2 4m

SH  ¦n,kz n (r) exp(ikz l). ¥0

 ,

(10.69)

n,kz

The factor SH/¥0 takes into account the degeneracy of Landau state n while kz is the momentum component along the direction of the magnetic field.Substituting this expansion into (10.69) one can find the LD free energy as a functional of the ¦n,kz coefficients: 

  ˛Tc + FLD ¦n,kz = n,kz

SH  T ln ¥0

(10.71)

n,kz

where ¦l is the order parameter of the l-th superconducting layer and the phenomenological constant J is proportional to the Josephson coupling between adjacent planes. The gauge with Az = 0 is chosen in (10.69). In the immediate vicinity of Tc the LD functional is reduced to the GL one with the effective mass M = (4J s 2)−1 along c-direction, where s is the inter-layer spacing. One can relate the value of J to the coherence length along the c-direction: J = 2˛Tc z2 /s 2. Since we are dealing with the GL region the fourth order term in (10.69) can be omitted. As it is well known the Landau representation is the most appropriate one for problems related with the motion of a charged particle in a uniform magnetic field. Basic wave functions can be chosen as the products of a plane wave propagating along the magnetic field direction and a Landau state wave functions n(r). Let us expand the order parameter ¦l (r) on the basis of these eigenfunctions: ¦l (r) =

F ( , H) = −

1 H  (10.70) n+ 2m¥0 2  + J (1 − cos(kz s)) |¦n,kz |2 .

In complete analogy with the case of an isotropic spectrum the functional integral over the order parameter configurations ¦n,kz in the partition function can be reduced to a product of ordinary Gaussian integrals, and the fluctuation part of the free energy in a magnetic field takes the form:

×

˛Tc +

H 2m¥0



T  . n + 12 + J (1 − cos(kz s))

The sum in (10.72) is evidently divergent and in order to regularize it we introduce a formal cut-off parameter nc , the number of the last Landau level, at which the summation is interrupted. Its value can be evaluated by appealing to the restrictions of the GL theory, which breaks down for short wave length fluctuations and is valid for k  kmax , where 2 kmax /4m ∼ ˛T .

(10.72)

In terms of the Landau level summation this condition can be rewritten as: nc ∼ ˛T/!c ∼ ˛Tm¥0 /H   2 ∼ ¥0/ xy H ∼ Hc2 (0)/H .

(10.73)

The formal divergence of the GL free energy does not effect on such observable properties as the heat capacity and the diamagnetic susceptibility. Being the second derivatives of the free energy these physical values are well defined, the momentum (or Landau levels) summation converges well for them. Nevertheless, the problem of regularization arises for example in calculation of the 3D fluctuation magnetization. Being only the first derivative of the free energy it turns out to be also divergent and must be cut off. Let us stress that the ultraviolet divergence of the fluctuation part of the GL free energy is the result of the limitedness of the GL functional approach, unapplicable to the description of short wavelength fluctuations. This problem does not appear at all in the consistent microscopic theory, which correctly accounts for the short wavelength fluctuations and does not require any renormalization procedure. In the limit of weak fields one can carry out the summation over the Landau states by means of the Euler–Maclaurin transformation

10 N 

N+1/2

f (n) =

n=0

f (n)dn −1/2

−

 1    f (N + 1/2) − f (−1/2) , 24

and obtain /s STH 2 N sdkz (10.74) F( , H) = F( , 0) + 2 24m¥02 −/s   1 . × ˛Tc +J (1 − cos(kz s)) Here N is the total number of layers. After the momentum integration one gets: F( , H) = F( , 0) +

VH 2 1 , √ 24m˛s¥02 ( + r)

with the anisotropy parameter defined as14 4 2(0) 2J = z2 . (10.75) ˛T s The magnetic susceptibility in a weak field turns out [28,29] to be r=

 xy e2T . √ 3s ( + r) 2

(layer,⊥) = −

(10.76)

These results confirm the qualitative estimation (10.61), additionally providing the exact value of the numerical coefficient and the temperature dependence in the crossover region. In the limit r  (10.76) transforms into the diamagnetic susceptibility of the 3D anisotropic superconductor [30]. For a film of thickness d the integral over kz in expression (10.75) has to be replaced by a summation over the discrete kz and when z (T)  d only the term with kz = 0 has to be taken into account: (film,⊥) = − 14

2 e2T  xy . 3d

(10.77)

Fluctuation Phenomena in Superconductors

387

Note that these formulas predict a nontrivial increase of diamagnetic susceptibility for clean metals [29]. The usual statement that fluctuations are most important in dirty superconductors with a short electronic mean free path does not hold in the particular case of susceptibility, because here  turns out to be in the numerator of the fluctuation correction. Now we will demonstrate that, besides the crossovers in its temperature dependence, the fluctuation induced magnetization is also a nonlinear function of magnetic field, and these nonlinearities, different for various dimensionalities, take place at relatively low fields. This manifestation of the nonlinear regime in fluctuation magnetization, which is strong in comparison with the expected scale of Hc2(0), and hence, field-dependent fluctuation susceptibility, was the subject of the intensive debates in early 1970s [30–39] (see also the older excellent review of W.J. Skocpol and M. Tinkham [40]) and after the discovery of HTS [41–44] (see also the very recent detailed essay of T. Mishonov and E. Penev [45] and references therein). We will mainly follow here the paper of Buzdin et al. [46], dealing with the fluctuation magnetization of a layered superconductor, which permits observing in a unique way all varieties of the crossover phenomena in temperature and the magnetic field. Let us go back to the general expression (10.72) and evaluate it without taking the magnetic field to small. The difficulty in dealing with it consists in the divergence of the sum over Landau levels n. This divergence can be regularized (see [45,47]), but let us observe that in order to calculate the magnetization we must know the magnetic field dependent part of the free energy only.A very convenient method to bypass the divergence problem [48] is to calculate the difference F(H) − F(0), turning the sum over Landau states in F( , 0) into an integral and then, in turn, turning this integral into a sum of integrals over the

Let us stress the difference between J and J in the two definitions (10.62) and (10.75) of the anisotropy parameter r. The first one was introduced as the electron tunneling matrix element, while the second one enters in the LD functional as the characteristic Josephson energy for the order parameter. Later on, in the framework of the microscopic theory, it will be demonstrated that, in accordance with our qualitative definition, r ∼ J 2 , while J turns out to be proportional to J 2 too. In the dirty case it depends on the relaxation time of the electron scattering on impurities: J ∼ ˛J 2 max{, 1/T}. Hence both definitions (10.62), appearing in the qualitative consideration, and (10.75), following from the LD model, are consistent.

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A. I. Larkin and A.A. Varlamov

Fig. 10.3. Schematic representation of the different regimes for fluctuation magnetization in ∗ the (H, T) diagram. The line Hc2 (T) is mirrorsymmetric to the Hc2 (T) line with respect to a y-axis passing through T = Tc . This line defines the crossover between linear and nonlinear behavior of the fluctuation magnetization above Tc

unit length intervals x ∈ [n − 1/2, n + 1/2]. Then VH F( , 0) = − lim T H→0 ¥0

/s −/s

1/2 ∞  dkz  dx 2 n=0 −1/2

T   × ln . H ˛Tc + 2m¥0 x + n + 12 +J (1 − cos(kz s)) and by introducing the dimensionless variable 15 H 2 , Hc2 (0) = 2m˛Tc /e = ¥0/2xy , Hc2(0) (10.78) one can write h=

TV F ( , H) − F( , 0) = − h 2 2sxy

 dz

−

1/2  

dx ln

n=0 −1/2

(2n + 1 + 2x)h + r/2(1 − cos z) + . × (2n + 1)h + r/2(1 − cos z) +

(10.79)

Performing the integrations over z and x in (10.79) and differentiating with respect to h we finally obtain a very convenient general expression for the fluctuation magnetization in a layered superconductor: ∞ T  ' (Rn + 1) M( , H) = n ln ¥0s n=0 ' (Rn )

+ ln 15

(10.80)

n + 1/2 ' (Rn + 1) − ' (Rn + 1/2) (Rn + 1/2)2 −  2

 with ' (x) = x + x 2 −  2, Rn = n + /2h +  and  = r/2h. The sum in (10.81) converges as 1/n2 and it provides a volume magnetization expression that can be compared with experiment. Let us comment on the different crossovers in the M( , H) field dependence analyzing the general formula (10.81). Let us fix the temperature  r. In this case the c-axis coherence length exceeds the interlayer distance (z  s) and in the absence of a magnetic field the fluctuation Cooper pairs motion has a 3D character. Supposing the magnetic field to be not too high (h  r) we may perform an expansion in (10.81) in 1/ and obtain √ T 2h1/2 M(3) (  r, h  r) = (10.81) ¥ 0 z   1/2 × (n + 1) n + 1 + 2h n 2    ) 1/2 3 (n + 1 + 3h . −n n + − 2h 2 n+ 1 + 2 2h For weak fields (h  ) the magnetization grows linearly with magnetic field, justifying our preliminary qualitative results. Nevertheless, this linear growth is changed to the nonlinear 3D high field regime √ M ∼ H already in the region of a relatively small fields Hc2( )  H (  h). The further increase of magnetic field leads to the next crossover in the magnetization field dependence at h ∼ r. However the expression (10.81) was obtained in the assumption

Let us recall that the exact definition of Hc2 (0) contains the numerical coefficient A(0) (see footnote 3).

10

h  r as an expansion over 1/,and it does not work any more. Handling with the Hurvitz zeta-function the summation in (10.81) for the 3D case can be carried out for an arbitrary field [34]:  1/2 √ T 2 M(3) ( ; h  r) = 3 h (10.82) ¥0 s r       1 1 1 1 − , + . ×  − , + 2 2 2h 2 2 2h 2h One can see from this formula that for large fields the magnetization saturates at the value M∞ [49]: T ln 2 T = −0.346 , M(h  r) → M∞ = − 2 ¥0 s ¥0 s (10.83) which is a typical for 2D superconductors. Therefore at h ∼ r we have a 3D → 2D crossover in M(H) behavior in spite of the fact that all sizes of fluctua-

Fluctuation Phenomena in Superconductors

389

tion Cooper pair considerably exceed the lattice parameters. The effective “bidimensionalization” of the fluctuations is related to the effect of a strong magnetic field which “freezes out” the rotations along its direction. Let us stress that this crossover occurs in the region of already strongly nonlinear dependence of M(H) and therefore for a rather strong magnetic field from the experimental point of view in HTS. Fixing the temperature  r in the formula (10.81) one can find the general formula for 2D fluctuation regime [45]: M(2) (  r, h) =    1 T 1 ln  + − ln (2) (10.84) ¥0 s 2 2h 2     1 + −1 , − 2h 2 2h

√ Fig. 10.4. Fluctuation magnetization of a YBCO123 normalized on H as the function of temperature in accordance with the described theory shows the crossing of the isofield curves at T = Tc (0) = 92.3 K. The best is fit obtained for anisotropy parameter r = 0.09. Inset: magnetization curves as the function of magnetic field

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A. I. Larkin and A.A. Varlamov

where  (x) is the Euler gamma-function and (x) = d ln  (x)/dx was already cited in the introduction digamma function. Using (10.84) one can directly pass from the linear regime in a weak magnetic field corresponding to (10.59) to the saturation of magnetization (10.83) in strong fields. Near the line of the upper critical field (hc2 ( ) = − ) the contribution of the term with n = 0 in the sum (10.81) becomes most important and for the magnetization the expression   T h M(h) = −0346 √ ¥0 s (h − hc2)(h − hc2 + r) can be obtained. It contains the already familiar to us “0D”regime (r  h−hc2  1),where the magnetization decreases as −M(h) ∼ h−h1 c2 (compare this with (10.66)), while for h − hc2  r the regime becomes “1D” and the magnetization decreases more slowly, 1 as −M(h) ∼ √h−h (see Fig. 10.3). c2 Such an analogy is observed in the next orders in Gi too. In [50] the analogy was demonstrated for the example of the first eleven terms for the 2D case and nine for the 3D case. Summation of the series of high order fluctuation contributions to the heat capacity by the Pade–Borel method resulted in a temperature dependence similar to the 0D and 1D cases without the magnetic field. Nevertheless, a considerable difference should not be forgotten: in the 0D and 1D cases no phase transition takes place, while in the 2D and 3D cases in a magnetic field a phase transition of first order to the Abrikosov vortex lattice state occurs. In conclusion, the fluctuation magnetization of a layered superconductor in the vicinity of the transition temperature turns out to be a complicated function of temperature and magnetic field, and it evidently cannot be factorized on these variables.The fit of the experimental data is very sensitive to the anisotropy parameter r and allows determination of the latter with a rather high precision [51,52]. In Fig.10.4 the successful application of the described approach to fit the experimental data on YBa2 Cu3 O7 is shown [53].

10.3

Fluctuations Below the Critical Temperature

10.3.1

Effect of Fluctuations on Superfluid Density and Critical Temperature

Superfluid Density Among the important thermodynamical properties of a superconductor is the magnetic field penetration depth (T).It is evidently asymmetric with respect to the critical temperature, growing when the temperature tends to Tc in the superconducting phase and is infinite in the normal phase.The simple London electrodynamics of a superconductor relates the current density j and (T) to the superfluid density ns : 1 2e2 A=− A, m 42 (T)

m . 8ns e2 (10.85) GL theory permits calculating the temperature dependence of (T) in the vicinity of the critical temperature, identifying the superfluid density with the average value of the order parameter j = −ns

2 = − ns =¦

2 (T) =

a b

and, hence, predicting its inverse square root divergence as a function of the reduced temperature:  (T) =

m b . 2 8e ˛Tc | |

(10.86)

Let us study how the appearance of fluctuation Cooper pairs affects the expulsion of the applied magnetic field. The superfluid density goes to zero at the transition point from below and its value determines (T), so the effect of fluctuations may be well pronounced. Let us first convince ourselves that the presence of fluctuation Cooper pairs above Tc does not result in the appearance of a superfluid density. One can calculate the fluctuation part of the supercurrent as ( [54,55] and a private communication from A. Koshelev in 1999)

10

9 : @Ffl (A) @ ln Z(A) =T jfl = − @A @A    ∗ D¦ (r)D¦ (r) @F [¦@A(r),A] exp − F [¦ T(r),A]   . =−  D¦ (r)D¦ ∗ (r) exp − F [¦T(r),0] This expression can be written in the form of two contributions: 9 : : 2e2 9 A |¦ (r)|2 A=0 jfl = j1 +j2 = − (10.87) m : e 9 Im[¦ (r)∇¦ ∗ (r)] A . − m The first one, j1 , just reproduces the London 9 :expression with the replacement of ns by |¦ (r)|2 . In the average the GL free energy with A = 0 can be used. The second term, j2 , has a more sophisticated nature. To get it different from zero one has to include the field dependent part of the GL free energy in the process of further averaging in fluctuation fields. Supposing the vector potential A to be weak enough one can expand the Gibbs exponent in it and find ' e2 j2 = − 2 Im[¦ (r)∇¦ ∗ (r)] (10.88) mT (  ∗ dV1 Im[¦ (r1 )∇¦ (r1)]A(r1 ) . A=0

In the following we will restrict ourselves to a superconductor of the strongly second order (  1). In this case the characteristic scale of the space variation of A(r1 ) is much larger than that of the order parameter, and the vector potential can be taken out of the integral in (10.88). Thinking about the GL region above Tc , one can expand the order parameter in a Fourier series and reduce the averaging to the expression   : 2e2  @ 9 j2 = |¦k |2 , k A m @k k

9 : which yields jfl = 0, as intuitively expected. Below Tc the situation is more cumbersome. In order to calculate j2 here let us separate the equilibrium value of the order parameter and its real and  + r + i i. imaginary fluctuating parts: ¦ (r) = ¦

Fluctuation Phenomena in Superconductors

391

This allows us to calculate the space integral in the expression (10.88):   k rk i,−k . dV1 Im[¦ (r1 )∇¦ ∗ (r1 )] = 2i k

The further functional integration can be carried out in the spirit of the previous calculations for the fluctuation contribution to the heat capacity below Tc resulting in $ &  9 : : 2e2 9 T 2 A |¦ (r)| A=0 − . jfl = − (2˛Tc | | + k 2/4m) m k : 9 2: 9 2 2 Let 9 2 :us calculate 9 : the remaining |¦ (r)| = ¦ + r +  r . The averaging of the first two terms i + 2¦ does not create any problem: T 1 , 2 2˛Tc | | + k 2/4m k 9 2: T  1 . i = 2 k 2/4m 9

2 r

:

=

(10.89)

k

9 : In order to carry out the r term the anharmonic contributions in the GL functional, originating from the fourth order term, have to be taken into account: 9 : r

=−

1  9 3  2¦

2 r

:

+

9

2 i

:

.

(10.90)

Finally one finds $ &  9 : 2e2 T 2  A ¦ −2 jfl = − m (2˛Tc | | + k2 /4m) k

and the superfluid density takes the form $ & ˛ 2b  1 Tc | | − 2 . ns (T)= b ˛ (2| | + k2 /4m˛Tc ) k (10.91) It is seen that in the 3D and 2D cases the sum over momenta in (10.91) is formally divergent at large k. This is not a first time when such problem has arisen and we know how to deal with it: this ultraviolet divergence is related to the restrictions on the applicability of the GL functional for |k|   −1 , so the integral has to be cut off at  · |k|= C ∼ 1.

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A. I. Larkin and A.A. Varlamov

For the 3D case the transparent way to cut off the sum is to separate the upper and lower limit contributions of the sum,which have different physical senses,  −2 by adding and subtracting the term ∼ |k| . As a result one obtains two contributions: the first one, | |-independent, originates from the upper limit cut off √ and the second turns out to be proportional to | | and appears from the lower limit. Using the microscopic relations between the GL parameters 8 2 ˛ 2 /b = 7(3)  and 4m˛Tc =  −2 one can finally find

In spite of the slower decrease of the fluctuation correction to the superfluid density when temperature tends to the critical√value, in comparison with the main contribution ( | | instead of | |), it becomes important only at | | ∼ Gi(3) . The 2D superfluid density (10.93) can be also written in terms of the 2D Ginzburg–Levanyuk number:

32m 2  Tc  2 (10.92) 7(3)   7(3) 7(3)  × 2| |Tc − C + 2| | . 3 16 3   3 16 2   3

and finally expressed in terms of the fluctuations renormalized critical temperature. It is enough to include the cut off parameter in the Tc -shift (compare with the RG estimation (10.47)):

ns3 (T) =

  mTc C2 ns2 (T) = | | − 2Gi(2) ln Gi(2) | |

In the 2D case the problem of the ultra-violet divergence is less important, since the cut off parameter turns out to be involved in a logarithm: 

ns2 (T) =



32m 2 7(3) C2  Tc  2 Tc | | − ln . 16 3   2 | | 7(3) (10.93)

(10.94)

C2 ıTc(2) = −2Gi(2) ln , Tc0 Gi(2) which results in16  ns2 (T) =

nBCS s2

Gi(2) Gi(2) ln 1−2 | | | |

 .

(10.95)

Fluctuation Shift of the Critical Temperature Let us discuss the second term appeared in (10.93). It evidently determines the fluctuation shift of the critical temperature ıTc = Tc − Tc0, where Tc0 is the mean field (BCS) value of the transition temperature. This shift ıTc(3) 7(3) 8 ∼− =− Gi(3) 3 3 Tc0 16  Tc0   coincides with that found in the framework of the renormalization group approach (see (10.49)).In further calculations we will include this shift in | | renormalizing the critical temperature to the value of Tc and identifying it with the experimentally observed transition temperature. The next order corrections demonstrate that the substitution of Tc0 → Tc is also necessary in the last term of (10.93), so for the superfluid density one can finally write  $ & 2Gi(3) BCS ns3 (T) = ns3 1 + 8 . | | 16

For sake of simplicity we still call the reduced temperature

10.3.2

Phase Fluctuations in 2D Systems

Let us discuss the results obtained for the superfluid density in the 2D systems. Being in the limits of the GL theory we cannot enter into the critical region, so the result (10.95) is valid for | |  Gi(2) only. At the same time the Berezinski–Kosterlitz–Thouless (BKT) theory of critical fluctuations in 2D systems establishes the value of superfluid density at the critical temperature (Nelson–Kosterlitz jump [56]) as: ns2 (Tc ) =

4mTc . 

(10.96)

An interpolation formula for the superfluid density of the 2D system can be written [13] by the unification of (10.95) and (10.96): ns2 (T) =

Tc −T Tc

as .

mTc 



 Gi(2) | | − 2 ln +4 . Gi(2) | | + Gi(2)

10

One can notice that the fluctuation to 9 :correction 9 : the average value of order parameter = r below Tc is divergent (see (10.89)). This means that the appropriate corrections cannot be supposed to be weak even relatively far from the transition | | : Gi(2) , 9 where the fluctuations of the value |¦ (r)|2 are still small. Neglecting the order parameter modulus fluctuations one can write the effective functional which describes the order parameter phase fluctuations only:   2 ns ns  2 2 F [' ] = d 2r ∇' = k 'k . 4m 4m k

Let us calculate now the average value of the order parameter using this functional without the assumption of weak fluctuations: %0 /   4mT 9 i' : ¦  = |¦ | e = |¦ | exp − . ns k 2 k (10.97) In the 2D case the sum in (10.97) diverges and ¦  = 0 [57]. Nevertheless a phase transition in such system exists. In order to see this let us study the behavior of the correlation function ¦ ∗ (0)¦ (r) at large distances. Above the mean field critical temperature the Fourier component |¦k |2  can be easily calculated with the use of expression (10.27): 

|¦k |  = 2

=

= > D¦k D¦k∗ |¦k |2 exp −˛( +  2k 2)|¦k |2  > = D¦k D¦k∗ exp −˛( +  2k 2)|¦k |2

1 , ˛( +  2k 2)

(10.98)

and the correlator ¦ ∗ (0)¦ (r) takes the form: 9 9 ∗ : : ¦ (0)¦ (r) (TTc0) = |¦k |2 exp (ikr) k

  1 r , (10.99) K0 = 4˛ 2 ( ) where K0 (x) is the modified function. At large  Bessel arguments K0(x  1) = 2x e−x and we find that in the normal phase this correlator decreases exponentially at distances r  ( ).

Fluctuation Phenomena in Superconductors

393

Below the transition point it is 9 ∗ : 9  : ¦ (0)¦ (r) = |¦ |2 exp i' (0) − i' (r) /  %0 

 (10.100) 1 − exp (ikr) 'k = |¦ |2 exp i k %  9 2: 1 2 2 = |¦ | exp − |1 − exp (ikr) | 'k . 2 k 9 : The average value 'k2 has already appeared in our calculations (for instance, in (10.89)) as the phase  −1 9 : fluctuation mode below Tc: 'k2 = 4mT ns k 2 .The last integral (sum) in the exponent of (10.100) evidently converges at small k. After the angular integration it can be expressed in terms of an integral of a Bessel function and it logarithmically diverges at the upper limit. This divergence 9must : be cut off at k ∼ −1 , where the expression for 'k2 is no longer valid. As a result below the transition point the correlator takes the form 9 ∗ : ¦ (0)¦ (r) |r| (T < Tc0) (10.101) mT   − n  s mT r r 2 2 = |¦ | = |¦ | exp − ln . ns  



These two (10.99) and (10.101), quite different, asymptotics of the correlator for the low and high temperature limits were obtained for the regions | |  Gi(2) . Nevertheless it is clear that there must be some point at which the high temperature exponential asymptotic changes to the low temperature power one. This temperature is reasonably called the point of the superconducting transition. In the 2D case such a transition is called the BKT transition.We do not discuss its properties in this review (see, for instance,the review articles of P.Minnhagen [58,59]). It is worth mentioning that, in spite of the fluctuation 9 : destruction of , in the low temperature phase the observable physical quantity ns is renormalized by fluctuations finitely and is not zero. In the GL region (| |  Gi(2) ) this renormalization even turns out to be weak. 10.3.3

Phase-Slip Events in 1D Systems

In the case of a 1D wire (10.91) also gives a small correction to ns . At the same time it is clear that at T = 0 the phenomenon of the superconductivity in

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1D does not exist. This results from: (a) the general statement about the absence of the phase transitions in 1D systems; (b) the evident exponential decrease of the correlator (10.100) for a 1D system below the transition point; and (c) the partition function in 1D can be calculated exactly.17 So, if the phase fluctuations are unable to destroy ns , what is the mechanism for killing superconductivity in 1D systems? It turns out that it is phaseslip events, which can also be called instantons or topological excitations. At temperatures beyond the critical region (| |  Gi(2) ) the probability of such excitations is exponentially small and they cannot be found by the methods of perturbation theory. There is a general statement of the absence of phase transitions in one-dimensional systems. The superconductive phase should be unstable and supercurrents cannot persist for a long time and must decay. Phase-slips are the mechanism of the supercurrent decay. Let us consider a closed loop made from a one-dimensional superconducting wire. The magnetic flux through this loop is a constant. If the phase of the wave function is a continuous function of the position, then it is not possible to change the supercurrent in the loop gradually. However, due to fluctuations, there is a finite probability to have the modulus of the order parameter vanish at some point. Thus, the phase is not defined at this point and a phase-slip may occur (a change of the phase along the loop by 2). Thus, the supercurrent that is determined by the gradient of the phase decays. Let us note that this reasoning works for one-dimensional systems only. In higher-dimensional systems, it is always possible to connect any two points by a path along which the phase is a continuous function. 17

18

With the aim of understanding the origin of phase-slip events let us start from a discussion of a small Josephson junction included in a superconducting ring of induction L. The potential energy related with the phase difference at the junction is the same as in a layered superconductor (10.69), so the energy of the ring can be expressed in terms of the magnetic flux ¥ 18 Er = EJ cos

2¥ ¥ 2 . + ¥0 2L

(10.102)

Besides the ground state with ¥ = 0 there are many metastable states with ¥ near n¥0 in such a system. The energies of such states are also the minima of the expression (10.102), but not absolute minima. In the limit of large EJ the energies of these minima are equal to ¥02n2 /2L. In order to transit from one metastable state to another one with lower energy, the system has to pass through a potential barrier. The heights of these barriers are determined by the maxima of Ex.(10.102) and are equal to 2EJ . The probability of such a “jump” (so-called “phase-slip process”) is proportional to exp{−2EJ /T}. The flux flow decay, and respectively, the voltage in the circuit are proportional to this probability: d¥ ∼ exp{−2EJ /T} . dt The positions of the maxima and minima of the function (10.102) are determined by the condition @Er /@¥ = 0. Let us consider the mechanism of the phase slip process in the circuit when the superconducting wire does not have a weak link and its free energy can be described by the GL functional (10.16). The minima of this expression are equal U=

It is possible to demonstrate [60] that the free energy of a D-dimensional classical field is equal to the ground state energy of the related D − 1-dimensional quantum mechanical system. Indeed, in the 1D case the functional integral turns out to be the same as the Feynman integral for a particle moving in the potential a|¦ |2 + b2 |¦ |4 . The ground state energy in this case is an analytical function of a and does not have a singularity at any temperature. The Josephson part of the Ginzburg–Landau energy, related to interlayer tunneling, was written as J |¦n − ¦n−1 |2 (see (10.69)). Supposing ¦n = |¦ |ei' n one can rewrite the corresponding energy as 2J |¦ |2 (1 − cos ' ). Applying this result to the superconducting ring with a weak link one can express the phase difference by means of the magnetic field flux:   ' = ∇' dl = 2e Adl = 2 ¥ /¥0 .

10

to −a2 U /2L+ ¥02 n2/2L and they correspond to the magnetic flux quantum ¥0. The transition between the neighboring minima can only occur through the saddle points of the functional (10.16). These saddle points correspond to the unstable solutions of the GL equation b ¦ + ¦ 3 +  2 (T)¦  = 0 . a The solutions of this equation are continuous in the same way as those corresponding to the minima of the functional, but they can pass through zero at some points where the modulus of the order parameter is equal to zero [61]. In the vicinity of these points the solution of the GL equation takes the form of an instanton: 7 x a ¦ (x) = − th . b 2(T) Substituting it into (10.16) one can find the energy of such an instanton in the wire of the cross-section S is [62]: √  √ 3/2 8 2 a2 2 | | F0 = S(T) = T. 3 2b 3 Gi(wire) So the probability for the superconducting ring to change its magnetic flux by one flux quantum only implies that, being thermically activated, it passes over such a barrier. It turns out to be exponentially small, and the wire resistance is equal to R ∼ exp{−F0 /T}. In the case of a thin film the topological defects carrying the flux quanta are given by fluctuation vortices. The energy of such a vortex is equal to E=

ns  ln . 4m 

In a thin film  is very large. In the framework of the BKT theory it is assumed to be infinite. Thus the film resistivity below the critical temperature turns out to be zero. 10.3.4

Fluctuations of Magnetic Field

In the above consideration we discussed fluctuations of the order parameter only supposing the magnetic

Fluctuation Phenomena in Superconductors

395

field to be a constant. At the same time the natural question arises: what is the effect of the usual blackbody radiation fluctuations on the superconductor properties? From the very beginning one can expect this effect to be small at low temperature. The order parameter fluctuation effects in bulk materials also turned out to be negligible. In the case of type I superconductors the coherence length  can noticeably exceed the magnetic field penetration depth . This means that the magnetic field fluctuations will take place on a scale less than the characteristic superconducting one, which can change some intrinsic superconducting properties. In this section we will show that magnetic field fluctuations, being small in amplitude, can change the order of phase transition of a type I superconductor. The same GL functional formalism permits us to take into account the fluctuations of a magnetic field [63]. Let us suppose that the external magnetic field H = 0, the system is below the critical temperature, the value of the equilibrium order parameter , and we neglect its fluctuations. Further calculais ¦ tions can be done in the spirit of the order parameter fluctuation calculations. One can rewrite (10.64) as the functional of the fluctuating vector potential, choosing the gauge  · A = 0 and writing it in the form of Fourier series:  F [A(r)] = FN + dV    |2 2 k 2 A 2 b 4 e 2 |¦  | 2 + |¦ | + × a|¦ A + 2 m 8    e 2 |¦  |2 k 2 , 0] + + A2k . = Fs [¦ m 8 k

The corresponding free energy can be written as   F = Fs [¦ , 0] + ln DAx DAy   %   |2 k 2 1  e 2 |¦ + A2k × exp − T m 8 k  T , 0] + T = Fs [¦ ln e2 |¦|2 . (10.103) k2 + 8 k m In the 3D case the sum in (10.103) evidently strongly diverges. The most divergent term is nothing else

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A. I. Larkin and A.A. Varlamov

than the consequence of the ultra-violet catastrophe and, as usual, it can simply be attributed to the background value of the free energy. The next, propor|2 , contribution to the sum in (10.103) tional to |¦ also turns out to be divergent. This divergence we treat in the same way as above: renormalize the critical temperature of the superconducting transition with respect to its mean field value. Regularized in this way free energy can finally be rewritten as √ 16T  e3 b 4 | .  |2 −  |3 + |¦ F = FN + a|¦ |¦ m3/2 2 One can see that an unusual cubic term appears in it due to the fluctuations of the electromagnetic field. This contribution, even being small in magnitude,changes the order of the superconducting phase transition. Indeed, putting a = 0 one can see that the minimum of the free energy takes place for a positive order parameter value. This means that already for positive a the free energy has two minima (one at ¦ = 0 and a second for positive ¦ ) appropriate to the normal and superconducting states. A first order phase transition takes place when the values of the free energy in both of these minima coincide, i.e. at  a = 128T 2e6 /bm3, or, using the microscopic expressions for  and ˛ 2 /b, at   ˛ 2 213T 4 e6 (T) 6 4  = 2 · Gi . = (2) (3) b (4˛mT)3 (T)

10.4

From this result one can see that for a type I superconductor this region noticeably exceeds the critical one and the phase transition, due to electromagnetic field fluctuations, turns out to be of first order. For type II superconductors the region of temperatures , where the electromagnetic field fluctuations are important, turns out to be inside the critical one. This means that these fluctuations have to be taken into account in the equations of the renormalization group, which also results in the conclusion of a first order phase transition in type II superconductors. Nevertheless for temperatures  Gi(3) these effects are negligible and in the following discussion we will not take them into account.

10.4.1

19

Ginzburg–Landau Theory of Fluctuations in Transport Phenomena

The appearance of fluctuating Cooper pairs above Tc leads to the opening of a “new channel” for charge transfer. In the Introduction the fluctuation Cooper pairs were treated as carriers with charge 2e while their lifetime GL was chosen to play the role of the scattering time in the Drude formula. Such a qualitative consideration results in the Aslamazov–Larkin (AL) pair contribution to conductivity (10.11) (socalled paraconductivity.19) Below we will present the generalization of the phenomenological GL functional approach to transport phenomena. Dealing with the fluctuation order parameter, it is possible to describe correctly the paraconductivity type fluctuation contributions to the normal resistance and magnetoconductivity, the Hall effect, thermoelectric power and thermal conductivity at the edge of transition. Unfortunately the indirect fluctuation contributions are beyond the possibilities of the description by the time-dependent GL approach and they will be calculated in the framework of the microscopic theory (see Sects. 10.6–10.8).

Time-Dependent GL Equation

In previous sections we demonstrated how the GL functional formalism allows one to account for fluctuation corrections to thermodynamical quantities. Let us discuss the effect of fluctuations on the transport properties of a superconductor above the critical temperature. In order to find the value of paraconductivity, some time-dependent generalization of the GL equations is required. Indeed, the conductivity characterizes the response of the system to the applied electric field. It can be defined as E = −@A/@t but, in contrast to the previous section, A has to be regarded

This term may have different origins. First of all, evidently, paraconductivity is analogous to paramagnetism and means excess conductivity. Another possible origin is an incorrect onomatopoeic translation from the Russian “paroprovodimost”, which means pair conductivity.

10

as time-dependent. The general nonstationary BCS equations are very complicated, even in the limit of slow time and space variations of the field and the order parameter. For our purposes it will be sufficient, following [64–70], to write a model equation in the vicinity of Tc , which in general correctly reflects the qualitative aspects of the order parameter dynamics and in some cases is exact. Let us keep in mind the GL functional formalism introduced above. If a deviation from equilibrium is assumed, then it is no longer possible to derive the GL equations starting from the condition that the variational derivative of the free energy is zero. At the same time, in the absence of equilibrium ¦ begins to depend on time.For small deviations from the equilibrium it is natural to assume that in the process of order parameter relaxation its time derivative @¦ /@t is proportional to the variational derivative of the free energy ıF /ı¦ ∗ , which is equal to zero at the equilibrium. But this is not all: side by side with the normal relaxation of the order parameter the effect of thermodynamical fluctuations on it has to be taken into account. This can be done by the introduction the Langevin forces (r, t) in the right hand side of equation describing the order parameter dynamics. Finally, gauge invariance requires that @¦ /@t should be included in the equation in the combination @¦ /@t + 2ie' ¦ , where ' is the scalar potential of the electric field. By concluding all these speculations one can write the model time-dependent GL equation (TDGL) in the form   ıF @ − GL + 2ie' ¦ = + (r, t) . (10.104) @t ı¦ ∗ with the GL functional F determined by (10.16), (10.64), and (10.69). The dimensionless coefficient

GL in the left hand-side of the equation can be related to pair lifetime GL (10.2): GL = ˛Tc GL = ˛/8 by the substitution in (10.104) of the first term of (10.16) only.20 Neglecting the fourth order term in the GL functional, (10.104) can be rewritten in operator form 20

Fluctuation Phenomena in Superconductors

397

as [B L−1 − 2ie GL ' (r, t)]¦ (r, t) = (r, t) ,

(10.105) B with the TDGL operator L and Hamiltonian C H defined as −1  @ B , B (10.106) L = GL + H @t

 C B − 2ieA)2 . H = ˛Tc − B2 (∇ We have introduced here the formal operator of the coherence length B  to have the possibility to deal with an arbitrary type of spectrum. For example, in the most interesting case for our applications to layered superconductors, the action of this operator is defined by (10.69). In the absence of an electric field one can write the formal solution of (10.105) as ¦ (0) (r, t) = B L(r, t) .

(10.107)

The Langevin forces introduced above must satisfy the fluctuation-dissipation theorem, which means that the correlator ¦p(0)∗ (t  )¦p(0) (t) at coinciding moments of time has to be the same as |¦p |2, obtained by averaging over fluctuations in thermal equilibrium (see (10.98)). This requirement is fulfilled if the Langevin forces (r, t) and  ∗ (r, t) are correlated by the Gaussian white-noise law 







 ∗ (r, t)(r , t ) = 2T Re GL ı(r − r )ı(t − t ) . (10.108) To show it let us restrict ourselves for sake of simplicity to the case of A = 0 and calculate the correlator  B∗B ¦ ∗ (r, t)¦ (r  , t) =  ∗ (r, t)L L(r , t)  dp ip(r−r ) e (10.109) = 2T Re GL (2)D ∞ d§ ∗ L (p, §)L(p, §) . × 2 −∞

L(p, §) can be found by performing the Fourier transform in (10.107): L(p, §) = (−i GL § + "p )−1 ,

(10.110)

It will be shown below that taking into account the electron–hole asymmetry leads to the appearance of an imaginary part of GL proportional to the derivative @ ln(v2 )/@E|EF ∼ O(1/EF ). This is important for such phenomena as the fluctuation Hall effect or fluctuation thermopower and, having in mind the most general formula, we will suppose

GL =  ˛/8 + i Im GL , where necessary.

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A. I. Larkin and A.A. Varlamov

resulting in 9

: ¦ ∗ (r, t)¦ (r  , t) p = 2T Re GL

∞ −∞

where

(10.111)

9 : d§ 1 = |¦p|2 , 2 2 2 2 GL § + "p

 2 p2 ) . "p = ˛Tc ( + B

(10.112)

is the Cooper pair energy spectrum.

cumbersome because it contains the Langevin force. To the first approximation it is possible to use here the order parameter ¦ (0) (r, t) (see (10.107)) unperturbed by the electric field as the convolution B L(r, t). In this way, using (10.108) and (10.109) we calculate the last average in (10.114)  9 :  2 d(r − r )e−ip(r−r ) Re  ∗ (r, t)B L(r, t)

GL ∞ 2T 2 d§ L(p, §) = = "p n(0) = 2T Re p 2

GL

GL −∞

10.4.2

Paraconductivity

Let us try to clear up the reason why the simple Drude formula works so well for complex phenomenon like paraconductivity.For this purpose let us try to derive the Boltzmann master equation for the fluctuation Cooper pair distribution function  9 : np (t) = ¦ (r, t)¦ ∗ (r, t) (10.113) × exp(−ip(r − r  ))d(r − r ) . Let us recall that in the state of thermal equilibrium 2 n(0) p = |¦p |  = T/"p .

In order to determine the electric field dependence of np let us write its time derivative using (10.104) @np(t)   = d(r − r )e−ip(r−r ) @t

( ( ' @¦ ∗ (r, t) @¦ (r, t) ∗  ¦ (r , t) + ¦ (r  , t) @t @t   e  −ip(r−r  )  2 [' (r) − ' (r )]¦ (r, t)¦ ∗ (r, t) . = d(r − r )e i ' (  9 : ıF  2 2 (r , t) + + Re ¦ (r, t) Re (r, t)¦ ∗ (r, t) ,

GL ı¦

GL

×

and obtain the transport equation   @np @np 2 + 2eE =− "p np − n(0) p @t @p

GL   2 =− np − n(0) . (10.115) p p In the absence of a magnetic field "p is determined by (10.112) and the momentum-dependent lifetime, corresponding to the Ginzburg–Landau one, can be introduced: GL ( ) p = GL /"p = . 1 +  2 ( )p2 Let us stress the appearance of the coefficient 2 on the right hand side of the equation (10.115). This means that the real Cooper pair lifetime, characterizing its density decay, is p /2 . The effect of a weak electric field on the fluctuation Cooper pair distribution function in the linear approximation is determined by

'

(10.114) where F is determined by (10.64). Expressing the scalar potential by the electric field E one can trans@n form the first term of the last integral into −2eE @pp . The term with the variational derivative can be evaluated by means of (10.27) and expressed in the form 2 − GL "p np . The evaluation of the last term is more

n(1) p = −

(0) @"p eT GL eE GL @np = . E· "p @p "p3 @p

(10.116)

Substituting this formula into the expression for the electric current (10.88) side by side with the Cooper pair velocity vp = @"p /@p one can find   j˛ = 2ev ˛ np = ˛ˇ Eˇ , p

where the paraconductivity tensor components are: ˛ˇ

(D) =

ˇ

 vp˛ vp  2 e ˛T . 4 "p3 p

In the case of an isotropic spectrum

(10.117)

10 ˇ

 vp˛ vp  2 e ˛T 4 "p3 p

Fluctuation Phenomena in Superconductors

399

It is worth mentioning that the results derived here (10.118) for paraconductivity are valid with the assumption of weak fluctuations: for the temperature range  ⎧ 2 e 1 ⎪ Gi (D) they are no longer applicable. Nevertheless, one ⎪ ⎪ 3D case, √ ⎪ ⎪ 32 can see that for not very dirty films, with p2F ld  1, ⎪ ⎪ ⎪ ⎨ 2 a wide region of temperatures Gi(2d)   1 exists . = ı ˛ˇ e 1 2D film, thickness: d  , where the temperature dependence of conductivity ⎪ 16d ⎪ ⎪ ⎪ ⎪ is determined by fluctuations and in this region the ⎪ ⎪ e2 1 ⎪ 2 ⎩ localization effects are negligible. 1D wire, cross-section: S   16S 3/2 The transport equation (10.115) was originally One can compare this result with that carried out derived many years ago by L.G. Aslamazov and in the Introduction from qualitative considerations, A.I. Larkin [71]. Recently T. Mishonov et al. [72] redbased on the Drude formula. Those simple specula- erived (10.115) and solved it for np in the case of an tions correctly reflect the physics of the phenomenon arbitrary electric field. but were carried out with the assumption of the momentum independence of the relaxation time p ,  taken as 0 = 8(T−T . As we have just seen, in reality 10.4.3 General Expression for Paraconductivity c) p decreases rapidly with increase of the momentum, the excess “2” appeared in (10.115) because of the Unfortunately the applicability of the master equawave nature of fluctuation Cooper pairs; accounting tion derived is restricted to weak magnetic fields for this circumstance results in the precise coeffi- (H  Hc2 ( )). For stronger fields Hc2( )  H  Hc2(0) the simple evaluation of averages in (10.114) cients of (10.119), which is different from (10.11). turns out to be incorrect; the density matrix has to be An especially simple form of the paraconductivintroduced and the master equation loses its attracity results in the 2D case, where, calculated per unit tive simplicity. At the same time, as we already know, square, it depends on the reduced temperature only: precisely these fields, which quantize the fluctuation e2 T Cooper pair motion, deserve special interest. This is  (T) = . (10.119) why in order to include in the scheme the magnetic 16 T − Tc field and frequency dependencies of the paraconducThe coefficient in this formula turns out to be a unitivity, we return to the analysis of the general TDGL 2 versal constant and is given by the value /e = equation (10.104) without the objective to reduce it 4.1 k§. For electronic spectra of other dimensionalito a Boltzmann type transport equation. ties this universality is lost, and the paraconductivity Let us solve it in the case when the applied eleccomes to depend on the electron mean free path. tric field can be considered as a perturbation. The Let us compare  (T) with the normal electron method will much resemble an exercise from a Drude part n = ne e2/m by writing the total concourse of quantum mechanics. To carry out the necductivity essary generality side by side with a formal simplicity   of expressions we will introduce some kind of sub1 e 2 pF l + . (10.120) script {i} which includes the complete set of quantum =  2 16 numbers and time. By a repeated subscript a summaOne sees that at cr = 0.4/(pF l) ∼ Gi(2c) the fluc- tion over a discrete and integration over continuous tuation correction reaches the value of the normal variables (time in particular) will be supposed. conductivity.Let us recall that the same order of magWe will look for the response of the order paramnitude for the 2D Ginzburg–Levanyuk number was eter to a weak electric field applied in the form obtained above from the heat capacity study. We will (0) (1) ¦kz (r,t) = ¦{i} + ¦{i} , (10.121) discuss the region of applicability of (10.120) below in Sect. 10.8. ˛ˇ

(D) =

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A. I. Larkin and A.A. Varlamov

(0) where ¦{i} is determined by (10.107). Substituting this expression into (10.105) and restricting our consideration to linear terms in the electric field we can write (1) (0) (B L−1 ){ik} ¦{k} = 2ie GL '{il} ¦{l} ,

with the solution in the form (1) = 2ie GLB ¦{i} L{ik} '{kl}B L{lm} {m} .

Let us substitute the order parameter (10.121) in the quantum mechanical expression for current   (0)∗ (1) (1)∗ (0) j = 2e Re ¦{i} , (10.122) B v{ik} ¦{k} + ¦{i} B v{ik} ¦{k} where B v{ik} is the velocity operator, which can be expressed by means of the commutator of r with Hamiltonian (10.107): B r}{ik} . B v{ik} = i{H,

(10.123)

The second term of (10.122) can be written by means of a transposed velocity operator (which is Hermitian) as the complex conjugated value of the first one: (1)∗ (0) ¦{i} B v{ik} ¦{k}

=

(0)∗ (1) ∗ (¦{k} ) B v {ik} ¦{i}

,

(10.124)

which results in j= =

(0)∗ (1) 2 Re{¦{i} (2eB v{ik})¦{k} } (10.125) 2 ∗ ∗ B B B −8e Im{ GL L{ki}B v{il} L{lm}'{mn} L{np}{k} {p}} .

Let us average now (10.126) over the Langevin forces moving the operator B L∗{ki} from the beginning to the end of the trace and using (10.108). One finds j = −16Te2 Re( GL ) (10.126) ∗ B B B v{il}L{lm} '{mn} L{np}L{pi} } . × Im{ GLB Now we choose the representation where the B L{lm} operator is diagonal (it is evidently given by the eigenfunctions of the Hamiltonian (10.107)): L{m} (§) =

1 , "{m} − i§ GL

(10.127)

where "{m} are the appropriate energy eigenvalues. Then we assume that the electric field is coordinate

independent but is a monochromatic periodic function of time: ' (r,t) = −Eˇ r ˇ exp(−i!t) .

(10.128)

In doing the Fourier transform in (10.127) one has to remember that the time dependence of the matrix elements '{mn} results in a shift of the frequency variable of integration § → § − ! in both L-operators placed after '{mn} or, what is the same, to a shift of the argument of the previous B L{lm} for !: j˛! = 16Te2 Re( GL ) ×



(10.129)



d§ ˇ ˛ B % GLB v{il} L{i} (§)B L∗{i} (§) Eˇ , L{l} (§ + !)[−ir{li} ]B 2

where %f (!) ≡ [f (!) + f ∗ (−!)]/2. Let us express the matrix element r{li} by means of B v{li} using the commutation relation (10.123). One can see that in the representation chosen ˇ

B r{li} = i

ˇ

B v{li} "{i} − "{l}

,

(10.130)

and, carrying out the frequency integration in (10.129), one finally gets for the fluctuation conductivity tensor (j˛! =  ˛ˇ (!)Eˇ ):  ˛ˇ ( , H, !) = 8e2T Re( GL ) $ ∞ ˛ ˇ  B v{il} B v{li} % GL × ∗ {i,l}=0

(10.131)

"{i} ( GL "{i} + GL "{l} − i| GL |2 !)("{l} − "{i} )

&

.

This is the most general expression describing the d.c., galvanomagnetic and high frequency paraconductivity contribution.When we are interested in diagonal effects only, where it is enough to accept GL as real ( GL = Re GL = ˛/8) omitting its small imaginary part, the last expression can be simplified by means of symmetrization of the summation variables:   ˛˛ ( , H, !) = ˛e2T (10.132) 2   ∞ ˛ ˛  B v{li} B v{il} × , % "{i} "{l}("{i} + "{l} − i GL !) {i,l}=0

Let us demonstrate the calculation of the d.c.paraconductivity in the simplest case of a metal with an

10

isotropic spectrum. In this case we choose a plane wave representation. By using "p defined by (10.112) one has B v{pp } = vp ıpp , vp =

@"p = 2˛Tc  2p . @p

(10.133)

We do not need to keep the imaginary part of GL , which is necessary to calculate particle-hole asymmetric effects only. Then the fluctuation conductivity calculated from (10.133) coincides exactly with (10.117). 10.4.4

Fluctuation Conductivity of Layered Superconductors

Let us return to the discussion of our general formula (10.133) for the fluctuation conductivity tensor. A magnetic field directed along the c-axis still permits separation of variables even in the case of a layered superconductor. The Hamiltonian in this case can be written as in (10.71), (10.107): 

B = ˛Tc − H

2 xy (∇xy

 r − 2ieAxy ) − (1 − cos(kz s) . 2 2

(10.134) It it is convenient to work in the Landau representation, where the summation over {i} is reduced to one over the ladder of the Landau levels i = 0, 1, 2 (each is degenerate with a density H/¥0 per unit square) and integration over the c-axis momentum in the limits of the Brillouine zone. The eigenvalues of the Hamiltonian (10.134) can be written in the form r "{n} = ˛Tc [ + (1 − cos(kz s)) + h(2n + 1)] 2 (10.135) = "kz + ˛Tc h(2n + 1) , eH was already defined by (10.78). For where h = 2m˛T c the velocity operators one can write

B v x,y =

Fluctuation Phenomena in Superconductors

First of all let us stress that the required matrix elements have to be calculated for the eigenstates of a quantum oscillator whose motion is equivalent to the motion of a charged particle in a magnetic field. The commutation relation for the oscillator’s velocity components is well known (see [73]): vy] = i [B v x ,B

eH i˛Tc h. = 2m2 m

One can check that the correct commutation relation (10.137) is fulfilled. Taking into account that √ l|B a|n = n|B a+ |l = nın,l+1 , it is seen that the only nonzero matrix elements of the velocity operator are 7  √  √ ˛Tc h x,y √lıl,n+1 + √nın,l+1 . l|B v |n = 2m i lıl,n+1 − i nın,l+1 Using these relations the necessary product of matrix elements can be calculated: v x |l = l|B v x |nn|B

˛Tc h (lıl,n+1 + nın,l+1) . 2m

Its substitution into (10.133) and accounting of the degeneracy of the Landau levels H/¥0 = 2m˛Tc h/ gives for the diagonal component of the in-plane conductivity tensor  xx ( , h, !) =

∞ ˛ 2 Tc2 e2  h % 4m

(10.138)

{n,l}=0

×

(lıl,n+1 + nın,l+1) 

"{l}"{n} "{l} + "{n} − i GL ! 

Let us start with the calculation of the in-plane components. The calculation of the velocity operator matrix elements requires some special consideration.

(10.137)

In order to calculate the necessary matrix elements let us present the velocity operator components in the form of boson-type creation and annihilation operatorsB a+ ,B a with the commutation relation [B a,B a+ ] = 1: 7  +  ˛Tc h B a +B a x,y B v = . iB a+ − iB a 2m

1 ˛rs (−i∇−2ieA)x,y ; B Tc sin(kz s) . vz = − 2m 2 (10.136)

In-Plane Conductivity

401

3

2

=

e (˛Tc ) h 2  ×

2

s − s

∞ dkz  % 2 n=0

 n+1 , "n+1 "n ("n+1 + "n − i GL !)

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A. I. Larkin and A.A. Varlamov

Table 10.2. Results for the magnetoconductivity h

 xx

 xx ( , h = 0) −

e2 [8 ( + r) + 3r 2 ] 2 h 28 s [ ( + r)]5/2

 zz

 zz ( , h = 0) −

e2 s r 2 ( + r/2) 2 h 2 [ ( + r)]5/2 28 xy

Expanding the denominator into simple fractions we reduce the problem to the calculation of the c-axis momentum integral, which can be carried out in the general case by use of the identity: 2 0

1 dx 1 = −√ , 2 cos x − z z2 − 1

(10.139)

 xx (", h, !) =  ∞ e2h  1 1 (n + 1) √ 8s n=0 h − i ! [ + h(2n + 1)][r + + h(2n + 1)] 1 1 √ h + i ! [ + h(2n + 3)][r + + h(2n + 3)]

(10.140) 

2h 1 √ . − 2 h +! 2 [ + h(2n + 2) − i ! ][r + + h(2n + 2) − i !]

where ! =

max{ , r}  h (2D)

e2 1 √ 4s 2hr 7 3.24e2 s r 2 xy h

e2 1 8s h 7(3)e2 s r 2 2 29 xy h2

 zz ( , H, !) =   ∞ ˛ ˛  B v{li} B v{il} 1 ˛e2T = % 2 "{i} "{l}("{i} + "{l} − 2i˛Tc ! ) {i,l}=0

e2 (˛Tc )3 32





sr xy

2  ∞ s dkz % h 2 n=0 $

which is valid for any complex parameter z = 1 with the proper choice of the square root branch. Using it we write the general expression for the in-plane component of the fluctuation conductivity tensor

+

hr (3D)

×

− s

sin2(kz s) "n2 (kz )["n (kz ) − i˛Tc ! ]

& .

The following transformations are similar to the calculation of the in-plane component: we expand the integrand into simple fractions and perform the kz integration by means of the identity (10.139). The final expression can be written as  zz ( , H, !) = 

e2 64s



sr xy

   2  ∞  @ 1 % − h @  + i ! n=0

1 ( + h(2n + 1) + )( + h(2n + 1) +  + r)

× √

(10.141)

 1 |=0 . ( + h(2n + 1) − i ! )( + h(2n + 1) − i ! + r)

−√

! . 16Tc

Analysis of the General Expressions Out-Of Plane Conductivity The situation with the out-of plane component of paraconductivity turns out to be even simpler z because of the diagonal structure of the B v{in} = ˛rs − 2 Tc sin(kz s) × ıin × ı(kz − kz  ). Taking into account that the Landau state degeneracy we write

In principle the expressions derived above give an exact solution for the a.c. (!  T) paraconductivity tensor of a layered superconductor in a perpendicular magnetic field H  Hc2 (h  1) in the vicinity of the critical temperature (  1). The interplay of the parameters r, , ! h entering into (10.141) and (10.142), as we have seen in the example of fluctua-

10

tion magnetization, yields a variety of crossover phenomena. 1. The simplest and most important results which can be derived are the components of the d.c. paraconductivity (! = 0) of a layered superconductor in the absence of magnetic field. Keeping ! = 0 and setting h → 0 one can change the summations over Landau levels into integration and find e2 1  xx (", h → 0, ! = 0) = , √ 16s [ (r + )]  zz ( , h → 0, ! = 0) =

e2s 2 32xy



 + r/2 − 1 . [ ( + r)]1/2

2. 2. The Aslamazov–Larkin contribution to the magnetoconductivity can be studied by putting ! = 0 and keeping magnetic field arbitrary. We will not go into details and just report the results (following [153] with some revision of the coefficient in the 3D case). Here it is worth making an important comment. The proportionality of the fluctuation magnetoconductivity to h2 is valid when using the parametrization = (T − Tc0)/Tc0 only. As is well known, a weak field shifts the critical temperature linearly, which often makes it attractive to analyze the experimental data by choosing as the reduced temperature parameter h = (T − Tc (H))/Tc (H). In this parametrization one can get a term in the magnetoconductivity linear in h, which was previously canceled out by the magnetic field renormalization of the critical temperature. So it is important to recognize that the effect of a weak magnetic field on the fluctuation conductivity cannot be reduced to a simple replacement of Tc0 by Tc (H) in the appropriate formula without the field. Vice versa, this effect is exactly compensated by the change in the functional dependence of the paraconductivity in the magnetic field,and finally it contains the negative quadratic contribution only. 3. Letting the magnetic field go to zero and considering nonzero frequency of the electromagnetic field one can find general expressions for the components of the a.c. paraconductivity tensor. They are cumbersome enough and in the

Fluctuation Phenomena in Superconductors

403

complete form can be found, for instance, in [76]. We recall here the simplified asymptotics for the Re  in the 2D regime only: xx Re (2D) (r  , ! ) = (10.142) $ $  &&  !   2 e2 1 2 !  2 , arctan − ln 1 + 16s !  !  zz Re (2D) (r  , ! ) = 2  2 $  2 & 2  e sr 1 !  . ln 1+ 8 2 s xy ! 

(10.143)

The general formulas (10.141) and (10.142) allow one to study the different crossovers in the a.c. conductivity of a layered superconductor in the presence of magnetic fields of various intensities.We leave this exercise for the reader with some practical interest in the problem. 10.4.5

Magnetic Field Angular Dependence of Paraconductivity

We have seen above that in the case of a geometry with a magnetic field directed along the z-axis many sophisticated fluctuation features of layered superconductors can be studied in the most general form. Nevertheless,even the attempt to explore the d.c.conductivity in a longitudinal magnetic field (directed in the xy-plane) [77] or, moreover, with the field directed at some arbitrary angle  with the z-axis leads to the appearance of the vector potential component in the argument of cos(kz s) and the problem requires a nontrivial calculation of the matrix elements over the Mathieu functions. We already learned that at temperatures very near to the critical one (  r) the 3D fluctuation regime takes place. Here the size of the Cooper pairs along the z-axis is so large that the peculiarities of the layered structure no longer play any role. This means that only small values of kz are important in the kz integrations, where the cos(kz s) in (10.71) can be expanded and the LD functional is reduced to its traditional GL form with an anisotropic effective mass tensor:

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A. I. Larkin and A.A. Varlamov

 b F [¦ ] = d r a|¦ |2 + |¦ |4 (10.144) 2    2 3   B2 H · B 1  1 d  − . + − 2eA ¦  + 4m  i dx

8 4

=1 

3

We will demonstrate below that in this case a scaling approach provides a direct access to the most general results by rescaling the anisotropic problem to the corresponding isotropic one on the initial level of the GL approach [78]. Let us suppose that the external field H is chosen to lie in the y − z plane and makes an angle  with the z-axis. For the sake of simplicity and because the oxide superconductors are within high accuracy uniaxial materials, we choose mx = my = m∗ , while 2 m−1 z = 2˛Ts r (compare with (10.69)). The effective anisotropy parameter a2 = m∗ /mz = 2˛Ts 2rm∗ < 1 is introduced. In (10.145) the anisotropy enters only in the gauge-invariant gradient term, so the simple rescaling of the coordinate axes: x =  x, y =  y, z = a z together with the scaling of the vector potential: A = ( Ax ,  Ay ,  Az / a) will render this term isotropic. The magnetic field evidently is rescaled to B = ( Bx / a ,  By / a,  Bz ) and the last three terms in (10.145), describing the magnetic field energy, are transformed to ıF [¦ ] =  2   3  

a 1    d   d3 r − 2e A

¦  8 4m =1 i d x

%  %&  2   Bxy Bxy · Hxy 2 . + + Bz − 2 + Bz Hz

a

a In short, we have removed the anisotropy from the gradient term but reintroduced it into the magnetic energy term. In general it is not possible to make both terms isotropic in the Gibbs energy simultaneously. However, depending on the physical question addressed, we can neglect fluctuations in the magnetic field, as was mostly done above. Let us demonstrate how the method works for the example of the d.c. fluctuation conductivity tensor calculated above for a magnetic field directed along the z-axis. We restrict our consideration to the 3D

region (  r). One can write the scaling relations between the electric field and current components before and after the scaling transformation by means of a conductivity tensor and the anisotropy parameter: jx,y =  jx,y , Ex,y , Ex,y = 

jz ∼ evz ∼ a jz , (10.145) 1 @' ∼  Ez ∼ Ez . @z

a

Now let us rewrite the relations between the current and electric field vectors before and after the scale transformation j˛ = ˛ˇ Eˇ ,  j˛ =  ˛ˇ Eˇ . Comparing them with (10.145) and introducing the operator of the direct scaling transformation T˛ˇ %  10 0 T˛ˇ = 0 1 0 , 0 0 a one can write j˛ = T˛  j , E˛ = (T −1 )˛  E and express the conductivity tensor as ˛ˇ = T˛    Tˇ . Now let us work in the already isotropic coordinate frame. We suppose that initially the magnetic field was directed perpendicular to layers and now we rotate it in the x–z plane by the angle   with respect to the initial direction. The conductivity tensor will be transformed by the usual matrix law:  ˛ˇ (  ) = R˛    (0)RTˇ = R˛ (T −1 ) & & (0)(T −1 )ı RTıˇ and ˛ˇ (  ) = T˛   ı (  )Tıˇ = T˛ R (T −1 ) & & (0)(T −1 )ı RTı Tˇ , ⎛

where R˛ˇ

⎞ cos   0 − sin   =⎝ 0 1 0 ⎠. sin   0 cos  

Finally the fluctuation conductivity tensor ˛ˇ () in the initial tetragonal system with the magnetic field

10

directed at the angle  with respect to the direction normal to layers can be expressed by means of the effective transformation operator M˛ˇ : T  )Mˇ ˛ˇ () = M˛& ( )& (0, H () ,

with ⎞  cos   0 − 1a sin  ⎠. =⎝ 0 1 0  0 cos  

a sin  ⎛

 ) = T˛ı Rı (T −1 )ˇ M˛ˇ (

The angle   can be expressed via therenormalized  = Hz2 + a2Hx2 : magnitude of the magnetic field H cos  =

z cos  H = ;  H cos2  + a2 sin2 

a sin  . sin  = cos2  + a2 sin2  In the case of the paraconductivity of a layered superconductor with the magnetic field applied at an arbitrary angle  the answer can be written in the general form by means of the three diagonal components of conductivity ii (0,  H ) in the perpendicular : field H ˛ˇ () =

1   R2 , a

(10.146)

 2   

a  − 12 zz sin 2 xx cos2  + zz sin2  R , a xy cos  2 xx ⎜       ⎜ R , a yx cos  R2 , a yy R , a yx a2 sin  ⎝ 2   

a  − 12 zz sin 2 R , a xy a2 sin  xx a4 sin2  + zz cos2  2 xx ⎛

×

⎞ ⎟ ⎟, ⎠

where   R , a = cos2  + a2 sin2 . In the simplest case of a longitudinal field  = 900 : ⎞ zz / a2 0 0 yy yx a ⎠ . ˛ˇ (900, H) = ⎝ 0 0 xy a xx a2 ⎛

Fluctuation Phenomena in Superconductors

10.5

Fluctuations Near the Superconductor–Insulator Transition

10.5.1

Quantum Phase Transition

405

It is usually supposed that the temperature of the superconducting transition does not depend on the concentration of nonmagnetic impurities (Anderson’s theorem [79, 80]). Nevertheless, when the degree of disorder is very high Anderson localization takes place,and it would be difficult to expect that under conditions of strong electron localization superconductivity can exist, even if there is inter-electron attraction. This means that at T = 0 the phase transition takes place with a change of the disorder strength or carrier concentration. Such a transition is called a quantum phase transition, since at zero temperature the classical fluctuations are absent. Indeed, one can see from (10.7) that in the limit T → 0 the thermal fluctuation Cooper pairs vanish. In the metallic phase of a disordered system the conductivity is mostly determined by the weakly decaying fermionic excitations,their dynamics yielding the familiar Drude formula (the method accounting for the fermionic excitations will be referred to as the Fermi approach later on). Inside the critical region the charge transfer due to fluctuation Cooper pairs turns out to be more important. In some approximation, the pairs may be considered as Bose particles. Therefore the approach dealing with the fluctuation pairs will hereinafter be called the Bose approach. Let us suppose that at temperature T = 0 the superconducting state occurs in a weakly disordered system. In principle, two scenarios of the development of the situation are possible with an increase of the disorder strength: the system at some critical disorder strength can go from the superconducting state to the metallic state or to the insulating state. The first scenario is natural and takes place in the following cases: if the effective constant of the interelectron interaction changes its sign with the growth of the disorder; if the effective concentration of magnetic impurities increases together with the disorder growth; if the pairing symmetry of superconducting state is nontrivial it can be destroyed even by the weak disorder level. We will study here the second scenario where the superconductor becomes an

406

A. I. Larkin and A.A. Varlamov

insulator with disorder increase. This means that at some disorder degree range,higher than the localization edge when the normal phase no longer exists at finite temperatures, superconductivity can still survive. At a first glance this statement seems strange: what does superconductivity mean if the electrons are already localized? And if it really can take place beyond the metallic phase, at what value of disorder strength and in which way does the superconductivity finally disappear? One has to bear in mind that localization is a quantum phenomenon in its nature and with the approach to the localization edge the coherence length of localization  grows. From the insulator side of the transition vicinity this means the existence of large scale regions where delocalized electrons exist. If the energy level spacing in such regions does not exceed the value of superconducting gap Cooper pairs still can be formed by the delocalized electrons of this region. The problem can be reformulated in another, already familiar, way: how does the critical temperature of the superconducting transition decrease with the increase of the disorder strength? In the previous sections we have already tried to solve it by discussing the critical temperature fluctuation shift. We have seen that the fluctuationshift of the critical temperature is proportional to Gi(3) for a 3D superconductor and to Gi(2) ln{1/Gi(2) } for a 2D superconductor. This means that the critical temperature is not changed noticeably as long as the Ginzburg– Levanyuk number remains small. So one can expect the complete suppression of superconductivity when Gi ∼ 1 only. For further consideration it is convenient to separate the 3D and 2D cases because their physical pictures of the superconductorinsulator transition are quite different. 10.5.2

3D Superconductors

As one can see from Table 10.1 in the 3D case the Ginzburg–Levanyuk   number remains small at pF l ∼ 1: Gi(3d) ≈ ETFc  1. Nevertheless, approaching the edge of localization, the width of the fluctuation region increases [81]. In the framework of the self-consistent theory of localization [82] such

growth of the width of the fluctuation region was found in [83]. Instead of the cited self-consistent theory let us make some more general assumptions concerning the character of the metal-insulator (M-I) transition in the absence of superconductivity [13].We suppose that in the case of very strong disorder and not very strong Coulomb interaction the M–I transition is of second order. The role of “temperature” for this transition is played by the “disorder strength”, which is pF l. . characterized by the dimensionless value g = 2 With its decrease the conductivity of the metallic phase decreases and at some critical value gc tends to zero as  = e2pF (g − gc ) .

(10.147)

This is the critical point of the Anderson (M-I) transition. We assume that the thermodynamic density of states remains constant at the transition point. The electron motion in metallic phase far enough from the M–I transition has a diffusion character and the conductivity can be related to the diffusion coefficient D = pF l/3m by the Einstein relation:  =  e2 D. One can say that diffusion-like “excitations” with the spectrum !(q) = iDq2 propagate in the system. At the point of the M–I transition normal diffusion terminates and conductivity, together with D, turns zero. In accordance with scaling ideas, the diffusion coefficient can be assumed here to be a power function of q: D(q) ∼ qz−2 , with the dynamical critical exponent z2. The anomalous diffusion excitation spectrum in this case would take the form ! ∼ qz . In the insulating phase (g < gc ) some local, anomalous diffusion, confined to regions of the scale , is still possible. It cannot provide charge transfer throughout the entire system, so D(q = 0) = 0, but for small distances (q  −1 ) anomalous diffusion takes place. Analogously, in the metallic phase (ggc ) the diffusion coefficient in the vicinity of the transition has an anomalous dependence on q for q  −1 and weakly depends on it for q  −1 . So one can conclude that the diffusion coefficient for q  −1 from both sides of the transition has the same qdependence as for all q in the transition point. It can be written in the form

10

  g ' (q) z−2 D(q) = , 3m pF  ⎧ ⎨ x, x  1 ' (x) = 1, x  1, g > gc , ⎩ 0, x  1, g < gc

(10.148)

where the localization length , characterizing the spatial scale near the transition, grows with the approach to the transition point like (g) =

1  (g − gc )− z−2 . pF

(10.149)

The critical exponent in this formula is found from the Einstein relation in the vicinity of the M–I transition. At finite temperatures, instead of the critical point gc , a crossover from metallic to insulating behavior of  (g) takes place. The width of the crossover region is  g − gc , where  g is determined from the relation D−2 ( g ) ∼ EF [pF ( g )]−z ∼ T (we have used the second asymptotic of (10.148)). In this region the diffusion coefficient is  2 T EF z D(T) ∼ T2 ∼ 2 (10.150) pF T and it depends weakly on g − gc . Beyond this region the picture of the transition remains the same as at T = 0. Let us consider now what happens to superconductivity in the vicinity of the localization transition. In the mean field approximation (BCS) the thermodynamic properties of a superconductor do not depend on the character of the diffusion of excitations. This should be contrasted with the fluctuation theory, where such a dependence clearly exists. We will show that the type of superconducting transition depends on the dynamical exponent z. If z3, the transition to superconductivity occurs on the metallic side of the localization transition (we will refer to such a transition as an S–N transition). If z < 3, the transition to superconductivity occurs from the insulating state directly (S–I transition). Let us study how the superconducting fluctuations affect the transition under discussion. In spirit of the GL approach fluctuation phenomena in the vicinity

Fluctuation Phenomena in Superconductors

407

of the transition can be described in the framework of the GL functional (10.27). The coherence length in the metallic region, far enough from the Anderson transition, was reported in Introduction to be equal  2 = c l = 0.42D/T. In the vicinity of the M–I transition we still believe in the diffusive character of the electron motion resulting in the pair formation. The only difference from the previous consideration is the anomalous character of the quasiparticle diffusion. So in order to describe the superconducting fluctuations simultaneously near superconducting (in temperature) and Anderson (in g) transitions let us use the GL functional (10.27) with the k-dependent diffusion coefficient (10.148). The value of Gi can be estimated from the expression for the fluctuation contribution to heat capacity (10.30) taken at ∼ Gi, where the fluctuation correction reaches the value of the heat capacity jump:  T d3 q 1∼ (10.151)  2 ,  TGi + D(q)q2 with T  Tc .Let us approach the M–I transition from the metallic side.If we are far enough from transition, Gi is small and the integral in (10.151) is determined by the region of small momenta D(q)q2  TGi: Gi ∼

T .  2 D3 (q = 0)

(10.152)

Two scenarios are possible: Gi becomes of the order of 1 in the metallic phase, or it remains small up to the crossover region, where finally reaches its saturation value. In the first case we can use the second asymptotic of (10.148) for D(q) and find: Gi ∼

Tc (pF )3z−6 . EF

(10.153)

One can see that Gi becomes of the order of 1 1 at pF M ∼ (EF /T) 3z−6 . Comparing this value with pF ( g ) ∼ (EF /T)1/z at the limit of the crossover region we see that for z3 the first scenario is realized. Concluding our discussion of the first scenario we can see that the superconducting critical temperature goes to zero at  = M , still in the metallic phase, so at T = 0 a superconductor-normal phase (S–N) type quantum phase transition takes place.

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A. I. Larkin and A.A. Varlamov

The second scenario takes place for z < 3 when Gi remains small even at the edge of crossover region, reaching there the value  Gi ∼

Tc EF

 2(3−z) z

 1.

(10.154)

In the crossover region the diffusion coefficient, and hence Gi, almost do not vary. This is why the temperature of superconducting transition remains almost frozen with further increase of disorder driving the system through the Anderson transition. The abrupt growth of Gi and decrease of Tc take place when the system finally goes from the crossover to the insulating region. In the insulator phase the diffusion coefficient D(q  l−1 ) = 0 and from (10.151) one can find for Gi:  EF 1 Gi ∼ . (10.155) Tc (pF )3/2 Comparing this result with the Table 10.1 it is easy to see that it coincides with the Ginzburg–Levanyuk number for a zero-dimensional granule of size   (T). Hence we see that in the second scenario the Ginzburg number reaches 1 and, respectively,  1/3 Tc → 0 at pF I ∼ ETFc , which is far enough from the M–I transition point. This is why in this case one can speak about the realization at T = 0 of a superconductor-insulator (S–I) type quantum phase transition. The scale I determines the size of the “conducting” domains in the insulating phase, where the level spacing reaches the order of the superconducting gap. It is evident that in the domain of scale   I superconductivity cannot be realized. In the vicinity of a quantum phase transition one can expect the appearance of nonmonotonic dependencies of the resistance on temperature and magnetic field. Indeed, starting from the zero resistance superconducting phase and increasing temperature from T = 0, the system passes through the localization region, where the resistance is high, to high temperatures where some hopping charge transfer will decrease the resistance again. The analogous speculations are applicable to the magnetic field effect: first the magnetic field“kills” superconductivity

Fig. 10.5. Phase diagram in the temperature-disorder plane for a 3D superconductor

and increases the resistance, then it destroys localization and decreases it. The phase diagram in the (T, g) plane has the form sketched in Fig. 10.5. For gI = gc − (Tc /EF )3(z−2)/ , an S–I transition takes place at T = 0. Increasing the temperature from T = 0 in the region 0  g  gI we remain in the insulating phase with exponential dependence of resistance on temperature. For gI  g   g− at low temperatures 0 ≤ T < Tc (g) the system stays in the superconducting state, which goes to the insulating phase at higher temperatures. In the vicinity of the Anderson transition ( g−  g   g+ ) the superconducting state goes with growth of the temperature to some crossover metal-insulator state which is characterized by a power decrease of the resistivity with the increase of temperature. Finally at gc  g the superconducting phase becomes of the BCS type and at T = Tc it goes to a metallic phase. The phase diagram in the magnetic field-disorder plane is similar to that in the (T, g) plane with the only difference that at T = 0 there is no crossover region, instead a phase transition takes place. 10.5.3

2D Superconductors

Preliminaries As was demonstrated, according to the conventional theory of paraconductivity, the sheet conductivity in the vicinity of the superconducting transition is given by a sum of the electron residual conduc-

10

tivity ge2 (Fermi part) and the conductivity of the Cooper pair fluctuations (Bose part) (see (10.120)). This expression is valid in the Ginzburg–Landau region when the second term is a small correction to the first one. The width of the critical region can be determined from the requirement of equality of both contributions in (10.120):21 cr =

1 = 1.3Gi(2d) . 16g

(10.156)

In accordance with general scaling ideas one can believe that inside the fluctuation region the conductivity should obey the form:   2  (T) = ge f . (10.157) Gi(2d) Concerning the scaling function f (x), we know its asymptotes in the mean field region (x  1) and just above the BKT transition [74,75]: 1 1 + x −1 , x1 f (x) =

 . −1/2 exp −b(x − xBKT ) ) , x → xBKT = −4 (10.158) The BKT transition temperature TcBKT is determined by expression (10.96) and one can find its value by comparing the superfluid density ns from (10.96) with that found in the BCS scheme accounting for GL fluctuations (see (10.94)): TcBKT = Tc (1 − 4Gi).

(10.159)

Here we assumed that the Ginzburg parameter is small, so that the BKT transition temperature does not deviate much from the renormalized by the long wave-length fluctuations BCS transition temperature Tc . Boson Mechanism of the Tc Suppression The classical and quantum fluctuations reduce ns and therefore, suppress TcBKT . At some g = gc ∼ 1, the superfluid density ns , and simultaneously TcBKT , go to 21

Fluctuation Phenomena in Superconductors

409

zero. In the vicinity of this critical concentration of impurities TcBKT  Tc0. Thus a wide new window of intermediate temperatures TcBKT  T  Tc0 opens up. In this window, according to the dynamical quantum scaling conjecture [84], one finds   T 2  =e ' . (10.160) TcBKT At T − TcBKT  TcBKT the Berezinski–KosterlitzThouless law (10.157)–(10.158) should hold, so ' (x) = f (x) and is exponentially small. In the intermediate region TcBKT  T  TcBCS the duality hypothesis gives ' (x) = /2. Let us derive this relation. We will start from the assumption that in the region TcBKT  T  Tc0 the conductivity is a universal function of temperature that does not depend on the pair interaction type. Being in the framework of the classical approach, let us suppose that in a weak electric field pairs move with the velocity v = F/, where F = 2eE is the force acting on the pairs. The current density j = 2env = E (here n is the pair density), so one can relate the conductivity with the effective viscosity :  =4e2n/. Let us recall that we are dealing with a quantum fluid, so another superconducting view on the problem of its motion near the quantum phase transition exists.One can say that with the increase of Gi the role of quantum fluctuations also grows and fluctuation vortices carrying the magnetic flux quantum ¥0 = /e are generated. With electric current flow in the system the Lorentz (Magnus) force acts on a vortex: F = j¥0 . The electric field is equal to the rate of magnetic flux transfer,i.e.to the density of the vortex current: E =¥0 nv vv =¥0 nv F/v , where nv is the density and v is the viscosity of the vortex liquid.As a result E =¥02nv j/v = j/ .So one can conclude that for vortices the velocity is proportional to the voltage, and the force is proportional to the current. For Cooper pairs (bosons) the situation is exactly the opposite. The duality hypothesis consists of the assumption that at the critical point the pair and the vortex liquid density flows are equal: nv vv = nv. Comparing

  (2d) = It is worth mentioning that this definition of the Ginzburg–Levanyuk number Gi   from the heat capacity fluctuations Gi(2d) = p0.3 . Fl

 8pF l

 agrees with that defined

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A. I. Larkin and A.A. Varlamov

these quantities,expressed in terms of the conductivity from the above relations, one can find a universal value for the conductivity at the critical point  =

2e 2e2 . = ¥0 

(10.161)

One can restrict oneself to a less strong duality hypothesis, supposing the product n = CT ı with a universal ı exponent both for the pair and the vortex liquids, while their constant C is different. In this case, based on duality,it is possible to demonstrate that ı = 0 and the conductivity is temperature independent up to Tc , but its value is no longer universal and can vary from one sample to another. To conclude, let us emphasize that in the framework of the boson scenario of superconductivity suppression, the BCS critical temperature is changed insignificantly, while the “real” superconducting transition temperature TcBKT → 0. The Fermion Mechanism of Tc Suppression Apart from the above fluctuation (boson) mechanism of the suppression of the critical temperature in the 2D case, there exists another, fermionic mechanism. The suppressed electron diffusion results in a poor dynamical screening of the Coulomb repulsion which, in turn, leads to the renormalization of the inter-electron interaction in the Cooper channel and hence to the dependence of the critical temperature on the value of the high-temperature sheet resistivity of the film.As long as the correction to the nonrenormalized BCS transition temperature Tc is still small, one finds [85–87]:   1 3 1 ln . (10.162) Tc = Tc0 1 − 12 2 g Tc0 At small enough Tc0 this mechanism of critical temperature suppression turns out to be the principal one. The suppression of Tc down to zero in this case may happen in principle even at g  1. A renormalization group analysis gives [88] the corresponding critical value of conductance   1 2 1 gc = ln . (10.163) 2 Tc0

Here we should recall that the typical experimental (see [89] and references therein) values of gc are in the region gc ∼ 1 − 2, and do not differ dramatically from the predictions of the boson duality assumption gc = 2 . If one attempts to explain the suppression of Tc within the fermion mechanism, one should assume that ln Tc01  5. Then, according to (10.163),gc 2/ and the boson mechanism is not important. On the contrary, if ln Tc01  < 4, then (10.162) gives a small correction for Tc even for gc = 2/ and the fermion mechanism becomes unimportant. The smallness of the critical temperature Tc compared to the Fermi energy is the cornerstone of the BCS theory of superconductivity and it is apparently satisfied even in high-Tc materials. Nevertheless, it is necessary to use the theoretically large logarithmic parameter with care, if one needs ln Tc01  to be as large as 4.

10.6

Microscopic Derivation of the TimeDependent Ginzburg–Landau Equation

10.6.1

Preliminaries

We have seen above how the phenomenological approach based on the GL functional allows one to describe fluctuation Cooper pairs (Bose particles) near the superconducting transition and to account for their contribution to different thermodynamical and transport characteristics of the system. Now we pass to the discussion of the microscopic description of fluctuation phenomena in superconductors. The development of the microscopic approach is necessary for the following reasons: • This description permits microscopic determination of the values of the phenomenological parameters of the GL theory. • This method is more powerful than the phenomenological GL approach and permits treatment of fluctuation effects quantitatively even far from the transition point and for magnetic fields strong as Hc2, taking into account the contributions of dynamical and short wavelength fluctuations. • The electron energy relaxation times in metals are relatively large ("  /T), which causes the electron low frequency dynamics to be sensitive

10

to the nearness to the superconducting transition. This is why the temperature dependence of fluctuation corrections can be determined, generally speaking, not only by the Cooper pair motion but also by changes in the single-electron properties. • There are some fluctuation phenomena in which the direct Cooper pair contribution is considerably suppressed or even absent altogether. Among them we can mention the nuclear magnetic relaxation rate, tunnel conductivity, c-axis transport in strongly anisotropic layered metals, thermoelectric power and heat conductivity where the fluctuation pairing manifests itself by means of the indirect influence on the properties of the single-particle states of electron system. Formally, in the above consideration averaging over the superconducting order parameter is accomplished by means of a functional integration over all its possible bosonic field configurations. In this description we have dealt with the fluctuation Cooper pair related effects only and the method of the functional integration turned out to be simple and effective for their description. In the following sections we will develop the diagrammatic method of Matsubara temperature Green functions, which is more adequate for the description of the properties of a Fermi system of interacting electrons.

10.6.2

The Cooper Channel of Electron–Electron Interaction

Let us start the microscopic description of fluctuation phenomena in a superconductor from the electron Hamiltonian.We will choose it in the simple BCS form:22 22 23

Fluctuation Phenomena in Superconductors

H=

 p,

−g

+ E(p)p, p,



411

(10.164)

+ + p+q, −p,− −p ,−  p +q,  .

p,p ,q, , 

The momentum conservation law and singlet pairing are already taken into account in the interaction term. Here E(p) is the quasiparticle spectrum of the normal metal; −g is the negative constant of electron-electron attraction which is supposed to be momentum independent and different from zero in a narrow domain of momentum space in the vicinity of the Fermi surface where !D !D pF − < |p|, |p| < pF + . vF vF + and p, are the creation and annihilation field p, operators in the Heisenberg representation, so the first term is just the kinetic energy of the noninteracting Fermi gas. The interaction term is chosen in the traditional form characteristic for the electron– phonon mechanism of superconductivity.23 For the description of the properties of an interacting electron system with the Hamiltonian (10.165) we will use the formalism of the Matsubara temperature diagrammatic technique. The state of a noninteracting quasiparticle is described by its Green function 1 G(p, "n ) = , (10.165) i"n − (p)

where "n = (2n + 1)T is a fermion Matsubara frequency and (p) = E(p)−EF is the quasiparticle energy measured from the Fermi level. As is well known, the effective electron–electron attraction leads to a reconstruction of the ground state of the electron system,which formally manifests itself by the appearance at the critical temperature of a pole in the two-particle Green function + L(p, p, q) = T [p+q, −p,− p+ +q,  −p  ,−  ] ,

We suppose that reader is familiar with the BCS formulation of the theory of superconductivity (see, for example, [90]). Fluctuations in the framework of more realistic Eliashberg [91] model of superconductivity were studied by B. Narozhny [92]. He demonstrated that the strong coupling does not drastically change the results of the weak coupling approximation. The critical exponents turn out to be exactly the same as in the framework of the GL theory, which provides an adequate description of paraconductivity in strong coupling superconductors. The robustness of the critical exponents and their dependence in the GL region on the space dimensionality was only stressed in [83] in relation to the discussion of the paraconductivity at the edge of the superconductor–insulator transition.

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A. I. Larkin and A.A. Varlamov

GL equation. In terms of the polarization operator introduced above it can be written as    ¢ ¢ dpdp L p, p, q = − = L. 1 − g¢ g

Fig. 10.6. The Dyson equation for the fluctuation propagator (wavy line) in the ladder approximation. Solid lines represent one-electron Green functions, bold points correspond to the model electron–electron interaction

where T is the time ordering operator and 4D vector notations are used [90]. As is well known, the twoparticle Green function can be expressed in terms of the vertex part [90]. In the case under consideration it is the vertex part of the electron–electron interaction in the Cooper channel L(q, §k ), which will hereinafter be called the fluctuation propagator. The Dyson equation for L(q, §k ), accounting for the e–e attraction in the ladder approximation, is represented graphically in Fig. 10.6. It can be written analytically as L−1 (q, §k ) = −g −1 + ¢(q, §k ) ,

(10.166)

where the polarization operator ¢(q, §k ) is defined as a loop of two single-particle Green functions: ¢(q, §k ) = T

 "n

d3p G(p + q, "n+k )G(−p, "−n ) . (2)3

(10.167) Let us emphasize that the two quantities intro  duced above, L p, p , q and L(q), are closely connected with each other. The former being integrated over momenta p and p becomes an average of the product of two order parameters:     1  dpdp L p, p, q = 2 q ∗q , (10.168) g where q is the superconducting gap proportional to the condensate wave function ¦ . Thus, this quantity represents the coefficient in the linear term in the

Comparing this equation with (10.166) for the fluctuation propagator, we see that the corresponding expressions are very similar. After analytical continuation to the real frequencies the fluctuation propagator L(q, i§) coincides with the quantity defined by (10.168) (up to a constant). One can calculate the propagator (10.166) using the one-electron Green functions of the normal metal (10.165). For the sake of convenience of future calculations let us define the correlator of two oneelectron Green functions P (q, "1 , "2 )  d3p = G (p + q, "1 ) G (−p, "2) (10.169) (2)3 0 / 1 , = 2 Ÿ(−"1 "2 ) |"1 − "2 | + i(q, p)|E(p)=EF F.S.

where Ÿ(−"1 "2 ) is Heavyside step function,  is the  d§p one-electron density of states, F.S. = 4 means the averaging over the Fermi surface, (q, p)| (p)=EF = [(q + p) − (−p)]|E(p)=EF ≈ (vp q)(p)=0 . The last approximation is valid not too far from the Fermi surface, i.e. when (vpq)(p)=0  EF . It is impossible to carry out the angular averaging in (10.169) for a general anisotropic spectrum. Nevertheless in the following calculations of fluctuation effects in the vicinity of critical temperature only small momenta vp q  T will be involved in the integrations, so we can restrict our consideration here to this region, where one can expand the integrand in powers of vp q. Indeed, the presence of Ÿ(−"1 "2 ) leaves the difference of the two fermionic frequencies in (10.169) to be of the order of the temperature that permits this expansion. The first term in vpq will ev   (vp q)2 F.S. Ÿ(−"1 "2 ) 1− . P(q, "1 , "2 ) = 2 |"1 − "2 | |"1 − "2 |2 (10.170)

10

Now one can calculate the polarization operator ¢(q, §k ) = T =

−

P(q, "n+k , "−n )

"n

 n≥0



1 n + 1/2 +

(10.171)

|§k | 4T

∞ (vp q)2 F.S.   (4T)2 n=0



1 n + 1/2 +

|§k | 4T

3 .

The calculation of the sums in (10.171) can be carried out in terms of the logarithmic derivatives of the  -function (n) (x). It worth mentioning that the first sum is well known in BCS theory, one can recognize in it the so-called “Cooper logarithm”; its logarithmic divergence at the upper limit ( (x  1) ≈ ln x) !D is cut off by the Debye energy (Nmax = 2T ) and one gets:  !D 1 |§k | + + 2 4T 2T   1 |§k | + (10.172) − 2 4T   (vp q)2 F.S.  1 |§k | + . − 2(4T)2 2 4T

1 ¢(q, §k ) = 

Fluctuation Phenomena in Superconductors

We find (10.175) for bosonic imaginary Matsubara frequencies i§k = 2iTk. These frequencies are necessary for the calculation of fluctuation contributions to any thermodynamical characteristics of the system. In the vicinity of the transition point one can restrict oneself to summations of the expressions with L(q, §k ) over Matsubara frequencies to the so-called static approximation, taking into account the term with §k = 0 only, which turns out to be the most singular term in  1. This approximation physically means that the product of Heisenberg field operators p, −p,− here appears like a classical field ¦ , which in the phenomenological approach describes the Cooper pair wave function and in the vicinity of critical temperature is proportional to the fluctuation order parameter.Having in mind namely this GL region of temperatures we restricted ourselves above to the assumption of small momenta vp q  T. For these conditions the static propagator reduces to



The critical temperature in the BCS theory is determined as the temperature Tc at which the pole of L(0, 0, Tc ) occurs L−1 (q = 0, §k = 0, Tc ) = g −1 − ¢(0, 0, Tc) = 0,   2 E 1 !D exp − , (10.173) Tc =  g where E = 1.78 is the Euler constant. Introducing the reduced temperature = ln( TTc ) one can write the propagator as L−1 (q, §k ) =      1 1 |§k | + − (10.174) − + 2 4T 2   (vp q)2 F.S.  1 |§k | + . − 2(4T)2 2 4T

413

L(q, 0) = −

1 1 .  +  2 q2

(10.175)

With an accuracy of a numerical factor and the total sign this 9correlator coincides with the expression : (10.98) for |¦q |2 . By this expression we have also finally obtained the microscopic value of the coherence length  for a clean superconductor with an isotropic D-dimensional Fermi surface, which was often mentioned previously (compare with (10.5)) 2 (D) =

7(3)vF2 . 16D 2T 2

(10.176)

In order to describe the fluctuation contributions to transport phenomena one has to start from the analytical continuation of the propagator (10.175) from the discrete set of §k ≥ 0 to the whole upper half-plane of imaginary frequencies. The analytical properties of (n) (x)-functions (which have poles at x = 0, −1, −2, . . .) permit one to obtain the retarded propagator LR (q, −i§) by simple substitution i§k → §. For small §  T the −functions can be expanded in −i§/4T and the propagator acquires the simple pole form:

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LR (q, §) = − =

1 1 i  − 8T § + +  2q2

8T 1  −1  i§ − GL +

8T 2 2   q

(10.177) .

This expression provides us with the microscopic  value of the GL relaxation time GL = 8(T−T , widely c) used above in the phenomenological theory. Moreover, a comparison of the microscopically derived (10.178) with the phenomenological expressions (10.107), (10.110) and (10.127) shows that ˛Tc =  and GL =  /8Tc . In evaluating L(q, §k ) we neglected the effect of fluctuations on the one-electron Green functions. This is correct when fluctuations are small, i.e. not too near to the transition temperature. The exact criterion of this approximation will be discussed in the following. 10.6.3

Superconductor with Impurities

Account for Impurities In order to study fluctuations in real systems like superconducting alloys or high temperature superconductors one has to perform an impurity average in the graphical equation for the fluctuation propagator (see Fig. 10.6). This procedure can be done in the framework of the Abrikosov–Gor’kov approach [90], which we briefly recall below. Let us start from the equation for the electron Green function in the potential of impurities U (r):   B GE (r, r ) = ı(r − r  ) . E − U (r) − H (10.178) If we solve this equation using the perturbation theory for the impurity potential and average the solution, then the average product of two Green functions can be presented as a series, each term of which is associated with a graph drawn according to the rules of diagrammatic technique (see Fig. 10.7). In this technique solid lines correspond to bare Green functions and dashed lines to random potential correlators.We assume that the impurity system random potential U (r) is distributed according to the Gauss ı-correlated law. Then all the correlators can be represented as the products of pair correlators 9 : 9 : U (r) = 0, U (r)U (r ) = U 2 ı(r − r  ) , (10.179)

where the angle brackets denote averaging over the impurity configuration. Equation (10.179) corresponds to the Born approximation for the electron interaction with short range impurities, and  2 9 2: U = Cimp V (r)dr where Cimp is the impurity concentration and V (r) is the potential of the single impurity. In conductors (far enough from the metal– insulator transition) the mean free path is much greater than the electron wavelength l   = 2/pF (which in practice means the mean free path up to tens of interatomic distances). As is well known [90] for the electron spectra with dimensionality D1 the angular integration in momentum space considerably reduces the contribution of the diagrams with intersecting impurity lines, which permits one to  −1 omit them to the leading approximation in pF l . For this approximation the one-electron Green function keeps the same form as the bare one (10.165) with the only substitution "n ⇒  "n = "n +

1 sign("n ) , 2

(10.180)

9 : where 1/ = 2 U 2 is the frequency of elastic collisions. Another effect of the coherent scattering on the same impurity by both electrons forming a Cooper pair is the renormalization of the vertex part (q, "1 , "2) in the particle–particle channel. Let us demonstrate the details of its calculation. The renormalized vertex (q, "1 , "2 ) is determined by a graphical equation of the ladder type (see Fig. 10.7). Here after the averaging over the impurity configurations 9 : 1 the value U 2 = 2  is associated with the dashed line. In the momentum representation this, generally speaking, integral equation is reduced to the algebraic one 1 −1 (q, "1 , "2 ) = 1 − P(q,  "1 ,  "2) , (10.181) 2  where P(q,  "1 ,  "2 ) was defined above by (10.169). Now one has to perform a formal averaging of the general expression (10.169) over the Fermi surface ( . . . F.S. ). Restricting the consideration to small momenta (q, p)||p|=pF  | "1 −  "2 | .

(10.182)

10

Fig. 10.7. The equation for the vertex part (q, !1 , !2 ) in the ladder approximation. Solid lines correspond to bare one-electron Green functions and dashed lines to the impurity random potential correlators

the calculation of (q, !1 , !2 ) for the practically important case of an arbitrary spectrum can be done analogously to (10.170). Indeed, expanding the denominator of (10.169) one can find (q, !1 , !2) =

|"1 − "2 | +

(10.183) | "1 −  "2 | ((q,p)||p|=pF )2 F.S. |! ˜ 1 −! ˜ 2 |2

Ÿ (−"1 "2 )

.

It is easy to see that assumed restriction on momenta is not too severe and is almost always satisfied in calculations of fluctuation effects at temperatures near Tc . In this region of temperatures the effective propagator momenta are determined by √ |q|eff ∼ [ GL (T)]−1 =  −1   −1 , while the Green function q-dependence becomes important for much larger momenta q ∼ min{ −1 , l−1 }, which is equivalent to the limit of the condition (10.182). The average in (10.184) can be calculated for some particular types of spectra. For example, in the cases of 2D and 3D isotropic spectra it is expressed in terms of the diffusion coefficient D(D) : v2 q2 ((q, p)||p|=pF )2 F.S.(D) =  −1 D(D) q2 = F . D (10.184) Another important example is already familiar case of quasi-two-dimensional electron motion in a layered metal: (p) = E(p ) + J cos(pz s) − EF ,

(10.185)

where E(p ) = p2 /(2m), p ≡ (p , pz ), p ≡ (px , py ), J is the effective nearest-neighbor interlayer hopping

Fluctuation Phenomena in Superconductors

415

Fig. 10.8. The Fermi surface in the form of a corrugated cylinder

energy for quasiparticles. We note that J characterizes the width of the band in the c-axis direction taken in the strong-coupling approximation and can be identified with the effective energy of electron tunneling between planes (see (10.62) and footnote 14). The Fermi surface, defined by the condition (p) = 0, is a corrugated cylinder (see Fig. 10.8). In this case the average (10.184) is written in a more sophisticated form: ((q, p)||p|=pF )2 F.S. = (10.186) 1 2 2 B 2, (v q + 4J 2 sin2 (qz s/2)) =  −1 Dq 2 F where we have introduced the definition of the genB in order to deal with eralized diffusion operator D an arbitrary anisotropic spectrum.

Propagator In Sect.10.4,in the process of the microscopic derivation of the TDGL equation, the fluctuation propagator was introduced. This object is of first importance for the microscopic fluctuation theory and it has to be generalized for the case of an impure metal with an anisotropic electron spectrum. This is easy to do using the averaging procedure presented in the previous section. Formally it is enough to use in (10.166) the polarization operator ¢(q, §k ) averaged over impurity positions, which can be expressed in terms of P(q,  "n+k ,  "−n ) introduced above:

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A. I. Larkin and A.A. Varlamov

¢(q, §k ) = T



(q, "n+k , "−n )P(q, ! n+k , ! −n )

!n

=T

 !n

1 "−n )]−1 − "n+k ,  [P(q, 

1 2 

.

(10.187) For relatively small q ((q, p)||E(p)|=EF  | "n+k −  "−n | ∼ max{T,  −1 }) and §  T one can find an expression for the fluctuation propagator, which can be useful in studies of fluctuation effects near Tc (  1) for the dirty and √ intermediate but not very clean case (T  1/ ). Expanding (10.184) in powers  2 of (q, p)||E(p)|=EF /|2 "n + §k | it is possible write LR (q, §) in a form that almost completely coincides with (10.178): LR (q, §) = −

1 1 . §  − i 8T +  2(T)q2

(10.188)

Let us stress that the phenomenological coefficient

GL turns out to be equal to the same value  as 8T in the clean case, and hence does not depend on the impurity concentration. The only difference in comparison with the clean case is the appearance of a dependence of the natural effective coherence length on the elastic relaxation time. In the isotropic D-dimensional case it can be written as 2 (D) (T) = (4m˛T)−1 = (D) (10.189)   1 1 1 1  2 v2  1 ( + )− ( )− =− F ( ) D 2 4T 2 4T 2

(we introduced here the parameter (D) frequently used in the microscopic theory).24 The generalization of (10.188) to the case of a layered electronic spectrum is evident: LR (q,§) = −

24

1 1 . 2 2  − i § +  q (2)  + r sin (qz s/2) 8T (10.190)

One has to remember that (10.188) was derived with the assumption of small momenta "n+k −  "−n | ∼ max{T,  −1 }, so (q, p)||E(p)|=EF  | the range of its applicability is restricted to the GL region of temperatures = ln( TTc )  1, where the integrands of diagrammatic expressions have singularities at small momenta of the center of mass of the Cooper pair. Finally let us express the Ginzburg–Levanyuk parameter for the important 2D case in terms of the microscopic parameter (2) . In accordance with (10.37) and the definition one has (10.189): Gi(2) (T) =

7(3) 1 . 2 16 mTc (2) (T)

(10.191)

One can see that this general definition in the limiting cases of a clean and dirty metal results in the same values Gi(2c) and Gi(2d) as was reported in Table 10.1.

10.7

Microscopic Theory of Fluctuation Conductivity of Layered Superconductors

10.7.1

Qualitative Discussion of Different Fluctuation Contributions

In Sect. 10.4 the direct fluctuation effect on conductivity, related with the charge transfer by means of fluctuation Cooper pairs, was discussed in detail. Nevertheless, in this section we return to its discussion and will demonstrate its calculation by means of the microscopic theory. This will be done for the purpose of preparing the basis for studies of the Aslamazov–Larkin contribution in the variety of physical values like magnetoconductivity near the upper critical field, conductivity far from transition point,fluctuation conductivity in the ultra-clean limit, Hall conductivity, etc.

Let us recall that its square determines the product of the GL parameter ˛ and the Cooper pair mass entering in the GL functional. In the clean case we supposed the letter equal to two free electron masses and defined ˛ in accordance with (10.18). As we just have seen in the case of the impure superconductor  depends on impurity concentration and this dependence, in principle, can be attributed both to ˛ or m. For our further purposes it is convenient to leave ˛ in the same form (10.18) as in the case of a clean superconductor. The Cooper pair mass in this case becomes dependent on the electron mean free pass, which physically can be attributed to the diffusion motion of the electrons forming the pair.

10

The microscopic approach also permits us to calculate the above cited indirect fluctuation effects like so called DOS and MT contributions. We will start now from their qualitative discussion. The important consequence of the presence of fluctuating Cooper pairs above Tc is the decrease of the one-electron density of states at the Fermi level. Indeed, if some electrons are involved in pairing they cannot simultaneously participate in charge transfer and heat capacity as single-particle excitations. Nevertheless, the total number of the electronic states cannot be changed by the Cooper interaction and only a redistribution of the levels along the energy axis is possible [93, 94]. In this sense one can speak about the opening of a fluctuation pseudo-gap at the Fermi level. The decrease of the one-electron density of states at the Fermi level leads to a reduction of the normal state conductivity. This, indirect, fluctuation correction to the conductivity is called the density of states (DOS) contribution and it appears side by side with the paraconductivity (or Aslamazov–Larkin contribution). It has the opposite (negative) sign and turns out to be much less singular in (T − Tc )−1 in comparison with the AL contribution, so that in the vicinity of Tc it was usually omitted. However, in many cases [29, 95–99], when for some special reasons the main, most singular, corrections are suppressed, the DOS correction becomes of major importance. Such a situation takes place in many cases of actual interest (quasiparticle current in tunnel structures, c-axis transport in strongly anisotropic high temperature superconductors, NMR relaxation rate, thermoelectric power). The correction to the normal state conductivity above the transition temperature related to the fluctuation DOS renormalization for the dirty superconductor can be evaluated qualitatively. Indeed, the fact that some electrons (Ne per unit volume) participate in fluctuation Cooper pairing means that the effective number of carriers taking part in oneelectron charge transfer diminishes, leading to a decrease of conductivity (here we are dealing with the longitudinal component):

DOS ıxx =−

2ns e2 Ne e2 =− , m m

(10.192)

Fluctuation Phenomena in Superconductors

417

where ns is the superfluid density coinciding with the Cooper pairs concentration. The latter can be identified with the average value of the square of the order parameter modulus already calculated as the correlator (10.99) with r ∼ . For the 2D case, which is of the most interest to us, one finds: ns =

√ 1 1 1 mT K0 ( ) = ln , 2 4˛ s s

(10.193)

where we have used the explicit relation (10.17) between ˛ and . As we will see the corresponding expression for the fluctuation DOS correction to conductivity (10.192) coincides with the accuracy of 2 with the microscopic expression (10.204) which will be carried out below. The third purely quantum fluctuation contribution is generated by the coherent scattering of the electrons forming a Cooper pair on the same elastic impurity. This is the so-called anomalous MakiThompson (MT) contribution [6, 7], which can be treated as the result of Andreev scattering of the electron by fluctuation Cooper pairs. This contribution often turns out to be important for conductivity and other transport phenomena. Its temperature singularity near Tc is similar to that of the paraconductivity, although being extremely sensitive to electron phase-breaking processes and to the type of orbital symmetry of pairing it can be suppressed.Let us evaluate it. The physical origin of the Maki–Thompson correction consists in the fact that the Cooper interaction of electrons with the almost opposite momenta changes the mean free path (diffusion coefficient) of electrons. As we have already seen in the previous section the amplitude of this interaction increases drastically when T → Tc : geff =

g 1 T 1 ≈ = . !D = 1 − g ln 2T T − Tc ln TTc

What is the reason of this growth? One can say that the electrons scatter one another in a resonant way with the virtual Cooper pairs formation. Or it is possible to imagine that the electrons undergo the Andreev scattering at fluctuation Cooper pairs binding in the Cooper pair themselves. The probability

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A. I. Larkin and A.A. Varlamov

of such induced pair irradiation (let us recall that Cooper pairs are Bose particles) is proportional to their number in the final state, i.e. n(p) (10.7). For small momenta it is n(p) ∼ 1/ . One can ask why such interaction does not manifest itself considerably far from the transition point. The matter of fact is that just a small number of electrons with the total momentum q   −1 (T) interact so intensively. In accordance with the Heisenberg principle the minimal distance between such electrons is of the order of ∼ (T). On the other hand, interacting electrons have to approach each other up to the distance of the Fermi length F ∼ 1/pF . The probability of such event may be estimated in the spirit of the self-intersection trajectories contribution evaluation in the weak localization theory [100,101]. In the process of diffusion motion the distance between two electrons increases with the time growth in accordance with the Einstein law: R(t) ∼ (Dt)1/2 . Hence the scattering probability tmax D−1 F vF dt . W∼ RD (t) tmin

The lower limit of the integral can be estimated from the condition R(tmin ) ∼ (T) (only such electrons interact in the resonant way). The upper limit is determined by the phase breaking time ' , since for larger time intervals the phase coherence, necessary for the pair formation, is broken. As a result the relative correction to conductivity due to such processes is equal to the product of the scattering probability on the effective interaction constant: ı MT / = W geff . In the 2D case ı MT ∼

D' e2 ln 2 . 8  (T)

This result will be confirmed below in the framework of the microscopic consideration. 10.7.2

Generalities

Let us pass to the microscopic calculation of the fluctuation conductivity of the layered superconductor. We begin by discussing the quasiparticle nor-

mal state energy spectrum. While models with several conducting layers per unit cell and with either intralayer or interlayer pairing have been considered [102], it has been shown [103] that all of these models give rise to a Josephson pair potential that is periodic in kz ; the wave-vector component parallel to the c-axis, with period s, the c-axis repeat distance. While such models differ in their superconducting densities of states, they all give rise to qualitatively similar fluctuation propagators, which differ only in the precise definitions of the parameters and in the precise form of the Josephson coupling potential. Ignoring the rather unimportant differences between such models in the Gaussian fluctuation regime above Tc (H), we therefore consider the simplest model of a layered superconductor in which there is one layer per unit cell with intralayer singlet s-wave pairing. These assumptions lead to the simple spectrum (10.185) and hence to a Fermi surface having the form of a corrugated cylinder (see Fig. 10.8). Some remarks regarding the normal-state quasiparticle momentum relaxation time are necessary. In the “old” layered superconductors the materials were generally assumed to be in the dirty limit (like TaS2 (pyridine)1/2). In the high-Tc cuprates, however, both single crystals and epitaxial thin films are nominally in the “intermediate” regime, with l/xy ≈ 2 − 5. In addition, the situation in the cuprates is complicated by the presence of phonons for T  Tc  100 K, the nearly localized magnetic moments on the Cu2+ sites, and by other unspecified inelastic processes. In this section we assume simple elastic intralayer scattering and restrict our consideration to the local limit in the fluctuation Cooper pair motion. This means that we consider the case of not too clean superconductors, keeping the impurity concentration ni and reduced temperature such that the resulting mean free path satisfies the requirement lxy (T) = √xy" and the impurity vertex can be taken in the local form (10.184) with ((q, p)|)2F.S. determined by (10.187). The phase-breaking time ' is supposed to be much larger than . The most general relation between the current density j(r,t) and vector-potential A(r , t  ) is given through the so-called electromagnetic response operator Q˛ˇ (r, r , t, t  ) [90]:

10

Fluctuation Phenomena in Superconductors

419

Fig. 10.9. Feynman diagrams for the leading-order contributions to the fluctuation conductivity. Wavy lines are fluctuation propagators, thin solid lines with arrows are impurity-averaged normal-state Green’s functions, shaded semicircles are vertex corrections arising from impurities, dashed lines with central crosses are additional impurity renormalizations and shaded rectangles are impurity ladders. Diagram 1 represents the Aslamazov–Larkin term, diagrams 2–4 represent the Maki–Thompson type contributions, diagrams 5–10 arise from corrections to the normal state density of states

 j(r,t) = −

Q˛ˇ (r, r , t, t  )A(r , t  ) dr dt  .

Assuming space and time homogeneity, one can take the Fourier transform of this relation and compare it with the definition of the conductivity tensor j˛ = ˛ˇ Eˇ .This permits us to express the conductivity tensor in terms of the retarded electromagnetic response operator ˛ˇ (!) = −

1 [Q˛ˇ ]R (!) . i!

(10.194)

The electromagnetic response operator Q˛ˇ (! ), defined for Matsubara frequencies ! = (2 + 1)T, can be presented as the correlator of two exact oneelectron Green functions [90] averaged over impurities and accounting for interactions, in our case the particle–particle interactions in the Cooper channel. The appropriate diagrams corresponding to the first order of perturbation theory in the fluctuation amplitude are shown in Fig. 10.9. With each electromagnetic field component A˛ we associate the external vertex ev˛ (p) = e @(p) . For the @p˛

longitudinal conductivity tensor elements (parallel to the layers, for which ˛ = x, y), the resulting vertex is simply ep˛ /m. For the c-axis conductivity, the vertex is given by evz (p) = e

@(p) = −eJs sin(pz s) . @pz

(10.195)

Each solid line in the diagrams represents a oneelectron Green function averaged over impurities (10.165), a wavy line represents a fluctuation propagator L(q, §k ) (10.188), and three-leg vertices were defined by (10.184).The four-leg impurity vertex,appearing in diagrams 3,4, 9 and 10 of the Fig. 10.9, is called the Cooperon in the weak localization theory (see, for example, [104]). It is easy to see that it differs from the above three-leg vertex only by the additional factor (2 )−1 . We do not renormalize the current vertices: it is well known (see [90]) that this renormalization only leads to the substitution of the scattering time  by the transport one tr . We integrate over the internal Cooper pair momentum q and electron momentum p and sum over the internal

420

A. I. Larkin and A.A. Varlamov

fermionic and bosonic Matsubara frequencies, with momentum and energy conservation at each internal vertex (fluctuation propagator endpoint) in the analytical expressions for the diagrams presented in Fig. 10.9. After these necessary introductory remarks and definitions we pass to the microscopic calculation of the different fluctuation contributions.

only). The remaining summation in (10.198) is performed in the same way as was done in (10.188) and gives: B˛ (q, §k , ! ) = − 8T × ! 

10.7.3

The Aslamazov–Larkin Contribution

We first examine the AL paraconductivity (diagram 1 of Fig. 10.9). Actually this contribution was already studied in the Sect. 10.4 in the framework of the TDGL equation but, in order to demonstrate how the method works, we will here carry out the appropriate calculations in the microscopic approach, as was originally done by Aslamazov and Larkin [5]. The AL contribution to the electromagnetic response operator tensor has the form: AL Q˛ˇ (! ) = −4e2 T

 d 3q B˛ (q, §k , ! )L(q, §k ) (2)3 § k

× Bˇ (q, §k , ! )L(q, §k + ! ) , (10.196)

+

$ 

: (2) 9 v˛ qˇ vˇ FS 2 vF

B 2 1 |§k | + ! + Dq + 2 4T

B 2 1 |§k+ | + ! + Dq + 2 4T

%

%

 −

 −

(10.198) Dq2 1 |§k | + B + 2 4T

B 2 1 ||§k+ | + Dq + 2 4T



%&

.

Now let us return to the general expression for AL Q˛ˇ (! ) and transform the §k − summation into a contour integral, using the identity [105]   1 z f (−iz) , f (§k ) = dz coth T 4i 2T C § k

where z = i§k is a variable in the plane of complex frequency and the contour C encloses all bosonic Matsubara frequencies over which the summation is carried out. In our case the contour C can be chosen as a circle with radius tending to infinity (see Fig. 10.10):

where the three Green function block is given by (10.197) B˛ (q, §k , ! ) =  3  d p T (q, "n+ , §k − "n )(q, "n , §k − "n ) 3 (2) " n

× v˛ (p)G(p, "n+ )G(p, "n)G(q − p, §k − "n ) . Expanding G(q − p, §k − "n ) over q one finds that the angular integration over the Fermi surface kills the first term and only leaves the second term of the expansion nonzero. Then the -integration is performed by means of the Cauchy theorem.The further summation over the fermionic frequency is cumbersome, so we will show it for the example of the simplest case of a dirty superconductor with T  1. In this case the main sources of the "n -dependence in (10.198) are the -vertices. Those originating from the Green functions can be neglected for the parameter T  1 (indeed, one can see that "n ∼ T are important in vertices,while in Green functions "n   −1

Fig. 10.10. The contour of integration in the plane of complex frequencies

10 AL Q˛ˇ (! )

e2 =− i × coth



d3 q (2)3

 dz

(10.199)

C

z B˛ (q, −iz + ! , −iz) 2T

× L(q, −iz)Bˇ (q, −iz + ! , −iz) × L(q, −iz + ! ) . One can see that the integrand function in (10.200) has ranges of analyticity at the lines Im z = 0 and Im z = −i! . Indeed, the fluctuation propagator L(q, §k ) and Green function blocks B˛ (q, §k , ! ) were defined on the bosonic Matsubara frequencies only, while now we have to use them as functions of the continuous variable z. As is well known from the properties of Green functions in the complex plane z, two analytical functions, related with L(q, §k ) can be introduced. The first one, LR (q, −iz) (retarded), is analytic in the upper half-plane (Im z 0), while the second one, LA (q, −iz) (advanced), has no singularities in the lower half-plane (Im z 0).As we have seen above the same lines separate the domains of the analyticity of the Green function blocks,so the functions BRR , BRA , BAA, analytic in each domain, can be introduced (with the appropriate choices of the |§k+ | and |§k | signs in the arguments of the -functions, see (10.199)). This means that by cutting the z-plane along the lines Im z = 0 and Im z = −i! we can reduce the calculation of the contour integral to the sum of three integrals along the contours C1 , C2 , C3 that enclose domains of well defined analyticity of the integrand function. The integral along the large circle evidently vanishes and the contour integral is reduced to four integrals along the cuts of the plane in Fig. 10.10: 

z I(q, ! ) = B˛ (q, −iz) dz coth 2T C1 +C2 +C3 × L(q, −iz)Bˇ (q, −iz)L(q, −iz + ! ) ∞ dz coth

= −∞

Fluctuation Phenomena in Superconductors

z R L (q, −iz + ! ) 2T

  RR R RA RA A × BRR ˛ Bˇ L (q, −iz) − B˛ Bˇ L (q, −iz)

∞−i!  

+

dz coth

421

z A L (q, −iz) 2T

−∞−i!





RA R AA AA A × BRA ˛ Bˇ L (q, −iz + ! ) − B˛ Bˇ L (q, −iz + ! ) .

Now one can shift the variable in the last integral to z = z − i! . We take into account that i! is the z period of coth 2T and get an expression analytic in i! → !. In the vicinity of Tc , due to the pole structure of the fluctuation propagators in (10.196), the leading contribution to the electromagnetic response operAL(R) ator Q˛ˇ arises from them rather than from the frequency dependence of the vertices B˛ . So we can neglect the §k -dependencies and ! -dependencies of the Green functions blocks and use the expression for B˛ (q, 0, 0) valid only for small in plane momenta projections (q˛ , ˛ = x, y): (2) B˛ (q) = −2 2 vF

1

vF2 q˛ , ˛ = x, y sJ 2 sin qz s, ˛ = z

.

Detailed calculations demonstrate that this result can be generalized to an arbitrary impurity concentration just by using (10.189) for (2) . Finally we have: AL(R) (!) Q˛ˇ

2e2 =− 

∞ ×

dz coth



d 3q B˛ (q)Bˇ (q) (2)3

 z 

LR (q, −iz − i!) 2T

−∞

 + LA (q, −iz + i!) Im LR (q, −iz) . Being interested here in the d.c. conductivity one can expand the integrand function in !. It is possible to show that the zeroth order term is canceled by the same type of contributions from all other diagrams (this cancelation confirms the absence of anomalous diamagnetism above the critical temperature). The remaining integral can be integrated by parts and then carried out taking into account that the contribution most singular in comes from the region z ∼  T:

422

A. I. Larkin and A.A. Varlamov AL xx

e2 = 2T ∞ ×



d3q 2 B (q, 0, 0) (2)3 x dz

z sinh2 2T −∞

=

= →

2  2 e2(2)



2

Im LR (q, −iz)



d2 q s (2)2 q2  ×

((2) q2 + )((2) q2 + + r) 3/2 e2 1 16s [ ( + r)]1/2 1 √ e2 1/ r,  r 16s

1/ ,  r

,

(10.200)

where the Lawrence–Doniach anisotropy parameter r [27] was already defined by (10.62). In the same way one can evaluate the AL contribution to the transverse fluctuation conductivity [95,106,107]:  d2 q e2sr 2 AL zz = (10.201) 32 (2)2 1  ×

2 ((2) q + )((2) q2 + + r) 3/2   e2s + r/2 = −1 32(2) [ ( + r)]1/2 1 √ r/ , for  r e2 s → . 64(2) (r/2 )2 , for  r Note that contrary to the case of in-plane conductivity, the critical exponent for zz above the Lawrence–Doniach crossover temperature TLD (for which (TLD ) = r) is 2 instead of 1, so the crossover occurs from the 0D to 3D regimes. This is related with the tunneling (so from the band structure point of view, effectively zero-dimensional) character of electron motion along the c-axis. 10.7.4

Contributions from Fluctuations of the Density of States

In the original paper of Aslamazov and Larkin [5] the most singular AL contribution to conductivity,

heat capacity and other properties of a superconductor above the critical temperature was considered. Diagrams of the type 5–6 were pictured and correctly evaluated as less singular in . Nevertheless, the specific form of the AL contribution to the transverse conductivity of a layered superconductor, which may be considerably suppressed for small interlayer transparency, suggested that the contributions from diagrams 5–10 of Fig.10.9 be re-examined, which are indeed less divergent in , but turn out to be of lower order in the transmittance and of the opposite sign with respect to the AL one [95,96]. These so-called DOS diagrams describe the changes in the normal Drude-type conductivity due to fluctuation renormalization of the normal quasiparticles density of states above the transition temperature (see Sect. 10.8). In the dirty limit, the calculation of contributions to the longitudinal fluctuation conductivity xx from such diagrams was discussed in [99,108]. Contrary to the case of the AL contribution, the inplane and out-of-plane components of the DOS contribution differ only in the square of the ratio of effective Fermi velocities in the parallel and perpendicular directions. This allows us to calculate both components simultaneously. The contribution to the fluctuation conductivity due to diagram 5 is 5 Q˛ˇ (! ) = 2e2T

 §k

×



d3 q L (q, §k )T (2)3

2 (q, "n , §k − "n )

"n

 ×

d3 p v˛ (p)vˇ (p)G2 (p, "n) (2)3

× G(q − p, §k − "n )G(p, "n+ ) , and diagram 6 gives an identical contribution. Evaluation of the integrations over the in-plane momenta p and the summation over the internal frequencies "n are straightforward. Treatment of the other internal frequencies §k is less obvious, but in order to obtain the leading singular behavior in the vicinity of transition it suffices to set §k = 0 [99]. After integration over qz , we have [95,97]:

10 5+6 ˛ˇ



e2 A˛ˇ 1 (2) =− 2s

|q|≤ −1

d2 q (2)2

(10.202)

1  ×

( + (2) q2 )( + r + (2) q2 ) 1/2   e21 2 A˛ˇ ln 1/2 , ≈− 8s + ( + r)1/2 where Axx = Ayy = 1, Azz = (sJ/vF ) , A˛=ˇ = 0 and 2

1 =

(vF )2  2  (2)







1 1 + 2 4T

 −

3 4T



  1 . 2

In order to cut off the ultra-violet divergence in q we have introduced here a cut off parameter qmax = −1/2  −1 = (2) in complete agreement with Sect.10.3.Let us stress that in the framework of the phenomenological GL theory we attributed this cut-off to the breakdown of the GL approach at momenta as large as q ∼  −1 . The microscopic approach developed here permits to see how this cut-off appears: the divergent shortwave-length contribution arising from GL-like fluctuation propagators is automatically restricted by the q-dependencies of the impurity vertices and Green functions, which appear on the scale q ∼ l−1 . In a similar manner, the equal contributions from diagrams 7 and 8 sum to  e2 d2 q 7+8 ˛ˇ = − A˛ˇ  2 (2) 2s (2)2 |q|≤qmax

1  ×

2 ( + (2) q )( +r + (2) q2 ) 1/2   e2  2 2 A˛ˇ ln 1/2 , ≈− 8s + ( +r)1/2   (vF )2  1 2 = . (10.203) 2 3 T 2 Comparing (10.203) and (10.203), we see that in the clean limit, the main contributions from the DOS fluctuations arise from diagrams 5 and 6. In the dirty limit, diagrams 7 and 8 are also important, having -1/3 the value of diagrams 5 and 6, for both xx and zz . Diagrams 9 and 10 are not singular in 1 at all

Fluctuation Phenomena in Superconductors

423

and can be neglected. The total DOS contribution to the in-plane and c-axis conductivity is therefore   e2 2 DOS , ˛ˇ = − (T)A˛ˇ ln 1/2 2s + ( + r)1/2 (10.204) where (T) = 1 + 2 = (10.205)   1   1  1 1 − + + 2T 2

 1 2 1 4T  1  1  →  1 2 + − − 4T 2 4T 2 2 

56(3)/ 4 ≈ 0.691, T  1

 √ 8 2 (T)2 / 7(3) ≈ 9.384 (T)2 , 1  T  1/

is a function of T only. As √ will be shown below at the upper limit T ∼ 1/ the DOS contribution reaches the value of the other fluctuation contributions and in the limit of T → ∞ exactly eliminates the Maki–Thompson one. 10.7.5

The Maki–Thompson Contribution

We now consider another quantum correction to fluctuation conductivity which is called the Maki–Thompson (MT) contribution (diagram 2 of Fig. 10.9). It was first discussed by Maki [6] in a paper which appeared almost simultaneously with the paper of Aslamazov and Larkin [5]. Both these articles gave rise to the microscopic theory of fluctuations in superconductors. Maki found that, in spite of the seeming weaker singularity of diagram 2 with respect to the AL one (it contains one propagator only, while the AL one contains two), it can contribute to conductivity comparably or even stronger than AL. From the moment of its discovery the MT contribution became the subject of intense controversy. In his original paper Maki found that in the 3D case this fluctuation correction is four times larger than the AL one. In the 2D case the result was striking: the MT contribution simply diverged. This paradox was, at least heuristically, resolved by Thompson [7]: he proposed to cut off the infra-red divergence in the Cooper pair center of mass momentum integration by the introduction of the finite length ls of inelastic scatterings of electrons on paramagnetic impurities. Further papers of Patton [109], Keller and Ko-

424

A. I. Larkin and A.A. Varlamov

renman [110] clarified that the presence of paramagnetic impurities or other external phase-breaking sources is not necessary: the fluctuation Cooper pairing of two electrons results in a change of the quasiparticle phase itself and the corresponding phasebreaking time ' appears as a natural cut off parameter of the MT divergence in the strictly 2D case. The minimal quasi-two-dimensionality of the electron spectrum, as we will show below, automatically results in a cut-off of the MT divergence. Although the MT contribution to the in-plane conductivity is expected to be important in the case of low pair-breaking,experiments on high-temperature superconductors have shown that the excess in-plane conductivity can usually be explained in terms of the fluctuation paraconductivity alone. Two possible explanations can be found for this fact. The first one is that the pair-breaking in these materials is not weak. The second is related to the d-wave symmetry of pairing that kills the anomalous Maki–Thompson process [111,112]. Below we will consider the case of s-pairing, where the Maki–Thompson process is well pronounced. The appearance of the anomalously large MT contribution is nontrivial and worth being discussed. We consider the scattering lifetime  and the pairbreaking lifetime ' to be arbitrary, but satisfying ' . In accordance with diagram 2 of Fig. 10.9 the analytical expression for the MT contribution to the electromagnetic response tensor can be written as   d3q MT Q˛ˇ (! ) = 2e2T L(q, §k )I˛ˇ (q, §k , ! ) , (2)3 §k (10.206) where I˛ˇ (q, §k , ! ) = T



(q, "n+ , §k−n− )(q, "n , §k−n )

"n

(10.207) 

d3 p (2)3

× v˛ (p)vˇ (q − p)G(p, "n+ )G(p, "n) × G(q − p, §k−n− )G(q − p, §k−n ) . In the vicinity of Tc , it is possible to restrict consideration to the static limit of the MT diagram, simply by setting §k = 0 in (10.206). Although dynamic

effects can be important for the longitudinal fluctuation conductivity well above TLD , the static approximation is correct very close to Tc , as shown in [98,113].The main q-dependence in (10.206) arises from the propagator and vertices . This is why we can assume q = 0 in Green functions and to calculate the electron momentum integral passing, as usual, to a (p) integration: 9 : I˛ˇ (q, 0, ! ) =  v˛ (p)vˇ (q − p) FS ×T

 "n

×



1 1   2 B B 2 |2"n+ | + Dq |2"n | + Dq

1 . | "n+ | + | "n |

(10.208)

In evaluating the sum over the Matsubara frequencies "n in (10.208) it is useful to split it into the two parts. In the first equation "n belongs to the domains [−∞, −! ] and [0, ∞], which finally give two equal contributions. This gives rise to the regular part of the MT diagram. The second, anomalous, part of the MT diagram arises from the summation over "n in the domain [−! , 0[. In this interval, the further analytic continuation over ! leads to the appearance of an additional diffusive pole: (1)

(2)

I˛ˇ (q, 0, ! ) = I˛ˇ (q, ! ) + I˛ˇ (q, ! ) = 9 :  v˛ (p)vˇ (q − p) FS $ ∞  1 × 2T   B 2 2"n+ + Dq n=0

1 1  × 2 B 2"n + Dq 2"n + ! +  −1 −1  T 1   B 2 ! +  −1 n=− 2"n+ + Dq & 1  . × B 2 −2"n + Dq

+

The limits of summation in the first sum do not depend on ! , so it is an analytic function of this argument and can be continued to the upper half-plane of the complex frequency by the simple substitution ! → −i!. Then, with ! → 0 one can expand the

10

and perform the summation in terms of digamma function: 9 : (1) I˛ˇ (q, −i!) =  v˛ (p)vˇ (q − p) FS (10.209) 1 %  i! @ @ @  +   × const. + × B 2 B 2 2 @ Dq @ ( −1 ) @ Dq 1 B 2  −1 − Dq

$ 

B 2 1 ! + Dq + 2 4T

%

 −

B 2 1 Dq + 2 4T

%&2

.

The values of characteristic momenta q  l−1 are determined by the domain of convergence of the final integral of the propagator L(q, 0) in (10.206) (analB 2 with ogously to (10.203)) and one can neglect Dq −1 respect to  . The result is 9 : (1)R (q, ! → 0) =  v˛ (p)vˇ (q − p) FS I˛ˇ  i! 2 (10.210) × const. + 8T $    &2     1 1  1  1 2 + − − . × 2 4T 2 2T The appearance of the constant in Q˛ˇ (! ) was already discussed in the case of the AL contribution and, as was mentioned there, it is canceled with the similar contributions of the other diagrams [98] and we will not consider it any more. (2) Now let us pass to the calculation of I˛ˇ (q, ! ). Expanding the summing function in simple fractions one can express the result of summation in terms of digamma functions: 9 : 1  v˛ (p)vˇ (q − p) FS 1 (2) I˛ˇ (q, ! ) = −1 B 2 4 ! +  ! + Dq %  %& $  B 2 B 2 1 Dq 1 2! + Dq + − + . × 2 4T 2 4T (10.211) Doing the analytical continuation i! → ! → 0 and taking into account that in the further q(2)R integration of I˛ˇ (q, ! → 0), due to the singularity B 2  T one can for small q, the important range is Dq

Fluctuation Phenomena in Superconductors

425

be expand the digamma functions in the numerator of (10.211) up to the second order. This results in ap(2) (q, ! ). The pearance of the two contributions in I˛ˇ first one, anomalous, 9 : i!  v˛ (p)vˇ (q − p) FS (an)R I˛ˇ (q, ! → 0) = − , B 2 16T −i! + Dq (10.212) while the second, regular, turns out to be exactly (1) (q, ! ), hence equal to I˛ˇ (reg)R

I˛ˇ

(1)R (q, !) = 2I˛ˇ (q, !) .

(10.213)

Because of the considerable difference in the angular averaging of the different tensor components we discuss the MT contribution to the in-plane and out of plane conductivities separately. 9 : Taking into account that vx (p)vx (q − p) FS = −v2F /2 one can find that the calculation of the regular part of MT diagram to the in-plane conductivity is completely similar to the corresponding DOS contribution. Here we list the final result [97] only:   e2 2 MT(reg) xx , = − ˜ ln 1/2 2s + ( + r)1/2 where (T) ˜ = 1 →

− 2





1 21 2



1 + 4T  1 + 4T −

+



1  21  2

1 + 2T

−

1 4T

56(3)/ 4 ≈ 0.346, for T  1 √ 8 T for 1  T  1/ 

1

 

21  2

(10.214)

is a function only of T.We note that this regular MT term is negative, as is the overall DOS contribution. For the anomalous part of the in-plane MT contribution we have:  d3 q MT(an) xx = 8e2(2) T (2)3 1 2 B [1/' + Dq ][ + (2) q2 + r2 (1 − cos qz s)] %  e2 1/2 + ( + r)1/2 , (10.215) ln = 1/2 4s( − ' )

' + ( ' + r)1/2 ×

426

A. I. Larkin and A.A. Varlamov

where, in accordance with [7], the infra-red divergence for the purely 2D case (r = 0) is cut off at Dq2 ∼ 1/' .25 The dimensionless parameter

' =

2  → 8T' v2F '

1

1, T  1  3  √ 7(3)/ 2 T , 1  T  1/

is introduced for simplicity. If r = 0 the MT contribution turns out to be finite even with ' = ∞. A comparison of (10.200) and (10.215) indicates that in the weak pair-breaking limit the MT diagram makes an important contribution to the longitudinal fluctuation conductivity: it is four times larger than the AL contribution in the 3D regime, and even logarithmically exceeds it in the 2D regime above TLD . For finite pair-breaking, however, the MT contribution is greatly reduced in magnitude. We now consider the calculation of the MT contribution to the transverse conductivity. The explicit expressions 9 for vz (p) and : vz (q − p) (see (10.195)) result in vx (p)vx (q − p) FS = 12 J 2 s 2 cos qz s. We take the limit J1 in evaluating the remaining integrals, which may then be performed exactly. The regular part of the MT contribution to the transverse conductivity is e2 s 2r (T) ˜ 4  cos qz s d3 q × (2)3 + (2) q2 + 2r (1 − cos qz s)  2 e2sr (T) ˜ ( + r)1/2 − 1/2 . =− 16(2) r 1/2

zzMT(reg) = −

This term is smaller in magnitude than is the DOS one, and therefore makes a relatively small contribution to the overall fluctuation conductivity. In the 3D regime below TLD it is proportional to J 2 , and in the 2D regime above TLD it is proportional to J 4 . 25

26

For the anomalous part of the MT diagram one can find  d3 q e2 J 2 s 2 =  MT(an) zz 4 (2)3 cos qz s × 2 B [1/' + Dq ][ + (2) q2 + 2r (1 − cos qz s)]  d 2q e2 s = (10.216) 4( − ' ) (2)2 $

' + (2) q2 + r/2  ×

( ' + (2) q2 )( ' + (2) q2 + r) 1/2 & + (2) q2 + r/2  −

( + (2) q2 )( + (2) q2 + r) 1/2  

' + r + e2s −1 . = 16(2) [ ( + r)]1/2 + [ ' ( ' + r)]1/2 In examining the limiting cases of (10.216),it is useful to consider the cases of weak ( ' r,⇐⇒ J 2 ' 1/2) and strong ( ' r, ⇐⇒ J 2 ' 1/2) pair-breaking separately.26 For weak pair-breaking, we have ⎧ r/ ' ,  '  r ⎪ ⎪ ⎨ √ 2 e s r/ , '   r . zzMT(an) → 16(2) ⎪ ⎪ ⎩ r/ (2 ) , '  r  In this case, there is the usual 3D to 2D dimensional crossover in the anomalous MT contribution at TLD . There is an additional crossover at T' (where Tc < T' < TLD ), characterized by (T' ) = ' , below which the anomalous MT term saturates. Below TLD , the MT contribution is proportional to J, but in the 2D regime above TLD it is proportional to J 2 . For strong pair-breaking it is 1

zzMT(an)

r/ ' ,  r  ' e2s → .   2 32(2) r / 4 ' , r  min{ ' , }

A detailed study of the phase-breaking time, its energy dependence and the effect on the MT contribution was done in [114]. Physically the value J 2  characterizes the effective interlayer tunneling rate [97,115]. When 1/ J 2 1/, the quasiparticles scatter many times before tunneling to the neighboring layers, and the pairs live long enough for them to tunnel coherently. When J 2 1/ , the pairs decay before both paired quasiparticles tunnel.

10

In this case,the 3D regime (below TLD ) is not singular, and the anomalous MT contribution is proportional to J 2 , rather than J for weak pair-breaking. In the 2D regime, it is proportional to J 4 for strong pairbreaking, as opposed to J 2 for weak pair-breaking. In addition, the overall magnitude of the anomalous MT contribution with strong pair-breaking is greatly reduced from that for weak pair-breaking. Let us now compare the regular and anomalous MT contributions. Since these contributions are opposite in sign, it is important to determine which will dominate. For the in-plane resistivity, the situation is straightforward: the anomalous part always dominates over the regular and the latter can be neglected. The case of c-axis resistivity requires more discussion. Since we expect ' ≥ , strong pair-breaking is likely in the dirty limit. When the pair-breaking is weak, the anomalous term is always of lower order in J than the regular term, so the regular term can be neglected. This is true for both the clean and dirty limits.The most important regime for the regular MT term is the dirty limit with strong pair-breaking. In this case, when ' T ∼ 1, the regular and anomalous terms are comparable in magnitude. In short, it is usually a good approximation to neglect the regular term, except in the dirty limit with relatively strong pair-breaking and only for the out-of-plane conductivity. Finally let us mention that the contributions from the two other diagrams of the MT type (diagrams 3 and 4 of Fig. 10.9) in the vicinity of critical temperature can be omitted: one can check that they have an additional square of the Cooper pair center of mass momentum q in the integrand of q-integration with respect to diagram 2 and hence turn out to be less singular in . 10.7.6

Discussion

Although the in-plane and out-of plane components of the fluctuation conductivity tensor of a layered superconductor contain the same fluctuation contributions, their temperature behavior may be qualitafl tively different. In fact, for xx , the negative contributions are considerably less than the positive ones in the entire experimentally accessible temperature

Fluctuation Phenomena in Superconductors

427

Fig. 10.11. The normalized excess conductivity for samples of YBCO-123 (triangles), BSSCO-2212 (squares) and BSSCO-2223 (circles) plotted against = ln T/Tc on a ln-ln plot as described in [124]. Dotted and solid lines are the AL theory in 3D and 2D, respectively. The dashed line is the extended theory of [113]

range above the transition, and it is a positive monotonic function of the temperature. Moreover, for HTS compounds, where the pair-breaking is strong and the MT contribution is in the overdamped regime, it is almost always enough to take into account only the paraconductivity to fit experimental data. Some examples of the experimental findings for in-plane fluctuation conductivity of HTS materials can be found in [116–123]. In Fig. 10.11 the fluctuation part of in-plane confl ductivity xx is plotted as a function of = ln T/Tc on a double logarithmic scale for three HTS samples (the solid line represents the 2D AL behavior √ (1/ ), the dotted line represents the 3D one: 3.2/ ) [124].One can see that paraconductivity of the less anisotropic YBCO compound asymptotically tends to the 3D behavior (1/ 1/2) for 0.1, showing the LD crossover at ≈ 0.07; the curve for the more anisotropic 2223 phase of BSCCO starts to bend for 0.03 while the most anisotropic 2212 phase of BSCCO shows a 2D behavior in the whole temperature range investigated.All three compounds show a universal 2D tem-

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A. I. Larkin and A.A. Varlamov

perature behavior above the LD crossover up to the limits of the GL region. It is interesting that around ≈ 0.24 all the curves bend down and follow the same asymptotic 1/ 3 behavior (dashed line). Finally at the value ≈ 0.45 all the curves fall down indicating the end of the observable fluctuation regime. Reggiani et al. [113] extended the 2D AL theory to the high temperature region by taking into account the short wavelength fluctuations.The following universal formula for 2D paraconductivity of a clean 2D superconductor as a function of the generalized reduced temperature = ln T/Tc was obtained:27 fl

xx =

e2 f ( ) , 16s

with f ( ) = −1 ,  1 and f ( ) = −3 ,  1. In the case of the out-of-plane conductivity the situation is quite different. Both positive contributions (AL and anomalous MT) are suppressed by the interlayer transparency, leading to a competition between positive and negative terms. This can lead to a maximum in the c-axis fluctuation resistivity which occurs in the 2D regime (in the case discussed J1, r1 and ' 1):   1 1 1 m/r ≈  ˜ − . − (8r)1/2 8 2 ' This nontrivial effect of fluctuations on the transverse resistance of a layered superconductor allows a successful fit to the data observed on optimally and overdoped HTS samples (see, for instance, Fig. 10.12) where the growth of the resistance still can be treated as the correction. The fluctuation mechanism of the growth of the transverse resistance can be easily understood in a qualitative manner. Indeed to modify the in-plane result (10.119) for the case of c-axis paraconductivity one has to take into account the hopping character of the electronic motion in this direction. If the probability of one-electron interlayer hopping is P1 , then the probability of coherent hopping for two electrons during the fluctuation Cooper pair lifetime GL is the conditional probability of these two events: 27

Fig. 10.12. Fit of the temperature dependence of the transverse resistance of an underdoped BSCCO c-axis oriented film to the results of the fluctuation theory [141]. The inset shows the details of the fit in the temperature range between Tc and 110 K

P2 = P1 (P1GL ). The transverse paraconductivity may thus be estimated as ⊥AL ∼ P2 AL ∼ P12 12 , in complete accordance with (10.202). We see that the temperature singularity of ⊥AL turns out to be stronger than that in AL , however for a strongly anisotropic layered superconductor ⊥AL is considerably suppressed by the square of the small probability of inter-plane electron hopping which enters in the pre-factor. It is this suppression which leads to the necessity of taking into account the DOS contribution to the transverse conductivity. The latter is less singular in temperature but, in contrast to the paraconductivity, manifests itself in the first, not the second, order in the interlayer transparency ⊥DOS ∼ −P1 ln "1 . The DOS fluctuation correction to the one-electron transverse conductivity is negative and, being proportional to the first order of P1 , can completely change the traditional picture of fluctuations just rounding the resistivity temperature dependence around transition. The shape of the temperature dependence of the transverse resistance is mainly determined by competition of the opposite sign contributions: the paraconductivity and MT term, which are strongly temperature-dependent but

  In Sect. 10.7 we will demonstrate how such a dependence 1/ ln3 (T/Tc ) appears by accounting for short wavelength fluctuations for the 2D fluctuation susceptibility.

10

are suppressed by the square of the barrier transparency and the DOS contribution, which has a weaker temperature dependence but depends only linearly on the barrier transparency.

10.8

Manifestation of Fluctuations in Various Properties

In this section we will demonstrate the applications of the microscopic theory of fluctuations.The limited volume does not permit us to deliver the systematic review of the modern theory here and we restrict ourselves by only presenting the several representative recent studies. It is necessary to underline that the comparison of the results offluctuation theory with the experimental findings on HTS materials has to be considered sooner in a qualitative context than in a quantitative context. Indeed, as is clear now, the superconductivity in the most of HTS compounds has the nontrivial symmetry. Moreover, as was discussed in the previous section, these compounds are rather clean than dirty. Both these complications can be taken into account (see for example [97, 111]), but this was not done in the majority of the cited papers.

10.8.1

The Effects of Fluctuations on Magnetoconductivity

The experimental investigations of the fluctuation magnetoconductivity are of special interest first because this physical value weakly depends on the normal state properties of superconductor and second due to its special sensitivity to temperature and magnetic field. The role of AL contribution for both the in-plane and out-of-plane magnetoconductivities was studied above in the framework of the phenomenological approach. The microscopic calculations of the other fluctuation corrections to the inplane magnetoconductivity conductivity show that the MT contribution has the same positive sign and temperature singularity as the AL one. In the case of weak pair-breaking it can even considerably exceed the latter. The negative DOS contribution, like

Fluctuation Phenomena in Superconductors

429

in the case of the zero-field conductivity, turns out to be considerably less singular and many authors (see, e.g. [125–136]) successfully explained the in-plane magnetoresistance data in HTS using the AL and MT contributions only [137–140]. Turning to the out-of-plane magnetoconductivity of a layered superconductor one can find a quite different situation. Both the AL and MT contributions here turn out to be of the second order in the interlayer transparency, and this circumstance makes a less singular DOS contribution, which however remains of first order in transparency, to be competitive with the main terms [142]. The large number of microscopic characteristics involved in this competition, like the Fermi velocity, interlayer transparency,phase-breaking and elastic relaxation times, gives rise to the possibility of occurrence of different scenarios for various compounds. The c-axis magnetoresistance of a set of HTS materials shows a very characteristic behavior above Tc0. In contrast to the ab-plane magnetoresistance which is positive at all temperatures, the magnetoresistance along the c-axis has been found in many HTS compounds (BSSCO [143,145–147],LSSCO [148],YBCO [149] and TlBCCO [150]) to have a negative sign not too close to Tc0 and turn positive at lower temperatures. We will show how this behavior finds its explanation within the fluctuation theory [97]. We consider here the effect of a magnetic field parallel to the c-axis. In this case both quasiparticles and Cooper pairs move along Landau orbits within the layers. The c-axis dispersion remains unchanged from the zero-field form. In the chosen geometry one can generalize the zero-field results reported in the previous section to finite field strengths simply by the replacement of the two-dimensional integration over q by a summation over the Landau levels 

d2 q H  h  → = 2 (2) ¥0 n 2(2) n

2 (let us recall that (2) = xy ). So the general expressions for all fluctuation corrections to the c-axis conductivity in a magnetic field can be simply written in the form [97]:

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A. I. Larkin and A.A. Varlamov

zzAL =

e2sr 2 h 2 64xy

(10.217)

∞ 

1

n=0

{[ + h(2n + 1)][r + + h(2n + 1)]}3/2

1

n=0

{[ + h(2n + 1)][r + + h(2n + 1)]}1/2

∞  

,

(10.218)

1/h 

zzMT(reg) = −

e2 s h ˜ 2 4xy

,

(10.219) 

+ h(2n + 1) + r/2 {[ + h(2n + 1)][r + + h(2n + 1)]}

1/2

n=0

zzMT(an) = ∞ 

×

e2sh − ' )

−1

,

(10.220)

2 (" 8xy



' + h(2n + 1) + r/2 >1/2 = [( ' + h(2n + 1)][ ' + h(2n + 1) + r)] n=0  + h(2n + 1) + r/2 . − {[ + h(2n + 1)][r + + h(2n + 1)]}1/2 For the in-plane component of the fluctuation conductivity tensor the only additional problem appears in the AL diagram, where the matrix elements of the harmonic oscillator type, originating from the B (q ) blocks, have to be calculated. How to do this was demonstrated in detail in Sect. 10.4. The other contributions are essentially analogous to their caxis counterparts: ∞

AL xx =



− +

e2 h( + ) ˜ 2s

(10.222)

1/h 

1

n=0

{[ + h(2n + 1)][r + + h(2n + 1)]}1/2

,

and

e2 srh =− 2 8xy

zzDOS

DOS MT(reg) + xx =− xx

e2  (n + 1) 4s n=0

(10.221) 1

{[ + h(2n + 1)][r + + h(2n + 1)]}1/2 2 {[ + h(2n + 2)][r + + h(2n + 2)]}1/2  1 {[ + h(2n + 3)][r + + h(2n + 3)]}1/2

,

e2h MT(an) (10.223) = xx 4s( − ' )  ∞  1 × >1/2 = [( ' + h(2n + 1)][ ' + h(2n + 1) + r)] n=0  1 . − {[ + h(2n + 1)][r + + h(2n + 1)]}1/2 These results can,in principle,already be used for numerical evaluations and fitting of the experimental data, which was indeed successfully done in a series of papers [149–151]. The detailed comparison of the cited results with the experimental data [149,152], especially in strong fields, raised the problem of regularization of the DOS contribution. If in the absence of the magnetic field its ultra-violet divergence was successfully cut off at q ∼  −1 , in the case under consideration the cut-off parameter depends on the magnetic field and makes the fitting procedure ambiguous. The solution of this problem was proposed in [153], where the authors calculated the difference zzDOS = zzDOS (h, ) − zzDOS (0, ) applying to (10.219) and (10.222) the same trick already used in Sect. 10.2 for the regularization of the free energy in magnetic field (10.79). The corresponding asymptotics for all out-of-plane fluctuation contributions are presented in Table 10.3: The procedure described gives an excellent fitting up to very high fields [154], which is shown in Fig. 10.13. Let us start the analysis from the 2D case (r  ). One can see that here the positive DOS contribution to magnetoconductivity turns out to be dominant. It grows as H 2 up to Hc2( ) and then crosses to a slow logarithmic asymptote. At H ∼ Hc2(0) the value of zzDOS (h ∼ 1, ) = −zzDOS (0, ), which means the total suppression of the fluctuation correction in such a strong field. The regular part of the Maki– Thompson contribution does not manifest itself in

10

Fluctuation Phenomena in Superconductors

431

Table 10.3. Results for the out of plane contribution of the fluctuations to the conductivity h

hr

(3D) 7 e2 s h 0.428 r 2 16xy r 7 e2 s˜ h 0.428 2 r 8xy r

e2 s r( + r/2) 2 h 2 [ ( + r)]3/2 3 25 xy

zzDOS zz

e2 s˜ r2 h2 2 6 3 2  xy [ ( + r)]3/2

−zzAL

e2 s r 2 ( + r/2) 2 h 2 [ ( + r)]5/2 28 xy

zzAL (0, ) −

e2s r2 h2 2 7 3 2  xy [ ( + r)]2

zzMT(an)(0, ) −

√ r 2 e2s h 3 27  2xy '3/2

zzMT(an)(0) −

MT(reg)

−zzMT(an) min{ , r}  ' −zzMT(an)

'  min{ , r}

zzDOS MT(reg)

MT(reg)

−zz

−zzAL

zzAL (0, ) −

min{ , r}  ' −zzMT(an)

'  min{ , r}

7

7

e2s 2 32xy

3.24e2 s 2 64xy

r h

7

r

'

r h

max{ , r}  h (2D) √ e2 s h r ln √ √ 2 8xy + +r

zz

−zzMT(an)

3.24e2 s 2 xy

(0, ) −

 2 e2 s˜ r 2 2 28 xy h

7(3)e2 s r 2 2 29 xy h2

zzMT(an)(0, ) −

3 2 e2 s max{r, ' } 2 28 xy h

zzMT(an)(0, ) −

3 2 e2 s (r + ) 2 28 xy h

this case while the AL term can compete with the DOS one in the immediate vicinity of Tc , where the small anisotropy factor r can be compensated by the additional 3 in the denominator.The anomalous MT contribution can contribute in the case of small pairbreaking only, which is opposite to what is expected in HTS. In the 3D case (  r) the behavior of the magnetoconductivity is more complex. In weak and inter-

mediate fields the main, negative, contribution to the magnetoconductivity occurs from the AL and MT terms. At H ∼ Hc2( )(h ∼ ) the paraconductivity is already considerably suppressed by the magnetic field and the h2 -dependenceof the magnetoconductivity changes through the hr tendency to the high (fl)

field asymptote −zz (0, ). In this intermediate region of fields(  h  r), side by side with the decrease (∼ hr ) of the main AL and MT contribu-

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A. I. Larkin and A.A. Varlamov

Fig. 10.13. Magnetoconductivity versus temperature at 27 T for an underdoped Bi-2212 single crystal. The solid line represents the theoretical calculation. The symbols are the experimental magnetoconductivity zz (B  c  I) [154]

tions,the growth of the still relatively small DOS term takes place. At the upper limit of this region (h ∼ r) its positive contribution is of the same order as the AL one and at high fields (r  h  1) the DOS contribution determines the slow logarithmic decay of the fluctuation correction to the conductivity, which is completely suppressed only at H ∼ Hc2(0). The regular part of the Maki–Thompson contribution is not of special importance in the 3D case. It remains comparable with the DOS contribution in the dirty case at fields h  r, but decreases rapidly (∼ hr ) at strong fields ( h  r), in the only region where the robust cDOS (h, ) ∼ ln hr shows up surviving up to h ∼ 1. The temperature dependence of the different fluctuation contributions to the magnetoconductivity calculated for an underdoped Bi-2212 single crystal at the magnetic field 27 T is presented in Fig. 10.14. The formulas for the in-plane magnetoconductivity are presented in Table 10.4: Analyzing the table one can see that in almost all regions the negative AL and MT contributions govern the behavior of inplane magnetoconductivity. Nevertheless, similar to the c-axis case, the high field behavior is again deterMT(reg) DOS mined by the positive logarithmic (xx + xx contribution, which is the only one to survive in strong field. It is important to stress that the suppression of the DOS contribution by a magnetic field takes place very slowly. Such robustness with respect to the magnetic field is of the same physical origin

as the slow logarithmic dependence of the DOS-type corrections on temperature. Another important problem that appears in the fitting of the resistive transition shape in relatively strong fields with the fluctuation theory is the much larger broadening of the transition than predicted by the Abrikosov–Gor’kov theory [155]. Kim and Gray [96] explained the broadening of the c-axis peak with increasing magnetic field in terms of Josephson coupling, describing a layered superconductor as a stack of Josephson junctions. In [69, 156] the selfconsistent Hartree approach was proposed for the extension of fluctuation theory beyond the Gaussian approximation. It results in the considerable shift of Tc (H) toward low temperatures with a corresponding broadening of the transition. The renormalized reduced temperature "˜h is determined according to the self-consistent equation [69]: "h = "˜h − 4 Gi(2) h ×

1/h  n=0

(10.224)

1 . [("˜h + hn)("˜h + h(n + 1) + r)]1/2

The authors of [157], following the procedure proposed by Dorsey and Ullah [69], modified (10.218)– (10.224) to account for (10.225). As a result they succeeded to fit quantitatively both in-plane resistivity transition and the transverse resistivity peak for BSCCO films strongly broadened by the applied magnetic field.

10

Fluctuation Phenomena in Superconductors

433

Fig. 10.14. Decomposition of the calculation of total theoretical magnetoconductivity for an underdoped Bi-2212 single crystal at 27 T. The inset shows the regular and anomalous parts of the MT contribution that are too small to be presented on the same scale as the AL and DOS contributions [154]

Table 10.4. In-plane magnetoconductivity contributions due to fluctuations h 2

AL xx

−

MT(an) xx

(min{ , r}  ' ) MT(an) xx

( '  min{ , r})

DOS + (xx MT(reg) xx

10.8.2

−

 h  r; 2

e [8 ( + r) + 3r ] 2 h 28 s [ ( + r)]5/2

( + r/2) 2 e2 h 5 3 2 s [ ( + r)]3/2

As was mentioned above the role of fluctuations is especially pronounced in the vicinity of the critical temperature. Nevertheless for some phenomena they

0.2e2 1 √ ; s hr

3 2 e2 1 32s h 7 e2 ( + ) ˜ h ; 0.428 26 s r √ e2 ( + ) ˜ h ln √ √ 32s ( + + r)

MT −xx (0, ) +

e2 ( + ) ˜ ( + r/2) 2 h 3 27 s [ ( + r)]3/2

Fluctuations Far from Tc or in Strong Magnetic Fields

e2 1 8s h

√

' e2 1 ln √ √ = 8s ' 2h + 2h + r

MT −xx (0, ) +

1 e h2 3 25 s 3/2 r 1/2

e2 1 ; √ 4s 2hr

AL −xx (0, ) +

MT −xx (0, ) +

2

−

AL −xx (0, ) +

max{ , r}  h

can still also be considerable far from the transition. In these cases the GL theory is certainly unapplicable since the short-wave and dynamical fluctuation contributions have to be taken into account. It can be done in the microscopic approach, which we will demonstrate by several examples.

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A. I. Larkin and A.A. Varlamov

Fluctuation Magnetic Susceptibility Far from Transition The given above qualitative estimations (10.57)– (10.64) for the fluctuation diamagnetic susceptibility, based on the Langevin formula, demonstrate that even at high temperatures T  Tc it turns to be of the order of P for clean 3D superconductors and noticeably exceeds this value for 2D systems. In order to develop the microscopic theory [29,158,159] let us start from the general expression for free energy in the one-loop approximation:  F=T Tr{ln[1 − g¢(§k , r, r )]} , (10.225) §k

where g is the effective interaction constant related with the transition critical temperature by (10.173). This approximation corresponds to the ladder one (see (10.166)) for the fluctuation propagator. The polarization operator ¢(§k , r, r ) is determined by expression (10.167). In the case of an applied magnetic field the homogeneity of the system is lost and ¢(§k , r, r) depends not on the space variable difference r− r, but on each one separately. Expanding ¢(§k , r, r) one can express the magnetic susceptibility of a layered superconductor in a weak magnetic field perpendicular to the layers in terms of the derivatives ¢x = @q@ x ¢(q) [29]: =−

@2 F @H 2

(10.226)

 2 = − e2T 3 § k



d3 q 3 L ¢x (¢x ¢yy − ¢y ¢xy ) . (2)3

The final expressions for the fluctuation diamagnetic susceptibility in the clean and dirty cases for wide range of temperatures can be written as: fl(2) (T) =

  0.05 1 EF P ,  −1  T  !D 3 pF s T ln (T/Tc ) P ,   ⎪ ⎪ 1 2 2 ⎪ −1 ⎩ ln ln − ln ln(T/Tc ) , Tc  T   3 Tc  ⎧ ⎪ ⎪ ⎪ ⎨

(10.227) 28

  0.05 1 EF P . (10.228) pF s T ln3 (T/Tc ) Let us stress that these results are also valid for the fluctuation diamagnetism of a normal metal with g0, if by Tc one uses the formal value Tc ∼ EF exp( 1g ). (2)

fl (T) =

Fluctuation Magnetoconductivity Far from Transition Let us discuss the conductivity of the 2D electron system with impurities in a magnetic field at low temperatures. Even in the absence of the field the effects of quantum interference of the noninteracting electrons in their scatterings on elastic impurities already results in the appearance of a nontrivial temperature dependence of the resistance. This result contradicts the statement of the classical theory of metals requiring the saturation of the resistance at its residual value at low temperatures. In a superconductor above the critical temperature this, so-called weak localization (WL), effect is amplified by the Andreev reflection of electrons on the fluctuation Cooper pair leading to appearance of the MT correction to the conductivity. The characteristic feature of both the MT and WL corrections is their extreme sensitivity to the dephasing time ' and to weak magnetic fields. Beyond the GL region (T  Tc ) the MT correction is determined by the same diagram 2 in Fig. 10.9 but now the dynamic (§k = 0) and short-wave-length (q ∼  −1 ) fluctuation modes have to be taken into account. The corresponding calculations were performed in [98, 160] and the result can be written in the form:    e2

ıWL+MT = ˛ − ˇ(T) Y §L ' , (10.229) 2 2 where we introduced the effective Larmour frequency for the diffusion motion §L = 4DeH with the diffusion coefficient D 28 and the function ⎧  ⎨ x2  1 1 , x1 . + = Y (x) = ln x + 24 ⎩ 2 x ln x, x  1 (10.230)

A comparison of the expressions (10.78), (10.135) and (10.229) relates the Larmour frequency with the dimensionless field: h = §L /2Tc introduced in Sect. 10.2 and the diffusion coefficient with the phenomenological GL constants D = 1/m˛.

10

The first term in this formula corresponds to the WL contribution (˛ = 1 if the spin-orbit interaction of the electrons with the impurities is small while in the opposite limiting case ˛ = −1/2), the second describes the MT contribution to magnetoconductivity. The function ˇ[ln(T/Tc )] was introduced in [160]. At T → Tc ˇ(x) = 1/x and (10.229) reduces to the already studied MT correction in the vicinity of critical temperature. For T  Tc ˇ(x) = 1/x 2 and the MT contribution gives a logarithmically small correction to the WL result. Its zero-field value, being proportional to ln−2 (T/Tc ), decreases with the growth of the temperature faster than both the AL contribution (in the dirty case ıAL ∼ 1/ ln(T/Tc )) and the especially slow DOS contribution (ıDOS ∼ ln ln(1/Tc ) − ln ln(T/Tc )) (see [98, 99]). It worth mentioning that for the region of temperatures T  Tc , analogous to (10.227)–(10.228), the result (10.229) can be applied both to superconducting and normal metals (g0), if in place of the critical temperature the formal value Tc ∼ EF exp( 1g ) is undermined. The interplay of the localization and fluctuation corrections was extensively studied (see, for example, [161–165]). Fluctuations in Magnetic Fields near Hc2 (0) As one can see from (10.222)–(10.224), in the vicinity of the upper critical field Hc2(T) the fluctuation cor−1/2 rections diverge as h−1 for the 2D case and as h for 29 the 3D case (it is enough to keep just the terms with n = 0 in these formulas). This behavior is also preserved in strong magnetic fields, but the coefficients undergo changes.A case of special interest is T  Tc (which means H → Hc2(0)), which represents an example of a quantum phase transition [166]. Microscopic analysis of the magnetoconductivity permits us to study the effect of fluctuations in magnetic fields of the order of Hc2 (0), where the GL functional approach is inapplicable. We restrict our study to the case of a dirty metal (T  1). In this limit |! ˜ n+ − ! ˜ −n | ≈  −1 and the Green function correlator (10.169) can be written in the form 29

Fluctuation Phenomena in Superconductors

435

P(q, "1 , "2 ) = 2  2 (−"1 "2 ) (10.231)   −1 2 B ×  − |"1 − "2 | − Dq . Expressing ¢(q, §k ) in terms of P(q, "1 , "2 ) by means of (10.187) and using the definition of the critical temperature one can find an explicit formula for the fluctuation propagator: L−1 (q, §k ) = −g −1 + ¢(q, §k ) $

= − ln

T + Tc



(10.232)

B 2 1 |§k | + Dq + 2 4T

% −

 & 1 . 2

The prominent characteristic of this expression is that it is valid even relatively far from the critical temperature (for temperatures T  min{ −1 , !D }) and for |q|  l−1 , |§k |  !D . One can rewrite this expression in a magnetic field applied along the c-axis  in the Landau representation B 2 ⇒ §L (n + 1/2) [158]: by simply replacing Dq  

L−1 n (qz , §k ) = − ln

T + Tc



1 |§k | + 2 4T

%  §L (n + 1/2) + 4J 2 sin2 qz s/2 − + 4T

(10.233)  & 1 . 2

In the case of arbitrary temperatures and magnetic fields the expression for the AL contribution to the conductivity takes the form: AL (! ) = Qxx ∞   Bn,m (§k + ! , §k )Lm (§k ) −4e2 T §k {n,m}=0

×Bm,n(§k , §k + ! )Ln (§k + ! )

(10.234)

(we have restricted our consideration to the 2D case). The expression for Bn,m (§k , ! ) can be rewritten as (10.235) Bn,m(§k + ! , §k ) =   √ −4  2 D(2) eH(n + 1)ım,n+1 + eHnım,n−1  n ("i + ! , §k − "i )m ("i , §k − "i ) ×T "i

with m ("1 , "2 ) =

h is the renormalized by the magnetic field reduced temperature h = + h

1 Ÿ (−"1 "2 ) . (10.236)  |"1 − "2 | + §L (m + 1/2)

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A. I. Larkin and A.A. Varlamov

The critical field Hc2(T) is determined by the equation L−1 0 (qz = 0, §k = 0) = 0. This is why in the vicinity of Hc2(T) the singular contribution to (10.234) originates only from the terms with L0 (0, §k ). The frequency dependencies of the functions Bn,m (§k + ! , §k ) and L1 (§k ) are weak although we cannot omit them. It is enough to restrict ourselves to the linear approximation in their frequency dependencies. If the temperature T  Tc0 the sum over frequencies in (10.235) can be approximated by an integral. Transforming the boson frequency §k summation to a contour integration as was done above and making the analytic continuation in the external frequency ! one can get an explicit expression for the d.c. paraconductivity. In the same spirit the contributionsof all other diagrams in Fig. 10.9 which contribute to fluctuation conductivity in the case under discussion are calculated side by side with the AL one. The final answer can be presented in the form: ıtot =

   Tc0 3 E T 2e2 Hc2 (T) + − ln 3 2 2 T Tc0 H − Hc2 (T)    Tc0 H − Hc2(T) (10.237) + 2 E T Hc2(T)  Hc2(T) Tc0  +4 2 E T H − Hc2 (T)     Tc0 H − Hc2 (T) −1 , × 2 E T Hc2 (T)

where E is the Euler constant. Let us consider some limiting cases. If the temperature is relatively high T/Tc0  (H − Hc2(T)) /Hc2(T), we obtain the following formula for the fluctuation conductivity: ı =

2 E e2 T  2 Tc0



Hc2(T) H − Hc2(T)

 .

(10.238)

If HHc2(0), we can introduce Tc (H) and rewrite (10.238) in the usual way ı =

3e2 Tc0 . 2 2 E  T − Tc (H)

(10.239)

If HHc2 (0), in the low-temperature limit T/Tc0  (H − Hc2(T)) /Hc2(T) we have ı = −

Hc2(T) 2e2 . ln 2 3 H − Hc2(T)

(10.240)

One can see, that even at zero temperature a logarithmic singularity remains and the corresponding correction is negative. It results from all three fluctuation contributions, although the DOS one exceeds the others by numerical factor. Let us recall that in the case of the c-axis conductivity of a layered superconductor, or in granular superconductors above Tc , the DOS contribution exceeds the MT and AL ones parametrically [167]. 10.8.3

The Effect of Fluctuations on the Hall Conductivity

Let us start with a discussion of the physical meaning of the Hall resistivity xy . In the case of only one type of carriers it depends on their concentration n and turns out to be independent of the electron diffusion coefficient: xy = H/ (en). The fluctuation processes of the MT and DOS types contribute to the diffusion coefficient, so their expected contribution to the Hall resistivity is zero. For the Hall conductivity in a weak field one can write 2 (n)2 (n) xy = xy xx = xy xx + 2xy xx ıxx   ıxx (n) 1+2 , (10.241) = xy xx

so, evidently, the relative fluctuation correction to Hall conductivity is twice as large as the fluctuation correction to the diagonal component. This qualitative speculation is confirmed by the direct calculation of the MT type diagram [168]. The AL process corresponds to an independent charge transfer which cannot be reduced to a renormalization of the diffusion coefficient. It contributes weakly to the Hall effect, and this contribution is related to the Cooper pair particle-hole asymmetry. This effect was investigated in a set of papers: [69, 168–173]. Let us recall that the proper general expression describing the paraconductivity contribution to the Hall conductivity in the general case

10

Fluctuation Phenomena in Superconductors AL xy =

of arbitrary magnetic fields and frequencies (in the TDGL theory limits) was already carried out above in the phenomenological approach (see (10.131)). The microscopic consideration of this value can be done AL in the spirit of the calculation of xx (see (10.200)) and after the analytical continuation results in  AL xy

=

2h (0) 

−/s



−∞

@ × Im LRn (z) Re LRn+1 (z) @z −

Im LRn+1 (z)

Im GL h 2 (0) /s dkz 1 × 

2 + r sin2 (kz s/2) 2 −/s %  + r sin2 (kz s/2) , ×F 2h

 (0) 2



 @ R Re Ln (z) , @z

@ ln Tc @E

where

 1 (x) + x (x) − 1 − ( + x) . 2 (10.245) For H → 0 the expression for the fluctuation Hall paraconductivity takes the form   + r/2 e2T Im GL AL xy = h . (10.246) 6s  (0) [ ( + r)]3/2 

F(x) = 4x 2

(10.242)

where dimensionless magnetic field h was introduced by (10.78). The phenomenological expression (10.131) can be obtained from this formula by carrying out the frequency integration in the same way as was done in the calculation of (10.200) (the essential region of integration is z  T). One can see from (10.242) that if Im LRn (−z) = − Im LRn (z) and Re LRn (−z) = Re LRn (z) the Hall conductivity is equal to zero, or, in terms of the phenomenological parametrization, the reality of GL results in a zero Hall effect. Physically it is possible to say that this zero is the direct consequence of electron-hole symmetry. However, from the formula (10.167) one can see that an energy dependence of the density of states or the electron interaction constant g immediately results in the appearance of an imaginary part of GL . In the weak interaction approximation

Im GL = −

(10.244)

e2 T

2  ∞

/s ∞ z dkz dz coth (n + 1) 2 2T n=0

437

 .

(10.243)

E=EF

Usually this value is small in comparison with Re GL by a ratio of the order of Tc /EF . Taking into account the terms of the order of Im GL in (10.242) and using the explicit form of the fluctuation propagator for layered superconductor (10.190) one can find

One can see that in the 2D case the temperature dependence of the AL fluctuation correction on the Hall conductivity   e2 Tc h AL xy ≈ 12s EF 2 turns out to be more singular than the MT one. 10.8.4

Fluctuations in the Ultra-Clean Case

When dealing with the superconductor electrodynamics in the fluctuation regime, it is necessary to remember that in the vicinity of the critical temperature the role of the effective size of a fluctuation Cooper pair √ is played by the GL coherence length GL (T) = 0 / . So, as has already been mentioned above,the case of a pure enough superconductor with electron mean free path   0 has to be formally subdivided into the clean (0    GL (T)) and ultra-clean (GL (T)  ) limits. The nontrivial cancelation of the contributions, previously divergent in T (see, for example, (10.206)), will be shown in this section. This results in a reduction of the total fluctuation correction in the ultra-clean case to the AL term only. We will base our considerations on [174], restricting our analysis to the case of a 2D electron system.

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A. I. Larkin and A.A. Varlamov

In terms of the parameter T, used in the theory of disordered alloys, three different domains of the metal purity can be√distinguished: T √1 (dirty case), 1  T  1/ (clean case), and 1/  T (ultra-clean case of nonlocal electrodynamics). The latter case has rarely been discussed in the literature [139,175,176] in spite of the fact that it becomes of primary importance for metals of very modest purity, let us say, with√T ≈ 10. Really, in this case the condition T ≥ 1/ , which in terms of the reduced temperature is read as 10−2 ≤  1, practically covers all the experimentally accessible range of temperatures for the fluctuation conductivity measurements. As regards the √ usually considered local clean case (1  T  1/ ) for the chosen value T ≈ 10, it would not have any range of applicability. Indeed, the equivalent condition for the allowed temperature interval is  (T)−2 and it almost contradicts the 2D thermodynamical Ginzburg–Levanyuk criterion for the mean field approximation applicability (Gi(2) = ETFc  ). Moreover, as we will show below, for transport coefficients the higher order corrections become comparable with the mean-field results much before they are importantfor thermodynamical quantities, namely at ∼ Gi(2) [177, 178]. So in practice one can speak about the dirty and the nonlocal ultra-clean limits only. As we saw above the 2D AL contribution turns out to be completely independent of the electron mean free path  [5].The anomalous Maki–Thompson contribution, being induced by the pairing on the Brownian diffusive trajectories [76], naturally depends on T, but in an indirect way. √It turns out to be independent up to T ∼ 1/ (see √ (10.215)) and diverges as T ln(T) for T  1/ [139, 175]. The analogous problem takes place in the case of the DOS contribution: its standard diagrammatic technique calculations lead to a negative correction (10.206) [95], which is evidently strongly divergent when T → ∞. In the derivation of all these results the local form of the fluctuation propagator and Cooperons were used. This is why√the direct extension of their validity for T  1/ → ∞ is incorrect. One can notice [174] that √at the upper limit of the clean case, when T ∼ 1/ , both the DOS and

anomalous MT (10.215) contributions turn out to be of the same order of magnitude but of opposite signs. So one can suspect that in the case of a correct procedure of impurity averaging in the ultra-clean case the large negative DOS contribution can be canceled with the positive anomalous MT one. In the case of a 2D electron spectrum the Cooperon can be calculated exactly for the case of an arbitrary electron mean free path: %−1  Ÿ(−"1 "2 ) . (q, "1 , "2 ) = 1 −   ( "1 −  "2 )2 + vF2 q2 (10.247) One can see that this expression can be reduced to (10.184) in the case of vF q  | "1 −  "2 |. Let us stress that this result was carried out without any expansion over the Cooper pair center of mass momentum q and is valid in the 2D case for arbitrary q. The fluctuation propagator in the 2D case of an arbitrary mean free path can be written as [174] ∞  T  1 −[ L(q, §k )]−1 = ln (10.248) + Tc n=0 n + 1/2  1 . −  2 §k 1 1 vF2 q2 1 + n+ + + − 2 4T 4T 16 2T 2 4T Near Tc ln TTc ≈ and in the local limit, when only small momenta q  1 are involved in the final integrations, (10.248) can be expanded in vF q/ max{T,  −1 } and reduces to the appropriate local expression. Let us demonstrate the specifics of the nonlocal calculations for the example of the Maki–Thompson contribution. We restrict our study to the vicinity of the critical temperature, where the static approximation is valid. Using the nonlocal expressions for the Cooperon and the propagator one can find after integration over electronic momentum:   d2 q Q(MT) (! ) = −4 vF2 e2 T 2 L(q, 0) (2)2 "n  ∼   ∼ ∼ ∼ × M n , n+ , q + M n+ , n , q , (10.249)

10

where   M ˛, ˇ, q = Rq (2˛)Rq(˛ + ˇ) − Ÿ(˛ˇ)Rq(2˛)Rq(2ˇ)    (ˇ − ˛)2 Rq(2˛) − 1 Rq(2ˇ) − 1 Rq (˛ + ˇ)  and Rq(x) = x 2 + vF2 q2 . The analogous consideration of the DOS diagrams 2 and 4 which are the leading ones in the clean case [95] results in similar expressions. One can see that, after analytical continuation with respect to the external frequency ! → −i! and ! → 0, each of the DOS or MT type diagrams is written in the form of a Laurent series of the type C−2 (T)2 +C−1 (T)+C0 +C1 (T)−1 +. . . and is divergent at T → ∞ in accordance with (10.206). Nevertheless, the expansion in a Laurent series of the sum of these nonlocal diagrams leads to the exact cancelation of all divergent contributions. The leading order of the sum of the MT and DOS contributions in the limit of T  1 turns out to be proportional to (T)−1 only and disappears in the ultra-clean limit. So the correct accounting of nonlocal scattering processes in the ultra-clean limit results in a total quantum correction negligible in comparison with the AL contribution. Nevertheless, its formal independence on impurities concentration (see (10.119)) was also re-examined for the ultra-clean case in [173]. There it was demonstrated that this statement is valid in a rigorous sense only in the case of direct current and absence of a magnetic field. Let us recall that the normal Drude conductivity in the ultra-clean case takes the form ± (!) = xx ± ixy =

e2 n/m , 1 − i(! ∓ !c )

(10.250)

where !c is the cyclotron frequency. When  → ∞ the real part of the conductivity vanishes. The analysis of the AL diagram in the ultra-clean case demonstrates that each of the Green functions blocks B acquires the same denominator. As a result the expression for the fluctuation conductivity contains the same Drude like-pole but it is of second order ±AL (!) =

AL(l) AL(l) ± ixy xx

(1 − i(! ∓ !c ))2

(10.251)

Fluctuation Phenomena in Superconductors

439

AL(l) is the component of the paraconductivity (˛ˇ tensor calculated above in the local limit (10.119)– (10.246)). The origin of this pole can be recognized by means of the following speculation. The electric field does not interact directly with the fluctuation Cooper pairs,but it produces the effect by interaction with the quasiparticles forming these pairs only. The characteristic time of the change of a quasiparticle state is of the order of . Consequently the singleparticle Drude type conductivity in an a.c. field has a first order pole, while in the AL paraconductivity it is of second order [173]. In spite of this difference one can see that the AL conductivity, like the Drude one, vanishes at ! = 0,  → ∞ because in the absence of impurities the interaction of the electrons does not produce any effective force acting on the superconducting fluctuations, while the d.c. paraconductivity conserves its usual -independent form. It is impossible to distinguish the motion of the electron liquid from the condensate motion in current experiments without additional scattering. The nonlocal form of the Cooperon and fluctuation propagator have to be taken into account not only for the ultra-clean case but in every problem where relatively large bosonic momenta are involved like dynamical and short wavelength fluctuations beyond the vicinity of critical temperature, the effect of relatively strong magnetic fields on fluctuations, etc. Recently such an approach was developed in a number of studies [174,175,179,180].

10.8.5

The Effect of Fluctuations on the One-Electron Density of States and on Tunneling Measurements

Density of States The appearance of nonequilibrium Cooper pairing above Tc leads to a redistribution of the oneelectron states around the Fermi level. A semiphenomenological study of the fluctuation effects on the density of states (DOS) of a dirty superconducting material was first carried out while analyzing the tunneling experiments of granular Al in the fluctuation regime just above Tc [181]. The second metallic electrode was in the superconducting regime and its well developed gap gave a bias voltage around

440

A. I. Larkin and A.A. Varlamov Fig. 10.15. The one-electron Green function with the first order fluctuation correction

which a structure, associated with the superconducting fluctuations of Al, appeared. The measured DOS energy dependence has a dip at the Fermi level,30 reaches its normal value at some energy E0 (T),shows a maximum at an energy value equal to several times E0 , and finally decreases towards its normal value at higher energies. The characteristic energy E0 was found to be of the order of the inverse of the GL relaxation time GL introduced above. The presence of a depression at E = 0 and of a peak at E ∼ (1/GL ) in the DOS above Tc are precursor effects of the appearance of the superconducting gap in the quasiparticle spectrum at temperatures below Tc . The microscopic calculation of the fluctuation contribution to the one-electron DOS can be carried out within the diagrammatic technique [93,94]. Let us start with the discussion of a clean superconductor. As is well known, the one-electron DOS is determined by the imaginary part of the retarded Green function integrated over momentum.This definition permits us to express the appropriate fluctuation correction in terms of the fluctuation propagator:  1 dD p ı (c) (E, ) = − Im ıGR (p, E)  (2)D

(c) ı(3) (E, )

(3) (0)

=

(10.254) √

Tc (4)3/2  Gi(3,c) Re  −1 7(3) GL − 2iE + 32 Tc 2 1 1 × 1/2 , −1/2 −1 −1 GL − iE + GL GL − 2iE + 32 Tc

−

(c) ı(2) (E, )

Tc2 (4)2  (10.255) Gi(2,c) 2 −1 (2) 7(3) E + 22Tc GL 2 1  −1 E + E2 + 22Tc GL E  , ln × 1−  −1 −1 E2 + 22 Tc GL 2 Tc GL  where D =  D/7(3). In a dirty superconductor the calculations may be carried out in a similar way with the only difference that the impurity renormalization of the Cooper vertices has to be taken into account [93]. The value of the fluctuation dip at the Fermi level can be written in the form: 1 Gi(3,d) −3/2, D = 3 ı (d) (0) ∼− . (10.256)  (0) Gi(2,d) −2 , D=2 =−

× G2(p, "n )G(q − p, §k − "n ) . (10.253)

−1 At large energies E  GL DOS recovers its normal value, according to the same laws (10.256) but with the substitution → E/Tc . It is interesting that the critical exponents of the fluctuation correction of the DOS change when moving from a dirty to a clean superconductor [94] (see Fig. 10.16). The analysis of (10.255)–(10.255) gives 1 Gi(3,c) −1/2, D = 3 ı (c) (0) ∼− . (10.257)  (0) D=2 Gi(2,c) −1 ,

The result of the integration of the last expression depends strongly of the electron spectrum dimensionality: for the two important cases of isotropic 3D and 2D electron spectra one finds [94]

The character of the DOS renormalization differs strongly for the clean and dirty cases for the energy scale at which this renormalization occurs. In the dirty case this energy turns out to be [93]

1 = − Im RR (E) , 

(10.252)

where RR (E) is the retarded analytical continuation of the expression corresponding to the diagram of Fig. 10.15:    dD q dD p R("n ) = T L(q, § ) k D (2) (2)D § k

30

Here we refer the energy E to the Fermi level, where we assume E = 0.

10

Fig. 10.16. The theoretical curve of the energy dependence for the normalized correction to the single-particle density of states vs. energy for a clean two-dimensional superconductor above Tc −1 E0(d) ∼ T − Tc ∼ GL , while in the clean case √ (c) E0 ∼ Tc (T − Tc ) [94]. To understand this important difference one has to study the character of the electron motion in both cases [94]. The relevant energy scale in the dirty case is the inverse of the time necessary for the electron to diffuse over a distance equal to the coherence length (T). This energy scale coincides with the inverse relax−1 ation time: t−1 = D −2 (T) ∼ GL ∼ T − Tc . In the clean case, the ballistic motion of the electrons gives rise to a different characteristic energy scale √ −1 1/2 t−1 ∼ vF  −1 (T) ∼ (Tc GL ) ∼ Tc (T − Tc ). One can check that the integration of (10.255) and (10.255) over all positive energies gives zero:

∞ ı (E)dE = 0 .

(10.258)

0

This “sum rule” is a consequence of a conservation law: the number of quasiparticles is determined by the number of cells in the crystal and cannot be changed by the interaction. So the only effect that can be produced by the inter-electron interaction is a redistribution of the energy levels near the Fermi energy. The sum rule (10.258) plays an important role in the understanding of the manifestation of the fluctuation DOS renormalization in the observable phenomena. As we will see in the next section the singularity in the tunneling current (at zero voltage),

Fluctuation Phenomena in Superconductors

441

due to the density of states renormalization,turns out to be much weaker than that in the DOS itself (ln instead of −1 or −2 , see (10.256)–(10.257)).A similar smearing of the DOS singularity occurs in the opening of the pseudo-gap in the c-axis optical conductivity, in the NMR relaxation rate etc. These features are due to the fact that we must always form the convolution of the DOS with some slowly varying function: for example, a difference of Fermi functions in the case of the tunnel current. The sum rule then leads to an almost perfect cancelation of the main singularity at low energies.The main nonzero contribution then comes from the high energy region where the DOS correction has its ‘tail’. Another important consequence of the conservation law (10.258) is the considerable increase of the characteristic energy scale of the fluctuation pseudo-gap opening with respect to E0 : this is eV0 = T for tunneling and ! ∼  −1 for the c-axis optical conductivity. The Effect of Fluctuations on the Tunnel Current It is quite evident that the renormalization of the density of states near the Fermi level, even of only one of the electrodes, will lead to the appearance of anomalies in the voltage-current characteristics of a tunnel junction. The quasiparticle current flowing through it may be written as a convolution of the densities of states with the difference of the electron Fermi distributions in each electrode (L and R): Iqp =

1 eRnL (0)R (0) ∞  ×

E E + eV − tanh tanh 2T 2T

(10.259) 

−∞

× L (E)R (E + eV )dE, where Rn is the Ohmic resistance per unit area and L (0) and R (0) are the densities of states at the Fermi levels in each of electrodes in the absence of interaction. One can see that for low temperatures and voltages the expression in parenthesis is a sharp function of energy near the Fermi level. Nevertheless, depending on the properties of the DOS functions, the convolution (10.259) may exhibit different properties. If the energy scale of the DOS correction is much larger

442

A. I. Larkin and A.A. Varlamov

than T, the expression in parenthesis in (10.259) acts as a delta-function and the zero-bias anomaly in the tunnel conductivity strictly reproduces the anomaly of the density of states around the Fermi level: ıG(V ) ı (eV ) = , Gn (0)  (0)

∞ ıGfl (V , ) 1 dE =   ı (2) (E, ) (10.261) 2 Gn (0) 2T cosh E+eV 2T −∞     ieV 1 2 ∼ Gi(2) ln √ − . √ Re  2 2T + +r

(10.260)

where G(V ) is the differential tunnel conductance and Gn (0) is the background value of the Ohmic conductance supposed to be bias-independent, ıG(V ) = G(V ) − Gn (0). This situation, for instance, occurs in a junction with one amorphous electrode [182], where the dynamically screened Coulomb interaction is strongly retarded, which leads to a considerable suppression of the density of states in the vicinity of the Fermi level, within  −1  T. It is worth stressing that the proportionality between the tunneling current and the electron DOS of the electrodes is widely accepted as an axiom, but generally speaking this is not always so. As one can see from the previous subsection, the opposite situation occurs in the case of the DOS renormalization due to the electron–electron interaction in the Cooper channel: in this case the DOS correction varies strongly already in the scale of E0 ∼ Eker  T and the convolution in (10.259) with the DOS (10.255) has to be carried out without the simplifying approximations assumed to obtain (10.260). We will show that the fluctuation induced pseudogap like structure in the tunnel conductance differs drastically from the anomaly of the density of states (10.255), both in its temperature singularity near Tc and in the energy range of its manifestation. Let us first discuss the effect of the fluctuation suppression of the density of states on the properties of a tunnel junction between a normal metal and a superconductor above Tc . The effect under discussion turns out to be most pronounced in the case of thin superconducting films (d  (T)) and layered superconductors like HTS cuprates. In order to derive the explicit expression for the fluctuation contribution to the differential conductance of a tunnel junction with one thin film electrode close to its Tc we differentiate (10.259) with respect to voltage, and substitute the DOS correction given by (10.255). This results in (see [178]):

It is important to emphasize several nontrivial features of the result obtained. First, the sharp decrease ( −2(1) ) of the density of electron states in the immediate vicinity of the Fermi level generated by fluctuations surprisingly results in a much more moderate growth of the tunnel resistance at zero voltage (ln 1/ ). Second, in spite of the manifestation of the DOS renormalization √ at the characteristic scales E0(d) ∼ T −Tc or E0(cl) ∼ Tc (T − Tc ),the energy scale of the anomaly developed in the I − V characteristic is much larger: eV = T  E0 (see Fig. 10.17). In the inset of Fig. 10.17 the result of measurements of the differential resistance of the tunnel junction Al-I-Sn at temperatures slightly above the critical temperature of the Sn electrode is presented. This experiment was done [183] with the purpose of checking the theory proposed [178]. The nonlinear differential resistance was precisely measured at low voltages, which permitted the observation of the fine structure of the zero-bias anomaly. The reader can compare the shape of the measured fluctuation part of the differential resistance (inset in Fig. 10.17) with the theoretical prediction. It is worth mentioning that the experimentally measured positions of the minima are eV ≈ ±3Tc ,while the theoretical prediction following from (10.262) is eV = ±Tc . Recently similar results on an aluminium film with two regions of different superconducting transition temperatures were reported [184]. The observations of the pseudogap anomalies in tunneling experiments at temperatures above Tc obtained by a variety of experimental techniques were reported in [185–189]. We will now consider the case of a symmetric junction between two superconducting electrodes at temperatures above Tc . In this case, evidently, the correction (10.262) has to be multiplied by a factor of “two”because of the possibility of fluctuation pairing in both electrodes. Furthermore, in view of the extraordinarily weak (∼ ln 1/ ) temperature dependence of the first order correction, different types of

10

Fluctuation Phenomena in Superconductors

443

Fig. 10.17. The theoretical prediction for the fluctuation-induced zero-bias anomaly in tunnel-junction resistance as a function of voltage for reduced temperatures = 0.05 (top curve), = 0.08 (middle curve) and = 0.12 (bottom curve). The inset shows the experimentally observed differential resistance as a function of voltage in an Al-I-Sn junction just above the transition temperature

high order corrections may manifest themselves on √ the energy scale eV ∼ T −Tc or Tc (T − Tc ).Among them are the familiar AL and MT corrections, which take place in the first order of Gi but in the second order of the barrier transparency. Another type of higher order correction appears in the first order of barrier transparency but in the second of fluctuation strength (∼ Gi2 ) [178]. Such corrections are generated by the interaction of fluctuations through the barrier and they can be evaluated directly from (10.259) applied to a symmetric junction.The second order correction in Gi can be written as [178]: ∞ ıGfl(2)

(0, ) ∼ −∞

2

2 Gi(2) dE  E  ı (2) (E, ) ∼ 3 . 2 cosh 2T

(10.262) This nonlinear fluctuation correction turns out to be small by Gi2 but its strong singularity in temperature and opposite sign with respect to ıGfl(1) makes it interesting. Apparently it leads to the appearance of a sharp maximum at zero voltage in G(V ) with a characteristic width eV ∼ T − Tc in the immediate vicinity of Tc (one can call this peak the hyperfine structure). This result was confirmed in [190], but to our knowledge such corrections were never observed in tunneling experiments.

One can see that ıGfl(1) and ıGfl(2) become of the √ 3 ∗ ∼ Gi, i.e. the critical region same order at cr where nonlinear fluctuations effects become important in the problem under consideration starts much before the thermodynamical criterion cr ∼ Gi. In the next section we will discuss this early manifestation of nonlinear fluctuation effects in transport phenomena.

10.8.6

Nonlinear Fluctuation Effects

As we have already seen in the temperature region Gi   1 the thermodynamic fluctuations of the order parameter ¦ can be considered to be Gaussian. Nevertheless, the example of the previous section demonstrates that in transport phenomena nonlinear effects,related with the interaction of fluctuations (higher order corrections) can manifest themselves much earlier. It has been found [177] that nonlinear fluctuation phenomena restrict the Gaussian region in the fluctuation conductivity of asuperconducting film to a new temperature scale: Gi(2d)   1 (see also [109, 110, 114,178, 191]). In this section we obtain expressions for the  conductivity in the temperature region Gi(2d)   Gi(2d) , where both the perturbation theory works well and the nonlinear fluctuation effects are important.

444

A. I. Larkin and A.A. Varlamov

Let us start from the correlator (10.188),which can be expressed by means of the Gi(2d) number: T 32 3 T2 1 Gi(2d) = . D 8T  + 8T k 2 7(3) k2 + D (10.263) 2 The long-wave-length fluctuations with k2 kmin = 8T /D can be considered as a local condensate. They lead to the formation of the pseudogap ¦k∗ ¦k  =

⎡ ⎢ pg = ⎣



2 k2 kmin

⎤1/2 2

dk (2)

⎥ ¦k∗ ¦k ⎦ 2

  T Gi(2d)

(10.264) in the single-particle spectrum of excitations.  Not very close to the transition ( > Gi(2d) ) only excitations with energies E  pg are important. The pseudogap does not play any role for them. Thus, in this region of temperatures it is sufficient to consider fluctuations in the linear approximation only  (see [5–7]). However, in the temperature region  Gi(2d) the nonlinear fluctuation contribution of the excitations with energies E < pg becomes essential. To take into account the spatial dependence of the order parameter we will use the results obtained in [192]. It was shown there that the spatial variations of pg act on single-particle excitations in the same way as magnetic impurities do (the analogy between the effect of fluctuations and magnetic impurities was observed in many papers, see for example, [193]). In this case, the total pair-breaking rate  can be written as a sum of the pair-breaking rate due to the magnetic impurities and the fluctuation term. Thus, the self-consistent equation for  can be written in the following form [192]:  ¦k∗ ¦k  d2 k 1  = + . (10.265) 2 (2) E + 12 Dk2 +  s In the region E   ,   T we obtain from (10.265) and (10.263): 1/2   pg , (10.266)  ∼ T Gi(2d) 31

which coincides with the results obtained in [109, 194]. Let us note that the pair-breaking rate  was found to be of the order of the pseudogap pg . Thus, a wide maximum appears in the density of states at E ∼ pg . As we have already seen, (10.215), in the purely 2D case the Maki–Thompson correction to the conductivity saturates for T  (where  = 8T ' /) and takes the form [14] ı MT T  . ∼ Gi(2d) ln n  8T

(10.267)

As can be seen from  (10.266) such a saturation takes place when < Gi(2d) . Similar results were obtained in [109, 110, 114], with slightly different numerical coefficients.31 However, its exact value is not very important since in the region T <  the Maki–Thompson correction is less singular than the Aslamazov–Larkin one and can be neglected.The latter does not saturate when T tends to Tc but becomes more and more singular. In the presence of the pseudogap if there is no equilibrium, the fluctuating Cooper pair lifetime increases with respect to the GL one: fl = aGL (a1). Recall that analogous changes in the coefficient a in the TDGL equations appear below the transition temperature (see, e.g. [64,70,195–197]). The growth of the coefficient a and, consequently, the increase of the fluctuation lifetime, is due to the fact that the quasiparticles require more time to attain thermal equilibrium (the corresponding time we denote as e ). A rough estimate gives a ∼ pg e . In the case of weak energy relaxation, e has to be determined from the diffusion equation taking account of the pseudogap (see [196–198]). Note that in this complicated case the coefficient a becomes a nonlocal operator. Rough estimates give the following value for the thermal equilibrium transition 2 time e ∼ (Dkmin )−1 ∼ (T )−1 . Taking into account (10.264) we obtain from (10.11) for the paraconductivity contribution in the discussed limit of the weak energy relaxation [14]:

Note that the numerical coefficient in (10.267) depends on the definition of Gi(2d) and how the summation of higher order diagrams is made.

10 3/2

Gi(2d) ı ∼ . n 2

(10.268)

Let us now discuss the role of the energy relaxation processes, characterized by a quasiparticle lifetime " . Nonelastic electron scattering off phonons and other possible collective excitations can decrease " significantly. These processes together with additional pair-breaking processes (due to magnetic impurities or a magnetic field) lead to a decrease of the nonlinear effects. In view of these processes, one can write the following interpolation formula for the nonlinear fluctuation conductivity [14]: ⎡ ⎤ ı 1 Gi(2d) Gi(2d) ⎣1 +   Im  ⎦ . = n + 1 + i T"

T'

(10.269) Note that (10.268)–(10.269) are valid only if the parameters  and " are such that the correction to conductivity ı is larger than the usual Aslamazov–  Larkin correction (10.119). If  > T, T" < Gi(2d) or if T 2 " / < Gi(2d) , then nonlinear effects are negligible and the usual result (10.119) is valid for all > Gi(2d) . We can see that the paraconductivity can exceed the value of the normal conductivity n in the region 3/4 Gi(2d) < < Gi(2d) . Let us recall that in this region corrections to all the thermodynamic coefficients are still small and the linear theory is well applicable. 10.8.7

The Effect of Fluctuations on the Optical Conductivity

The optical conductivity of a layered superconductor can be expressed by the same analytically con(R) tinued electromagnetic response operator Q˛ˇ (!) (see (10.194)) but in contrast to the d.c. conductivity case, calculated without the assumption ! → 0. Let us recall that the paraconductivity tensor in an a.c. field was already studied in Sect. 10.4 in the framework of the TDGL equation [31] and the most interesting asymptotics for our discussion, (10.143) and (10.144), valid for !  T in the 2D regime, were calculated there. The microscopic calculation of the AL diagram [98] shows that in the vicinity

Fluctuation Phenomena in Superconductors

445

of Tc and for !  T the leading singular contriAL (R) arises from bution to the response operator Q˛ˇ the fluctuation propagators rather than from the B˛ blocks, which confirms the TDGL results. Nevertheless, the DOS and MT corrections can be calculated only by the microscopic method as was done in [98,199]. Let us note that the external frequency ! enters in the expression for the DOS contribution to Q˛ˇ (!) only by means of the Green’s function G(p, !n+ ) and it is not involved in q integration. So, near Tc , even in the case of an arbitrary external frequency, we can restrict consideration to the static limit, taking into account only the propagator frequency §k = 0, and get [199]: e2 ˆ (!, T, ) A˛ˇ 2s   2 × ln √ √ , +r+

DOS Re ˛ˇ (!) = −

where the anisotropy tensor A˛ˇ was introduced in (10.203). Let us stress that, in contrast to the AL frequency-dependent contribution, this result has been found with only the assumption  1, so it is valid for any frequency and impurity concentration. The function ˆ (!, T, ) was calculated in [199] exactly, but we present here only its asymptotics for the clean and dirty cases: ⎧ 7(3) ⎪ ⎪ , !  T   −1 ⎪ 2 ⎪ 2 ⎪ ⎪     8 ⎨ T 2 ˆd !, T   −1 = , T  !   −1 , ⎪ ! ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ − T , T   −1  ! !3    −1 ˆcl !, T   ⎧ (T)2 , !   −1  T ⎪ ⎪ ⎪   ⎪ ⎪ T 2 −1 3 ⎨ , !T . = ! 28(3) ⎪  ⎪ 3 ⎪ T ⎪ ⎪ ⎩ −4 ,  −1  T  ! ! The general expression for the MT contribution is too cumbersome, so we restrict ourselves here to the important 2D overdamped regime (r  ≤ ' ):

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zzMT(an)(2D) (!)

⎧ ⎪ ⎨

1, ! ˜  '−1 es r 2  = 7 , 8Tc ' 2 (2) ' ⎪ ,! ˜  '−1 ⎩ ! 2

2

⎧

' 1 ⎪ ⎪ ln , !  '−1 ⎪ e 2 ⎨ ' MT(an)(2D) xx (!) = . 2  8s ⎪ 8T

⎪ c ' −1 ⎪ ⎩ , !  ' ! Let us discuss the results obtained. Because of the large number of parameters entering the expressions we restrict our consideration to the most interesting c-axis component of the fluctuation conductivity tensor in the 2D region (above the Lawrence– Doniach crossover temperature). The AL contribution describes the fluctuation condensate response to the applied electromagnetic field. The current associated with it can be treated as the precursor phenomenon of the screening currents in the superconducting phase. As was demonstrated above the characteristic“binding energy” of the fluctuation Cooper pair is of the order of T − Tc , so it is not surprising that the AL contribution decreases when the electromagnetic field frequency exceeds this value. Indeed ! AL ∼ T − Tc is the only relevant scale for  AL : its frequency dependence does not contain T, ' and . The independence from the latter is due to the fact that elastic impurities do not present obstacles for the motion of Cooper pairs.The interaction of the electromagnetic wave with the fluctuation Cooper pairs resembles, in some way, the anomalous skin-effect where the reflection is determined by the interaction with the free electron system. The anomalous MT contribution is also due to fluctuation Cooper pairs, but this time they are formed by electrons moving along self-intersecting trajectories. Being the contribution related with the Cooper pair electric charge transfer it does not depend on the elastic scattering time, but it turns out to be extremely sensitive to the phase-breaking mechanisms. So two characteristic scales turn out to be relevant in its frequency dependence: T − Tc and '−1 . In the case of HTS, where '−1 has been estimated as at least 0.1Tc , for temperatures up to 5–10 K above Tc the MT contribution is overdamped; it is deter-

mined by the value of ' and is almost temperatureindependent. The DOS contribution to Re  (!) is quite different from those above. In the wide range of frequencies !   −1 the lack of electron states at the Fermi level leads to the opposite sign effect in comparison with the AL and MT contributions: Re  DOS (!) turns out to be negative and this means an increase of the surface impedance,or,in other words,decrease of the reflectance. Nevertheless, the applied electromagnetic field affects the electron distribution and at very high frequencies ! ∼  −1 the DOS contribution changes its sign. It is interesting that the DOS contribution, as a one-electron effect, depends on the impurity scattering in a similar manner to the normal Drude conductivity. The decrease of Re  DOS (!) starts at frequencies ! ∼ min{T,  −1 } which for HTS are much higher than T − Tc and '−1 . The !-dependence of Re zztot with the most natural choice of parameters (Tc r  Tc ≤ '−1  min{T,  −1 }) is presented in Fig. 10.18. Let us discuss this referring to a strongly anisotropic layered superconductor. The positive AL and MT contributions to zztot , being suppressed by the square of the interlayer transparency, are small

Fig. 10.18. The theoretical dependence [199] of the real part of the conductivity, normalized by the Drude normal conductivity, on !/T, % [  (!)] = Re [ (!)] / n . The dashed line refers to the ab-plane component of the conductivity tensor whose Drude normal conductivity is n = N(0)e2 vF2 . The solid line refers to the c-axis component whose Drude normal conductivity is ⊥n = n J 2 s2 /vF2 . In this plot we have put T = 0.3, EF /T = 50, r = 0.01, = 0.04, T' = 4

10

in magnitude and they vary in the low frequency region ! ∼ min{T − Tc , '−1 }. The DOS contribution is proportional to the first order of transparency and in this region remains almost invariable. With a further increase of frequency min{T − Tc , '−1 }  ! the AL and MT contributions decay; Re ⊥ remains negative up to ! ∼ min{T,  −1 }, then it changes its sign at ! ∼  −1 , reaches maximum and rapidly decreases. The following high frequency behavior is governed by the Drude law. So one can see that the characteristic pseudogap-like behavior in the frequency dependence of the c-axis optical conductivity takes place: a transparency window appears in the range ! ∈ [T − Tc ,  −1 ]. In the case of the ab-plane optical conductivity the two first positive contributions are not suppressed by the interlayer transparency, and exceed considerably the negative DOS contribution in a wide range of frequencies. Any pseudogap like behavior is theretot fore unlikely in xx (!); the reflectivity will be of the metallic kind. 10.8.8

Thermoelectric Power above the Superconducting Transition

Thermoelectric effects are difficult both to calculate and to measure if compared with electrical transport properties. At the heart of the problem lies the fact that the thermoelectric coefficients in metals are the small resultant of two opposing currents that almost completely cancel. In calculating the thermoelectric power one finds that the electrons above the Fermi level carry a heat current that is nearly the negative of that carried by the electrons below EF . In the model of a monovalent metal in which band structure and scattering probabilities are symmetric about EF , this cancelation would be exact; in a real metal a small asymmetry survives. Because of their compensated nature,thermoelectric effects are very sensitive to the characteristics of the electronic spectrum, presence of impurities and peculiarities of scattering mechanisms. The inclusion of many-body effects, such as electron–phonon renormalization, multi-phonon scattering, drag effect, adds even more complexity to the problem of calculating the thermoelectric power. Among such effects there is also the influence of

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thermodynamical fluctuations on the thermoelectric transport in a superconductor above the critical temperature. This problem has been attracting the attention of theoreticians for more than twenty years, since the paper of Maki [200] appeared, where the logarithmically divergent AL contribution was predicted for the two-dimensional case. So the AL term turns out to be less singular compared with the corresponding correction to conductivity. In every case where the main AL and MT fluctuation corrections are suppressed for some reason, the contribution connected with fluctuation renormalization of the one-electron density of states (DOS) can become important. The analogous situation also occurs in the case of the thermoelectric coefficient [201,202].Although the DOS term has the same temperature dependence as the AL contribution [200,203], it turns out to be the leading fluctuation contribution in both the clean and dirty cases, due to its specific dependence on the electron mean free path. We introduce the thermoelectric coefficient # in the framework of linear response theory as # =

1 Im[Q(eh)R (!)] lim , T !→0 !

where Q(eh)R (!) is the Fourier representation of the retarded correlation function of electric J e and heat J h current operators in Heisenberg representation: 99

:: Q(eh)R (X − X  ) = −Ÿ(t − t  ) J h (X), J e (X  ) . Here, X = (r, t) and · · ·  represents both thermodynamical averaging and averaging over random impurity positions.The correlation function Q(eh)R in the diagrammatic technique is represented by a bubble with two exact electron Green’s functions and two external field vertices, the first, ev, associated with the electric current operator and the second, i 2 ("n + "n+ )v, associated with the heat current operator ("n is fermionic Matsubara frequency) [171]. The first order fluctuation corrections to Q(eh) (! ) are represented by the same diagrams as for conductivity (see Fig. 10.9). The first diagram describes the AL contribution to thermoelectric coefficient and was calculated in

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[200, 203] with the electron-hole asymmetry factor taken into account in the fluctuation propagator. Diagrams 2–4 represent the Maki–Thompson contribution, neither anomalous nor regular parts of these diagrams contribute to # in any order of electronhole asymmetry [171, 203]. The contribution from diagrams 5–10 describes the correction to # due to fluctuation renormalization of the one-electron density of states.Evaluating it in the same way as (10.206) but with one heat current vertex one obtains a vanishing result if electron-hole asymmetry is not taken into account. The first possible source of this factor is contained in the fluctuation propagator; it was used in [203] for the AL diagram but for the DOS contribution this correction results in nonsingular contributions to # only and can be neglected. Another source of electron-hole asymmetry is connected with expansion of energy-dependent functions in powers of /EF near the Fermi level:   @( ()v 2 ()) .  ()v 2() =  (0)v 2 (0) +  @ =0 (10.270) Only the second term in (10.270) contributes to the thermoelectric coefficient. Performing the integration over , summations over fermionic frequencies and analytical continuation of the result obtained we find that the contribution to the thermoelectric coefficient associated with the DOS renormalization takes the form   1 eTc @( v 2 ) DOS #2D = (10.271) 4 2  (0)vF2 @ =0 2 ], ×  ∗ (T) ln[ √ √ + +r where  ∗ (T) = −

1

 1 + 8T  1 1 + 4T − − 2

  1  1 T 2 4T 2 ⎧ 2 8 ⎨ T ≈ 9, 4T T  1 . (10.272) = 7(3) ⎩ (T)−1 T  1

Summing (10.272) with the AL contribution [203] one can find the total correction to the thermoelectric coefficient in the case of a 2D superconducting film of thickness d:

  # DOS + # AL 1 1 Tc ln = −0.02 #0 EF  pF d T − Tc   !D ∗ . ×  (Tc ) + 5.3 ln Tc Assuming ln(!D /Tc ) ≈ 2 one finds that the DOS contribution dominates the AL one for any value of impurity concentration:  ∗ has a minimum at T ≈ 0.3 and even at this point the DOS term is twice as large. In both limiting cases T  1 and T  1 this difference strongly increases. In practice, although the Seebeck coefficient S = −# / is probably the easiest to measure among the thermal transport coefficients, the comparison between experiment and theory is complicated by the fact that S cannot be calculated directly; it is rather a composite quantity of the electrical conductivity and thermoelectric coefficient.As both # and  have corrections due to superconducting fluctuations, the total correction to the Seebeck coefficient is given by   #  S = S0 . (10.273) − #0 0 We see that the fluctuations result in a decrease of the absolute value of the overall Seebeck coefficient as the temperature approaches Tc . The situation is complicated additionally in HTS materials, where the temperature behavior of the background value of the thermoelectric power remains unknown. This does not permit us to extract precisely from the experimental data the fluctuation part # and to compare it with the theoretical prediction.Nevertheless,the very sharp maximum in the Seebeck coefficient experimentally observed in a few papers [204–206] seems to be unrelated to the fluctuation effects. This conclusion is supported by recent analysis of the temperature dependence of the thermoelectric coefficient close to the transition in [208]. 10.8.9

The Effect of Fluctuations on NMR Characteristics

Preliminaries In this section we discuss the contribution of superconducting fluctuations to the spin susceptibility

10

and the NMR relaxation rate. For both these effects the interplay of different fluctuation contributions is unusual with respect to the case of the conductivity. Like in the case of the optical conductivity, the fluctuation contributions to the spin susceptibility and the NMR relaxation rate can manifest themselves as the opening of a pseudogap already in the normal phase, a phenomenon which is characteristic to HTS compounds. We begin with the dynamic spin susceptibility (R) ± (k, !) = ± (k, i! → ! + i0+ ) where 1/T    ± (k, ! ) = dei!  Tˆ  Sˆ + (k, )Sˆ − (−k, 0) .

(10.274)

0

Here Sˆ ± are the spin raising and lowering operators, Tˆ  is the time ordering operator, and the brackets denote thermal and impurity averaging in the usual way. The uniform, static spin susceptibility is given by s = ±(R) (k → 0, ! = 0),while the dynamic NMR relaxation rate is given by  d3 k 1 A = lim Im ±(R) (k, !) , (10.275) T1T !→0 ! (2)3 where A is a positive constant involving the gyromagnetic ratio. For noninteracting electrons ±0 (k, ! ) is determined by the usual loop diagram. Simple calculations lead to the well known results for T  EF : s0 =  (Pauli susceptibility) and (1/T1T)0 = A 2 (Korringa relaxation).We will present the fluctuation contributions in a dimensionless form by normalizing to the above results. To leading order in Gi the fluctuation contributions to ± can be discussed with the help of the same diagrams drawn for the conductivity in Fig. 10.9. It is important to note that the role of the external vertices (electron interaction with the external field) is now played by the Sˆ ± (k, ) operators. This means that the two fermion lines attached to the external vertex must have opposite spin labels (up and down). Consequently,the Aslamazov–Larkin diagram for ± does not exist since one cannot consistently assign a spin label to the central fermion for spin-singlet pairing.The next set of diagrams to consider is the Maki– Thompson contribution. While the MT diagrams for

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± appear to be identical to those for the conductivity, there is an important difference in the topology that arises from their spin structure. It is easy to see, by drawing the fluctuation propagator explicitly as a ladder of attractive interaction lines, that the MT diagram is a nonplanar graph with a single fermion loop. In contrast the MT graph for the conductivity is planar and has two fermion loops. The number of loops, in accordance with the rules of diagrammatic technique [90], affects the sign of the contribution. Diagrams 5 and 6 in Fig.10.9 represent the effect of fluctuations on the single-particle self energy, leading to a decrease in the DOS. The DOS diagrams 7 and 8 include impurity vertex corrections (note that these have only a single impurity scattering line as additional impurity scattering in the form of a ladder has a vanishing effect). Finally diagrams 9 and 10 are the DOS diagrams with the Cooperon impurity corrections. Spin Susceptibility We note that when the external frequency and momentum can be set to zero at the outset, as is the case for s , there is no anomalous MT piece (which as we shall see below is the most singular contribution to 1/T1). The MT diagram 2 then yields a result that is identical to the sum of the DOS diagrams 5 and 6. In the clean limit (Tc   1) the fluctuation contribution is given by sfl = s 2 + s 5 + s 6 ; all other diagrams turn out to be negligible. In the dirty case (Tc   1), the DOS diagrams 5 and 6, together with the regular part of the MT diagram (2),yield the same result as in the clean limit (of the order O(Tc /EF )). One can see that this contribution is negligible in comparison with the expected dominant one for the dirty case of the order O(1/EF ).A thorough study of all diagrams shows that the important graphs in the dirty case are those with the Cooperon impurity corrections MT 3 and 4, and the DOS ones 9 and 10. This is the unique example known to us where the Cooperons, which play a central role in the weak localization theory, give the leading order result in the study of superconducting fluctuations. Diagrams 3 and 4 give one half of the final result given below and diagrams 9 and 10 provide the other half. The total fluctuation

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susceptibility is sfl = s 3 + s 4 + s 9 + s 10 . Interestingly,in both the clean and dirty cases sfl / s (0) can be expressed by the same formula if one expresses the coefficient in terms of the GL number Gi(2) (10.191):   sfl 2 √ . (10.276) = −2Gi ln √ (2) s (0) + +r It is tempting to explain the negative sign of the fluctuation contribution to the spin susceptibility in (10.276) as arising from a suppression of the DOS at the Fermi level. But one must keep in mind that only the contribution of diagrams 5 and 6 can strictly be interpreted in this manner; the MT graphs and the coherent impurity scattering described by the Cooperons do not permit such a simple interpretation. Relaxation Rate The calculation of the fluctuation contribution to 1/T1 requires rather more care than s because of the subtleties of analytic continuation. Let us define the local susceptibility  K(! ) = (dk) +− (k, ! ) . In order to write down the fluctuation contribution to 1/T1 for the case of an arbitrary impurity concentration including the ultra-clean case let us start from the anomalous MT contribution and evaluate it using the standard contour integration techniques   2 1 (an)R lim Im K (dq)L(q, 0)K(q) , (!) = − !→0 ! 8 (10.277) ∞ K(q) = 2 −∞

× 

dz cosh2 (z/4T) 1

l2 q2 − (z − i)2 − 1

 

(10.278) 1 l2 q2 − (z + i)2 − 1

.

We have used the impurity vertices in the general form (10.247). The first simple limiting case for (10.279) is lq  1, when the square roots in the denominator can be expanded and K(q) = 2/Dq2 .As

we already know from Sect. 10.8 this corresponds to the usual local √ approximation and covers the domain Tc   1/ . Introducing the pair-breaking rate ' as an infrared cut-off one can find: ı (1/T1)MT(an)

28(3) 1 Gi(2,d) ln( / ' ) . 4  − ' (1/T1) (10.279) The other limiting case is the“ultra-clean limit”when the characteristic q-values √ satisfy lq  1. This is obtained when T  1/  1. From (10.279) we then find K(q) = 4 ln(lq)/vq, which leads to 0

=

√ 3 ı(1/T1)MT(an) 1 = Gi(2,cl) √ ln(T ) . 0 (1/T1) " 14(3) (10.280) We note that in all cases the anomalous MT contribution leads to an enhancement of the NMR relaxation rate over the normal state Korringa value. In particular, the superconducting fluctuations above Tc have the opposite sign to the effect for T  Tc (where 1/T1 drops exponentially with T). One might argue that the enhancement of 1/T1 is a precursor to the coherence peak just below Tc . Although the physics of the Hebel–Slichter peak (the pile-up of the DOS just above gap edge and coherence factors) appears to be quite different from that embodied in the MT process, we note that both effects are suppressed by strong inelastic scattering. We now discuss the DOS and the regular MT contributions,which are important when strong dephasing suppresses the anomalous MT contribution discussed above. The local susceptibility arising from diagrams 5 and 6 can be easily evaluated. The other remaining contribution is from the regular part of the MT diagram. It can be seen that this regular contribution is exactly one half of the total DOS contribution from diagrams 5 and 6. All other diagrams either vanish (as is the case for graphs 7 and 8) or contribute at higher order in 1/EF  (this applies to the graphs with the Cooperon corrections). The final results can be presented in a unique way for the clean (but not ultra-clean) and dirty cases by means of the Gi(2) number:

10

 2 . √ √ + +r (10.281) The negative DOS contribution to the NMR relaxation rate is evident from the Korringa formula and it signs seems very natural while the sign of the positive Maki–Thompson contribution can generate a questions about its physical origin. Let us consider a self-intersecting trajectory and the motion of the electron along it with fixed spin orientation (let us say “spin up”). If, after passing a full turn, the electron interacts with the nucleus and changes its spin state and momentum to the opposite value it can pass again along the previous trajectory moving in the opposite direction. Interaction of the electron with itself on the previous stage of the motion is possible due to the retarded character of the Cooper interaction and such a pairing process, in contrast to the AL one, turns out to be an effective mechanism for relaxation near Tc . This purely quantum process opens a new mechanism of spin relaxation, and so contributes positively to the relaxation rate 1/T1. In the case of the nuclear magnetic relaxation rate calculations, the electron interaction causing nuclear spin flip is considered. If one would try to imagine an AL process of this type one would be in trouble, because the electron-nuclei scattering with spin-flip evidently transforms the initial singlet state of the fluctuation Cooper pair into a triplet-one, which is forbidden in the scheme discussed. So the formally discovered absence of the AL contribution to the relaxation rate is evidence enough. It is worth mentioning that the cancelation of the MT and DOS contributions to conductivity found in Sect. 10.8 is crucial for the fluctuation contributions to the NMR relaxation rate. In fact, the MT and DOS contributions here have the same structure as in the conductivity while the AL contribution is absent. So the full fluctuation correction to the NMR relaxation rate in a clean superconductor simply disappears. ı(1/T1)DOS = −16Gi(2) ln (1/T1)0



Discussion The main results of this section, valid for  1, can be summarized as follows:

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• Fluctuations lead to a suppression of the spin susceptibility s , due to the combined effect of the reduction of the single particle density of states arising from the self energy contributions, and of the regular part of the MT process. • “Cooperon” impurity interference terms, involving impurity ladders in the particle–particle channel, are crucial for the s suppression in the dirty limit. • The processes dominating the results in (1) and (2) above have usually been ignored in fluctuation calculations (conductivity, 1/T1, etc.). The spin susceptibility is unusual in that the AL and the anomalous MT terms, which usually dominate, are absent. • For weak pair-breaking (1/'  Tc ), an enhancement of 1/T1T, coming from the positive anomalous MT term, takes place [175,207]. • Strong dephasing suppresses the anomalous MT contribution, and 1/T1 is then dominated by the less singular DOS and the regular MT terms. Being negative, these contributions lead to a suppression of spectral weight and a decrease in 1/T1. An intensive controversy existed in recent years in relation to the magnetic field dependence of the fluctuation contribution to 1/T1. The situation here much resembles the situation with the magnetoconductivity: a positive MT contribution is suppressed by the magnetic field, while the magnetic field dependent part of the DOS contribution increases with the growth of the field. But in contrast to the magnetoconductivity, which can be measured extremely precisely, the NMR relaxation rate measurements are much more sophisticated. The result of this delicate competition, depending on many parameters (r, ' , ), was found in HTS materials to be qualitatively different in experiments of various groups. The absence of a strong positive AL contribution, possible d-pairing,killing the MT contribution [207], small magnitude of the sum of MT and DOS effects even in the case of s-pairing, lack of the precise values of r, ' , , leading to contradictive theoretical predictions [112,175,179,209], the dispersion in the quality of samples and experimental methods were the reason for this discussion [44,112,210–213].

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10.9

Conclusions

Several comments should be made in conclusion. As was mentioned in the Introduction the first “fluctuation boom” took place at the end of 1960s to the beginning of the 1970s, just after the discovery of the fluctuation smearing of the superconducting transition and formulation of the microscopic theory of fluctuations. The discovery of HTS reanimated this interest and, in order to account for the specifics of these layered structures with high critical temperatures, low charge carrier concentration and other particularities, considerable progress in studies of fluctuation phenomena was achieved (see, for instance, the conference proceedings [214, 215] and the extensive review article [76]). As it is recognized now the optimally or overdoped phases of HTS compounds present an example of a “bad” Fermi liquid. The accounting for superconducting fluctuations is identical to including of the electron–electron interaction beyond the Fermi-liquid approximation. As a result many anomalies of the normal state properties of such HTS compounds can be explained. The situation was found to be much more sophisticated in underdoped phases where the quasiparticle approach, from which we have started this chapter, fails. In the fluctuation theory discussed above, as in modern statistical physics in general, two methods have mainly been used: the diagrammatic technique and the method of functional (continual) integration over the order parameter. As we have seen, each has its own advantages and disadvantages, and in different parts of this review we used the former or the latter. The years of the fluctuation boom coincided with the maximum development of the diagrammatic methods of many-body theory in condensed matter theory. This methods turns out to be extremely powerful: any physical problem,after its clear formulation and writing down the Hamiltonian, can be reduced to the summation of some classes of diagrams. The diagrammatic technique is especially comfortable for problems containing some small parameter. In the theory of superconducting fluctuations such a small parameter exists: as we have seen, this is the Ginzburg–Levanyuk number Gi(D) , which is ex-

pressed as some powers of the ratio max{Tc ,  −1 }/EF . This is why superconducting fluctuations led to the appearance of the small corrections to different physical values in a wide range of temperatures, and due to this smallness these corrections can be evaluated quantitatively. On the other hand, their specific dependence on proximity to the critical temperature T − Tc permits one to separate them in experiment from other effects. In those cases when fluctuations are small it is possible to restrict their summation to the ladder approximation only. The diagrammatic technique gives in a unique way to describe the quantum and classical fluctuations, and the thermodynamical and transport effects. In the description of thermodynamic fluctuations the method of functional integration turns out to be simpler. The ladder approximation in the diagrammatic approach is equivalent to the Gaussian approximation in functional integration. The method of functional integration also turns out to be more effective in the case of strong fluctuations, for instance, in the immediate vicinity of the phase transition. The final equations of the renormalization group carried out by means of functional integrations turn out to be equivalent to the result of the summation of the parquet diagrams series. Nevertheless the former derivation is much more simple. There is one another reason why we have tried to use both methods and even to carry out some results in both ways.In its explosive development of the last decades physics became an “oral science”. In the process of such direct communication near a blackboard it is difficult to write and to read some cumbersome formulas.The language of diagrams is much more comprehensive: by drawing them the speaker demonstrates that this one is small and that one has to be taken into account for this and that reason, which clear for the experienced listener. The success of the diagrammatic technique in some sense is similar to the success of geometry in Ancient Greece, where the science was also “oral”. This advantage of the diagrammatic technique transforms into its disadvantage when there is no direct communication between the speaker and the listener. It is difficult to learn the diagrammatic tech-

10

nique by a textbook on your own, when no one helps you to find the necessary insight into a complex graph. Maybe for similar reasons geometry disappeared in Middle Ages when direct communications between scientists was minimal, while “written” algebra continued to develop. Operating with Osvald Spengler “prosymbols” we can say that the diagrammatic technique belongs more to the Ancient Greece culture style with its “finite body” prosymbol, while functional integration, side by side with the travels of the Vikings to unknown lands and Leibnitz’ analysis of infinitesimals, is an obvious modern contribution to West-European culture with its “infinite space” prosymbol. This is why, suspecting that the modern physics in the near future can fall to a “New Middle Ages” period, we have carried out some results by means of functional integration instead of the diagrammatic technique.

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Acknowledgments In the first place we would like to express our deep gratitude to R.S. Thompson and T. Mishonov, who were the first readers of the manuscript and made a lot of valuable comments. We are grateful to our colleagues and friends G. Balestrino, A. Buzdin, F. Federici, V. Galitski, D. Geshkenbein, A. Koshelev, D. Livanov,Yu.N. Ovchinnikov,A.Rigamonti, G. Savona, collaboration and discussions with whom helped us in writing this work. A.A. Varlamov acknowledges the financial support of COFIN-MURST 2000 and the Scientific Exchange Program of the University of Minnesota.A considerable part of this work was written during the visits in their framework. A.I. Larkin acknowledges the financial support of the NSF Grant No. DRM-0120702.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

V.L. Ginzburg, L.D. Landau, Zh. Exp. Teor. Fiz. 20, 1064 (1950) J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys.Rev. 106, 162; ibid. 108, 1175 (1957) V.L. Ginzburg, Soviet Solid State 2, 61 (1960) A.P. Levanyuk, Zh. Exp. Teor. Fiz. 36, 810 (1959) L.G. Aslamazov, A.I. Larkin, Soviet Solid State 10, 875 (1968) Phys. Lett. 26A, 238 (1968) K. Maki, Progr. Theor. Phys. (Kyoto) 39, 897 (1968); ibid. 40, 193 (1968) R.S. Thompson, Phys. Rev. B 1, 327 (1970) R.E. Glover, Phys. Rev. Lett. A 25, 542 (1967) J. Low Temp. Phys. 6, (1970) K. Maki, In: Superconductivity, edited by R.D. Parks (Dekker, New York, 1969) L.P. Gor’kov, Zh. Exp. Teor. Fiz. 37, 1407 (1959) V.V. Schmidt, In: Proceedings of the 10th International Conference on Low Temperature Physics (VINITI, Moscow) C2, 205 (1967) Tsuboi Taketumi, Suzuki Takao, J. Phys. Soc. Jpn. 42 (1977) A.I. Larkin, Ann. Phys. (Leipzig) 8, 785 (1999) A.I. Larkin, Yu. N. Ovchinnikov. Zh. Exp. Teor. Fiz.119, 595 (2001); cond-mat 0010014 (2000) L.D. Landau, A.A. Abrikosov, I.M.Khalatnikov, DAN USSR. 95, 437; ibid. 773; ibid 1177 (1954) I.T. Dyatlov, V.V. Sudakov, K.A. Ter-Martirosyan, Zh. Exp. Teor. Fiz. 32, 767 (1957) V.V. Sudakov, DAN USSR. 111, 338 (1956) A.I. Larkin, D.E. Khmelnitski, Zh. Exp. Teor. Fiz. 56, 2087 (1969) M.Gel Man, F.E. Low, Phys. Rev. 95, 1300 (1954) E.S. Stueckelberg, A. Peterman, Helv. Phys. Acta 24, 153 (1951) C. Di Castro, G. Jona-Lasinio, Phys. Lett. A29, 322 (1969) K.G. Wilson, Phys. Rev. B 4, 3174, ibid. 3184 (1971) A.S. Patashinski, V.L. Pokrovski, In: The Fluctuation Theory of Phase Transitions (Nauka, Moscow, 1982) A.A. Varlamov, L. Yu, Phys. Rev. 44, 7078 (1991)

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11

Universal Properties of Cuprate Superconductors: Evidence and Implications

Toni Schneider

Physik-Institut der Universit¨at Z¨urich, Switzerland

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 11.2 Critical Behavior at Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11.2.1 Sketch of the Scaling Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11.2.2 Evidence for Finite Temperature Critical Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 11.3 Quantum Critical Behavior and Crossover Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 11.3.1 Sketch of the Scaling Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 11.3.2 Evidence for Doping Tuned Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 11.3.3 Evidence for Substitution Tuned Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 11.3.4 Evidence for Magnetic Field Tuned Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 483 11.4 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 11.5 Concluding Remarks and Comparison with Other Layered Superconductors . . . . . . . . . . . . . . . . . . . . 487 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

11.1

Introduction

Establishing and understanding the phase diagram of cuprate superconductors in the temperaturedopant concentration plane is one of the major challenges in condensed matter physics.Superconductivity is derived from the insulating and antiferromagnetic parent compounds by partial substitution of ions or by adding or removing oxygen. For instance, La2 CuO4 can be doped either by alkaline earth ions or oxygen to exhibit superconductivity. The empirical phase diagram of La2−x Srx CuO4 [1–9] depicted in Fig. 11.1 shows that after passing the so called underdoped limit (xu ≈ 0.047),Tc reaches its maximum value Tcm at xm ≈ 0.16. With further increase of x, Tc decreases and finally vanishes in the overdoped limit xo ≈ 0.273.This phase transition line is thought to be a generic property of cuprate superconductors [10] and is well described by the empirical relation





x Tc (x) = Tc (xm ) 1 − 2 −1 xm

2 % (11.1)

2Tc (xm ) (x − xu ) (xo − x) , 2 xm proposed by Presland et al. [11], with xm = 0.16. Approaching the endpoints along the axis x, at zero temperature La2−x Srx CuO4 undergoes doping tuned quantum phase transitions. As far as their nature is concerned, resistivity measurements reveal a quantum superconductor to insulator (QSI) transition in the underdoped limit [3, 12–15] and in the overdoped limit a quantum superconductor to normal state (QSN) transition [15]. Another essential experimental fact is the doping dependence of the anisotropy. In tetragonal cuprates it is defined as the ratio   = ab /c of the corre  lation lengths parallel ab and perpendicular c to CuO2 layers (ab-planes). In the superconducting =

460

T. Schneider

T = 0 ( T=0 ).As the dopant concentration is reduced,

Tc and T=0 increase systematically, and tend to diverge in the underdoped limit. Thus the temperature range where superconductivity occurs shrinks in the underdoped regime with increasing anisotropy. This competition between anisotropy and superconductivity raises serious doubts whether 2D mechanisms and models, corresponding to the limit T = ∞, can explain the essential observations of superconductivity in the cuprates. From Fig. 11.1 it can also be seen that T (x) is well described by Fig. 11.1. Variation of Tc [1–9] and T with x for La2−x Srx CuO4 . Filled circles correspond to 1/ Tc [1,2,4,6,7] and filled triangles to 1/ T=0 [8,9]. The solid curve is (11.2) with Tcm = 39K. The dashed and dotted lines follow from (11.2) with Tc ,0 = 2 and T=0,0 = 1.63

state it can also be expressed as the ratio = c /ab of the London penetration depths due to supercurrents flowing perpendicular (c ) and parallel (ab ) to the ab-planes. Approaching a nonsuperconductor to superconductor transition  diverges, while in a superconductor to nonsuperconductor transition  tends to infinity. In both cases, however, remains finite as long as the system exhibits anisotropic but genuine 3D behavior. There are two limiting cases:

= 1 characterizes isotropic 3D and = ∞ 2D critical behavior. An instructive model where can be varied continuously is the anisotropic 2D Ising model [16]. When the coupling in the y direction goes to zero, = x /y becomes infinite, the model reduces to the 1D case and Tc vanishes. In the Ginzburg– Landau description of layered superconductors the anisotropy is related to the interlayer coupling. The weaker this coupling, the larger . The limit = ∞ is attained when the bulk superconductor corresponds to a stack of independent slabs of thickness ds . With respect to experimental work,a considerable amount of data is available on the chemical composition dependence of . At Tcit can be inferred from resistivity ( = ab /c = ab /c ) and magnetic torque measurements, while in the superconducting state it follows from magnetic torque and penetration depth ( = c /ab ) data. In Fig.111.1 we included the doping dependence of 1/ T evaluated at Tc ( Tc ) and

T (x) =

T,0 . x − xu

(11.2)

Having also other cuprate families in mind, it is convenient to express the dopant concentration in terms of Tc .From (11.2) and(11.2) we obtain the correlation between Tc and T :  2 Tc

T (xm ) =1− −1 , Tc (xm )

T

T,0

T (xm ) = . (11.3) xm − xu Provided that this empirical correlation is not merely an artefact of La2−x Srx CuO4 , it gives a universal perspective on the interplay of anisotropy and superconductivity, among the families of cuprates, characterized by Tc (xm ) and T (xm ). For this reason it is essential to explore its generic validity. In practice, however, there are only a few additional compounds, including HgBa2 CuO4+ı [17] and Bi2Sr2 CuO6+ı , for which the dopant concentration can be varied continuously throughout the entire doping range. It is well established, however, that the substitution of magnetic and nonmagnetic impurities, depress Tc of cuprate superconductors very effectively [18, 19]. To compare the doping and substitution driven variations of the anisotropy, in Fig. 11.2 we depicted the plot Tc /Tc (xm ) versus T (xm) / T for a variety of cuprate families. The collapse of the data on the parabola, which is the empirical relation (11.3), reveals that this scaling form appears to be universal. Thus, given a family of cuprate superconductors, characterized by Tc (xm ) and T (xm ), it gives a universal perspective on the interplay between anisotropy and superconductivity.

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Universal Properties of Cuprate Superconductors

461

Fig. 11.2. Tc /Tc (xm ) versus T (xm ) / T for La2−x Srx CuO4 (•, Tc (xm ) = 37K, Tc (xm ) = 20) [1,2,4,6,7] , (', Tc (xm ) = 37 K, T=0 (xm ) = 14.9) [8, 9], HgBa2 CuO4+ı (, Tc (xm ) = 95.6K, Tc (xm ) = 27) [17], Bi2 Sr2 CaCu2 O8+ı (, Tc (xm ) = 84.2K, Tc (xm ) = 133) [20], YBa2 Cu3 O7−ı (, Tc (xm ) = 92.9K, Tc (xm ) = 8) [21], YBa2 (Cu1−y Fey )3 O7−ı (, Tc (xm ) = 92.5K, Tc (xm ) = 9) [22], Y1−y Pry Ba2 Cu3 O7−ı (, Tc (xm ) = 91 K, Tc (xm ) = 9.3) [23], BiSr2 Ca1−y Pry Cu2 O8 (, Tc (xm ) = 85.4K, T=0 (xm ) = 94.3) [24] and YBa2 (Cu1−y Zny )3 O7−ı (, Tc (xm ) = 92.5K, T=0 (xm ) = 9) [25]. The solid and dashed curves are (11.18), marking the flow from the maximum Tc to QSI and QSN criticality, respectively

Fig. 11.4. Tc (16 O ) and Tc (18 O ) versus x for La2−x Srx CuO4 . From Guo–Meng Zhao et al.

Fig. 11.3. Phase diagram of La2−x Srx Cu1−y Zny O4 . The blue solid curve corresponds to yc (x), a line of quantum phase transitions. The pink arrow marks the doping tuned insulator to metal crossover and the green arrow marks a path where a QSI and QSN transition occurs. Experimental data taken from Momono et al. [15]

The effect of a substitution for Cu by other magnetic or nonmagnetic metals has also been investigated extensively [15,18,19]. A result common to all of these studies is that Tc is suppressed in the same manner, independently of whether the substituent is magnetic or nonmagnetic. For this reason, the phase diagram of La2−x Srx Cu1−y Zny O4 , depicted in Fig. 11.3, applies quite generally. Apparently, the sub-

stituent axis (y) extends the complexity and richness of the phase diagram considerably. The blue curve corresponds to a line of quantum phase transitions, given by yc (x). The pink arrow marks the doping tuned insulator to metal crossover, and the green arrow corresponds to a path along which a QSI and QSN transition occurs. From Fig. 11.4 it can be inferred that isotope substitution, though much less effective, has essentially the same effect. Tc is lowered and the underdoped limit xu shifts to some yc (x). This suggests that substitution induced local distortions, rather than magnetism, is the important factor. For y > yc (x) superconductivity is suppressed due to the destruction of phase coherence. The point of reference for magnetic field tuned transitions is embodied in the schematic phase dia-

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Fig. 11.5. Schematic (x, H, T)-phase diagram. There is the superconducting phase (S), bounded by the zero-field transition line, Tc (x, H = 0), the critical lines of the vortex melting or vortex glass to vortex fluid transitions, Tm (x = fixed, H) and the line of quantum critical points, Hm (x, T = 0). Along this line superconductivity is suppressed and the critical endpoints coincide with the 2D– QSI and 3D–QSN critical points at xu and xo , respectively

gram shown in Fig. 11.5. It is strongly affected by the combined effect of pinning, thermal and quantum fluctuations, anisotropy and dimensionality [26]. In clean cuprates and close to Tc (x, H = 0) thermal fluctuations are thought to be responsible for the existence of a first-order vortex melting transition. In the presence of disorder, however, the long-range order of the vortex lattice is destroyed and the vortex solid becomes a glass [27]. Since a sufficiently large magnetic field suppresses superconductivity, due to the destruction of phase coherence, there is a line Hm (x) of quantum phase transitions, connecting the zero field QSI and QSN transitions. Indeed, recent experiments revealed that sufficiently-intense magnetic fields suppress superconductivity and mediate a metal to insulator (MI) crossover [28–31]. The QSI transition can also be traversed in films by changing their thickness [13]. An instructive example is the measurements on YBa2 Cu3 O7−ı slabs of thickness d separated by 16 unit cells (≈ 187 A) of PrBa2 Cu3 O7 . Due to their large separation the YBa2 Cu3O7−ı slabs are essentially uncoupled. As is shown in Fig. 11.6, Tc was found to vary with the thickness d of the YBa2 Cu3O7−ı slabs as

Fig. 11.6. Zero resistance Tc versus 1/d of YBa2 Cu3 O7−ı layers of thickness d separated by 16 PrBa2 Cu3 O7 unit cells. Taken from Goodrich et al. [32]. The straight line is a linear fit to (11.4)

Tc (d) ds = Tcbulk d



 d −1 , ds

(11.4)

with Tcbulk = 91 K and ds = 10.1 A [32]. ds is the critical film thickness below which superconductivity is lost. Although the decrease of Tc is partially due to the 3D–2D crossover, the occurrence of the QSI transition points to the dominant role of disorder and quantum fluctuations. Nevertheless, it is conceivable that in sufficiently clean films superconductivity may also occur at and slightly below ds . This review aims to analyze the empirical correlations and phase diagrams from the point of view of thermal and quantum critical phenomena, to identify the universal properties, the effective dimensionality and the associated crossover phenomena. In view of the mounting evidence for 3D-XYuniversality close to optimum doping [7, 13, 33–39], we concentrate here on the thermodynamic and ground state properties emerging from the QSI and QSN-transitions, including the associated crossover phenomena. For this purpose we invoke the scaling theory of quantum critical phenomena [13,40]. Zero temperature phase transitions in quantum systems differ fundamentally from their finite temperature counterparts in that their thermodynamics and dynamics are inextricably mixed. Nevertheless, by means of the path integral formulation of quantum mechanics, one can view the statistical mechan-

11

ics of D-dimensional T = 0 quantum system as the statistical mechanics of a D + z-dimensional classical system with a fake temperature that is some measure of the dynamics, characterized by the dynamic critical exponent z. This allows one to apply the scaling theory developed for classical critical phenomena to quantum criticality. In particular this leads to an understanding of the low T and crossover behavior close to quantum phase transitions and to universal relations between various properties. Evidence for power law behavior should properly consist of data that cover several decades in the parameters to provide reliable estimates for the critical exponents.In cuprate superconductors,the various power laws span at best one decade. Accordingly, more extended experimental data are needed to determine the critical exponents of the quantum phase transitions.Nevertheless,irrespective of their precise value, the evidence for scaling and with that for data collapse exists.It uncovers the relationship between various properties and the significance of the empirical correlations and offers an understanding of the doping, substitution and magnetic field tuned quantum phase transition points and lines (see Figs. 11.1, 11.3 and 11.5). Evidently, the anisotropy, the associated dimensional crossover and the scaling relations between various properties close to the OSI and QSN criticality provide essential constraints for the understanding of the phase diagrams and the microscopic theory of superconductivity in these materials. Note that this scenario is not incompatible with the zoo of microscopic models, relying on competing order parameters [41–51]. Here it is assumed that in the doping regime where superconductivity occurs, competing fluctuations, including antiferromagnetic and charge fluctuations, can be integrated out. The free-energy density is then a functional of a complex scalar, the order parameter of the superconducting phase, only. Given the generic phase diagrams (Figs. 11.1, 11.3 and 11.5) the scaling theory of finite temperature and quantum critical phenomena leads to predictions, including the universal properties, which can be confronted with experiment. As it stands, the available experimental data appears to be fully consistent with a single complex scalar order parameter, a doping tuned dimensional crossover

Universal Properties of Cuprate Superconductors

463

and a doping, substitution or magnetic field driven suppression of superconductivity, due to the loss of phase coherence. When the evidence for this scenario persists, antiferromagnetic and charge fluctuations turn out to be irrelevant close to criticality. Moreover, it implies that a finite transition temperature and superfluid aerial superfluid density in the ground state require a finite anisotropy in chemically doped systems. The important conclusion there is that a finite superfluid density in the ground state of bulk cuprates oxides is unalterably linked to an anisotropic but 3D condensation mechanism. Thus despite the strongly two-dimensional layered structure of cuprate superconductors, a finite anisotropy associated with the third dimension, perpendicular to the CuO2 planes, is an essential factor in mediating superfluidity. The paper is organized as follows. Section 11.2 is devoted to the finite temperature critical behavior. Since a substantial review on this topic is available [13], we concentrate on the specific heat. In Sect. 11.1 we sketch the scaling theory of finite temperature critical phenomena in anisotropic superconductors falling into the 3D-XY universality class. This leads naturally to universal critical amplitude combinations, involving the transition temperature and the critical amplitudes of specific heat, correlation lengths and penetration depths. The universality class to which the cuprates belong is thus not only characterized by its critical exponents but also by various critical-point amplitude combinations that are equally important. Indeed, though these amplitudes depend on the dopant concentration, substitution, etc., their universal combinations do not. Evidence for 3D-XY universality and their implication for the vortex melting transition is presented in Sect.11.2.Here we also discuss the limitations arising from the inhomogeneities and the anisotropy, which render it difficult to observe 3D-XY critical behavior along the entire phase transition  line Tc (x) (Fig.11.1) or on the entire surface Tc x, y (Fig. 11.3). In Sect. 11.3 we examine the quantum phase transitions and the associated crossover phenomena.The scaling theory of quantum phase transitions [40], extended to anisotropic superconductors [13], is reviewed in Sect. 11.1. Essential predictions include a

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universal amplitude relation in D = 2 involving the transition temperature and the zero temperature inplane penetration depth, as well as a fixed value of the in-plane sheet conductivity.Moreover,we explore the scaling properties of transition temperature,penetration depths, correlation lengths, anisotropy and the specific heat coefficient at 2D-QSI and 3D-QSN criticality. In Sect. 11.2 we confront these predictions with the empirical correlations (11.2), (11.2) and (11.3) and pertinent experiments. Although the experimental data are rather sparse, in particular close to the 3D–QSN transition, we observe a flow pattern pointing consistently to a 2D–QSI transition with z = 1 and  ≈ 1, and 3D–QSN criticality with z = 2 and  ≈ 1/2. z is the dynamic critical exponent and  the correlation length exponent. The estimates for the 2D–QSI transition coincide with the theoretical prediction for a 2D disordered bosonic system with long-range Coulomb interactions [52,53]. This reveals that in cuprate superconductors the loss of phase coherence, due to the localization of Cooper pairs, is responsible for the 2D–QSI transition. On the other hand, z = 2 and  ≈ 1/2 point to a 3D-QSN critical point, compatible with a disordered metal to d-wave superconductor transition at weak coupling [54]. Here the disorder destroys superconductivity, while at the 2D–QSI transition it destroys superfluidity. A characteristic feature of the 2D–QSI transition is its rather wide and experimentally accessible critical region. For this reason we observe consistent evidence that it falls into the same universality class as the onset of superfluidity in 4He films in disordered media, corrected for the long-rangeness of the Coulomb interaction. As also discussed in this section, the existence of 2D–QSI and 3D–QSN critical points implies a doping and substitution tuned dimensional crossover. A glance at Fig. 11.2 shows that it is due to the dependence of the anisotropy on doping and substitution. An important implication is that despite the small fraction, which the third dimension contributes to the superfluid energy density in the ground state, a finite transition temperature and superfluid density in bulk cuprates are unalterably linked to a finite anisotropy. Thus, despite their strongly two-dimensional layered structure, a finite anisotropy associated with the third dimension, per-

pendicular to the CuO2 planes, is an essential factor in mediating pair condensation. This points unambiguously to the conclusion that theories formulated for a single CuO2 plane cannot be the whole story. Moreover, the evidence for the flow to 2D–QSI criticality also implies that the standard Hamiltonian for layered superconductors [55] is incomplete. Although its critical properties fall into the 3D-XY universality class, disorder and quantum fluctuations must be included to account for the flow to 2D–QSI and 3D–QSN criticality. Section 11.4 is devoted to the magnetic field tuned quantum phase transitions. Contrary to finite temperature, disorder is an essential ingredient at T = 0. It destroys superconductivity at 3D–QSN criticality and superfluidity at 2D–QSI critical points. On the other hand, superconductivity is also destroyed by a sufficiently large magnetic field. Accordingly, one expects a line Hm (x) of quantum phase transitions, connecting the zero field 2D–QSI and 3D–QSN transitions (see Fig. 11.5). The relevance of disorder at this critical endpoints suggests a line of quantum vortex glass to vortex fluid transitions. Although the available experimental data is rather sparse, it points to the existence of a quantum critical line Hm (x) and 2D localization, consistent with 2D–QSI criticality. In Sect. 11.5 we treat cuprates with reduced dimensionality. Empirically it is well established that a quantum superconductor to insulator transition in thin films can also be traversed by reducing the film thickness. There is a critical film thickness (ds ) where Tc vanishes and below which disorder destroys superconductivity [13]. In chemically doped cuprates the critical thickness is comparable to the c-axis lattice constant.Moreover,the empirical correlation (11.3), displayed in Fig. 11.2, implies that in the bulk superconductivity disappears in the 2D limit. Thus, the combined effect of disorder and quantum fluctuations appears to prevent the occurrence of strictly 2D superconductivity. For this reason it is conceivable that in sufficiently clean films superconductivity may also occur at and below this value of ds . Since chemically doped materials with different carrier densities also have varying amounts of disorder, the third dimension appears to be needed to delocalize the carriers and to mediate superfluidity.

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A comparison with other layered superconductors, including organics and dichalcogenides is made in Sect. 11.5.

11.2

Critical Behavior at Finite Temperature

11.2.1

Sketch of the Scaling Predictions

In superconductors the order parameter is a complex scaler, but it can also be viewed as a twocomponent vector (XY). Supposing that sufficiently close to the phase transition line, separating the superconducting and non superconducting phase, 3DXY-fluctuations dominate. The scaling form of the singular part of the bulk free energy density then adopts the form [13,56]  −1 fs = −kB TQ3± x± y± z± , (11.5) where i± is the correlation length diverging as ± i± = i,0 |t|− , i = x, y, z, ± T − Tc = sign (t) , t = , Tc

(11.6)

and Q3± are universal constants. In this context it should be kept in mind that in superconductors the pairs carry a nonzero charge in addition to their mass and the charge (¥0 = hc/2e) couples the order parameter to the electromagnetic field via the gradient term in the Ginzburg–Landau Hamiltonian. In extreme type II superconductors, however, the coupling to vector potential fluctuations appears to be weak [57], but nonetheless, in principle, these fluctuations drive the system very close to criticality, to a charged critical point [59]. In any case, inhomogeneities in cuprate superconductors appear to be prevented from entering this regime, due to the associated finite size effect [13]. For these reasons, the neglect of vector potential fluctuations appears to be justified and the critical properties at finite temperature are then those of the 3D-XY-model, reminiscent of the lambda transition in superfluid helium, extended to take the anisotropy into account [13,33]. In the superconducting phase the order parameter ¦ can be decomposed into a longitudinal (¦0 + ¦lo ) and transverse (¦tr ) part:

Universal Properties of Cuprate Superconductors

¦ = ¦0 + ¦lo + i¦tr ,

465

(11.7)

where ¦0 = ¦  is chosen to be real. At long wavelengths and in the superconducting phase the transverse fluctuations dominate and the correlations do not decay exponentially, but according to a power law [13, 56]. This results in an inapplicability of the usual definitions of a correlation length below Tc . However, in terms of the helicity modulus, which is a measure of the response of the system to a phasetwisting field, a phase coherence length can be defined [58]. This length diverges at critical points and plays the role of the standard correlation length below Tc . In the presence of a phase twist of wavenumber ki , the singular part of the free energy density adopts the scaling form kB TQ3− ¥ (11.8) xtr ytr ztr    × kx ytr ztr , ky xtr ztr , kz xtr ytr ,

fs = −

yielding for the helicity modulus the expression  @2 fs  ¤i = − = @ki2 k=0  kB TQ3− @2 ¥  = , (11.9) itr @ki2 k=0   where the normalization, Q3− @2 ¥ /@ki2 k=0 = 1, has been chosen.At Tc this leads to the universal relation tr tr tr y0 z0¤x0 ¤y0¤z0 (kB Tc )3 = x0

 =

¥02 16 3

3

tr tr tr x0 y0 z0 2 2 2 x0 y0 z0

,

(11.10)

and the definition of the phase coherence lengths, also referred to as the transverse correlation lengths. The critical amplitudes of the transverse correlation length, i0tr , helicity modulus, ¤i0 and penetration depth, x0 , are then defined as itr = i0tr |t|− , ¤i = ¤i0 |t| , i = i0 |t|− /2 . (11.11) The relationship between helicity modulus and penetration depth, used in (11.10), is obtained as follows. From the definition of the supercurrent

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T. Schneider

ji = c

ıfs , ıAi

(11.12)

where A is the vector potential and c the speed of light, we obtain for the magnetic penetration depth the expression    1 16 3 @2fs  16 3 4ji = − − = ¤i cAi A=0 i2 ¥02 @k2 k=0 ¥02 (11.13) by imposing the twist ki = 2Ai /¥0 . Noting then that the transverse correlation function decays algebraically, Str (Ri ) = ¦tr (Ri ) ¦tr (0) ∝ ¦02 jtr jtr × , j = j = i , (11.14) Ri it is readily seen that the length scales  − correspond to the real space counterparts of the transverse correlation length defined in terms of the helicity modulus. These length scales are related by i− = jtr jtr , j = j = i , (11.15) so that, − − − tr tr tr x0 y0z0 = x0 y0 z0 .

(11.16)

From the singular behavior of the specific heat T @ 2 fs A± −˛ C |t| , =− ≈ VkB kB @t 2 ˛

(11.17)

it the follows that the combination of critical amplitudes  ± 3 tr tr tr R = A± x± y± z± = A± x0 y0z0 = −Q3± ˛ (1 − ˛) (2 − ˛)

(11.18)

is universal, provided that 3 = 2 − ˛

(11.19)

Thus, the critical amplitudes are expected to differ from system to system and to depend on the dopant concentration, the universal combinations (11.10), (11.18) and (11.20) should hold for all cuprates and irrespective of the doping level, except at the critical endpoints of the 3D-XY critical line. A characteristic property of cuprate superconductors is their anisotropy. In tetragonal systems, where a = b = ab , it is defined as the ratio = ab /c , of the correlation length parallel and perpendicular to the ab-planes. Noting that according to (11.9) and (11.13) ztr ytr ¤x  tr 2 = ztr = tr tr = z2 = ¤z x x y x



x− z−

2 (11.21)

holds, we obtain for the relation

Tc =

− ab0 c0 . − = c0 ab0

(11.22)

The universal relation (11.10) can then be rewritten in the form ab,0 ¥02 . 2 16 3 ab,0

Tc −

kB Tc =

(11.23)

− Clearly, Tc , ab,0 , ab,0 and depend on the dopant concentration, but universality implies that this relation applies at any finite temperature, irrespective of the doping level. Another remarkable consequence follows from the universal relation 3  ¥02 (R− )3 kB Tc3 = , (11.24) 2 2 2 16 3kB A− x0 y0z0

which follows from (11.10) and (11.18). Indeed, considering the effect of doping, substitution and pressure, denoted by the variable y, we obtain   2 1 dA−  2 d 1/i0 3 dTc =− − + . (11.25) i0 Tc dy A dy dy i

holds. Moreover, additional universal relations include − − − tr tr tr y0z0 x0 y0 z0 x0 A− = R , = = R . A + + + + + + + A x0 y0z0 x0y0 z0

(11.20)

Thus, the effect of doping, substitution and pressure on transition temperature, specific heat and penetration depths are not independent, but related by this law.

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In an applied magnetic field the singular part of the free energy density adopts the scaling form [13,38] kB TQ3± ± G ( z) , x± y± z± 3

G3± (0) = 1 ,

(11.26)

1 2 2 2 Hx y z + Hy2x2 z2 + Hz2 x2y2 , ¥0

(11.27)

fs = − where  z=

and is a universal scaling function of its argument. Note that in the isotropic case, where  = x = y = z , this scaling form is identical to that of uniformly rotating 4 He near the superfluid transition. Magnetic field and rotation frequency are related by H → m4 c§/e [60]. In the presence of a magnetic field and T < Tc the correlations no longer decay algebraically. The Fourier transform of Str (R) behaves for small wavenumbers q as 1 . q2 + 1/ 2

(11.28)

Supposing then that there is a phase transition in the (H, T)-plane for T < Tc the scaling function must have a singularity at some value zc . Examples are the vortex melting and vortex glass transition. Since the vortex melting transition is first order,  does not diverge but is bounded by 1 2 2 2 Hx y z + Hy2 x2z2 + Hz2 x2 y2 =  zm . ¥0

(11.29)

Invoking (11.6) we obtain for the first-order transition line with H cz the expression, Hcm Tc − Tcm Tc

  ¥0 zm Tc − T 2 = , a,0 b,0 Tc   a,0 b,0 1/2 1/2 = Hc . (11.30)  zm ¥0

In the interval Tcm < T < Tc one expects a remnant of the zero field specific heat singularity. Because the correlation length is bounded, there is a magnetic field induced finite size effect. At the melting transition the limiting length is L2 = ( zm ¥0 ) /Hc

467

(see (11.29). Close to Tc on dimensional grounds one expects L2 ≈ (¥0 ) /Hc to hold. Since the correlation length cannot exceed L, the zero field singularity, i.e. in the specific heat, is removed. As a remnant of this singularity, the specific heat will also exhibit a maximum at Tp , which is located below Tc according to   Tc − Tp a,0b,0 1/2 1/2 ≈ Hc . (11.31) Tc ¥0 11.2.2

G3± ( z)

  Str q ∝

Universal Properties of Cuprate Superconductors

Evidence for Finite Temperature Critical Behavior

Provided that the thermal critical behavior is fluctuation-dominated (i.e. nonmean-field) and the fluctuations of the vector potential can be neglected, cuprate superconductors fall into the 3D-XY universality class. We have seen that the universality class to which a given system belongs is not only characterized by its critical exponents but also by various critical-point amplitude combinations. The implications include: (i) The universal relations hold irrespective of the dopant concentration and material; (ii) given the nonuniversal critical ampli± tudes of the correlation lengths, i,0 , and the uni± versal scaling function G3 ( z), universal properties can be derived from the singular part of the free energy close to the zero field transition. These properties include the specific heat, magnetic torque, diamagnetic susceptibility, melting line, etc. Although there is mounting evidence for 3D-XY-universality in cuprates [7,13,33–39], it should be kept in mind that evidence for power laws and scaling should properly consist of experimental data that covers several decades of the parameters. In practice, there are inhomogeneities and cuprates are homogeneous over a finite length L only. In this case, the actual correlation length (t) ∝ |t|− cannot grow beyond L as t → 0, and the transition appears rounded. Due to this finite size effect, the specific heat peak occurs at a temperature TP shifted from the homogeneous system by an amount L−1/ , and the magnitude of the peak located at temperature TP scales as L˛/ . To quantify this point in Fig. 11.7 we show the measured specific heat coefficient of YBa2 Cu3 O7−ı [61]. The rounding and the shape of the specific heat coefficient clearly exhibits the characteristic behavior

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Fig. 11.7. Specific heat coefficient C/T (mJ/gK2 ) versus T (K) of YBa2 Cu3 O7−ı (sample YBCO3 [61])

Fig. 11.8. Specific heat coefficient C/T (mJ/(gK2 )) versus log10 |t| for YBa2 Cu3 O7−ı with Tc = 92.12 K (sample YBCO3 [61])

of a system in confined dimensions, i.e. rod or cube shaped inhomogeneities [62]. A finite-size scaling analysis [13] reveals inhomogeneities with a characteristic length scale ranging from 300 to 400 A, in the YBa2 Cu3 O7−ı samples YBCO3, UBC2 and UBC [61]. Note that recent measurements by a variety of techniques suggest that superconductivity is not homogeneous in cuprates [63–65]. For this reason, deviations from 3D-XY critical behavior around Tp do not signal the failure of 3D-XY universality, as previously claimed [61], but reflect a finite-size effect at work. Indeed, from Fig.11.8 it can be seen that the finite-size effect makes it impossible to enter the asymptotic critical regime. To set the scale we note that in the -transition of

He the critical properties can be probed down to |t| = 10−9 [66, 67]. In Fig. 11.7 we marked the intermediate regime where consistency with the 3DXY-critical behavior, C/T =  A± 10−˛ log10 |t| +  B± for + − ˛ = −0.013 and  A / A = 1.07, can be observed in terms of full circles. The upper branch corresponds to T < Tc and the lower one to T > Tc . The open circles closer to Tc correspond to the finite-size affected region,while further away the temperature dependence of the background, usually attributed to phonons, becomes significant. Hence, due to the finite size effect and the temperature dependence of the background the intermediate regime is bounded by the temperature region where the data depicted in Fig. 11.8 fall nearly on straight lines. In this context it should be kept in mind that the effect of disorder and inhomogeneities is quite different. Since the critical exponent ˛ of the specific heat is negative at the 3D– XY transition, the Harris criterion implies that disorder is irrelevant so that the critical behavior will be that of the pure system [68]. To provide quantitative evidence for 3D–XY universality in this regime, we invoke the universal relations (11.10) and (11.18) to calculate Tc from the critical amplitudes of specific heat and penetration depth. Using A+ = 8.4 1020 cm3 , derived from the data shown in Fig. 111.8 for sample YBCO3 with Tc = 92.12 K,a,0 = 1153 A,b,0 = 968 A and c,0 = 8705 A, derived from magnetic torque measurements on a sample with Tc = 91.7 K [38], together with the universal numbers A+ /A− = 1.07 and R− ≈ 0.59 [13], we obtain Tc = 88.2 K . Hence, the universal 3D–XY-relations (11.10) and (11.18) are remarkably well satisfied. Another difficulty in observing 3D–XY critical properties stems from the fact that most cuprates are highly anisotropic. A convenient measure of the anisotropy is Tc (see (11.22)), which depends on the dopant concentration (see Figs. 11.1 and 11.2). Although the strength of thermal fluctuations grows with increasing ,they become essentially 2D slightly away from Tc .Accordingly,the 3D–XY-critical regime shrinks and the corrections to scaling become significant. To document this point, in Fig. 11.9 we depicted estimates for the derivative of the universal scaling function G3± ( z) derived from magnetic torque measurements [7]. Even though the quali-

11

Universal Properties of Cuprate Superconductors

469

Fig. 11.9. Scaling function dG± z)/d z derived from 3 ( the angular dependence of the magnetic torque for YBa2 Cu3 O6.93 , La1.854 Sr0.146 CuO4 , HgBa2 CuO4.108 , HgBa2 CuO4.096 , La1.914 Sr0.086 CuO4 and La1.920 Sr0.080 CuO4 (taken from [7])

tative behavior is the same for all samples, the deviations increase with rising Tc . This systematics cannot be attributed to the experimental uncertainties of about 40%. It is more likely that it reflects the reduction of the temperature regime where 3Dfluctuation dominates so that corrections to scaling become important [13].With this view, it is clear that due its moderate anisotropy [13,38],optimally doped YBa2 Cu3O7−ı is particularly suited for observing and checking the consistency with 3D–XY-criticalbehavior. In contrast to this, in highly anisotropic cuprates like Bi2Sr2 CaCu2 O8−ı , it will be difficult to enter the regime where 3D fluctuations dominate [69]. Nevertheless, since the critical behavior of the charged fixed point is the only alternative left, it becomes clear that even the intermediate critical behavior of highly anisotropic cuprates like Bi2Sr2 CaCu2 O8−ı falls into the 3D–XY universality class. Fluctuations grow with increasing , they become essentially 2D slightly away from Tc . . The melting transition of the vortex lattice was discovered in 1993 in Bi2.15Sr1.85CaCu2 O8−ı using the

SR technique [70]. An anomaly attributed to this transition was observed in specific heat measurements of YBa2 Cu3O7−ı [71]. In Fig. 11.10 we depicted the temperature dependence of the specific heat coefficient of an untwined YBa2 Cu3 O7−ı single crystal for various applied fields Hc. Of particular inter-

Fig. 11.10. Temperature dependence of the specific heat coefficient for various applied magnetic fields (Hc) of an untwined YBa2 Cu3 O7−ı single crystal. The numbers on the top of the peak like features denote the applied field strength. The inset shows representative data for H = 1 T (Hc) and H = 8 T (H(a, b)), data shifted vertically by 10 mJ/molK2 (taken from [71])

est in this context is the small anomaly below the main peak, marked by the strength of the applied field. It is attributed to the vortex melting transition. Evidence for the first order nature of the transition stems from magnetization measurements, revealing a jump at Hm [72], which signals the singularity in the scaling function G3± ( z) at z = zm (see (11.26)). Due to the first order nature of the transition, the correlation lengths remain bounded,so that the melting line Hcm and the temperature TP , where the broad peak in the specific heat coefficient adopts its maximum value, should scale according to (11.30) and (11.31). A glance at Fig. 11.11 shows that this behavior is experimentally well confirmed. Note that the variation of the amplitudes Acm and Acp is due to the doping dependence of a,0b,0 ( see (11.30) and (11.31)). From (11.29) it can be seen that the melting line also yields useful information on the anisotropy. As an example we consider the angular dependence in the (c, a) plane. Equation (11.29) yields

470

T. Schneider

Fig. 11.11. Melting line Hc,m and Hc,max versus T − Tc for a YBa2 CuO7−ı single crystal with ı = 0 (), ı = 0.03 () and ı = 0.053 (•) for Hc. Taken from [73].  2 The curves correspond to Hcm = Acm TcT−T and c  2 with  = 2/3 Hcp = Acp TcT−T c

Fig. 11.12. Left panel: Melting line for YBa2 Cu3 O7−ı with Tc = 93.25 K derived from field (') and angular-dependent ( ) torque curves. The solid line corresponds to Hm ∝ (1 − T/Tc )2 with  = 2/3. Data taken from [74]. Right panel: Melting lines Hm (ı, T) for YBa2 Cu3 O7−ı with Tc = 92 K, detected by a direct measurement of the entropy change. The symbols denote experimental data taken at different magnetic field orientations in the (c, a) plane, measured by the angle ı [75]. The solid lines are (11.33) with ca = 7.6 and  = 2/3

Hm (ı) = Hm (ı = 0) ca

(11.32)

−1/2  , × sin2 (ı) + ca2 cos2 (ı) where

ca =

a , Hm (ı = 0) ∝ (1 − T/Tc )2 . c

From Fig. 11.12 it can be seen that this behavior is well confirmed. In this context the question arises as to whether or not phase coherence and with that superfluidity persists above the melting line. Recently,

numerical simulations revealed that the vortex liquid is incoherent, i.e. phase coherence is destroyed in all directions, including the direction of the applied magnetic field, as soon as the vortex lattice melts [76,77]. To summarize, there is considerable evidence that the intermediate finite temperature critical behavior of cuprate superconductors, when attained, is equivalent to that of superfluid helium. Moreover, we have seen that finite size scaling is a powerful tool to identify and characterize inhomogeneities.

11

11.3

Quantum Critical Behavior and Crossover Phenomena

11.3.1

Sketch of the Scaling Predictions

Given the empirical phase transition line Tc (x) or  surface Tc x, y with critical endpoints or lines (see Figs. 11.1 and 11.3) doping and substitution tuned quantum phase transitions can be expected. To invoke and sketch the scaling theory of quantum critical phenomena we define ı, measuring the relative distance from quantum critical points, in terms of ⎧ y=0 : ı = (x − xu ) /xu ⎨ ı= . y=0 : ı = (x  o − x) /xo ⎩ y = 0, xu ≤ x ≤ xo : ı = yc (x) − y /yc (x) (11.33) At T = 0 and close to quantum criticality one has two kinds of correlation lengths [13, 40]. The usual spatial correlation length in direction i − i

=

−  i,0 ı −

(11.34)

and the temporal one −

−

  =  ,0ı −  , where the dynamic critical exponent is defined as the ratio  . (11.35) z=  The singular part of the free energy density then adopts the scaling form [13,14,40]  −

fs (ı, T) = QD  ,0

D A

%−1 −

 i,0

  ı  (D+z) FD y ,

i=1

y = kB T  = kB T ,0 ı −z , (11.36)     where FD y with FD y = 0 = 1 is a universal scaling function and QD a universal constant. Another quantity of interest is the helicity modulus, which adopts the scaling form [13,14] ¤iD (ı, T) = QD (11.37)  % −1 D  − 2 − A   − ×  i,0  ,0  i,0 ı  (D−2+z) YD y , i=1

Universal Properties of Cuprate Superconductors

471

    where YD y with YD y = 0 = 1 is a universal scaling function of its argument. As a first application we consider a line of finite temperature transitions Tc (ı) ending at a quantum critical point at T = 0 and ı = 0. The scaling forms then require that kB Tc =

yc z − ı ,  ,0

(11.38)

where yc is the universal value of the scaling function argument at which the scaling functions exhibit a singularity at finite temperature. Combining (11.38) and (11.38) we obtain in D = 2 kB Tc D=2 ¤ab (ı, 0)

= R2 , R2 =

y  c , Y2 y c Q 2

(11.39)

yielding the universal relation 2 Tc ab (0) =

¥02R2 ds 16 3 kB

(11.40)

between transition temperature and zero temperature in-plane penetration depth.ds denotes the thickness of the superconducting slab and R2 is a universal dimensionless constant. Analogously, in D = 3, (11.38) and (11.38) yield kB Tc D=3 ¤ab (ı, 0)

−

= R3  c , R3 = −

y  c , Y3 y c Q 3

(11.41)

−

2 so that Tc , ab (0),  c and  ab are related by −

2 Tc ab (0) =

¥02 R3  ab ¥02 R3 −  = . 16 3kB c 16 3kB T=0

(11.42)

This is just the quantum analogy of the universal finite temperature relation (11.23). When there is an anisotropy tuned 3D–2D crossover, where T=0 → ∞, matching of (11.40) and (11.42) requires that −

−

R3  c = R3

 ab = R2ds .

T=0

(11.43)

Noting then that the universal relation (11.42) applies close to both, the 2D–QSI and 3D–QSN transitions it is expected to provide useful information on the cor− − relation lengths ( c , ab ),given experimental data for 2 Tc ab (0) and T=0 . Moreover, useful scaling relations for the specific heat coefficient c are readily derived from the singular part of the free energy density (11.36), namely:

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T. Schneider

 c 

c T=0 =  = (11.44) T T=0  @2fs  ∝ ı  (D−z) ∝ Tc(D−z)/z ∝ Hc(D−z)/2 , @T 2  T=0

 − −2 since Hc scales as Hc ∝  ab [13], when applied parallel to the c-axis, and   d  c  d c  = = (11.45) dT T=0 dT T T=0  @3 fs  ∝ ı  (D−2z) ∝ Tc(D−2z)/z . @T 3 T=0 Finally, considering 2D quantum critical points resulting from a 3D–2D crossover in the ground state, (11.43) implies that close to 2D quantum criticality the anisotropy diverges as −

−

T=0 =

 ab R3  ab,0 − ı ∝ Tc−1/z . − = c R2ds

(11.46)

Combining the scaling relations (11.38), (11.40), (11.43) and (11.46) a 2D quantum critical point resulting from a 3D–2D crossover is then characterized by −2 −z z Tc ∝ ab (0) ∝ n s (0) ∝ T=0 ∝ ı , 

c T=0 ∝ Tc(2−z)/z ∝ Hc(2−z)/2 . (11.47)   2 n s (0) = ds /ab (0) denotes aerial superfluid density. In particular it relates the superconducting properties to the anisotropy parameter, fixing the dimensionality of the system. It reveals that an anisotropy driven 3D–2D crossover destroys superconductivity even in the ground state. From (11.42), rewritten in the form −

16 3 kB 2 Tc ab (0) , ¥02 R3 16 3 kB 2 = 2 Tc ab (0) T=0 , ¥0 R 3

−

(11.48)

2 2 it can be seen that Tc ab (0) and Tc ab (0) T=0 are appropriate indicators for the occurrence of quantum − phase transitions.Close to 3D–QSN criticality c and −  ab diverge according (11.34) as −

−

3D-QSN:  c ∝  ab ∝ ı − ∝ Tc−1/z ,

R2 ds , (11.50) R3 R2 = ds T=0 ∝ ı − ∝ Tc−1/z R3

−

2D-QSI:  c = −

 ab −

applies. Here  c tends to a finite value, proportional to the thickness ds of the sheets. Another quantity of interest is the zero temperature condensation energy. Close to 2D–QSI and 3D–QSN criticality it scales according to (11.36) and (11.38) as −E (T = 0) ∝ ı  (D+z) ∝ Tc(D+z)/z .

(11.49)

(11.51)

As expected, when superconductivity disappears, the condensation energy vanishes. Next we turn to the critical behavior of the conductivity. In the normal state and close to a 3D–XY DC critical point, the DC conductivities, parallel (ab ) DC and perpendicular (c ) to the layers, scale as [13] DC ab ∝

 c  , cDC ∝ 2 , c ab

(11.52)

where  is the correlation length associated with the finite temperature critical dynamics. At Tc the ratio is then simply given by the anisotropy cDC = DC ab



c ab

2

 =

1

Tc

2 .

(11.53)

Approaching 2D–QSI criticality, the scaling relation (11.47) implies

Tc = Tc ,0 ı − .

c =  ab

because T=0 remains finite.This differs from the 2D– QSI transition, where according to (11.46)

(11.54)

Another essential property of this critical point is that for any finite Tc the in-plane areal conductivity is always larger than [13,40] DC ab ds = 0

4e2 . h

(11.55)

This follows from the fact that close to 2D–QSI the in-plane resistivity adopts the scaling form [13,40]

11

ab =

  h F y , y = kB T  . 2 4e 0

h h = T2c ,0ı −2 2 . 4e2 0 4e 0

(11.57)

  F y is a scaling function of its argument and F (0) = 1. This behavior uncovers the 3D–2D crossover associated with the flow to 2D–QSI criticality in the normal state. To establish a relation between normal and superconducting properties, we express ı in terms of c (0). Using (11.47) we obtain   −(2+z)/4 , c (0) = §s cDC Tc+  %(2+z)/4 4e20 §s = 0,0ab,0 (0) , (11.58) h T2c ,0 where

T = T,0 ı − ,

ab,0 (0) = ab,0 (0) ı −z /2 (11.59)

are the zero temperature critical amplitudes. The scaling relation (11.58) differs from the mean-field prediction for bulk superconductors in the dirty limit [78] and layered BCS superconductors, treated as a weakly coupled Josephson junction [79–81]:   −1/2 c (0) = §s cDC Tc+ ,

(11.60)

where  §s =

c 2 4 2  (0)

1/2 ,

cDC ∝

(11.56)

Thus, close to a 2D–QSI transition    the normal state resistivity cDC Tc+ = 1/cDC Tc+ , evaluated close but slightly above Tc , is predicted to diverge as   cDC Tc+ = T2c

Universal Properties of Cuprate Superconductors

 (0) = 1.76kB Tc , (11.61)

and  (0) denotes the zero temperature energy gap. Approaching 3D–QSN criticality, the finite temperature relations (11.52) and (11.53) still apply, but DC both cDC and ab diverge at Tc , while the anisotropy

Tc remains finite. For this reason  , the correlation length associated with the finite temperature critical dynamics, cannot be eliminated. Nevertheless, since zcl  scales as, ∝ ab ,where zcl is the critical exponent of the finite temperature dynamics, from (11.52) we obtain the relation

473

c   zcl −1 ∝ ∝ ab 2

Tc ab ab

zcl −1 ∝ ab,0 |t|− (zcl −1) ,

(11.62)

valid close to 3D–XY critical points. Since close to a 3D–QSN criticality, the doping dependence of the finite temperature critical amplitude ab,0 is given by   ab (see (11.23) and (11.42)), with c (0) ∝ ı − 2 (1+z) ((11.38)) we finally   obtain the relationship between c (0) and c Tc+ the relationship    1+z c (0) ∝ c Tc+ 2(zcl −1) ,

(11.63)

characterizing flow to a 3D–QSN critical point in  the  the c (0) , c Tc+ plane. Noting that the in-plane resistivity tends close to 2D–QSI criticality to a fixed value (see (11.56)), ab can also be used as a parameter measuring the distance from the critical point. This is achieved by set 0  0 −z ting y = kB T  ∝ Tı −z ∝ T ab − ab /ab . Given then a transition line Tc (ı) ending at the 2D– QSN critical point the scaling form (11.56) requires that z   0 ab − ab h 0 , ab = 2 (11.64) Tc ∝ 0 4e 0 ab   because the scaling function F y exhibits a singularity at yc ,signaling the finite temperature transition line. 11.3.2

Evidence for Doping Tuned Quantum Phase Transitions

The empirical correlations between Tc , dopant concentration x and anisotropy T (see (11.2)–(11.3)) clearly point to the existence of quantum critical endpoints. A glance at Fig. 11.2 shows that when Tc vanishes in the underdoped limit the anisotropy T tends to infinity. Accordingly a 2D–QSI transition is expected to occur. In the overdoped limit, Tc vanishes again but the finite anisotropy implies a 3D– QSN transition. Since the aforementioned empirical correlations turned out to be remarkably generic (see Fig. 11.2) they appear to reflect universal properties characterizing these quantum phase transitions. Indeed, the empirical correlation (11.2) points with the

474

T. Schneider

Fig. 11.13. Tc versus ab (T = 0)−2 for La2−x Srx CuO4 (': [84], : [9], : [83]) YBa2 Cu3 O7−ı ((: [85]) and Y1−x Prx Ba2 Cu3 O6.97 (⊗: [86]). The dashed and solid lines correspond to (11.68)

scaling law (11.38) to z = 1 in both transitions. Moreover, the empirical relation between Tc and T (11.3), according to the scaling law (11.46) implies a 2D–QSI transition with  = 1. Thus, the universality classes emerging from the empirical relations are characterized by the critical exponents: 2D-QSI: z = 1,  = 1 ,

(11.65)

3D-QSN: z = 1 .

(11.66)

These 2D–QSI exponents are consistent with the theoretical prediction for a 2D disordered bosonic system with long-range Coulomb interaction. Here the loss of superfluidity is due to the localization of the pairs, which is ultimately responsible for the transition [52, 53]. A potential candidate for the 3D–QSN transition is the Ginzburg–Landau theory proposed by Herbut [54]. It describes a disordered d-wave superconductor to metal transition at weak coupling and is characterized by the critical exponents z = 2 and  = 1/2, except in an exponentially narrow region. Since the resulting z coincides with the value implied by the empirical correlation (11.2), with the estimate (11.66) one expects, 3D-QSN: z = 2,  = 1/2 .

(11.67)

A characteristic property of a 2D–QSI transition, irrespective of the value of z , is the universal relation (11.40) between transition temperature and

zero temperature penetration depth. An instructive example is the onset of superfluidity in 4 He films adsorbed on disordered substrates, where the linear relationship between Tc and aerial superfluid 2 density n s (0) ∝ ds /ab (0) is well confirmed [82]. In cuprates, a nearly linear relationship between Tc and ab (T = 0)−2 was established some time ago by Uemura et al. [83]. In the present context, it is not strictly universal, because ds , the thickness of the independent slabs, is known to adopt familydependent values [13, 87]. This fact can be anticipated from Fig. 11.13, showing experimental data for 2 Tc versus 1/ab (T = 0) of La2−x Srx CuO4 [9, 83, 84], YBa2 Cu3 O7−ı [85] and Y1−x Prx Ba2 Cu3O6.97 [86].With Tc in K and ab (T = 0) in A, the dashed and solid straight lines correspond to Tc ≈

2.5 108 3.2 108 , . 2 2 ab (T = 0) ab (T = 0)

(11.68)

Invoking then the universal relation (11.40) we obtain the estimate, 3.2 ds (YBa2 Cu3 O7−ı ) ≈ 1.3 , ≈ ds (La2−x Srx CuO4 ) 2.5

(11.69)

which is consistent with ds (YBa2 Cu3 O7−ı )/ ds (La2−x Srx CuO4 ) ≈ 10.1 A/7.6 A ≈ 1.33, derived from the thickness tuned QSI transition and the crossing point phenomenon, respectively [13, 87]. Consequently, dTc /d(1/2(T = 0)) is not strictly

11

Universal Properties of Cuprate Superconductors

475

Next we turn to the behavior of the resistivity close to 2D–QSI criticality. In Fig. 11.14 we displayed the data of Semba and Matsuda [88] for the temperature dependence of the in-plane resistivity ab of YBa2 Cu3 Oy at various dopant concentrations y. Below y ≈ 6.3 the resistivity increases with decreasing temperature, signaling the onset of insulating behavior in the zero temperature limit. Above y ≈ 6.3 and as the temperature is reduced, the resistivity drops rapidly and vanishes at and below Tc . Thus, for y  6.3 there is a superconducting phase and the 2D– QSI transition occurs at y ≈ 6.3. Moreover, the temperature dependence of c /ab,depicted in Fig.11.14, is in accord with the scaling relations (11.38) and 2/z (11.53), yielding c /ab = T2c ∝ ı −2 ∝ Tc . Indeed, the anisotropy increases by approaching the 2D–QSI transition. In contrast, near the 2D–QSI transition (y ≈ 6.3), there is a finite threshold inth plane resistivity ab ≈ 0.8m§cm (see Fig. 11.14). According to the scaling relation (11.55) this is a characteristic feature of a 2D–QSI transitions. For ds ≈ 11.8A and 0 ≈ 1 this leads to the sheet resisth tance ab /ds ≈ h/ 4e2 ≈ 6.5 k§. Since c ∝ 2 ab   th and ab → ds h/ 4e2 it also becomes evident that the rise of c for T > Tc simply reflects the increasing anisotropy and with that the flow to 2D–QSI criticality. Together with the doping dependence of Tc (see Figs. 11.1 and 11.2), these features clearly confirm the occurrence of a 2D–QSI transition. According to (11.40) and (11.64)a 2D–QSI transition is also characterized by the scaling relation 

Fig. 11.14. Upper panel: Temperature dependence of the inplane resistivity ab of YBa2 Cu3 Oy at various dopant concentrations y. Taken from Semba and Matsuda [88]. The threshold resistivity is indicated by an arrow. Above panel: c /ab versus T of underdoped YBa2 Cu3 Oy (taken from Semba and Matsuda [88])

universal. Nevertheless, due to the small variations of ds within a family of cuprates it adopts there a nearly unique value. For this reason the empirical proportionality of Tc and ab (T = 0)−2 for underdoped members of a given family confirms the flow to 2D–QSI criticality.

z 

¥02 ds , 2 (T = 0) 16 3Q2 ab (11.70) where 0 and 0c denote the residual and critical residual sheet resistivity, respectively. This prediction is well confirmed by the data of Fukuzumi et al. for Zn-substituted La2−x Srx CuO4 and YBa2 Cu3 O7−ı [3] displayed in Fig. 11.15. Approaching the underdoped limit the data merge on a straight line. With the scaling relation (11.70) this points to z = 1 consistent with the value emerging from the empirical correlations (Eq. (11.65)). Next we turn to the isotope effect. Since universal relations like (11.18), (11.23) and (11.40) should apkB Tc = c

0c − 0 0c

=

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T. Schneider

ˇTc = ˇds + ˇ1/ab2 (0) ,

(11.71)

where

m B , (11.72) B m where B denotes the shift of B upon isotope substitution. Although the available experimental data on identical samples are rather sparse, the results shown in Fig. 11.16 for the oxygen isotope effect in La2−x Srx Cu1−x O4 [89, 90] and Y1−x Prx Ba2 Cu3 O7 clearly reveal the crossover to the asymptotic 2D– QSI behavior marked by the dashed straight line.The solid curve is a fit to (11.71), yielding the estimate ˇB = −

ˇds ≈ −0.2 , Fig. 11.15. Residual in-plane resistance versus normalized critical temperature for Zn-substituted La2−x Srx CuO4 and YBa2 Cu3 O7−ı . Tc0 is the transition temperature of the Zn free compound (taken from Fukuzumi et al. [3])

(11.73)

for the isotope coefficient of ds . An essential result is that the flow to 2D–QSI criticality implies that the isotope coefficients ˇTc and ˇ1/ab2 (0) diverge.Some insight is obtained by noting that in the doping regime of interest, isotope substitution does not affect the dopant and substitution concentrations [89]. In contrast it lowers the transition temperature and shifts the underdoped limit xu [34, 92]. From the relation Tc = a ı z (see (11.38)), yielding z xu Tc =− , Tc x/xu − 1 xu

(11.74)

we obtain for the isotope coefficient the expression   1 Tc (xm ) 1/z ˇTc = , (11.75) r Tc  1/z m xu 1 a = z , r m xu Tc (xm ) Fig. 11.16. Oxygen isotope effect for underdoped La2−x Srx CuO4 in terms of 1/ˇ1/2 (0) versus 1/ ˇTc . The ab dashed line marks the critical behavior at the 2D–QSI transition (ˇ1/2 (0) → ˇTc → ∞), while the solid curve ab is (11.71) with ˇds = −0.2. Experimental data taken from • [89],  [90] and  [91]

ply irrespective of the doping and substitution level, the isotope effects on the quantities involved are not independent. As an example, we consider the universal relation (11.40), predicting that close to the 2D– QSI transition the isotope effect on transition temperature and zero temperature in-plane penetration depth are related by

applicable close to the 2D–QSI transition. Here we rescaled Tc by Tc (xm ), the transition temperature at optimum doping, to reduce variations of Tc between different materials [34]. In Fig. 11.17 we show the experimental data for Y1−x Prx Ba2 Cu3 O7 [92], La1.85 Sr0.15Cu1−x Nix O4 [93] and YBa2−x Lax Cu3O7 [94] in terms of 1/ˇTc versus Tc /T (xm ). As predicted by (11.76), approaching the 2D–QSI transition Tc /Tc (xm ) = 0, the data collapse on a straight line, pointing again to z ≈ 1 (see 11.65)). Accordingly, the strong doping dependence of the isotope coefficients of transition temperature and zero temperature in-plane penetration depth in

11

Fig. 11.17. Inverse isotope coefficient 1/ˇTc versus Tc /Tcm for Y1−x Prx Ba2 Cu3 O7 [92], La1.85 Sr0.15 Cu1−x Nix O4 [93] and YBa2−x Lax Cu3 O7 [94]. The straight line corresponds to 1/ˇTc = r Tc /Tcm (see (11.76)) with z = 1 and r = 6

Universal Properties of Cuprate Superconductors

477

coefficient in the limit temperature. From the  of zero(D−z)/2  scaling relation c T=0 ∝ Hc (see (11.45)) it is  1/2 seen that for both transitions c T=0 ∝ Hc holds, provided that z = 1 and z = 2 at 2D–QSI and 3D– QSN criticality, respectively. The experimental data displayed in Fig. 11.18 shows that in La2−x Srx CuO4 (D − z) /2 = 1/2 holds irrespective of the doping level. Thus these data provide rather unambiguous evidenced for a 2D–QSI transition with z = 1 and a 3D–QSN criticality with z = 2. This implies that  1/2

c T=0 ∝ Hc is not a characteristic feature of dwave pairing, as proposed by Volovik [97] and Won and Maki [98]. The above comparison with experiment provides rather clear evidence for the occurrence of a 2D– QSI transition with z = 1 and  = 1 in the underdoped and a 3D–QSI critical point z = 2 and  = 1/2 in the overdoped limit. Next, to substantiate this scenario further, we consider the crossover between these quantum critical points. Noting that 2 1/ab (0) scales close to these critical points as (see (11.38)) 1 2 ab (0)

D ∝ ¤ab (0) ∝ ı  ((D−2+z)) ,

(11.76)

b a + , x − xu (xo − x)3/2

(11.77)

we invoke 2 ab (0) =

to interpolate between the 2D–QSI and 3D–QSN transition. A fit to the experimental data of La2−x Srx CuO4 , yielding the parameters Fig. 11.18. Magnetic field dependence of the specific heat  coefficient at zero temperature c T=0 for La2−x Srx CuO4 at x=0.1, 0.16 and 0.22 (taken from Chen et al. [96])

underdoped cuprates follows naturally from the doping tuned 3D–2D crossover and the associated 2D– QSI transition in the underdoped limit. One might hope that this novel point of view about the isotope effects in cuprate superconductors [95] will stimulate further experimental work to obtain new data to confirm or refute these predictions. A property suited to shed light on the critical behavior of both, the 2D–QSI and 3D–QSN transition is the magnetic field dependence of the specific heat

a = 5.42 105 A2 , b = 6.9 103 A2 ,

(11.78)

is shown Fig. 11.19. It is remarkable that this simple interpolation scheme, reducing in the underdoped and overdoped limit to the expected asymptotic behavior, describes the data so well. Due to this agreement, this interpolation function provides in conjunction with the empirical law for Tc (x) (see (11.2)), a realistic description of the doping dependence of the zero temperature out-of-plane correlation − length  c , given by (11.48), yielding  c ∝ 2 Tc ab (0), close to 2D–QSI and 3D–QSN criticality. 2 The doping dependence of Tc ab (0) is displayed in

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T. Schneider

2 Fig. 11.20. (a) Tc ab (0) versus x for La2−x Srx CuO4 . •: taken from [9] and [83]. The solid line is (11.2) and (11.77) with 2 Tc (xm ) = 39.8 K and the parameters listed in (11.78). Note that close to the 2D–QSI and 3D–QSN transition Tc ab (0) ∝  c . −2 −2 (b) Tc ab (0) T=0 versus x for La2−x Srx CuO4 . •: Tc and ab (0) taken from [9, 83] and T=0 from (11.2) with T=0,0 = 2. −2 Note that Tc ab (0) T=0 ∝  ab (11.48). The solid curve indicates the crossover from 2D–QSI (z = 1,  = 1) to 3D–QSN = 2,  = 1/2) criticality according to (11.2), (11.2), (11.76) and (11.77) (z

Fig. 11.19. (0) versus x for La2−x Srx CuO4 . •: experimental data taken from [9, 83]. The solid curve is a fit to (11.77) with the parameters listed in (11.78). It indicates the crossover from a 2D–QSI transition with z = 1 and  = 1 to a 3D–QSN criticality with z = 2 and  = 1/2 2 1/ab

2 Fig. 11.20 for La2−x Srx CuO4 . Since Tc ∝ ds /ab (0) holds in the underdoped limit (see (11.40) and 2 Fig. 11.13), it is clear that Tc ab (0) tends to a constant value,proportional to ds ,the thickness thein of 2 dependent sheets. Since initially d/dx Tc ab (0) ≈ −Tc (xm ) a xo , the linear decrease simply reflects the parabolic form of the empirical law (11.2). Finally, the upturn close to the overdoped limit, a regime which experimentally has not yet been attained, sig2 nals the 3D–QSN transition, where  c ∝ Tc ab (0) diverges.In this context it is also instructive to consider the zero temperature in-plane correlation length. By

 Fig. 11.21. Tc d c /dT T=0 versus x for La2−x Srx CuO4 . Data taken from [15] and [99]

definition it diverges at a 2D and 3D quantum phase transition. According to the scaling relation (11.48) 2 it can be measured in terms of  ab ∝ Tc ab (0) T=0 . In Fig. 11.20 we displayed the experimental data for 2  ab ∝ Tc ab (0) T=0 versus x. For comparison we included the behavior resulting from (11.2), (11.2), (11.76) and (11.77) in terms of the solid curve. While the rise of  ab in the underdoped regime, signaling the occurrence of a 2D–QSI transition, is well confirmed, it does not extend sufficiently close to the overdoped limit to detect 3D–QSN criticality. From Fig. 11.21, showing the doping dependence of the T-linear term of the specific heat coefficient of La2−x Srx CuO4 , it is seen that this quantity tends to a finite value in the underdoped limit and increases in the overdoped regime. This behavior is fully consis-

11

 tent with the scaling relation d c /dT T=0 ∝ ı  (D−2z) (see (11.46)). Indeed at 2D–QSI criticality with z = 1,  d c /dT T=0 tends to a constant value and diverges close to the 3D–QSN critical point for z > 3/2.Unfortunately, the data does not extend sufficiently close to the overdoped limit to provide reliable estimates of the exponent combination  (3 − 2z). 2 Provided that the linear T-term of 1/ab (T) in the zero temperature limit exists, the scaling relations (11.38) and (11.38) yield close to quantum criticality the universal relation    d dYD  i (0) 2  Tc = y , (11.79)  c dT  (T)  dy  i

T=0

y=0

where D = 2 and D = 3 for the 2D–QSI and 3D–QSN transition, respectively. In Table 11.1 we collected additional estimates for various cuprates and doping levels. The rise of the magnitude of Tc d/dT (ab (ı, 0)/i (ı, T))2T=0 with increasing doping level reflects the 2D–3D-crossover in the scaling function YD . Noting that most compounds are close to optimum doping it is not surprising that the listed values scatter around –0.61, the value of nearly optimally doped La2−x Srx CuO4 . Before turning to the substitution tuned quantum transitions it is useful to express the doping dependence in the interpolation formula (11.77) in terms of the transition temperature. This is achieved by invoking the empirical correlation 11.2 between Tc and doping concentration x. In Fig. 11.22 we dis2 played the resulting Uemura plot, Tc versus 1/ab (0) for La2−x Srx CuO4 in terms of the solid and dashed curves, resembling the outline of a fly’s wing. The solid curve marks the flow from Tc (xm ) to the 2D– QSI critical point and the dashed one the flow to 3D–QSN criticality. The dotted line indicates the universal 2D-behavior (see (11.40)) and (11.68)). For comparison we included the experimental data of Panagopoulos et al. [9] and Uemura et al. [83]. Although the data does not attain the respective critical regimes, together with the theoretical curves, they provide a generic perspective of the flow to 2D– QSI and 3D–QSN criticality. Indeed, convincing evidence for these flows emerges from the SR data displayed in Fig. 11.23 for Y0.8Ca0.2 Ba2 (Cu1−y Zny )O7−ı (Y0.8 Ca0.2-123), Tl0.5−y Pb0.5+y Sr2 Ca1−xYx Cu2 O7 (Tl-

Universal Properties of Cuprate Superconductors

479

2 Fig. 11.22. Tc versus 1/ab (0) for La2−x Srx CuO4 . •: experimental data taken from [9,83]. The solid and dashed curves result from the empirical law (11.2) and the interpolation function (11.77) with Tc (xm ) = 39.8 K and the parameters listed in (11.78). The solid line indicates the flow from optimum doping to 2D–QSI criticality and the dashed line to the 3D–QSN critical point

1212) [103] and TlBa2 CuO6+ı (Tl-2201) [104]. In analogy to Fig. 11.22, the solid curves mark the flow from Tc (xm) to the 2D–QSI critical point and the dashed curves the approach to 3D–QSN criticality. Additional confirmation of this flow emerges from Fig. 11.24, showing the condensation energy versus hole concentration p for Y0.2Ca0.2 Ba0.2 Cu3 O7−ı taken from Tallon and Loram [105]. The solid and dashed curves are (11.51), in terms of  3 −E (T = 0) ∝ ı (z+D)/z ∝ p − pu ∝ Tc3 , 5/2  ∝ Tc5/2 , (11.80) −E (T = 0) ∝ po − p with (z + D) /z = 3 and (z + D) /z = 5/2, the values appropriate for the 2D–QSI (see (11.65)) and 3D– QSN (see (11.67)), respectively. A generic flow to the 2D–QSI critical  point also emerges from the plot c (0) versus c Tc+ displayed in Fig. 11.25 for YBa2 Cu3 O7−ı , La2−x Srx CuO4 and Bi2Sr2 CaCu2 Oy at various doping levels. An empirical, nearly linear, correlation between these quantities has been proposed by Basov et al. [106]. From the scaling relation (11.58) it can be seen that  the systematic rise of c (0) with decreasing c Tc+ , tuned by decreasing dopant concentration and the associated rise of the anisotropy T , again reflects the flow to 3D–QSI criticality. The straight line marks

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 Table 11.1. Estimates for Tc d/dT (ab (0) /ab (T))2 T=0 derived from the experimental data of: slightly underdoped HgBa2 Ca2 Cu3 O8+ı , slightly overdoped HgBa2 CuO4+ı [100], under-, optimally- and over-doped La2−x Srx CuO4 [9], Bi2 Sr2 CaCu2 O8+ı [101], Y0.94 Ca0.6 Ba2 Cu4 O8 , YBa2 Cu4 O8 , Y1.925 La0.075 Cu4 O8 and Y1.9 La0.1 Cu4 O8 [102]  2  i (ı,0) d  Tc (K) ab (0) (A) Tc dT i (ı,T)  T=0

HgBa2 Ca2 Cu3 O8+ı HgBa2 CuO4+ı La1.9 Sr0.1 CuO4 La1.85 Sr0.15 CuO4 La1.8 Sr0.2 CuO4 La1.78 Sr0.22 CuO4 La1.76 Sr0.24 CuO4 Bi2 Sr2 CaCu2 O8+ı Y0.94 Ca0.06 Ba2 Cu4 O8 YBa2 Cu4 O8 YBa1.925 La0.075 Cu4 O8 YBa1.9 La0.1 Cu4 O8

1770 1710 2800 2600 1970 1930 1940 2600 1361 1383 1521 1593

135 93 30 39 35 27.5 19 91 88 81 74 72

–0.59 –0.65 –0.49 –0.61 –0.72 –0.72 –0.94 –0.61 –0.51 –0.58 –0.61 –0.63

−2 Fig. 11.23. Tc versus 0 ∝ ab (0) for Y0.8 Ca0.2 Ba2 (Cu1−y Zny )O7−ı (Y0.8 Ca0.2 -123), Tl0.5−y Pb0.5+y Sr2 Ca1−x Yx Cu2 O7 (Tl-1212) [103] and TlBa2 CuO6+ı (Tl–2201) [104]. The solid and dashed curves result from the empirical law (11.2) and the interpolation function (11.77). The solid curves indicate the flow from optimum doping to 2D–QSI criticality and the dashed curves to the 3D–QSN critical point

the asymptotic behavior La2−x Srx CuO4 for z = 1 and §s ≈ 24 m(§cm)3/4 . Since §s depends on the critical amplitudes 0,0, ab,0 (0), Tc ,0 and the thickness ds ((11.58) and (11.59)), its value is unique within a family of cuprates. §s ≈ 24 m(§cm)3/4 follows from 0,0 ≈ 1.63, Tc ,0 ≈ 2 (see Fig. 11.1), ds ≈ 6.6 A, 0 ≈ 1 and ab,0 (0) ≈ 736 A (see (11.78). Although the experimental data is still quite far from the underdoped limit, the flow to 2D–QSI criticality

with family-dependent values of §s can be anticipated. This differs from the mean-field prediction for bulk superconductors in the dirty limit and layered BCS superconductors,treated as weakly coupled Josephson junctions (see (11.60) and (11.61)), where −1/2 §s ∝ Tc . Moreover, as the optimally doped and underdoped regimes are approached, systematic deviations from the straight line behavior appear, indicating the flow to 3D–QSN criticality (see (11.63)).

11

Fig. 11.24. Condensation energy −E (T = 0) versus hole concentration p for Y0.2 Ca0.2 Ba0.2 Cu3 O7−ı . The experimental data is taken from Tallon and Loram [105]. The solid and dashed lines are (11.80), indicating the approach to 2D–QSI and 3D–QSN criticality

Universal Properties of Cuprate Superconductors

a rather wide and experimentally accessible critical region. For this reason we observed considerable and consistent evidence that it falls into the same universality class as the onset of superfluidity in 4He films in disordered media, corrected for the long-rangeness of the Coulomb interaction. The resulting critical exponents, z = 1 and  ≈ 1, are also consistent with the empirical relations ((11.2), (11.2) and (11.3)) and the observed temperature and magnetic field dependence of the specific heat coefficient in the limit of zero temperature. These properties also point to a 3D–QSN transition with z = 2 and  ≈ 1/2, describing a d-wave superconductor to disordered metal transition at weak coupling. Here the disorder destroys superconductivity, while at the 2D–QSI transition it localizes the pairs and destroys superfluidity. Due to the existence of the 2D–QSI and 3D–QSN critical points,the detection of finite temperature 3D–XY critical behavior will be hampered by the associated crossovers that reduce the temperature regime where thermal 3D–XY fluctuations dominate. In any case, our analysis clearly revealed that the universality of the empirical correlations reflect the flow to 2D–QSI and 3D–QSN criticality.Moreover, the doping tuned superconductivity in bulk cuprate superconductors turned out to be a genuine 3D phenomenon, where the interplay of anisotropy and superconductivity destroys the latter in the 2D limit. 11.3.3



 +

Fig. 11.25. c (0) versus c Tc for YBa2 Cu3 O7−ı (), La2−x Srx CuO4 (• ) [107] and Bi2 Sr2 CaCu2 Oy () [108]. The straight line is (11.58) with z = 1 and §s ≈ 24 m (§cm)3/4 , the estimate for La2−x Srx CuO4

According to the empirical correlation between Tc and anisotropy T (see (11.3 and Fig. 11.2), the initial value of the dopant concentration determines whether the flow leads to 2D–QSI or 3D–QSN criticality. For initially underdoped cuprates, the rise of

T , tuned by the doping induced reduction of Tc , directs the flow to 2D–QSI criticality. Conversely, in initially overdoped cuprates, the fall of T to a finite value drives the flow to the 3D–QSN critical point.As the nature of the quantum phase transitions is concerned we have seen that the 2D–QSI transition has

481

Evidence for Substitution Tuned Quantum Phase Transitions

It has been well established, both experimentally and theoretically, that in conventional superconductors (e.g., A15 compounds or the Chevrel phases) the presence of magnetic impurities depresses the superconducting transition temperature more efficiently than does the introduction of nonmagnetic ions [109], and this has been ascribed to the breaking of pairs by the magnetic impurities.To determine whether the cuprates behave similarly with respect to magnetic impurities, extensive studies involving the substitution for Cu by other 3d metals have been performed [18,19]. A result common to all these studies is that Tc is depressed in the same manner, independent of whether the substituent is magnetic or non-

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Fig. 11.26. Variation of Tc with substitution for La1.95 Sr0.15 Cu1−y Ay O4 ( A=Fe, Cu, Ni, Zn, Ga and Al ) (taken from Xiao et al. [18])

Fig. 11.27. Temperature dependence of the resistivity for La2−x Srx Cu1−y Zny O4 at yc (x) (taken from Momono et al. [15])

magnetic, and in contrast to that observed in conventional superconductors. To illustrate this point in Fig. 11.26 we depicted the variation of Tc with substitution for La1.95 Sr0.15Cu1−y Ay O4 (A=Fe, Cu, Ni, Zn, Ga and Al) [18]. Tc is seen to vanish at a critical substitution concentration yc , where a quantum phase transition occurs. The experimental data for La2−x Srx Cu1−y Zny O4 , Y0.8 Ca0.2 Ba2 Cu3−3y Zn3y O7−ı [110] and Bi2Sr2 CaCu2−2y Co2y O8+ı [111] shows that this behavior is not restricted to optimum doping.

Thus Tc depends on both the dopant (x) and substitution concentration (y). Figure 11.3 shows the resulting (Tc , x, y) diagram for La2−x Srx Cu1−y Zny O4 . The blue curve, yc (x), is a line of quantum phase transitions. At the corresponding critical endpoints the system undergoes a 2D–QSI and 3D–QSN transition. Along the path marked by the green arrow, a QSI and QSN transition is expected to occur. The pink arrow indicates the crossover from insulating to metallic behavior. This scenario, emerging from the temperature dependence of the resistivity of La2−x Srx Cu1−y Zny O4 [15], is shown in Fig. 11.27. Along the phase transition line yc (x) superconductivity disappears and for x  0.16 metallic behavior sets in, while for x  0.16 the resistivity exhibits insulating behavior at low temperatures. For fixed x and close to yc (x), Tc scales according to (11.33) and (11.38) as z    y . (11.81) Tc y ∝ 1 − yc (x) Since z = 1 is expected to hold in both the doping tuned 2D–QSI and 3D–QSN transition ((11.65) and (11.67)), this combination should hold along the entire line yc (x). Although the data shown in Fig. 11.26 is remarkably consistent with z = 1 it remains to be understood why the linear relationship applies over almost the entire substitution regime. Close to the 2D–QSI and 3D–QSN transitions,the singular part of the free energy density scales in analogy to (11.36) as   fs ∝ ı  (D+z) F yı − , ı = x/xu − 1, 1 − x/xo . (11.82) F is a scaling function of its argument.A phase transition is signaled by a singularity of the scaling function at some value of its argument. Thus, yc (x) ∝ ı  ,

(11.83)

with  = 1 close to the 2D–QSI (see (11.65)) and  = 1/2 near the 3D–QSN transition (see (11.67)).The resulting phase transition line is shown in Fig. 11.28 for La2−x Srx Cu1−y Zny O4 , where we included the experimental data for comparison. As expected from the doping dependence of the correlation lengths (see Fig. 11.20) close to the 3D–QSN transition yc (x) exhibits a very narrow critical regime. For this reason the data considered here is insufficient to provide an

11

Fig. 11.28. yc versus x for La2−x Srx Cu1−y Zny O4 . • : Taken from Momono et al. [15]. The solid curve interpolates between the critical behavior of a 2D–QSI transition with  = 1 (dotted line: yc (x) = 0.03 (x − 5)) and a 3D–QSN transition with  = 1/2 (dashed line: yc (x) = 0.46 (27 − x)1/2 )

estimate for  . Nevertheless,  = 1/2 yields a reasonable qualitative description of the quantum critical line yc (x). The picture we now have is summarized in the phase diagram shown in Fig. 11.3. The blue curve corresponds to the line of quantum phase transitions yc (x) shown in Fig. 11.28. Along this line, z = 1 is expected to hold ((11.65) and (11.67)), while  ≈ 1 (see (11.65)) at the 2D–QSI (y = 0 and x = xu ) and  = 1/2 at the 3D–QSN ((y = 0 and x = xo ) transition. According to the empirical correlation between Tc and anisotropy T (see (11.3) and Fig. 11.2), the initial value of the dopant concentration determines whether the flow upon substitution leads to 2D– QSI or 3D–QSN criticality. For initially underdoped cuprates, the rise of T , tuned by the substitution induced reduction of Tc , drives the flow to 2D–QSI criticality. Conversely, in initially overdoped cuprates, the fall of T to a finite value directs the flow to 3D– QSN criticality. The mechanism whereby the substitution of Cu leads to a reduction of Tc and finally to the quantum critical line yc (x) appears to be not well understood. When the aforementioned universality classes of the 2D–QSI and 3D–QSN transitions hold true ((11.65) and (11.67)), disorder plays an essential role.At 2D–QSI criticality it localizes the pairs and destroys superfluidity [52,53] and at the 3D–QSN

Universal Properties of Cuprate Superconductors

483

transition it destroys superfluidity and the pairs [54]. In this context we also note that the phase diagram of La2−x Srx CuO4+ı shown in 11.4,reveals that isotope substitution leads to identical behavior,although less pronounced (taken from unpublished work of GuoMeng Zhao et al.. Since isotope substitution is accompanied by local lattice distortions, we conclude that in addition to disorder, lattice degrees of freedom tune superconductivity in an essential manner. Another essential facet emerges from the fact that the doping and substitution tuned flow to the 2D–QSI critical point is associated with an enhancement of T . Thus, the fact that the fraction  despite  2  = 1/c2 (0) / 1/ + 1/b2 (0) + 1/c2 (0) = (0) a  2 1/ 1 + 2 T=0 [13], which the third dimension contributes to the superfluid energy density in the ground state, is very small, this implies that a finite Tc is inevitably associated with an anisotropic but 3D condensation mechanism, because T is finite whenever superconductivity occurs (see Fig. 11.2). This points unambiguously to the conclusion that theories formulated for a single CuO2 plane cannot be the whole story. It does not imply, however, a 3D pairing mechanism because in the presence of fluctuations pairing and superfluidity occur separately. 11.3.4

Evidence for Magnetic Field Tuned Quantum Phase Transitions

We have seen (Sect. 11.2) that a strong magnetic field destroys superconductivity at finite temperature. In sufficiently clean systems this destruction occurs at the first-order vortex melting transition. However, in the presence of disorder, the long-range order of the vortex lattice is destroyed and the vortex solid becomes a glass [27]. The vortex fluid to glass transition appears to be a second-order transition,signaled by the vanishing of the zero-frequency resistance in the vortex glass phase. Since disorder plays an essential role at 2D–QSI and 3D–QSN criticality, one expects a line Hm (x) of vortex glass to fluid quantum phase transitions,leading to the schematic phase diagram depicted in Fig. 11.5. There is the superconducting phase (S), bounded by the zero-field transition line, Tc (x, H = 0), the critical lines of the vortex melting or vortex glass to vortex fluid transitions,

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T. Schneider

Fig. 11.29. Left panel: Inplane resistivity ab versus T at H = 0, 4, 8, 16 and 18 T for YBa2 Cu3−z Znz Oy . a: y = 6.7, z = 2.7% and b: y = 6.7, z = 2.3%. Taken from Segawa et al. [30]. Right panel: Schematic (H, T)-phase diagram of YBa2 Cu3−z Znz Oy derived from the data shown in the left panel

Fig. 11.30. Left panel: Phase diagram of Bi2 Sr2−x Lax CuO6+ı in the doping concentration (x or p)-temperature plane. Taken from Ono et al. [31]. Right panel: Temperature dependence of ab for a Bi2 Sr2−x Lax CuO6+ı crystal in H = 0 and 60 T for various dopant concentrations x. The inset shows a clearer view of the low temperature behavior for x = 0.66 and 0.73 (taken from Ono et al. [31])

  Tm x = fixed, H and the line of quantum critical points, Hm (x, T = 0). Along this line superconductivity is suppressed and the critical endpoints coincide with the 2D–QSI and 3D–QSN critical points at xu and xo , respectively. To fix the critical line Hm (x, T = 0) close to the 2D–QSI and 3D–QSN transitions, we note that the singular part of the ground

state energy density scales in analogy to (11.36) as  fs (ı, T) = QD

−  ,0

D A

%−1 −  i,0

ı  (D+z) GD (z) ,

i=1

z=

H a  b , ¥0

(11.84)

11

where H corresponds to a field applied parallel to the c-axis. Supposing that in the (H, ı) plane there is a continuous transition first order melting transition. Then the scaling function GD (z) will exhibit a singularity at some universal value z = zm . Thus close the 2D–QSI and 3D–QSN critical endpoints the quantum critical line (see Fig. 11.5) is given by Hm (ı) =

z m ¥0 2 ı , a,0b,0

(11.85)

with  ≈ 1 (see (11.65)) and  ≈ 1/2 (see (11.67)) close to 2D–QSI and 3D–QSN criticality, respectively. Moreover, the singular behavior of the scaling function GD (z) must be such as to enable the correlation length when H = 0 to correspond to the vortex glass transition, so that    H − Hm (ı) − m    H =  H,0  . (11.86) Hm (ı)   m is the correlation length exponent of the vortex glass to fluid transition. Noting that Tc ∝ ı  z ∝  −z (see (11.34) and (11.38), with (11.86) we obtain for the magnetic field induced reduction of Tc the relation Tc (H) ∝ (Hm (ı) − H) m zm .

(11.87)

zm is the dynamic critical exponent of the vortex glass to fluid transition. The existence of a quantum phase transition line Hm (ı) can be anticipated from the in-plane resistivity data for YBa2 Cu3−z Znz Oy of Segawa et al. [30] shown in the left panel of Fig. 11.29. With increasing magnetic field strength Tc is depressed. In the sample with y = 6.7 and z = 2.7% the temperature dependence of ab clearly points to a magnetic field tuned QSI transition around 16 <  Hm y = 6.7, z = 2.7%  18T.Since Hm (ı) scales as   ı 2 ∝ (zc − z)2 (see (11.85)), Hm y = 6.7, z should increase with reduced Zn concentration z. This behavior  is consistent with  the data for z = 2.3%, where Hm y = 6.7, z = 2.3% > 18T. The emerging (H, T) phase diagram, consisting of finite temperature transition lines (vortex glass to fluid transitions) with 2D–QSI critical endpoints is displayed in the right panel of Fig. 11.29. In the schematic (x, H, T)-phase

Universal Properties of Cuprate Superconductors

485

diagram shown in Fig. 11.5, these lines result from cuts near the zero field 2D–QSI transition with x replaced by z. Related behavior was also observed in La2−x Srx CuO4 [29] and Bi2 Sr2−x Lax CuO6+ı [31]. In the hole doped La2−x Srx CuO4 [29] and the electron doped Pr2−x Cex Cu4+ı a magnetic field tuned metal to insulator crossover was observed close to optimum doping, while in Bi2Sr2−x Lax CuO6+ı [31] the crossover sets in well inside the underdoped regime. The phase diagram of Bi2 Sr2−x Lax CuO6+ı is displayed in the left panel of Fig. 11.30. In the underdoped limit (x ≈ 0.84) the transition temperature vanishes and a 2D–QSI transition is expected to occur. Evidence for this transition emerges from the temperature dependence of ab shown in the right panel of Fig. 11.30 [31], taken at H = 0 and 60T for six doping levels x. To strengthen this point in Fig. 11.31 we plotted the resulting Tc versus 0,c − 0 for x = 0.76, 0.73 and 0.66 and 0,c ≈ 0.5 m§cm at H = 0. Apparently there is suggestive consistency with the 2D–QSI scaling relation (11.42) with z ≈ 1, as well as with the corresponding data for La2−x Srx CuO4 and YBa2 Cu3 O7−ı shown in Fig. 11.15. As the 60T data (symbols) are concerned, the samples closest to the QSI transition (x = 0.76 and 0.84) exhibit a pronounced upturn in ab at low temperatures. This points to an insulating normal state. Indeed, the weak upturn below 6 K at x = 0.73, visible in the inset of Fig. 11.30, signals the proximity to the onset of insulating behavior, while the low T behavior of the x = 0.66 sample with Tc (H = 0) = 23 K exhibits metallic behavior. Thus, there is a metal to insulator (MI) crossover. The emerging (x, T, H)-phase diagram is displayed in Fig. 11.32. The arrows mark the flows emerging from the experimental data displayed in Fig. 11.30 for x = 0.84 (1) and x = 0.66 (2). This differs from the behavior observed in La2−x Srx CuO4 where the detectable MI-crossover appears to set in close to optimum doping. In Fig. 11.33 we depicted the logarithmic plot of ab (T) for Bi2Sr2−x Lax CuO6+ı and La2−x Srx CuO4 crystals for H = 0 and 60T and various doping concentrations [31]. Concentrating on the presence or absence of an upturn,it is clearly seen that in La2−xSrx CuO4 the MI-

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Fig. 11.32. Schematic (x, T, H)-phase diagram close to the 2D–QSI transition. There is the superconducting phase (S), bounded by the zerofield transition line, Tc (x, H = 0), the lines of the vortex melting or  vortex glass  to vortex fluid transitions Tm x = fixed, H and the quantum critical line Hm (x). The critical lines Tc (x, H = 0) and Hm (x) merge in the underdoped limit (x = xu ) where a doping driven quantum superconductor to insulator (QSI) transition (•) occurs. The arrows mark the flows emerging from the experimental data shown in Fig. 11.31 at x = 0.84 (1), x = 0.66 (2)

Fig. 11.31. Tc versus 0,c − 0 for Bi2 Sr2−x Lax CuO6+ı . Deduced from the data shown in Fig. 11.30 for x=0.76, 0.73 and 0.66 and 0,c = 0.5 m§ cm. The straight line is the scaling relation (11.70) with z ≈ 1

crossover occurs around x ≈ 0.15, corresponding to optimum doping. However, it should be kept in mind that this crossover is nonuniversal. For this reason, the dopant concentration where insulating behavior is detectable is expected to be material-dependent. The magneto resistance ab (H, T) of underdoped La2−x Srx CuO4 films with x ≈ 0.048 (Tc = 0.45K) and 0.051 (Tc = 4K) was also studied in magnetic fields up to 0.5 T and at temperatures down to 30mK [112]. The temperature dependence of ab (H, T) of the film closest to the 2D–QSI tran-

Fig. 11.33. Logarithmic plot of ab (T) for Bi2 Sr2−x Lax CuO6+ı and La2−x Srx CuO4 crystals in H = 0 T (solid lines) and 60 T (filled circles), labeled by La concentration, x. The straight line indicates consistency with the log(1/T) behavior at x = 0.84. and open circles are H = 30 T data. La2−x Srx CuO4 data in H = 0 T (dashed lines) and in 60 T (open squares), labeled by the Sr concentration x (taken from Ono et al. [31])

11

sition (x ≈ 0.048, Tc = 0.45K) was  found to be consistent with ab ∝ exp (T/T0)1/3 below 1 K and for fields from 4 to 6 T. Thus this study points to 2Dlocalization. On the other hand, the data of Ando et al. [28] (Fig. 11.33) appears to be consistent with a logarithmic divergence of the normal-state resistivity down to 0.7K for x = 0.08 and H = 60 T, suggesting 2D weak localization. In any case, the evidence for 2D localization confirms an essential property of 2D–QSI criticality: disorder localizes the pairs and destroys superfluidity [52,53].

11.4

Thin Films

There is considerable evidence that the quantum superconductor to insulator transition in thin films can also be traversed by changing a parameter such as film thickness, disorder, etc. [13]. As far as cuprates are concerned, an instructive example are the measurements of YBa2 Cu3 O7−ı slabs of thickness d separated by 16 unit cells (≈ 187 A) of PrBa2 Cu3O7 . Due to the large separation the YBa2 Cu3 O7−ı slabs can be considered to be essentially to be uncoupled. As shown in Fig. 11.6 in YBa2 Cu3O7−ı slabs of thickness d the transition temperature was found to vary according to (11.4). Together with the scaling relation (11.38) this points to a 2D–QSI transition with z = 1, in agreement with our previous estimate (see (11.65)). Since quantum fluctuations alone are not incompatible with superconductivity, this 2D– QSI transition must be attributed to the combined effect of disorder and quantum fluctuations. Nevertheless, it is conceivable that in cleaner films superconductivity may also occur at and below this value of ds , which is close to the estimate derived from the bulk (see (11.69)). What is seen is then superconductivity in 2D and due to the reduced dimensionality fluctuations will be enhanced over the full range of dopant concentrations. As it stands, in the bulk (see, e.g. Fig. 11.2) and chemically doped films superconductivity disappears in the 2D-limit.Since chemically doped materials with different carrier densities also have varying amounts of disorder, the third dimension is apparently needed to delocalize the carriers and to mediate superfluidity.

Universal Properties of Cuprate Superconductors

11.5

487

Concluding Remarks and Comparison with Other Layered Superconductors

Evidence for power laws and scaling should properly consist of data that cover several decades in the parameters. The various power laws that we have exhibited span at best one decade and the evidence for data collapse exists only over a small range of the variables. Consequently, though the overall picture of the different types of data is highly suggestive, it cannot really be said that it does more than indicate consistency with the scaling expected near a quantum critical point or a quantum critical line. Nevertheless, the doping, substitution and magnetic field induced suppression of Tc clearly reveal the existence and the flow to 2D–QSI and 3D–QSN quantum phase transition points and lines (see Figs. 11.1, 11.2, 11.3 and 11.5). In principle, their universal critical properties represent essential constraints for the microscopic theory and phenomenological models. As it stands, the experimental data is fully consistent with a single complex scalar order parameter, a doping induced dimensional crossover, a doping, substitution or magnetic field driven suppression of superconductivity, due to the loss of phase coherence. When the evidence for this scenario persists, antiferromagnetic and charge fluctuations are irrelevant close to criticality. Moreover, given the evidence for the flow to 2D–QSI criticality, the associated 3D– 2D crossover,tuned by increasing anisotropy,implies that a finite Tc and superfluid density in the ground state of bulk cuprates is unalterably linked to a finite anisotropy. This raises serious doubts that 2D models are potential candidates to explain superconductivity in bulk cuprates. Thus, there is convincing evidence that the combined effect of disorder and quantum fluctuations plays an essential role and even destroys superconductivity in the 2D limit of chemically doped cuprates. Thus, as far as the nature of the quantum phase transitions in chemically doped cuprates is concerned, disorder is an essential ingredient. We have seen that the 2D–QSI transition has a rather wide and experimentally accessible critical region. For this reason we observed considerable and consistent evidence that it falls into the same universality class as the onset of superfluid-

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ity in 4 He films in disordered media, corrected for the long-rangeness of the Coulomb interaction. The resulting critical exponents, z = 1 and  ≈ 1 are also consistent with the empirical relations ((11.2), (11.2) and (11.3)) and the observed temperature and magnetic field dependence of the specific heat coefficient in the limit of zero temperature. These properties also point to a 3D–QSN transition with z = 2 and  ≈ 1/2, describing a d-wave superconductor to disordered metal transition at weak coupling. Here the disorder destroys superconductivity, while at the 2D–QSI transition it localizes the pairs and destroys superfluidity. Due to the existence of the 2D–QSI and 3D–QSN critical points, the detection of finite temperature 3D–XY critical behavior will be hampered by the associated crossovers that reduce the temperature regime where thermal 3D–XY fluctuations dominate. In any case, our analysis clearly revealed that superconductivity in chemically doped cuprate superconductors is a genuine 3D phenomenon and

that the interplay of disorder (anisotropy) and superconductivity destroys the latter in the 2D limit.As a consequence, the universality of the empirical correlations reflects the flow to 2D–QSI and 3D–QSN criticality, tuned by chemical doping, substitution and an applied magnetic field. A detailed account of the flow from 2D–QSI to 3D–QSN criticality is a challenge for microscopic theories attempting to solve the puzzle of superconductivity in these materials. Although the mechanism for superconductivity in cuprates is not yet clear, essential constraints emerge from the existence of the quantum critical endpoints and lines. However, much experimental work remains to be done to fix the universality class of the 2D–QSI and particularly of the 3D–QSN critical points unambiguously. In conclusion we note that universal properties emerging from thermal and quantum fluctuations are not restricted to cuprate superconductors. Potential candidates are the highly anisotropic organic

Fig. 11.34. c (0) versus cDC for a variety of superconductors. YBa2 Cu3 O7−ı (YBCO) [114–118], La2−x Srx CuO4+ı (LSCO) [116, 119, 120], HgBa2 CuO4+ı (Hg1201) [121], Tl2 Ba2 CuO6+ı (Tl2201) [122–124], Bi2 Sr2 CaCu2 O8+ı (Bi2212) [125, 126], Nd2−x Cex CuO4+ı (NCCO) [127]. Blue points: underdoped (UD), green points: optimally doped (OpD) and red points: overdoped (OD). Transition metal dichalcogenides [144,145]; (ET)2 X compounds [135–143]; (TMTSF)2 ClO4 [128–134]; Sr2 RuO4 [146,147]; MgB2 [148–150]; Nb [151,152]; Pb [152]; Nb Josephson junctions [153]; ˛Mo1−x Gex [154]. The dotted line is (11.88) and the dashed lines (11.58). The red arrows indicate the flow to 2D–QSI and for overdoped YBa2 Cu3 O7−ı to 3D–QSN criticality, respectively

11

superconductors that are close to a metal–insulator boundary.A glance at Fig. 11.34  shows  that the correlation between c (0) and cDC Tc+ is particularly rewarding. In various classes of layered superconductors, including organics, transition metal dichalcogenides and cuprates, c (0) is systematically suppressed  with increase of normal state conductivity cDC Tc+ . Without thermal and quantum fluctuations the DC conductivity is given by cDC =

ne2 c 2 tr . tr = mc 4c2 (0)

(11.88)

tr is the mean scattering time of the normal electrons in transport properties with number density per unit volume n and effective mass mc . Provided that the variations of tr are small, superconductors where fluctuations can be neglected (mean-field superconductors) are then characterized by the dotted line in Fig. 11.34, which is −1/2  c (0) ∝ cDC .

(11.89)

This agrees rather well with the data of mean-field superconductors. Interestingly enough, MgB2 and Sr2 RuO4 appear to fall into this class as well. Prominent and systematic deviations from this behavior occur for the highly anisotropic organics and underdoped cuprates. As c (0) increases with the reduction of the normal state conductivity cDC,we observe that the data tends to scatter around −3/4two branches, consistent with c (0) = §s cDC , which is the scaling relation (11.58) with z = 1, describing the critical behavior close to 2D–QSI critical points. The most prominent deviations from these trends occur for overdoped cuprates (see also Fig. 11.23). In particular the data for overdoped YBa2 Cu3O7−ı (YBCO) exhibit an upturn, signaling the crossover to 3D–

Universal Properties of Cuprate Superconductors

489

QSN criticality (see (11.63)). Since organics undergo superconductor to insulator transitions [155], are highly anisotropic and can be varied over a rather extended interval, i.e. Tc = 180 for -(ET)2 Cu[N(CS)2 ]Br, Tc = 350 for -(ET)2 Cu(NCS)2 [156] and Tc = 2 × 103 for ˛- (BEDT-TTF)2 NH4 Hg(SCN)4 , it becomes clear that the organics line in Fig. 11.34 indicates the flow of this class of superconductors to 2D–QSI criticality, where the scaling relation (11.58) applies. Thus unlike the organics the cuprates undergo a doping tuned crossover 2D–QSI to 3D–QSN  from  criticality, where cDC Tc+ and c (0) tend to infinity, while Tc vanishes. These critical points are attained in the underdoped and overdoped limit, respectively, where Tc vanishes. In this crossover the cuprates either pass the organics or the dotted mean-field line, spanned by conventional superconductors (see Fig. 11.34). YBa2Cu3 O7−ı (YBCO) and La2−x Srx CuO4+ı (LSCO) cross the dotted meanfield line in the neighborhood of TaS2, Sr2 RuO  4 and MgB2 . In this region of the c (0)-cDC Tc+ plane fluctuations do not play a dominant role. Indeed, Sr2 RuO4 exhibits Fermi liquid normal state properties [157]. However, overdoped cuprates flow to 3D– QSN criticality,where disorder destroys the phase coherence and the pairs. On the contrary, the organics evolve rather smoothly from the mean-field regime, to which Pb, Nb, Nb-junctions, ˛-Mo1−x Gex and the dichalcogenides belong. Thus, these flows provide a perspective of the regimes where fluctuations and disorder are essential. Acknowledgements It is a pleasure to thank H. Keller and K.A. M¨uller for numerous helpful discussions.

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135. X. Su, F. Zuo, J.A. Schlueter, A.M. Kini, and J.M. Williams, Phys. Rev. B 58, 2944 (1998) 136. A. Carrington, I.J. Bonalde, R. Prozorov, R.W. Giannetta, A.M. Kini, J. Schlueter, H.H. Wang, U. Geiser, and J.M. Williams, Phys. Rev. Lett. 83, 4172 (1999) 137. T. Shibauchi et al., Phys. Rev. B 55, 11979 (1997) 138. M. Dressel, O. Klein, G. Gruener, K.D. Carlson, H.H. Wang, and J.M. Williams, Phys. Rev. B 50, 13603 (1994) 139. H. Taniguchi, H. Sato, Y. Nakazawa, and K. Kanoda, Phys. Rev. B 53, 8879 (1996) 140. S. Wanka, D. Beckmann, J. Wosnitza, E. Balthes, D. Schweitzer, W. Strunz, and H.J. Keller, Phys. Rev. B 53, 9301 (1996) 141. K. Kajita, Y. Nishio, S. Moriyama, W. Sasaki, R. Kato, H. Kobayashi, andA. Kobayashi, Sol. Stat. Comm. 64, 1279 (1987) 142. X. Su, F. Zuo, J.A. Schlueter, J.M. Williams, P.G. Nixon, R.W. Winter, and G.L. Gard, Phys. Rev. B 59, 4376 (1999) 143. R. Prozorov, R.W. Giannetta, J. Schlueter, A.M. Kini, J. Mohtasham, R.W. Winter, and G.L. Gard, Phys. Rev. B 63, 052506 (2001) 144. H. Schwenk, K. Andres, F. Wudl, and E. Aharon-Shalom, Solid State Commun. 45, 767 (1983) 145. K. Murata et al., J. Phys. Soc. Jpn. 50, 3529 (1981) 146. Y. Maeno et al., Nature 372, 532 (1994) 147. K. Yoshida, Y. Maeno, T. Fujita, and S. Nishizaki, Physica C 263, 519 (1996) 148. C. Panagopoulos, B.D. Rainford, T. Xiang, C.A. Scott, M. Kambara, and I.H. Inoue, Phys. Rev. B 64, 094514 (2001) 149. M.Angst, R. Puzniak,A.Wisniewski, J. Jun, S.M. Kazakov, J. Karpinski, J. Roos, and H. Keller, Phys. Rev. Lett. 88, 167004 (2002) 150. D.K. Finnemore, J.E. Ostenson, S.L. Bud’ko, G. Lapertot, and P.C. Canfield, Phys. Rev. Lett. 86, 2420 (2001) 151. A.V. Pronin, M. Dressel, A. Pimenov, A. Loidl, I.V. Roshchin, and L.H. Greene, Phys. Rev. B 57, 14416 (1998) 152. O. Klein, E.J. Nicol, K. Holczer, and G. Gruener, Phys. Rev. B 50, 6307 (1994) 153. E. Goldobin, M.Yu. Kupriyanov, I.P. Nevirkovets, A.V. Ustinov, M.G. Blamire, and J.E. Evetts, Phys. Rev. B 58, 15078 (1998) 154. S.J. Turneaure, Th.R. Lemberger, J.M. Graybeal, Phys. Rev. Lett. 84, 987 (2000) 155. Y. Nakazawa, H. Taniguchi, A. Kawamoto, and K. Kanoda, Phys. Rev. B 61, 16295 (2000) 156. S. Kawamata et al., J. Low. Temp. Phys. 105, 1721 (1996) 157. C. Bergemann, S.R. Julian, A.P. Mackenzie, S. NishiZaki, and Y. Maeno, Phys. Rev. Lett. 84, 2662 (2000), and references therein

12

Vortex Matter

G. Blatter Theoretische Physik, ETH Z¨urich, Switzerland V. B. Geshkenbein Theoretische Physik, ETH Z¨urich, Switzerland

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 12.2 Ginzburg–Landau and London Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 12.3 Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 12.3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 12.3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 12.3.3 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 12.3.4 Free Energy and Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 12.4 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 12.4.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 12.4.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 12.4.3 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 12.4.4 Free Energy and Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 12.5 Layered Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 12.5.1 Lawrence–Doniach Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .519 12.5.2 Josephson Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 12.5.3 Pancake Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 12.5.4 Renormalized Superfluid Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 12.5.5 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 12.6 Anisotropic Scaling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 12.7 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 12.7.1 Fluctuations in 2D Thin Film and Layered Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 12.7.2 Lindemann Analysis of Vortex Lattice Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 12.7.3 Lindemann Analysis of Layer Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 12.7.4 Vortex Lattice Melting in 2D Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 12.7.5 Electromagnetically Coupled Layered Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 12.7.6 Characteristics of First-Order Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 12.7.7 Vortex Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 12.7.8 Constitutive Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 12.8 Quenched Disorder: Pinning and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 12.8.1 Strong Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567 12.8.2 Weak Collective Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 12.8.3 Weak Versus Strong Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 12.8.4 Vortex Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573 12.9 Uncorrelated Disorder: Collective Pinning and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .575 12.9.1 Dirty Elastic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 12.9.2 Single Vortex Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 12.9.3 Vortex Lattice Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 12.9.4 Collective Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

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12.11

12.12

12.13

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12.9.5 Thermal Depinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 12.9.6 Anisotropic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .601 12.9.7 Layered Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 12.9.8 Pinning and Creep Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 Correlated Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605 12.10.1 Columnar Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 12.10.2 Twin Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 Surface and Geometrical Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 12.11.1 Surface Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .611 12.11.2 Geometrical Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Vortex Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 12.12.1 Disorder-Induced Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .616 12.12.2 Vortex-Glass Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

Introduction

The discovery of high-temperature superconductivity by Bednorz and M¨uller in 1986 [1] has triggered an enormous research activity with various goals: unraveling the microscopic mechanism producing such high transition temperatures, understanding the phenomenological properties of these layered and strongly type II superconductors, and developing materials ready for technological applications. This chapter deals with the phenomenological aspect of strongly fluctuating type II superconductors in general and materials with high transition temperatures in particular; as such the main goal is to develop the physics of vortices as a new state of matter,hence the title“Vortex Matter”.In fact,thanks to a set of lucky circumstances the new high-temperature superconductors set the stage for the physical realization and experimental accessibility of the entire statistical mechanics of vortices and thus, from a more general point of view, of interacting elastic strings. Furthermore, the coupling of the vortices to external currents and fields allows to exert forces on these strings, opening up the wide field of dynamical behavior of the vortex system. Finally, vortices

are susceptible to disorder and thus the statistical and dynamical properties of this system have to be investigated in the presence of a random potential environment. The phenomenology of superconductivity is based on the Ginzburg–Landau theory [2] which provides the starting functional (free energy) for the charged superfluid coupled to electromagnetism. The two fields determining the physics of the system are the superconducting order parameter ¦ and the vector potential A; the complex order parameter ¦ = |¦ | exp(i' ) describes the density and the phase of the quantum fluid. The resulting mean-field version of the H–T phase diagram has been derived by Abrikosov back in 1957 [3] and has proved to describe accurately the phenomenological behavior of all the conventional low-temperature superconductors. This mean-field H–T phase diagram, shown in Fig. 12.1, comprises a Meissner–Ochsenfeld phase characterized by a complete flux expulsion at low magnetic fields H < Hc1 , separated from the mixed (or Shubnikov) phase [4] at higher fields H > Hc1 , where the magnetic field penetrates the superconductor in the form of flux lines (or vortices). The lower critical field Hc1 is mainly determined by the

12 Vortex Matter

Fig. 12.1. Mean-field phase diagram for a type II superconductor: below the lower critical field Hc1 the magnetic field is expelled from the sample; this is the Meissner– Ochsenfeld effect. In the intermediate field regime Hc1 < H < Hc2 the magnetic field penetrates the sample via flux lines or vortices; this is the Shubnikov phase with vortices arranged in a triangular Abrikosov lattice. Above the upper critical field Hc2 superconductivity is destroyed and the sample is in the normal metallic phase

penetration depth , which is the length scale associated with the electromagnetic response of the superconductor.Since the superconducting state is a quantum fluid, the magnetic flux enclosed in a vortex is quantized in units of ¥0 = hc/2e ≈ 2.068 10−7 Gcm2 , the flux quantum.With increasing field the density of flux lines (which form a triangular lattice) increases until the vortex cores overlap when the upper critical field Hc2 is reached; beyond this field we recover the normal metallic state. The upper critical field Hc2 is determined by the coherence length  of the superconductor which represents the second fundamental length scale in the system associated with the response of the macroscopic quantum fluid ¦ . Applying an external current density j to the vortex system, the flux lines start to move under the action of the Lorentz force F L = j ∧ B/c. Within a homogeneous system the driving Lorentz force is

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counteracted by the friction force F  = −v alone, where v is the steady state velocity of the vortex system, v = j ∧ B/c, and  is the Bardeen–Stephen viscous drag coefficient [5]. The flux motion produces an electric field E = B ∧ v/c directed parallel to the driving current density j; as a result the power P = (j ∧ B)2 /c 2 is dissipated in the system and the superconducting property of dissipation free current flow is lost. Hence, while the Meissner–Ochsenfeld phase is a true superconductor, the Shubnikov phase exhibits a normal response under the application of a transverse current j ⊥ H. On the other hand, the longitudinal (j  H) response in the Shubnikov phase is superconducting. Finally, the normal-metal phase exhibits an ohmic response in all directions. In order to recover the desired property of dissipation-free current flow the flux lines have to be pinned such that v = 0 in the presence of a force field F L = 0. In this case the driving Lorentz force is counteracted by the pinning force Fpin [6–10].Fortunately, any static disorder affecting the superconducting order parameter will contribute to a finite pinning force density F pin and thereby the technological usefulness of the type II superconductors is reestablished. However, dissipation-free flow now has become a matter of optimizing the pinning force density Fpin since increasing the current density j beyond jc = cFpin/B (we assume j ⊥ B) leads to the depinning of the vortices and hence to the reappearance of dissipation. The current density jc is the critical depinning current density, which is always bounded by the depairing current density j0 ∼ cHc / determined by the thermodynamic critical field Hc . The dimensionless critical current density ratio jc /j0 is a measure of the strength of the pinning force density Fpin and as such can be conveniently used as the fundamental parameter in the phenomenology of type II superconductors characterizing the strength of the quenched disorder in the system. Going beyond the traditional mean-field picture, thermal fluctuations have to be taken into account and the statistical physics of vortex matter has to be developed. Large thermal fluctuations (of the order of the lattice constant a0 ≈ (¥0/B)1/2) of the vortex positions will lead to a melting transition and produce a new vortex-liquid phase [11–13], see Fig. 12.2.

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Fig. 12.2. Schematic phase diagram for a strongly fluctuating type II superconductor. The vortex lattice undergoes a first-order melting transition transforming the vortex solid into a vortex liquid. Transverse screening produces an exponentially weak interaction between the vortex lines at low magnetic fields, leading to a reentrance of the melting line. The flux lines in the vortex-liquid are entangled resulting in an ohmic longitudinal response, hence the vortexliquid and normal metallic phases are separated by a mere crossover at Hc2

Near the normal-metal–superconductor transition not only the phase but also the modulus of the order parameter is subject to large thermal fluctuations, resulting in a sizeable critical region. Besides these thermodynamic consequences, the inclusion of thermal fluctuations into the phenomenology of type II superconductors also affects the dynamical behavior of the vortex system in various ways. First of all, the vortex lines can move at small current densities j < jc due to thermally activated jumps over the pinning barriers [14], leading to the famous creep phenomenon in type II superconductors. Creep of vortex lines produces a small but finite directed motion of the vortex lines and thus reestablishes dissipation in the system.One then could ask if true “transverse” superconductivity exists at all in the thermodynamic sense (note that longitudinal superconductivity persists as current den-

sities j  B produce a force-free configuration). The crucial question is whether creep persists down to the limit of zero driving force: Does the dissipation and hence the resistivity  vanish in the limit j → 0 or does a superconductor in the mixed phase finally end up in a resistive state (note that probing the superconductor with a vanishing current density j → 0 amounts to investigating the thermodynamic state of the vortex system). The kind of response the system develops for j → 0 then determines whether the vortex system constitutes a new vortex-glass phase [15,16], characterized by infinite barriers against creep [17] and hence (j → 0) → 0, or whether the vortex system is in a liquid state with finite creep barriers allowing vortex motion at any nonzero temperature producing a finite resistivity (j → 0) > 0. Another hint to the appearance of a glassy phase is the destruction of the triangular vortex lattice:As has been shown by Larkin [18], in less than four dimensions quenched disorder always destroys the translational long-range order of the Abrikosov vortex lattice,giving way to a glassy/liquid phase.As long as the elastic properties of the vortex lattice are preserved, i.e., in the absence of dislocations, the destruction of the lattice-order is weak: with a displacement field growing only logarithmically with distance [19–21] the vortex lattice remains quasi-ordered and the resulting phase has been termed a “Bragg-glass” [21]. Increasing the effect of disorder the quasi-ordered lattice will undergo a disorder-driven melting transition [22] taking the vortex system into a more disordered glassy or frozen liquid state, cf. Fig. 12.3. The interplay between quenched disorder and thermal fluctuations not only manifests itself in the phenomenon of creep. Thermal fluctuations of the individual vortex lines lead to a dynamical sampling, and hence thermal averaging, of the disorder potential over the spatial extent of the mean thermal displacement u2 1/2 th of the vortex lines. Thermal disorder hence opposes quenched disorder also via a smoothing of the disorder potential and thereby reduces the critical current density jc in the system. The smoothing of the quenched disorder potential due to thermal fluctuations has become known under the name of thermal depinning [23], of thermal

12 Vortex Matter

Fig. 12.3. In a defected crystal the vortex lines are pinned by the disorder potential. The vortex solid is transformed into a vortex-glass, while the vortex-liquid remains a (viscous) liquid. At low fields the solid is only weakly disordered; the Bragg-glass is free of dislocations and exhibits a superconducting response in all directions. The more strongly disordered vortex-glass (possibly a frozen vortex-liquid) carries a finite density of dislocations

though care should be taken that the phenomenon depinning is not an abrupt transition but rather a continuous crossover from a pinned to an unpinned situation. It turns out that the fundamental parameter governing the strength of thermal fluctuations in all of the above phenomena is the Ginzburg number Gi = (Tc /Hc2Vcoh )2 /2, which measures the relative size of the (T = 0) condensation energy Hc2 Vcoh within a coherence volume Vcoh with respect to the critical temperature Tc (we set the Boltzmann constant to unity, kB = 1). The layered structure of the copper-oxide superconductors reduces the coherence volume by the anisotropy factor " = (m/M)1/2 , Vcoh = " 3 , where m and M ( m) denote the effective masses of the charge carriers parallel and perpendicular to the superconducting planes.

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At low temperatures the role of thermal fluctuations is taken over by their quantum analog. Similar to thermal fluctuations,quantum fluctuations can affect superconductivity via fluctuations in the order parameter modulus and phase,e.g.,via quantum motion of vortices (quantum creep [24]). Again, the importance of quantum fluctuations can be expressed by a dimensionless parameter; in the limit of dissipative vortex motion this parameter is determined by the normal state resistivity  n of the material, Qu = (e2/)( n /"). With the three fundamental parameters quantifying quenched disorder (jc /j0 ), thermal fluctuations (Gi), and quantum fluctuations (Qu) we are in a position to express the difference between the conventional low-Tc superconductors and the hightemperature superconductors in a compact way: In conventional superconductors pinning is usually strong, jc /j0 ∼ 10−2–10−1 , whereas both thermal and quantum fluctuations are weak, Gi ∼ 10−8 and Qu ∼ 10−3 . In generic high-Tc superconductors such as YBa2 Cu3 O7−y (YBCO) or the Bi- and Tl-based materials (e.g., Bi2Sr2 Ca1 Cu2 O8+y (BiSCCO)) pinning is usually weak, jc /j0 ∼ 10−3 –10−2 , whereas thermal and quantum fluctuations are large, Gi ∼ 10−2 and Qu ∼ 10−1 . This change in the relative importance of static and dynamic disorder is a consequence of the extreme material parameters characterizing the oxide superconductors: First of all, the transition temperature Tc ≈ 100 K is very large and hence the coherence length  ∼ vF /Tc is small,  ≈ 16 Å in YBCO and  ≈ 25 Å in BiSCCO (vF denotes the Fermi velocity). Since the oxide superconductors are doped insulators rather than generic metals the electronic density is small, hence vF is small and the resistivity  n as well as the penetration depth  tend to be large (provided we can still use Fermi-liquid concepts for these crude estimates): the penetration depth  ≈ 1400 Å and  ≈ 2000 Å in YBCO and BiSCCO, respectively. In addition, the layered structure of the oxides (with layer separation d ≈ 12 Å and d ≈ 15 Å in YBCO and BiSCCO, respectively) introduces a large uniaxial anisotropy in the system which reduces the coherence length along the anisotropy axis (the c-axis of the material) by an additional factor " = (m/M)1/2 < 1; " ≈ 1/8-1/5 in YBCO while

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" < 1/50 in BiSCCO (depending on oxygen doping the anisotropy may be much larger with " of order 10−3 [25]). The anisotropy strongly promotes thermal and quantum fluctuations, since Gi ∝ 1/" 2 and Qu ∝ 1/". The above material parameters tend to increase the importance of the fluctuations in the high-temperature superconductors. Whereas the mean-field description was quite sufficient for the understanding of the conventional low-temperature superconductors this is no longer the case for the new materials. In particular, the increase of Gi by roughly six orders of magnitude implies that the vortex lattice is melted over a large portion of the H–T phase diagram; similarly, the increase of Qu by two orders of magnitude renders macroscopic quantum effects relevant. We wish to point out that these novel effects are not unique to high-temperature superconductors, they are present as well in the conventional low-Tc materials. However, it is the special set of material parameters in the oxides which renders these phenomena physically important and accessible to experimental investigations. The special material properties of the hightemperature superconductors not only push the importance of fluctuations, they also tend to decrease pinning in these materials. The weak pinning (small jc /j0 ) is not only a consequence of the small coherence length , but also follows from the absence of generic extended pinning sites in these materials such as precipitates or grain boundaries occurring in many conventional superconductors. Again, this circumstance is a consequence of the oxides being doped insulators rather than conventional metals, hence pinning centers are mainly provided by point-defects, e.g., oxygen vacancies [26]. An important exception are the twin boundaries in YBCO which provide extended pinning sites [27, 28] upon proper alignment of the magnetic field. Furthermore, in order to boost the critical current density for technological applications strong pinning centers can be introduced artificially through irradiation of the material with heavy ions producing columnar tracks [29,30]. The generic weakness of pinning in the hightemperature superconductors is interesting from a theoretical point of view as it allows to treat the ef-

fects of quenched disorder perturbatively. In particular, weak pinning implies the existence of a well defined starting Hamiltonian in the form of a continuum elastic description of the vortex lattice, upon which effects due to disorder can be studied in a well defined fashion. The dynamical behavior of the vortex system in the presence of quenched disorder then can be studied within the framework of the weak collective pinning theory introduced by Larkin and Ovchinnikov [8,10]. Furthermore, if pinning is weak enough,the melting transition separating the vortexsolid and liquid phases will be only little perturbed by the presence of quenched disorder, allowing for the observation of a generic melting transition in the vortex system. The above discussion shows that we can expect a wealth of novel phenomena in type II superconductors; at the same time these should be experimentally accessible and theoretically tractable in a well defined fashion, thus allowing for a mutual exchange of experimental and theoretical results. On the experimental side such novel phenomena include the broadening of the resistive transition in a finite magnetic field [31–33], the existence of a distinct irreversibility line far below Hc2 [34], often associated with a vortex-glass transition [35],the appearance of a first-order vortex lattice melting transition in high quality crystals [36–43], the presence of giant creep [44–46] even at temperatures far below Tc [47, 48], a rapid decrease in the critical current density jc with increasing temperature [49], and the appearance of quantum effects on approaching zero temperature [50–52]. On the theoretical side the statistical mechanics and the dynamics of individual vortices and of the vortex lattice has been studied in homogeneous and in disordered environments. These investigations have led to the postulation of new phase transitions in the vortex system, such as the vortex-lattice melting [11–13] and vortex-glass transitions [15,16]. New vortex phases such as the entangled and the disentangled vortex-liquids [11,53,54], the vortex-glass [15–17], the Bragg-glass [19–21], and the Bose-glass phase [55] have been proposed. The presence of thermal fluctuations leads to thermal depinning of vortices [23]. The dynamical response of the vortices is very different in these var-

12 Vortex Matter

ious phases and thus can be used for the characterization of the new vortex-liquid [56] and vortexglass phases [15–17,19–21]. Also, the study of classical vortex motion has been extended to the quantum regime [24,57]. A second source of novel phenomena is associated with the layered structure of the new oxides. Whereas the YBCO compound still can be described reasonably well within a continuum anisotropic model, the layered Bi- and Tl-based compounds have to be described by a discrete Lawrence–Doniach model [58]. Superconductivity in these materials then assumes features of quasi-two-dimensionality and many experiments exhibit a characteristic Berezinskii–Kosterlitz–Thouless (BKT) type behavior [59,60].Depending on the physical situation,the simple rectilinear vortex has to be replaced by a more complex object consisting of an array of 2D-pancake vortices interconnected by (coreless) Josephson vortices running parallel in between neighboring superconducting planes [61–65]. The layered structure also introduces new features in the dynamical response of the vortices due to the appearance of intrinsic pinning and creep [66–69]. The statistical mechanics of the vortex system is affected by the layering via the presence of large (quasi-2D) fluctuations [70] and also through the appearance of new phase transitions, e.g., a layer decoupling transition in a field H directed perpendicular to the layers [64,71–76]. When the field H is directed parallel to the layers commensuration effects appear between the crystal and the magnetic structure [77–80]; the prediction of a decoupling transition in this geometry and an associated smectic transition [62,79–84] is much disputed [85–88]. Much interest has concentrated on attempts to increase the critical current density via the artificial introduction of strong pinning centers (e.g., columnar defects [29, 30, 89–91]) and the investigation of the macroscopic properties of the new materials [92–95]. While research on these topics is driven by their relevance for technological applications of hightemperature superconductors, it is exciting to notice that they bear a wealth of fundamental questions, such as the substitution of the vortex-glass phase by the Bose-glass phase in the case where the disorder

501

is correlated [55], or the relation of the macroscopic behavior of a type II superconductor in a magnetic field with ideas of self-organized criticality [96,97]. In order to give a theoretical description of these very diverse phenomena one has to gather various concepts and results from different branches of modern theoretical physics, such as the theory of elastic manifolds in quenched random media, polymer physics, spin-glass theory, fluctuational theory of phase transitions, strongly correlated quantum systems, disordered quantum Bose liquids, macroscopic quantum tunneling, hopping conductivity in semiconductors, and self-organized criticality. The above discussion then illustrates the richness and complexity we encounter when dealing with the physics of Vortex Matter. The main philosophy followed below is to base all the discussions on some “basic Hamiltonian” with the requirement that it shall describe the vortex system accurately. In particular, all the internal structure existing in the vortex system (e.g., the vortex lattice) as well as the interactions present in the system (e.g., between the vortices and the static disorder potential) shall be accurately described by the model Hamiltonian. Examples for such “microscopic Hamiltonians” used below are the (continuous anisotropic) Ginzburg–Landau and London free energy functionals, the discrete Lawrence–Doniach model, and the continuum elastic free energy functional for the vortex lattice. We emphasize that the theory of Vortex Matter developed below applies equally well to conventional low-Tc and to unconventional copper-oxide high-Tc superconductors,the only requirement being that the order parameter describing the superfluid is a scalar quantity; given the dx2 −y 2 -symmetry of the order parameter [98–100] this condition is satisfied. In our terminology we use the term “microscopic” when referring to length scales ≥ , with the coherence length  denoting the smallest scale in our problem. The calculation of  itself is the subject of truly microscopic considerations involving the solution of a very complicated many-body problem; here, we will treat  as a phenomenological parameter. Also, we will require our starting Hamiltonians to describe a minimal model. In particular, we will introduce the simplest possi-

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ble type of disorder in the system, usually described by short-ranged Gaussian correlations. It turns out that the physics described by such a minimal model is already quite rich and accounts for a wealth of experimental observations in a consistent manner. Of course, starting out with these still very complex Hamiltonians we cannot hope to develop exact answers to our questions. We thus will have to resort to approximate methods in dealing with the vortex system, such as scaling estimates or perturbative analysis. The vortex system subject to quenched disorder, thermal and quantum fluctuations, and possibly driven by external forces is a complex object, and the structure of the answers we will derive below is not simple but involves many regimes where the behavior of the system is different. In what follows, we first discuss the basic framework for the discussion of Vortex Matter, the Ginzburg–Landau, London, Lawrence–Doniach and continuum elastic theories. We discuss single vortex properties in Sect. 12.3, the vortex lattice in Sect. 12.4, pancake and Josephson vortices in layered superconductors in Sect. 12.5, and the general scaling theory used in the description of anisotropic material in Sect. 12.6. Turning to the statistical physics of vortex systems (Sect. 12.7) we discuss the various fluctuation regimes, the vortex lattice melting and decoupling transitions, and the nature of the resulting vortex liquid phase.We then concentrate on the properties of vortex matter in the presence of quenched disorder,starting with an overview (Sect.12.8) on different types of pinning, e.g. strong versus weak pinning, pinning by uncorrelated/correlated disorder, and a general discussion of vortex creep. In Sect. 12.9 we present a detailed discussion of the theory of weak collective pinning and creep: we analyze singlevortex pinning and creep, its generalization to vortex bundles, the phenomenon of quantum creep at low temperatures and thermal depinning at high temperatures.Aspects of anisotropy and of the layered structure are dealt with as well.We proceed (in Sect. 12.10) with a discussion of strong pinning due to correlated disorder (e.g.,due to columnar tracks) and then concentrate on the effects of surface and geometrical barriers (Sect. 12.11). Turning back to thermodynamic aspects, but now in the presence of disorder,

we discuss the vortex-glass phases (see Sect. 12.12) and the possibility of a field-driven disorder-induced melting transition at low temperatures.We close with a discussion of Vortex Matter phase-diagrams. Unfortunately, this program leaves out a number of prominent topics: among them are the c-axis transport [101] and Josephson plasma physics [102, 103] in layered superconductors, the physics of moving vortex systems [104], or the structure of individual vortex lines in d-wave superconductors [105–107]. An introduction to the phenomenology of type II superconductors can be found in [96,108,109]; several aspects of Vortex Matter physics have been discussed in recent reviews [38,110] and the statistical physics of dirty elastic manifolds has been reviewed in [111,112].

12.2

Ginzburg–Landau and London Theories

The phenomenological description of superconductivity is based on the Ginzburg–Landau (GL) free energy functional [15]: The quantum liquid is described by a complex order parameter field ¦ (r) = |¦ (r)| exp[i' (r)].Furthermore,if the fluid is charged, as is the case in superconductors, the order parameter field ¦ (r) couples to the vector-potential field A(r) in a gauge invariant way. The free energy functionals describing the superconductor take the form  H·B G[¦ , H] = F[¦ , B] − d3 r , 4   ˇ F[¦ , B] = d3 r ˛|¦ |2 + |¦ |4 (12.1) 2    2 3   B2 2e 1   d  + A ¦  + + 2m  i dx

c 8

=1 and have to be completed by the Maxwell equations. The magnetic field B = ∇ ∧ A is driven by the applied external field H. The Ginzburg–Landau (GL) parameter ˛ = −|˛(0)|(1 − T/Tc) changes sign at the transition temperature Tc while ˇ assumes a constant value. We choose the charge unit e to be positive, e > 0. The parameters m , = 1, 2, 3, denote the effective masses along the main axes of the crystal. Here, we are mainly interested in isotropic and

12 Vortex Matter

503

in uniaxially anisotropic material described by effective masses mx = my = m and mz = M and a mass anisotropy ratio " 2 = m/M < 1 ;

(12.2)

while the isotropic case (m = m, = 1, 2, 3) mainly serves to introduce the basic ideas and concepts,many of the interesting strongly fluctuating superconductors, such as the copper-oxide or organic materials [113], are layered crystals and hence uniaxially anisotropic. We choose a coordinate system with the z-axis parallel to the axis of the material (the c-axis) and the x and y-axes in the isotropic ab-plane, see Fig. 12.4. An additional parameter appearing in the discussion of anisotropic superconductors is the direction of the external magnetic field H with respect to the crystal axes: the external field H is chosen to lie in the yzplane and to enclose an angle H with the c-axis (an angle #H = /2 − H with the ab-plane, see Fig. 12.4; below we consistently use  and # = /2 −  to denote angles measured from the c-axis and from the ab-planes). In the following, we will deal with the isotropic case first and discuss the generalization to the anisotropic situation whenever appropriate. The Ginzburg–Landau energy (12.1) describes a one-component superfluid and applies to conventional (isotropic) s-wave superconductors.More generally, the functional (12.1) with a scalar order parameter field ¦ (r) also describes unconventional superconductors with order parameters belonging to other symmetry classes, with the constraint that the symmetry representation is one-dimensional. E.g., in the high-Tc superconductors the l = 2 angular momentum channel splits under the tetragonal/orthorhombic crystal symmetry and the relevant irreducible representation (of dx2 −y 2 symmetry) is one-dimensional. The phenomenological description of Vortex Matter in type II superconductors developed below thus equally describes the situation in conventional low-Tc and in unconventional high-Tc (copper-oxide) materials. In particular, the parametric dependence of the phenomenological parameters entering the Ginzburg–Landau functional (12.1) on the underlying microscopic parameters remain the same, while numericals may be different. Examples

Fig. 12.4. Coordinate systems used in the description of anisotropic and layered superconductors. The axes x, y, and z are aligned with the crystal axes a, b, and c; c is the axis of uniaxial anisotropy and the ab-plane contains the superconducting layers. The magnetic field H lies in the yzplane and encloses an angle H =  /2 − #H with the c-axis. The induction B is in general not aligned with the magnetic field H and encloses an angle # with the ab-plane. The coordinate system x  , y , z is aligned with the induction B and has a common x axis with the crystal system, x = x

where the d-wave symmetry of the high-Tc ’s leads to strong (parametric) modifications of phenomenological results are rare; they appear in the context of low-energy quasi-particle excitations in the gap nodes, e.g., in connection with the field-dependence of the low-temperature specific heat [114,115] scal√ ing like cv ∝ H instead of the conventional behavior cv ∝ H, the low-temperature dependence of the superfluid density ns (0) − ns (T) ∝ T [116], or the non-linear Meissner effect (H) − (0) ∝ H [117], which appears difficult to confirm experimentally, though [118,119]. The homogeneous solutions of (12.1) are ¦0 = 0 (normal state) and |¦0 |2 = |˛|/ˇ (superconducting state). The energy gain in the superconducting state (˛ < 0) is due to the condensation energy

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Hc2 /8 = ˛ 2 /2ˇ,with Hc the thermodynamic critical field.In the inhomogeneous situation the variation of (12.1) with respect to the fields ¦ ∗ and A furnishes the equations determining the spatial variations of the order parameter ¦ = |¦ | exp(i' ) and the vector potential A (we concentrate on the isotropic situation here),     2i 2 |¦ |2 2  ∇+ ¦ = 0 , (12.3a) A ¦ + 1− ¥0 |¦0 |2 ¥0 |¦0 |2 ∇' , (12.3b) 2 ∇ ∧ (∇ ∧ A) + A = − |¦ |2 2 where ¥0 = hc/2e ≈ 2.068 10−7 Gcm2 denotes the flux unit. Equation (12.3b) is conveniently rewritten with the help of Amp`ere’s equation ∇ ∧ B = (4/c)j and provides us with an expression for the current density e ∗ 4e2 ∗ [¦ ∇¦ − ¦ ∇¦ ∗ ] − ¦ ¦A , (12.4) mi mc   2 2e 2 2e 2 |¦ | ∇' + |¦ | ∇ '˜ , =− A ≡− m ¥0 m

j=−

where we have defined the gauge-invariant phase gradient ∇ '˜ . The characteristic lengths entering the GL equations are the coherence length (T) and the penetration depth (T), 2 1 = 02 , (12.5a) 2m|˛(T)| 1 − T/Tc mc 2 1 2 (T) = = 02 , (12.5b) 16e2|¦0 (T)|2 1 − T/Tc  2(T) =

determining, respectively, the scale of variations in the order parameter ¦ and in the potential A (or magnetic induction B,current density j = (c/4)∇ ∧ B). In the following, we use the zero temperature values of the length scales  and  for the characterization of the superconductor. The thermodynamic critical field Hc is conveniently expressed through these two basic scales, ¥0 Hc = √ . (12.6) 2 2 The maximal dissipation free current density the superconductor can carry (the depairing current density) takes the value

cHc ; j0 = √ 3 6

(12.7)

for larger current densities the kinetic energy of Cooper pairs is larger than the energy gap  and the pairs unbind. The Ginzburg–Landau theory derives from an expansion in the small order parameter ¦ and hence its applicability is restricted to temperatures close to the superconducting transition temperature Tc . Alternatively, we can ignore spatial variations in the order parameter and construct a free energy from the kinetic energy of the currents j = (c/4)∇ ∧ B and the magnetic field energy B2 /8. The resulting London free energy functional takes the form 

 1 d3 r B2 + 2 (∇ ∧ B)2 FL [B] = (12.8) 8 and is valid within the whole temperature range 0 < T < Tc as long as the magnetic field does not lead to a significant suppression of the order parameter. For the strong type II superconductors with large Ginzburg–Landau parameter  = /  1 we are (mainly) interested in here, the London theory 12.8 provides a good approximation to the phenomenological description of superconductivity over a large portion of the H-T phase diagram. The penetration depth  appearing in (12.8) relates to the density ns of superconducting electrons via 2 (T) =

mc 2 4ns (T)e2

(12.9)

and the supercurrent density j can be written in the alternative forms   c ¥0 ∇' + A = −ens v s , j=− 42 2   2  ∇' + A , vs = 2m ¥0

(12.10)

with v s denoting the superfluid velocity (in general, the total current density j can pick up an additional normal component jn , in which case we will denote the supercurrent density (12.10) by js ). Variation of (12.8) with respect to B provides us with the London equation

12 Vortex Matter

[1 − 2 ∇ 2]B = 0 .

(12.11)

Note that we distinguish between the extrapolated zero-temperature parameter 0 of the Ginzburg– Landau theory, cf. (12.5b), and the zero-temperature BCS expression [120] for this length, (0) ≈ 1.41 0 .

(12.12)

A similar relation holds for the coherence length, (0) ≈ 1.36 0 . Below we will make use of the interpolation formula (T)2 ≈ (0)2 /(1 − T 2/Tc2 ); also, in going from a Ginzburg–Landau to a London description we will use the definition |¦0 |2 = (ns (0)/4)(1 − T 2 /Tc2 ) with the T = 0 superfluid density equal to the electronic density n, ns (0) = n. The above results apply to a clean material with a mean free path l > (0); for a dirty material (l < (0)) we have to replace  2 → () 2 and 2 → 2 / () with ≈ 1.17/,  = vF /2Tc l; going over to a dirty material increases the GL parameter  and the superconductor becomes more strongly type II. In unconventional superconductors described by a scalar order parameter the GL parameters involve additional angular averages over products involving the structure of the gap function (k) (of dx2 −y 2 symmetry in the copper-oxide high-Tc superconductors) and the Fermi velocity v F (k); these angular averages depend on the details of the Fermi surface and gap shapes and can modify the numerical factors relating the GL parameters, and hence also  and , to the underlying microscopic parameters [121,122]. In an anisotropic material the London free energy and the London equations take the form (in the vortex frame of reference (x  , y , z ), cf. Fig. 12.4)  1 FL [B] = d3 r  (12.13) 8 

2 × B + (∇  ∧ B) ·   · (∇  ∧ B) and

[B + ∇  ∧ [  · (∇  ∧ B)] = 0 ;

(12.14)

here, ˛ˇ = 2 ı˛ˇ +(c2 −2 )n˛ nˇ denotes the screening tensor with  and c = /" the in-plane and out-of-plane screening lengths and n˛ are the components of the unit vector pointing along the crystal axis (as seen from the vortex frame of reference; for a system with B  zˆ we have  11 =  22 = 2 = " 2  33 and 0 else).

12.3

505

Vortex Lines

The Ginzburg–Landau equations (12.3a)–(12.3b) admit topological solutions, these flux lines, or Abrikosov vortex solutions determine the rich phenomenological phase diagram of √ type II superconductors with  = / > 1/ 2. The cylindrical vortex- or flux-lines exhibit a non-trivial structure in the order parameter field ¦ and in the magnetic field B: the phase of the order parameter turns by 2 and drives a rotating current (note that with the present definition of the charge (e > 0) the phase field generating a vortex line rotates clockwise, ∇'v = −ˆz ∧ R/R2). Near the center of the phase-turn the currents become large and suppress the order parameter to zero, leading to a vortex core of size . At the same time, the rotating currents generate a magnetic field and thus bind a flux to the vortex extending a distance  away from the vortex core, where it saturates to the flux unit ¥0 = hc/2e ≈ 2.068 10−7 Gcm2 . To be more quantitative (see [96, 108, 109] for a standard derivation), let us integrate the cylindrical current flow (12.5), generated by a wave function ¦v = |¦v | exp(−in' ) with n phase turns, along a loop

R of radius R,    2e ¥ (R) 2 |¦v | 2 n − . dl · j = 2Rj' (R) = m ¥0

R

(12.15)  Here,¥ (R) = R dl·A is the flux penetrating the loop. The vortex solution (n = 1) then involves a circular current flow j' (R) = (2e|¦v |2 /mR)[1 − ¥ (R)/¥0]. The apparent divergence of this current density upon approaching the origin is cutoff through the suppression of the order parameter ¦v on the scale R ∼  as the current density j' becomes of the order of the depairing current density j0 ; a convenient expression for this suppression as obtained via a variational Ansatz [123,124] has the form |¦v |2 (R) ≈

R2 |¦0 |2 . R2 + 2 2

(12.16)

Going to large distances, the screening currents assemble a flux ¥ (R) of the order of the flux unit ¥0 on a scale R ∼ ; increasing R beyond  the drive

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∝ [1 − ¥ (R)/¥0] vanishes and the current density j' drops to zero rapidly: this is the phenomenon of transverse screening. The precise shape of the trapped magnetic field B(R) follows from the London equation (12.11) completed with a source term ¥0 zˆ ı 2(R) describing a vortex line placed at the origin, [1 − 2 ∇ 2 ]B = ¥0zˆ ı 2 (R) ;

(12.17)

integration provides us with the magnetic field (for R > )   ¥0zˆ K0 R/ (12.18) B(R) = 22 (K0 () is the MacDonald function, K0 (  1) ∼ √ − ln ; K0 (  1) ∼ e− / ). The kinetic energy of the screening currents j' gives rise to the line energy l of the vortex, 

l =

=

(12.19)

Going beyond Hc1 , vortices rush into the superconductor to form a triangular lattice (named after Abrikosov [3]) with the lattice constant a and the density n0 = a−2 0 determined by the induction B,  1/2 1/2 ¥0 ¥0 2 a = √ . (12.23) and a20 = B B 3

(12.20)

In a large external field H the vortex cores start overlapping when a0 ≈  and superconductivity disappears above the upper critical field Hc2 ,

  2   2i 2  2 |¦0|2 dR  ∇+ dR A ¦v  ≈ 2m  ¥0 m R 2



¥0 4



2 ln

 . 

The prefactor

 "0 =

¥0 4

or dxy -type [105],however,their impact on the vortex shape is usually small. These topological solutions of the Ginzburg– Landau equations are crucial in determining the (mean-field) H–T phase diagram of a type-II superconductor: At small magnetic fields the superconductor is strong enough to expel the magnetic field and this defines the Meissner–Ochsenfeld state. Increasing the field, the free energy of the vortex state drops below that of the Meissner state when the vortex line energy l matches up with the magnetic energy ¥0 H/4 and we find the lower critical field Hc1 (see Fig. 12.1),    4 l ¥0 (12.22) ln + 0.497 . = Hc 1 = ¥0 42 

2

defines the basic energy scale of Vortex Matter. The above estimate for the line energy involves only the kinetic energy of the currents and has to be completed by the magnetic field- and normal-core condensation energies; the magnetic energy contributes a term "0 /2 and a similar term C"0 with C a numerical small compared to the logarithm ln(/) has to be added for the core. The numerical solution of the Ginzburg–Landau equation (with   1) provides the result [125]      ¥0 2 l = (12.21) ln + 0.497 4  (note that the constant in (12.21) differs from usual values cited in the literature [96,108,109]). In unconventional d-wave superconductors the currents and magnetic fields appearing in the vortex core can induce other order parameter components, e.g. of the s

Hc 2 =

¥0 , 2 2

(12.24)

giving way to the normal-metallic phase. In an anisotropic material the line energy depends on the angle # between the vortex and the ab-plane of the material [126,127], see Fig. 12.4,    l (# ) = "0 "# ln , (12.25) "#  with the angle-dependent anisotropy parameter "#2 = " 2 cos2 # + sin2 #

(12.26)

and "0 = (¥0/4)2 ,  denoting the in-plane screening length. The lower critical field Hc1 is obtained by minimizing the Gibbs free energy G; at small values of B we have G ≈ B2 /8 + (B/¥0 ) l (# ) − (BH/4) cos(# − #H ), where # (#H ) denotes the angle enclosed by B (H) and the ab-plane. Minimizing

12 Vortex Matter

G with respect to By and Bz and setting B to zero we find the lower critical field [126] Hc1 (H ) ≈ =

¥0 "  ln 2 4 "H "#  Hcc1

" ln(/"# ) , "H ln()

(12.27)

where H denotes the angle enclosed between the magnetic field H and the c-axis, hence "2H = cos2 #H + " 2 sin2 #H . The angle # at Hc1 is given by tan2 # = " 2 tan2 #H and we can reexpress "# = " 2 (1 + " 2 tan2 #H )/(1 + " 4 tan2 #H ) in (12.27). Minimizing G with respect to # we obtain the relation between #H and # away from Hc1 (note that # < #H and the vortex lines tend to align with the superconducting planes), Hcc1 (1 − " 2 ) sin # cos # ln(/"# ) . H "# ln() (12.28) The expression for the upper critical field follows from the anisotropic vortex core with dimensions  and "#H  along x and y and takes the form sin(#H − # ) =

Hc2 (#H ) =

¥0 1 = Hcc2 . 2 2"#H  "#H

(12.29)

For large fields we have to replace the logarithm ln(/"# ) in (12.28) by its counterpart ln[˛Hc2 (# )/B)1/2] (cf. (12.54) below); for H  Hcc1 /"# the misalignment between the magnetic field H and the induction B becomes small. 12.3.1

Elasticity

Within a phenomenological or statistical mechanics description a vortex is usually modeled as an elastic string which, when compared with a point-like object, exhibits additional static and dynamic properties due to its one-dimensional extended nature. The elastic response of the vortex line is determined by its line tension: We consider a vortex in an isotropic material aligned with the z-axis of our coordinate system. The elastic energy of a deformed vortex is determined by the increase of the vortex length,

507



> = dz l [1 + (@z u)2 ]1/2 − 1 ,    "l @u 2 , (12.30) ≈ dz 2 @z

Fel =

where u(z) = (ux , uy ) denotes the displacement field of the vortex. In an isotropic material the long wavelength line tension "l is equal to the line energy l ,    (12.31) "l = "0 ln + 0.497 .  In general, however, the vortex line tension is dispersive; solving the London equation for a deformed vortex s(z) = [u(z), z],  ds [1 − 2 ∇ 2]B = ¥0 dz ı 3 (r − s(z)) (12.32) dz (conveniently done in Fourier space) and inserting the result back into the London functional (12.8)  rewritten in the form FL = (1/8) d3 rB(r)[1 − 2 ∇ 2]B(r) we obtain the energy of the distorted vortex line    "0 du(z) FL [s(z)] = ,1 (12.33) dzdz 2 dz   du(z ) × , 1 V [s(z) − s(z )] , dz with the interaction potential between vortex segments  d3 k ikr e V (k) , V (r) = 42 (2)3 2 2

e− k . (12.34) 1 + 2 k2 Expanding (12.34) to second order in the distortion u(z) one arrives at the dispersive line tension [173], 1/2  2 "l (kz ) ≈ "0 ln (12.35) 1 + 2 kz2 1 + "0 2 2 ln(1 + 2 kz2 )1/2 .  kz V (k) =

This expression is usually dominated by the first term with its weak logarithmic dispersion, "l (kz > 1/) ≈ "0 ln(1/kz ). The result (12.36) is easily understood in the following terms: the line energy (12.19) of a vortex mainly arises from the currents circulating

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the core, which usually are cut off at short (the core radius ∼ ) and long (the screening length ∼ ) distances. In a short wavelength distortion only currents on scales smaller than 1/kz are distorted and we have to replace the cutoff  by 1/kz .Similarly,the second term originates from distortions in the magnetic field: a short wavelength distortion (kz > 1/) involves only a fraction 1/kz22 of the total field energy "0 /2 and we obtain the strongly dispersive electromagnetic component of the line tension. In (12.19) we have ignored a further term accounting for the deformation of the vortex core which is dispersive on the scale . Below we will deal with various phenomena involving short wavelength distortions kz ∼ 1/L of the vortex line with  < L   (e.g., thermal fluctuations with L ∼ a0   at high fields or vortex pinning). In this case we will often drop the logarithmic correction ln(1/kz ) in the line tension and use the approximation "l (kz ∼ 1/L) ≈ "0 . In an anisotropic material the angular dependence of the vortex line energy produces two dif ferent angle-dependent elasticities "l (# ) and "l⊥ (# ) for in- and out-of-plane distortions: we start with the energy l (L, # ) = L l (# ) of a vortex segment of length L enclosing an angle # with the ab-plane (see (12.25)) and determine the increase in energy due to a symmetric distortion Lı on a distance 2L; ı l =

@ l 1 @ l @ l ıL + ı# + (ı# )2 . @L @# 2 @# 2

(12.36)

An out-of-plane tilt by an angle ı = ı# increases the length of the segment by ıL ≈ L(ı# )2 /2; similarly,an in-plane distortion gives rise to an additional length ıL ≈ L(ı)2/2 and produces a change in # determined by tan(# + ı# ) = L sin # /[(L cos # )2 + (Lı)2 ]1/2 , hence ı# = −(tan # )(ı)2/2. The outof-plane and in-plane elasticities are defined by the relations @ l L(ı# )2 = ıL + 2 @L 2 @ l  L(ı) = ıL + "l 2 @L

" 2 "0 "#3

and

1 @ 2 l (ı# )2, 2 @# 2 @ l ı# ; (12.37) @#

a few algebraic manipulations then produce the expressions for the out-of-plane and in-plane elasticities (we ignore logarithmic corrections)

"l ≈

" 2 "0 . "#

(12.38)

A second contribution to the tilt modulus arising from electromagnetic interactions is discussed in Sect. 12.4.1. 12.3.2

Dynamics

A number of forces attack the vortex line (here parallel to zˆ ) in a type-II superconductor in the dynamical situation and their correct addition (without omissions or double counting) is far from trivial. We distinguish between hydrodynamic forces arising at large distances and forces due to quasi-particle scattering in the vortex-core region; the most well known example for a vortex equation of motion is given through the compensation of the Lorentz force (from hydrodynamics) and the Bardeen–Stephen friction force (from dissipative scattering in the vortex core) [5], f L + f BS =

¥0 j ∧ zˆ − l v L = 0 , c

(12.39)

resulting in a vortex velocity v L transverse to the driving current density j; the viscous drag coefficient l as derived by Bardeen and Stephen reads l =

2

"l⊥

"l⊥ ≈

¥02 , 2c 2  n  2

(12.40)

where  n denotes the normal-state resistivity of the material. In order to discuss the limitations of this equation and the appearance of various other forces attacking the individual vortex line, we consider a simple phenomenological model providing us with some insight into the origin of the vortex equation of motion; an accurate derivation starting from microscopic theory has been given by Kopnin and Kravtsov [128], see also [129,130].We model the vortex core as a sharp normal cylinder with an electron density nc which in general may differ from the asymptotic density n. The average force density acting on the electronic system involves contributions from the Lorentz force j ∧ B/c, the electric field force enE, and the scattering term mnc v c f (B/Hc2 )/r (v c is the carrier velocity in the vortex core and r is the transport relaxation time),

12 Vortex Matter

j ∧ B/c + enE − mnc v c f (B/Hc2 )/r = 0 .

(12.41)

The normal metallic state is uniform and we have nc = n, v c = v, and f = 1; solving the equation of motion (12.41) for the current density (velocity) j = env we obtain the well known expressions for the longitudinal and transverse (Hall) conductivities in the normal state,  =  n /(1 + !c2 r2 ) and ⊥ =  n !c r /(1 + !c2 r2 ), with the normal state conductivity  n = e2 nr /m and the cyclotron frequency !c = eB/mc. On the other hand, the superconducting state is inhomogeneous and scattering is limited to the vortex cores; correspondingly we introduce the reduction factor f ≈ B/Hc2 . Furthermore, the electric field is now determined by the vortex velocity via E = B ∧ v L /c. Usually, the carrier density inside the core is assumed to be equal to the asymptotic electron density, nc = n, and we can write the transport current density in the form j = enc v c = ens vs , i.e., the current density flows through the system in a homogeneous manner (ns and v s denote the superfluid density and its velocity). The force equation (12.41) then relates the carrier velocity v c in the core to the vortex velocity v L , (12.42) v c = !0 r (v c − v L ) ∧ zˆ , with !0 ≈ eHc2 /mc (in a clean material, !0 ≈ 2 /2m 2 ≈ 2 /"F corresponds to the level splitting of the quasi-particles trapped in the vortex core [131]). Solving for the velocity v c we obtain vc =

!0 r !02 r2 zˆ ∧ v L + vL , 2 2 1 + !0 r 1 + !02 r2

(12.43)

and taking the cross product of (12.43) with en(¥0 /c)ˆz we obtain the force equation [132] ¥0 j ∧ zˆ = l v L + ˛l v L ∧ zˆ , c

(12.44)

with the response parameters !0 r !02 r2 l = n and ˛l = n , 2 2 1 + !0 r 1 + !02 r2 (12.45) consistent with the result derived from a microscopic analysis [128]. The two terms in (12.44) then push the vortex either in a direction transverse to

509

the driving current (dissipative motion governed by l ) or drag the vortices along the current direction (Hall motion governed by ˛l ). The resulting vortex motion produces longitudinal and transverse (Hall) flux-flow conductivities of the form FF = (Hc2 /B) n /(1 + !02 r2 ) and ⊥FF = !0 r FF . Usually,the parameter !0 r is small,!0 r  1,i.e., the mean free path l is not large,l  "F / ≈ (kF ); we then recover the dissipative equation of motion (12.39). The vortex exhibits a fast (vL ≈ vs /!0r ) motion transverse to the flow j = ens v s . The resulting large electric field E ≈ vL Hc2 /c in the core region provides the necessary drive to push the normal current density jn through the core such that the overall current flow is divergence free [123], ∇·(jn +j s ) = 0.The friction coefficient agrees with the expression derived by Bardeen and Stephen [5]; the flux-flow conductivity is given by the well known expression FF = (Hc2 /B) n (note that corrections to the Bardeen–Stephen result become important near the transition temperature and for high magnetic fields [133,134]). Expressions similar to (12.45) are obtained in the quasi-classical analysis of flux flow in d-wave superconductors [135]; as expected the results keep their parametric form but involve additional angular averages of the gap structure over the Fermi surface, at least for moderate disorder. Modifications due to the gap nodes appear in very clean material only (the so-called “superclean” limit with a large mean free path l > "F /, see below); these are, an unusual magnetic-field dependence of FF ∝ B−3/2 , universal limits (independent of r ) of the flux-flow conductivities for low temperatures and fields, and a residual (in the limit r → ∞) finite dissipation due to Landau damping on zero-frequency vortex modes. In an anisotropic material the viscous drag coefficients depend on the direction # of the flux line and on the direction of motion: the two effects of an angle-dependent vortex core size "#  2 and an angle-dependent electronic mass me /"#2 for the outof-plane (electronic) motion along y compete with each other and produce the dissipative response coefficients for the moving vortex in the form 

l = "# lc ,

l⊥ =

lc , "#

(12.46)

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G. Blatter and V. B. Geshkenbein

with lc = ¥02/2c 2  2 n the viscous drag coefficient for a vortex aligned with the c-axis of the crystal ( n the usual in-plane resistivity). A material with a parameter !0 r  1 is classified as belonging to the“superclean”limit; in this case the level broadening /r due to quasi-particle scattering is smaller than the level separation !0 . The vortex equation of motion then is determined by the compensation of the Lorentz force f L = ns v s ∧ zˆ and the transverse force f⊥ = −nv L ∧ zˆ . We split the transverse force into two terms involving the superfluid density ns and the normal component nn = n − ns , f⊥ = −ns v L ∧ zˆ − nn v L ∧ zˆ , and combine the term ∝ ns with the Lorentz force into the Magnus force f M = ns [v s − v L ] ∧ zˆ . The second term is identified as the Iordanskii force [491] fI = nn [v n − v L ] ∧ zˆ and we obtain the equation of motion in the superclean limit f M + f I = ns [v s − v L ] ∧ zˆ + nn [v n − v L ] ∧ zˆ = 0 (12.47) (in our derivation above we have assumed a zero velocity v n = 0 for the normal component). As a result, the vortex line is dragged along with the center of mass motion of the liquid, as is usually the case in bosonic superfluids [136]. In such superclean material the Hall field E = [B ∧ v L ]/c ⊥ j is expected to be large as the vortices move along the direction of the current; assuming the normal fluid is at rest, v n = 0, the transverse flux-flow conductivity takes the form ⊥FF = enc/B. The transverse force f ⊥ acting on a vortex is the result of a subtle addition/cancellation of various terms: when dragging the vortex line with a velocity v L through the fluid which is at rest in the laboratory frame of reference (vs = 0 = vn ) the transverse force −nv L ∧ˆz acting on the vortex combines components of the Magnus and Iordanskii forces and involves the total density n; on the other hand, assuming the normal component to be at rest in the lab frame (vn = 0), the Lorentz force acting on a vortex line at rest is given by ns v s ∧ zˆ and involves only the superfluid component ns . Furthermore, when quasi-particles in the vortex core are scattered they transfer momentum to the underlying lattice. On the one hand, this produces the dissipa-

tive force l v L , but at the same time the transverse force f sf = n[1/(1 + !02 r2 )]v L ∧ zˆ is generated, the so-called spectral-flow force [137]. Hence the scattering of quasi-particles in the vortex core not only produces dissipation but is also responsible for the cancelation of the Magnus force; the vanishing of the Hall component ∝ ˛l in the equation of motion (12.44) in dirty and clean (but not superclean) superconductors then is the consequence of the mutual cancelation of the hydrodynamic Magnus force and the spectral-flow force due to quasi-particle scattering. Therefore, the conventional situation in typeII superconductors involves a large dissipative and only a small Hall component in the vortex motion, resulting in a small Hall angle 

tan Hall =

vL ˛l = , l vL⊥

(12.48)



where vL and vL⊥ denote the vortex velocity components parallel and perpendicular to the current density j. The Hall component is substantially modified when particle-hole symmetry is broken [138, 139]. Then the quasi-particle density nc in the core is different from the asymptotic density n. Breaking the particle-hole symmetry through a non-constant density of states with @E N(E) = 0 produces a (small) vortex charge [140–142] of order ekF (TF /)2 (TF denotes the Thomas Fermi screening length) and a correction to the Hall coefficient ˛l which might take either sign, a circumstance which may be related to a peculiar sign change in the Hall effect observed in some conventional and most high temperature superconductors [143–145]. The correction to the Hall coefficient ˛l is easily derived via the above Drude-like analysis based on the force equation (12.41) [146]: in this case the transport current density j has to be evaluated in the vortex co-moving system, ˜j = enc v˜ c where ˜j = j − env L and v˜ c = v c − v L . The additional term ∝ ın = nc − n in the expression for the transport current density j = enc v c − eınvL

(12.49)

then compensates for the asymptotic density n in the electric field force, see (12.41), replacing it by the

12 Vortex Matter

density nc in the core. As a result, the force density equation (12.41) involves only the core density nc and the velocity relation (12.43) remains unchanged. Substituting (12.43) into (12.49) and taking again the cross product with (¥0 /c)ˆz (note the substitution n → nc ) we recover the force equation (12.44) with two modifications: i) the relevant density entering the dynamical coefficients l and ˛l now is the density nc in the core, and ii) the correction term ∝ ın = nc − n in the expression for the transport current density j adds to the Hall coefficient, ˛l = nc [!02 r2 /(1 + !02 r2 ) − ın/nc ]. Hence, depending on the sign of the density modulation ın in the vortex core, the Hall coefficient ˛l may reverse its sign. Note that it is the unscreened density modulation ın/n ∝ 2 / F2 which enters the expression for the Hall coefficient [146]. Let us then have a closer look at the origin of the density modulation ın: Starting from microscopic (weak coupling) BCS theory, ın is related to the particle-hole asymmetry parameter @E N(E) via ın ∼ −

dN(E) 2 !D @ ln Tc  ln ∼ −N2 dE Tc @

(12.50)

and is of order n 2 / F2 (the Debye frequency !D provides the high energy cutoff, N(E) denotes the normal electron density of states). Here, we have used the relation @E N ln(!D /Tc ) = N@ ln Tc deriving from the BCS expression Tc ∼ !D exp(−1/NV ) in order to express the result through the phenomenological parameter @ Tc . Alternatively, the density modulation ın can be found using phenomenological considerations via the appropriate derivative of the free energy density, ın = −@ f |∞ 0 with f = −˛0 (Tc − T)|¦ |2 + . . . , hence ın = −(Hc2 /4)@ Tc = −N2 @ ln Tc , and we recover the result (12.50). The phenomenological parameter @ Tc also enters the derivation of the Hall conductivity via the time-dependent Ginzburg–Landau (TDGL) approach [138, 139]: The non-dissipative dynamical term can be derived from the free energy functional by exploiting the dependence of Tc ( + e¥ ) on the local electro-chemical potential

+ e¥ ; we write the free energy density in the form f = −˛0 (Tc −T +e¥ @ Tc )|¦ |2 +. . . and going over to the gauge invariant combination e¥ → e¥ − i@t /2

511

the variation ıf /ı¦ ∗ produces the dynamical term i  @t ¦ with  = −(˛0 /2)@ Tc = ın/n [147]. The first-order time derivative @t ¦ then involves both a real and an imaginary part =  + i  governing the dissipative and the Hall part of the vortex motion. The corresponding Hall conductivity as derived via the TDGL approach then coincides with the Drude type derivation discussed above combined with the weak coupling BCS result (12.50) for the density modulation ın in the vortex core. A word of caution is appropriate regarding the mutual interrelation between parameters: on a phenomenological level we deal with the dependence of Tc on the chemical potential ; the quantity @ Tc enters the expression for the vortex charge, the Hall dynamics of the vortex, and is related to the particlehole symmetry breaking parameter @E N, at least within weak coupling BCS theory. The question then arises, whether the knowledge of the doping dependence of Tc or of the bandstructure allows for the determination of the vortex charge and of the vortex Hall dynamics — the answer is “no”, in general, as the critical temperature Tc may depend separately on the Fermi energy "F ≈ and on the particle density n. In this situation the gauge invariant extension Tc ("F − i@t /2, n) involves only the dependence via the Fermi energy "F and the vortex charge and Hall conductivity may involve different parameters [148]. Hence the full derivative dTc /d determining the vortex charge is different from the partial derivative @Tc /@ determining the Hall coefficient  and the partial derivative @Tc /@n describing the doping dependence of Tc . An attempt to explain the doping dependence of the sign change in the high-Tc cuprates through the behavior of the parameters @E N (as measured in photoemission and tunneling experiments) or @n Tc (doping dependence) predicts a behavior different from what is found in the experiments; the latter show a change of sign in the underdoped region and no such Hall anomaly in overdoped samples (these are experiments on Y-,Bi-,La-, and Tl-based samples [144,145]). Deriving ın from @E N, the hole-like Fermi-surface geometry predicts no sign change at all. On the other hand, extracting ın from the doping dependence of Tc , a Hall anomaly is predicted in the

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overdoped regime, while none should occur in underdoped samples — this is exactly the opposite from what is found in the experiment [145]. However, the discussion on the interrelation of various parameters given above warns us that in general we cannot hope to extract ın directly from @E N or @n Tc . A consistent explanation of the Hall anomaly has been attempted in [148] on the basis of a microscopic mechanism involving preformed pairs, with negatively charged Cooper pairs; this theory can explain the observed change in sign of the Hall conductivity in the underdoped regime [148]. On the other hand, the overdoped regime is expected to behave more regularly, in which case the hole-like Fermi surface @E N correctly predicts the absence of a Hall anomaly. 12.3.3

Green’s Functions

Completing the free energy (12.30) with a force term −f (z) · u(z), a variation with respect to the displacement field u(z) provides us with the differential equation "l @z2 u = −f and the static Green’s function G, after Fourier transformation, G(q) = 1/"l q2 . As discussed above, the vortex dynamical response in type II superconductors is usually dissipative and the full dynamical Green’s function takes the form G(q, !) = 12.3.4

1 . "l q2 − i!l

(12.51)

Free Energy and Action

When dealing with classical or quantum statistical aspects of individual vortices we need appropriate expressions for the free energy and for the imaginary time Euclidean action describing the vortex line. While the former has been derived above, see (12.30), the latter follows via completion with an appropriate dynamical term. The vortex dynamics in type II superconductors is usually dominated by the dissipative term producing a non-local in time contribution to the (Euclidean) action [149],     "l @u(z, ) 2 SE = ddz (12.52) 2 @z    2  l  u(z, ) − u(z,  ) d , + 4  − 

where the displacement field u now additionally depends on the imaginary time argument . The generalization to the anisotropic case is straightforward [150].

12.4

Vortex Lattice

Increasing the external field beyond Hc1 , see (12.22) and (12.27), the formation of vortices turns energetically favorable and we enter the Shubnikov phase [3,4,96,108,109]. The pairwise interaction between parallel straight flux lines is repulsive, V (R > ) = 2"0 K0 (R/), with the MacDonald function K0 describing a 2D Coulomb interaction screened on a scale . The Ginzburg–Landau free energy is minimized by√a triangular lattice with a lattice constant a = (2/ 3)1/2(¥0 /B)1/2 and (planar) lattice coordinates ( = (m, n))  √ 2m + n 3 R = na , a , 2 2   2n − m m , K  = 2 √ , 3a a

(12.53)

in real and reciprocal space (the Abrikosov parameter takes the value ˇA ≡ |¦ |4 /|¦ |2 2 = 1.1595953). The constitutive relation B(H) is determined by the free energy density (the term fs0 = −Hc2 /4 accounts for the condensation energy)

f (B) ≈

⎧     1/2 B ⎪ ⎪ l + 6"0 e−a/ , ⎪ fs0 + ⎪ ⎪ ¥0 2a0 ⎪ ⎪ ⎪ ⎨ B B2 ˛Hc2 fs0 + + , "0 ln 8 2¥ B 0 ⎪ ⎪ ⎪ 2 2 ⎪ 1 B (Hc2 − B) ⎪ ⎪ ⎪ − , ⎪ ⎩ 8 8 1 + (2 2 − 1)ˇA

Hc 1 ≤ H , H c 1  H  Hc 2 , H ≤ Hc 2 ,

(12.54) via the derivative H = 4@B f and the relation B = H + 4M, 2¥0 1 B(H → Hc1 ) = √ , 2 (H − H )]}2 2 {ln[3¥ /4 3 0 c1 (12.55)

12 Vortex Matter

Hc1 ln(˛Hc2 /H) , (12.56) 4 2 ln  1 Hc 2 − B = M (H → Hc2 ) = − 4 1 + (2 2 − 1)ˇA

M (Hc1  H  Hc2 ) = −

=−

1 Hc 2 − H . 4 (2 2 − 1)ˇA

(12.57)

The numerical ˛ is of order unity, ˛ ≈ 0.35 as calculated theoretically in [151] and ˛ ≈ 1.2-1.5 when comparing to data [152, 153]. Thermal fluctuations and quenched disorder modify these results (see Sect. 12.7.8); in anisotropic superconductors the Hc1 transition may be driven first-order at low temperatures. In anisotropic superconductors the free energy density picks up an additional dependence on the angle # , B2 B"# ˛Hc2 + . (12.58) "0 ln 8 2¥0 "# B As a result,the angles # and #H of B and H differ from one another (see (12.28)) and the sample is exposed to a torque T = M ∧ H, f (# , B) ≈

HHcc1 (1 − " 2 ) sin # cos # ln(˛Hcc2 (# )/B) ; 4 "# 2 ln(/) (12.59) measuring the maximum torque at ⎧ ˛Hcc2 √ ⎪ ⎪ , " 1, " , 1  " ln ⎨ "B  #max = c 1/2 c ˛Hc2 ˛Hc2 ⎪ ⎪ 1, , " ln ⎩ " ln "B "B T=

(12.60) allows for the determination of the anisotropy parameter " [154]. Fermi surface effects in the underlying crystal may distort the hexagonal vortex lattice, e.g., the strong Fermi surface anisotropy in the Borocarbides produces a square vortex lattice [155, 156] and a hexagonal-to-square vortex lattice transition with decreasing magnetic field [157]; these effects can be understood when non-local corrections are included in the London model [158]. Similarly, the nodes in the d-wave order parameter of the copper-oxides favor a square vortex lattice [159, 160]: at low magnetic fields the repulsive interaction between vortices dominates and the vortex lattice exhibits the

513

usual hexagonal symmetry, as observed in numerous Bitter decoration experiments [161] and in small angle neutron scattering experiments [162] (note that both Bitter decoration [163] and small angle neutron scattering [164] experiments measure the magnetic field distribution and thus are limited to low magnetic fields B < ¥0 /2 ). On the other hand, scanning tunneling experiments on YBCO carried out at large fields (6 Tesla; this technique allows to measure the vortex cores and hence is not limited to low fields) reveal an oblique lattice [165] which is compatible with a square lattice [166] distorted due to the (small) ab-anisotropy within the superconducting planes [167, 168]. Note, that twin boundary pinning is a competing mechanism also favoring a square lattice [169,170]. 12.4.1

Elasticity

In general, the vortex lattice is not in its equilibrium configuration but in some distorted state characterized by the two-component displacement field u (z) or its Fourier transform u(k) [r  ≡ (R , z)],   2 u(k) = a0 dz e−ikr  u (z),  u (z) =

 3

d k ikr  e u(k), (2)3

(12.61)

BZ

where the integration runs over the two-dimensional Brillouin zone (BZ) of the vortex lattice and the integration along kz is cut off at the inverse core radius, |kz | < / (a convenient approximation is to replace the hexagonal Brillouin √ zone by a circular geometry with radius KBZ = 4/a0). The London free energy of a distorted configuration with vortex positions s = r  + u takes the form [171,173] (cf. (12.34); here, i, j ∈ {x, y, z}; see [10,172] for an analysis describing the situation close to Hc2 )  "0  dsi, dsj, Vij (s − s ) ; (12.62) FL [{s  }] = 2 , in the isotropic situation the interaction potential Vij (r) = ıij V (r) between vortex segments is given by (12.34). Note that the sum over ,  in (12.62) should

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G. Blatter and V. B. Geshkenbein

be understood as a sum over segments; for an individual vortex line the integral goes over all pairs of line segments ds and ds , cf. (12.34). Expanding (12.62) to second order in the displacement field u provides us with the elastic energy (indices appearing twice are summed over)   1 d3 k

F[u] = u˛ (k)¥˛ˇ (k)uˇ (−k) . (12.63) 3 2 (2) BZ

The elastic matrix ¥˛ˇ (k) of the vortex lattice is characterized by the symmetries ∗ ¥˛ˇ (k) = ¥ˇ˛ (k) = ¥˛ˇ (k) ∗ ∗ = ¥˛ˇ (−k) = ¥˛ˇ (k + K  ) (12.64)

and relates to the interaction potential V (k) via ¥˛ˇ (k) =

 B2 

f˛ˇ (k + K  ) − f˛ˇ (K  ) , (12.65) 4 

f˛ˇ (k) = (k˛ kˇ + ı˛ˇ kz2 )V (k) .

(12.66)

Within the nonlocal continuum limit the elastic matrix ¥˛ˇ (k) takes the form ¥˛ˇ (k) = [c11 (k) − c66] K˛ Kˇ

 + ı˛ˇ c66 K 2 + c44(k) kz2 , (12.67) with c11 (k) and c44(k) denoting the dispersive compression and tilt moduli and c66 is the non-dispersive shear modulus. Within the continuum isotropic approximation we restrict the sum over reciprocal lattice vectors in (12.65) to the  = 0 term and obtain for the compression and tilt moduli c11(k) ≈ c44 (k) ≈

B2 1 , 4 1 + 2 k2

(12.68)

while the shear modulus c66 ≈ 0 vanishes (within this rough approximation we cannot decide whether to write c11 or c11 − c66 in (12.68); the accurate analysis [13,173] shows that c11−c66 ≈ (B2 /4)/(1+2 k2 )). The same result can be derived in a hydrodynamic description and hence the expressions (12.68) describe well the vortex-liquid phase [174].At the same

time the relations (12.68) provide convenient expressions for the compression and tilt moduli in the vortex solid phase, while the calculation of the lattice shear modulus requires summation of all the terms  = 0 in (12.65), a somewhat non-trivial task: we express the elastic matrix ¥˛ˇ through the real-space sum  1 

¥˛ˇ (K) = 2 1 − exp(iK · R ) @˛ @ˇ V (R ) , a0  (12.69) with the interaction potential V (R) = 2"0K0 (R/) acting between a pair of (straight) vortex lines. The long wavelength limit of the transverse projection defines the shear modulus, c66 = (ı˛ˇ − K˛ Kˇ /K 2 )¥˛ˇ (K)/K 2 |K→0 . Expanding the exponential in (12.69) to order K 2 one arrives at  (K · R )2  (K · R )2 V  (R ) c66 = K 2 R2 R 2a20 K 2  =0  (K ∧ R )2  + V (R ) . (12.70) K 2 R2 Using the symmetry of the triangular lattice, we perform a partial resummation over stars {R }61 and thereby eliminate the dependence on the direction K/K,  1 

c66 = 3R V (R ) + R2 V  (R ) . (12.71) 2 16a0  =0 In the dilute limit B < Hc1 / ln  the vortex separation increases beyond the screening length; the sum in (12.71) is dominated by the six nearest neighbors and the shear modulus is exponentially small [18,175],    a  3   1/2 "0  c66 = . (12.72) exp − 2 6 a 2  The dense limit with a0   is puzzling: at short distances we can approximate V (R) ≈ −2"0 ln(R/); with 3RV (R) = −6"0 and R2V (R) = 2"0 we find that each “shell” in (12.71) contributes the negative value −4"0 . On the other hand, large distances R >  contribute with a positive sign and eventually overcompensate the negative terms from small distances R < . Technically, the final result is obtained by replacing the sum by an integral while subtracting the contribution from the origin; the integral vanishes

12 Vortex Matter

(after partial integration) and the remaining term provides the desired answer c66 ≈

¥0 B "0 = 2 . 2 (8) 4a0

(12.73)

The shear modulus of the dense vortex lattice appears as a subtle cancelation of contributions from small and large distances. Similar results are obtained when calculating the “cage potential” Vcage (u) =  =0 V (R − u) acting on the vortex at the origin and produced by all other vortices in its environment: choosing an unscreened interaction V (R) = −2"0 ln(R/L) (with L denoting the system size) the cage potential vanishes, while the screened potential Vs (R) = 2"0 K0 (R/) results in a finite cage potential Vcage(u) ≈ "0 (u/a0 )2 . A similar addition of terms  = 0 in (12.65) produces (non-isotropic) corrections to the tilt and compression moduli important at the BZ boundary and reflecting the hexagonal symmetry of the vortex lattice (this“geometric” dispersion depends only on the product a0 k and thus is weaker than the “essential” dispersion originating from the  = 0 term and involving the parameter k). Furthermore, close to the Brillouin zone boundary the tilt modulus c44(k) crosses over to the single vortex result [173] "0 1 ln . 2 kz a0 (12.74) A convenient interpolation formula is given by 0 c 0 c44 (k) = c44 (k) + c44 (k) with c44 (k) given by (12.68) and  2 "0 c c44 (kz ) ≈ 2 ln 2 2a0 1 + 2 (KBZ + kz2 )  1 + 2 2 ln(1 + 2 kz2) . (12.75) kz  c44(K > KBZ /[ln(a0 /)]1/2, kz ) ≈

The kz → 0 limit of this expression takes the form c c44 (0) = −BM = B(H − B)/4 (cf. (12.56)) and we obtain the correct expression c44 (0) = BH/4 for the uniform tilt modulus as determined from thermodynamic considerations [7]. At small field values B < Hc1 / ln  the compression modulus becomes exponentially small [18,175], c11 = 3c66 ,

(12.76)

515

whereas the tilt modulus goes over into the single vortex expression "l /a20 with "l given by (12.36). In the opposite situation of large inductions the renormalization of the length scales due to the suppression of the order parameter should be taken into account. The results for the compression and tilt moduli (12.68) are obtained through the substitution  →  = /(1 − b)1/2 with the reduced field b = B/Hc2 (T) [171]; the suppression of the shear modulus (12.73) near to the upper critical field Hc2 is more pronounced and involves the factor (1 − b)2 , c66 ≈ "0 (1 − b)2 /4a20 . The dispersive behavior of the compression and tilt moduli c11 (k) and c44 (k) leads to a considerable softening of√the vortex lattice. Near the BZ boundary KBZ ≈ 4/a0 a suppression factor (KBZ )2 ≈ B ln  /Hc1 is obtained with respect to the value at k = 0 describing a uniform distortion. On the other hand, the shear modulus is essentially free of dispersion and always small, in fact, c66 ≈ c11 (KBZ )/4. The physical origin of the dispersion in c11 and in c44 is found in the long-range interaction potential (12.34): For fields B > Hc1 / ln  the nearest neighbor distance a0 drops below the screening length  and the vortex-vortex interaction extends beyond the nearest neighbors. The local limit described by the functional  1 F[u] = d3 r [c11(0) (∇ · u)2 + c66 (∇⊥ · u)2 2 + c44 (0) (@z u)2 ] , (12.77) with ∇⊥ = (@y , −@x ) and c11 (0) ≈ c44(0) ≈ B2 /4, should therefore be used with caution.Since c11 (0)  c66 the functional (12.77) essentially describes an incompressible solid. The generalization of the theory of elasticity to anisotropic superconductors is somewhat cumbersome; we summarize the main results below and add a few remarks on a simplified derivation via the scaling approach [176,177] in the end. As the (internal) magnetic field is tilted away from the c-axis the (equilateral) triangular lattice defining the equilibrium configuration is deformed and the new basis vectors (in the vortex frame of reference, see Fig. 12.4; we remind that  = (m, n)) become [178] 

  R = a n 3/4"# , (2m + n) "# /4 , (12.78)

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in space, K  a /2 = [(2n − m) √ reciprocal √  "# /3, m/ "# , with the angle-dependent anisotropy parameter (12.26). Note that the equilibrium lattice configuration is characterized by a short lattice vector R(1,0) pointing along the y -axis and the planar rotational degeneracy present for H  c is removed as the field is tilted away from the c-axis. The configuration (12.78) describes the equilibrium state of the vortex lattice at large magnetic fields H > Hcc1 /"# ; for small fields H < Hcc1 /"# the equilibrium state for the tilted lattice is realized by the so called chain state [179–181] where the vortex lines rearrange themselves into parallel running chains defining planes of high flux density oriented parallel to the yz-plane containing the c-axis and the magnetic field vector. This new state is a consequence of the attractive vortex-vortex interaction experienced by the tilted vortices in anisotropic superconductors [179,182,183] and has been observed via Bitter decoration experiments on YBCO surfaces [184]. The elastic properties of the vortex lattice derive from the elastic matrix ¥˛ˇ (k), see (12.63). Within London theory we have to replace the expression for the potential (12.34) and (12.66) by (i, j ∈ {x = x  , y , z }; ˛, ˇ ∈ {x = x , y }) 2

2

,⊥



⊥ + c44(k)(kz  ux )2 + c44 (k)(kz  uy  )2 ⊥ + 2c14 (k)(ky  kz  u2y  ) ,⊥

(12.80)

with K = (kx , ky  ), K ⊥ = k ∧ cˆ and cˆ the unit vector along the c-axis (as seen from the vortex frame of reference; we remind the definitions c = ", the coherence length along the c-axis, and c = /", the screening length for currents flowing parallel to the c-axis). In an anisotropic material the number of elastic moduli increases dramatically: the linear elastic energy within the nonlocal continuum approximation takes the form

(12.81) 

+ 2c14 (k)(kx ux kz  uy  ) + c66(ky  ux )2   ⊥ + c66 (kx uy  )2 + 2c64(ky  kz  u2x ) . In addition, the vortex lattice in the anisotropic material exhibits rotational modes [185]. The shear deformations involve the soft in-plane and hard out-ofplane moduli [185] 

c66 (# ) = c66 "#3 ,

⊥ c66 (# ) =

c66 , (12.82) "#

with c66 = ¥0 B/(8)2 the isotropic result evaluated with the planar screening parameter . The tilt  ⊥ moduli c44 and c44 for the (soft) in-plane and for the (hard) out-of-plane tilt modes can be written in a 0 c 0 form c44 (k) = c44 (k) + c44 (k), with c44 (k) the strongly dispersive contribution arising from the term  = 0 c in (12.65) and c44 (k) denoting the correction due to the remaining terms in the sum [12,71,186,187] (note that  = /2 − # , "2 = " 2 sin2 # + cos2 # ), B2 1 , 2 4 1 + c K 2 + 2 kz2 1 + (" c k)2 0, 0,⊥ (k) = c (k) . c44 1 + 2 k2 44

2 2

+ kz2 V˛ˇ (k) − 2kz  kˇ Vz  ˛ (k) ,

d3 k  c (k)(kx ux )2 (2)3 11

⊥ + c11 (k)(ky  uy  )2 + 2c11 (k)(kx ux ky  uy  )

e−( K +c kz ) Vij (k) = (12.79) 1 + 2 k2   (c2 − 2 ) K⊥i K⊥j , × ıij − 1 + 2 k2 + (c2 − 2 )K 2 f˛ˇ (k) = k˛ kˇ Vz  z  (k)



1 F= 2

0,

c44 (k) =

(12.83)

c The correction term c44 (k) becomes important at large k-vectors and describes the crossover from the lattice modulus to the (dispersive) single vortex line tension as the interaction between neighboring vortices becomes small, for B  zˆ ,

  "0  2 c2 1 2 kz2 " ln + 2 2 ln 1+ , 2 2 2 2 2 2 2 1+c KBZ + kz  kz 1+ KBZ 2a0 (12.84) where c = /". In layered material, the first term in (12.84) is due to the Josephson coupling between the superconducting planes; it is only weakly dispersive, depends strongly on the anisotropy ",and vanishes in the limit " → 0. The second term is due to the electromagnetic interaction and is strongly dispersive c c44 (kz ) ≈

12 Vortex Matter

but independent of the anisotropy parameter " and hence always finite.A particularly interesting limit is the single vortex line tension which becomes strongly dispersive as a consequence of the anisotropy, 1/2  c2 2 "l (kz ) ≈ " "0 ln 1 + 2 kz2 1 + "0 2 2 ln(1 + kz2 2 )1/2 . (12.85)  kz For large wave-vectors kz > 1/" the line tension is small, "l ∼ " 2 "0 ; in the intermediate regime 1/ < kz < 1/" the line tension rapidly increases, "l ∼ "0 /kz22 , and reaches the long-wavelength limit "l ∼ "0 for kz < 1/. This separation into a small line tension due to the Josephson coupling and a strongly dispersive component due to the electromagnetic coupling is specific to the anisotropic situation. In the isotropic situation there is no suppression of the first term and the dispersion is always weak. Note that the expression (12.85) is valid in the elastic regime; for large displacements ukz > 1 the logarithm in the second term of (12.85) should be cut on 2/u rather than kz [65]. It is interesting to note that the ratio of the bulk tilt 0,⊥ 0, moduli c44 /c44 ≈ "2 /" 2 shows a different angular  dependence as compared to the ratio "l⊥ /"l ≈ 1/"#2 of the single vortex tilt moduli, see (12.38); this difference is due to the admixing of the bulk modulus 0,⊥ to the tilt mode c44 .  ,⊥ ⊥ The compression moduli c11, c11 , and c11 take the form [12,186,187] 1 + ("# c k) 0, c44 (k) 1 + 2 k2 (12.86) ,⊥ ⊥ and the mixed compression-tilt moduli c14 and c14 read [187] "#2 "2 − " 2 c2 k2  0, ⊥ c14 (k) = c14 (k) = c44 (k) . (12.87) 1 + 2 k2 

In addition, corrections to the above moduli due to the admixing of the shear mode to the tilt and compression modes can only be obtained by considering terms  = 0 in the sum over reciprocal lattice vectors in (12.65). The complete analysis of these terms has not yet been done. Using the anisotropic scaling approach (cf. Sect. 12.6 and [176]) the above results are easily rederived provided that the condition k˜ > 1, with k˜ the rescaled k-vector modulus in the isotropized system, is satisfied [177]. The elastic theory for anisotropic superconductors for angles # away from the main axes is quite cumbersome since the elastic energy is not any more diagonal in the vortex frame of reference; a more convenient approach is to use the anisotropic scaling theory (see [176] and the summary in Sect. 12.6) and study the problem within the isotropized system instead of carrying out the analysis in the anisotropic vortex frame of reference where the elastic energy is not diagonal. 12.4.2

,⊥



The mixed shear-tilt modulus c64 arises from terms  = 0 in the sum (12.65) and has been obtained using anisotropic scaling theory [177], "#2 "2 − " 2   c64(k) = c66 . (12.88) "#2

Dynamics

The dynamical response of the vortex lattice follows trivially from that of the individual vortex lines, see Sect. 12.3.2: In type II superconductors the vortex dynamics is usually dissipative and the viscous drag coefficient  for the lattice follows from the Bardeen Stephen expression (12.40) by going over to the appropriate density, ≡

2

⊥ c11(k) = c11 (k) = c11 (k) =

517

12.4.3

¥0 B l = . 2 a0 2c 2  2 n

(12.89)

Green’s Functions

The Green’s function for the vortex lattice is derived from the free energy (12.63) completed with a force term −f (k) · u(−k). The variation with respect to the displacement field u(−k) provides us with the differential equation [¥˛ˇ (k) − i!ı˛ˇ ] u˛ (k) = fˇ (k) ,

(12.90)

where we have already completed the expression with the dissipative dynamical term −i!u(k). The Green’s function for the vortex lattice takes the form G˛ˇ (k, !) = [¥ (k) − i!1]−1 ˛ˇ .

(12.91)

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Expressing the elastic matrix ¥ through the elastic moduli with the help of (12.67) the Green’s function takes the form  P˛ˇ (K) G˛ˇ (k, !) = c11(k)K 2 + c44 (k)kz2 − i! ⊥ (K) P˛ˇ (12.92) + 2 c66K + c44 (k)kz2 − i! with the projection operators K˛ Kˇ K˛ Kˇ  ⊥ P˛ˇ (K) = , P˛ˇ (K) = ı˛ˇ − . K2 K2 (12.93) 12.4.4

Free Energy and Action

The statistical physics of Vortex Matter including thermal and quantum fluctuations is determined by the appropriate partition function requiring knowledge of the free energy and the imaginary time Euclidean action of the vortex lattice. The fully nonlocal expression for the free energy takes the form  1 d3 k

F= (12.94) c11(k)|K · u(k)|2 2 (2)3  + c66|K ⊥ · u(k)|2 + c44 (k)|kz u(k)|2 , with K ⊥ = (ky , −kx ), while completion with a nonlocal in time dynamical term describing the viscous motion of the vortex lattice provides us with the Euclidean action (in Fourier space; f denotes the free energy density in (12.95))   d3 k d! 1 SE = (12.95) 3 2 (2) 2

 × f u(k, !) +  |!||u(k, !)|2 .

12.5

Layered Materials

The most extreme limit of an anisotropic superconductor is a layered material made from a stack of uncoupled individual superconducting planes; the phenomenology of such a material exhibits interesting properties due to the electromagnetic coupling between the layers. A finite Josephson coupling between the layers establishes a bulk superconducting

response and increasing the Josephson coupling one may scan all values of the anisotropy parameter ", thus covering the physics of layered and continuous anisotropic up to isotropic materials. Many superconductors fall into the class of layered materials: the oxide superconductors are layered compounds with building blocks made from conducting (metallic) Cu-O planes separated by buffer layers which serve as charge reservoirs. The transport properties are roughly uniaxial, with a large anisotropy between the c-axis and the ab-planes due to the layered structure and essentially isotropic behavior within the Cu-O planes. The layered superconductors of the BEDT-TTF family [113] are other candidates requiring a discrete description and the most extreme members of this class of materials are the artificially grown multi-layer structures, among them the (Y/Pr)Ba2 Cu3O7−y [188] and Mox Ge1−x /Ge [189] systems.While for not too large anisotropy a description in terms of a continuous anisotropic Ginzburg– Landau or London theory is applicable, for very large anisotropy the discreteness of the structure becomes relevant and a description in terms of a set of weakly coupled superconducting layers, the basic starting point of the discrete Lawrence–Doniach model [58], is more appropriate. The criterion defining the crossover from a continuous anisotropic to a discrete layered description depends on the specific physical question at hand. Examples for such criteria are known from the context of critical fluctuations or the behavior of the parallel upper critical field: in both cases the 2D behavior is established when the coherence length c (T) along the c-axis drops below the layer separation d with decreasing temperature. The precise criterion involves the dimensionless ratio cr = 2c2(0)/d 2: for a large coherence length c (0), i.e., cr  1, the continuous anisotropic description is always appropriate. On the other hand, for small cr  1 a crossover will take place at a temperature Tcr = (1−cr ) Tc < Tc ; for temperatures T < Tcr the fluctuations in the magnetic and electric response behave 2D-like [190] and the parallel upper critical field is only limited through pair breaking via Pauli paramagnetism or spin-orbit coupling [191]. Other physical phenomena, however, involve different criteria: e.g., vortex lattice melting

12 Vortex Matter

√ involves the length scale " ¥0/B, while pinning is characterized by the pinning length " 4/3Lc as determined by the roughness of the disorder landscape (cf. Sect. 12.9.2); the 2D–3D crossover takes place when these lengths drop below the interlayer distance d. Among the important class of copper oxide based materials a continuous anisotropic description is usually applicable to YBa2Cu3 O7−ı [192],whereas the more strongly layered Bi2Sr2 CaCu2 O8+x compound belongs to the class of materials with cr  1 and hence 2D-like behavior is abundant; examples are the thermodynamic properties of the superfluid or of the vortex lattice which more closely resemble those of a 2D superconducting film rather than of a 3D bulk material. Still, the continuous anisotropic Ginzburg– Landau or London based analysis often can provide a rather good description of the physics of layered materials. For example, this is the case in the discussion of the elastic properties of the vortex lattice within a wide angular regime, the reason being that the nonlinear term in the Lawrence–Doniach model depending on the gauge invariant phase difference between the layers can be linearized in many situations [193]. 12.5.1

Lawrence–Doniach Model

The basis for the phenomenological description of layered superconductors is given by the LawrenceDoniach model [58]. The free energy functional describes a discrete set of superconducting layers with order parameters ¦n , separated by a distance d and coupled via a Josephson term,   ˇ 2 F[¦n, A] = d d2 R ˛|¦n |2 + |¦n |4 + 2 2m n    (2) 2i (2)  2 2  ∇ + A ¦n + (12.96)  ¥0 2Md 2 (n+1)d  2i  2   dz Az B2 ¥0   . − ¦n  + d 3 r  ¦n+1 e nd 8 The above formulation allows to make direct contact with the continuous-anisotropic Ginzburg-Landau functional (12.1) upon approximating the discrete (non-linear) coupling term as a derivative along z; the effective masses along the c-axis and in the abplanes are denoted by M and m, respectively, and

519

" 2 = m/M is the usual anisotropy parameter.The full functional (12.97) can be treated in a London approximation assuming a constant modulus |¦n | within the planes and allowing only for phase ('n ) degrees of freedom,    2 (2) 2 2 "0 d ∇ (2) 'n + A F['n , A] = d R 2 n ¥0   2  2m

3 B , 1 − cos ¥ + d + r n+1,n Md 2 8 (12.97) with 2 |¦n|2 /2m = "0 /2 and ¥n+1,n the gauge invariant phase difference between the layers n and n + 1, ¥n+1,n

2 = 'n+1 − 'n + ¥0

(n+1)d 

dz Az ;

(12.98)

nd

here, we express the (superconducting) properties of the layers by the period d of the layer structure and the planar bulk penetration depth . An alternative description is based on the thickness ds of the superconducting layers combined with the penetration depth s of the planes — the two descriptions are related through the equality d/2 = ds /s2 , guaranteeing equal superfluid sheet densities. The variation of (12.97) with respect to the vector potential A and the phases 'n define the differential equations describing spatial variations of A and 'n , 2 A(2) = d

 n

 ¥0 (2)  ∇ 'n , (12.99) ı(z − nd) A(2) + 2

4 jJ sin ¥n+1,n , Az = c  2 (2) (2)   2 (2) 'n + ∇ A ¥0 = sin ¥n,n−1 − sin ¥n+1,n ,

(12.100)

(12.101)

with ¥n+1,n the gauge invariant phase difference between the layers n and n + 1 and jJ the Josephson current density coupling the layers, c¥0 " " ≈ j0 . (12.102) 8 2 2     The parameter   = d/" denotes the in-plane screening length where non-linearities in the coupling between the layers are important (we have used a gauge jJ =

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with ∇ · A = 0; the expressions ∇ (2) , (2) , and A(2) denote the planar components of ∇,,and A).Equation (12.101) is equivalent to the current conservation law ∇ · j = 0. 12.5.2

Josephson Vortices

We consider a magnetic field in the ab-plane pointing along the y-direction and investigate the structure of an individual Josephson vortex aligned with the yaxis [61–63,194]. We choose a gauge with Ay = 0 and Ax , Az depending only on x and z. Subtracting the phase equations (12.101) for n + 1 and for n from each other provides a coupled set of equations for the gauge invariant phase differences ¥n+1,n [78],  2 @x2 ¥n+1,n = 2 sin ¥n+1,n − sin ¥n+2,n+1 − sin ¥n,n−1 d2 + 2 sin ¥n+1,n . (12.103)  Outside a core region with dimensions   and d the non-linearity and discreteness of the above equations are irrelevant and (12.103) can be approximated by the continuum differential equation   "2 2 2 2 @x + " @z − 2 ¥ (r) = 0 . (12.104)  Indeed, it is easy to see that the screening current density along the z-axis in the continuum anisotropic description, jz = −(2e/m)|¦ |2 "@z '˜ ≈ c¥0 "/82 x, becomes equal to the Josephson current density jJ at a distance x ∼  , hence for x >   the linearized

continuum approximation provides a good description of the situation. On the other hand, for small distances x <   a further increase in the current density jz is cut off by the condition |jz | < jJ and the discreteness of the problem becomes relevant. Equation (12.104) tells us that the driving (gauge invariant) phase differences vanish due to screening on a length /" along the x-axis and on a scale  along the z direction.Since the phase difference is the quantity driving the currents we obtain a magnetic size /" (along x) and  (along z) for the Josephson vortex, see Fig. 12.5. On these magnetic length scales the Josephson and Abrikosov vortices in an anisotropic superconductor are roughly equivalent. Within the phase core [63] with dimensions   (along x) and d (along z) we have to take the non-linearity and the discreteness of (12.103) into account. On these scales the phase is changing rapidly and the current density jz reaches its maximum value jJ at a distance ∼   away from the axis of the vortex [194]. Furthermore, within this region the order parameter is weakly suppressed in the layers adjacent to the axis of the Josephson vortex: Ignoring the coupling to other layers, the Ginzburg–Landau equation for the order parameter ¦n (¦n∞ = 1) in the n-th layer reads,  2e−i'n ∇ 2 (|¦n|ei'n ) + |¦n | − |¦n|3 = 0 ;

(12.105)

separating into real and imaginary parts we obtain for the real part = 2 (2) >  [ − (∇ (2) 'n )2 ] + 1 − |¦n |2 |¦n | = 0 . (12.106) If a Josephson vortex directed along the y-axis is present between the n-th and the n + 1-th layer

Fig. 12.5. Josephson vortex in a strongly layered superconductor. The usual normal core of the Abrikosov vortex (dimensions " and  along z and x, respectively) is replaced by the phase core (dimensions d and   = d/" along z and x, respectively) within which the non-linearity and the discreteness of the problem is relevant. The region outside the phase core is roughly equivalent to the corresponding regime in an Abrikosov vortex with screening currents extending a distance  along z and /" along x

12 Vortex Matter

the phase 'n (x) will rapidly change on the scale   and we can approximate the derivative of the phase @x 'n ≈ 1/ . Combining this estimate with (12.106) we obtain a suppression of the order parameter ¦n of the order of ı|¦n| ≈ (/ )2 at the center (x = 0) of the Josephson vortex: contrary to the Abrikosov vortex where the large current flow near the core leads to a complete suppression of the order parameter, the corresponding order parameter suppression in the superconducting layers is only weak in the presence of a Josephson vortex. Note that the set of equations (12.103) for the gauge-invariant phase differences ¥n+1,n can be generalized to an arbitrary field direction by substituting the derivative @x2 with the planar Laplacian @x2 +@y2 and completing the resulting equations with appropriate boundary conditions describing the singularities in the phase field produced by vortices penetrating the layers [195] ∇ (2) ∧ (∇ (2) ¥n+1,n ) ≡ @x @y ¥n+1,n − @y @x ¥n+1,n   ı(r − sn+1  ) − ı(r − sn ) , (12.107) = 2 

where sn denotes the position of the singularity in the phase 'n corresponding to the  -th vortex in the n-th layer. Note that a straight vortex with sn = sn+1  produces no driving singularity for the phase differences ¥n+1,n , i.e., no screening currents crossing the layers are set up. It is quite remarkable that such a closed set of equations involving only the gauge invariant phase differences ¥n+1,n can be found. The Josephson vortex in a layered superconductor involves two length scales along the x-axis, the core size   = d/" and the magnetic size c = /". This has to be contrasted with the Josephson vortex in the junction between two weakly coupled bulk superconductors, for which there is only one characteristic length scale J = [c¥0/16 2jJ ]1/2, with  the penetration depth of the bulk superconductors and jJ the coupling current density of the junction. The difference can be understood in the following way: Consider two bulk superconductors joined via a junction with a coupling jJ . We place a vortex line into the junction and analyze its evolution from an Abrikosov vortex at strong coupling jJ ≈ j0 to a Josephson vor-

521

tex at weak coupling jJ  j0 [196,197].As soon as the translational symmetry is broken by the junction,the structure of the Abrikosov vortex is modified as the zero in the order parameter disappears and is replaced by a finite jump across the junction (this is easily understood by solving the GL equation ¦ ≈ 0 near the singularity with the boundary condition ¦+ − ¦− = l@x ¦+ = l@x ¦− , ¦± = (x + iy) ± l/2; the length l is related to the transparency of the junction, here chosen along the y-axis). Decreasing the coupling jJ below j0 the normal core of the Abrikosov vortex is transformed into a phase core extending over a distance J ≈ (j0/jJ ) >  along the junction where the phase of the order parameter changes by ∼ 2. As the current across the junction is limited to a value jJ < j0 the order parameter suppression is reduced, ı¦ ∼ (jJ /j0)2 , as compared to the complete order parameter suppression in the Abrikosov vortex core. With decreasing coupling jJ the phase core expands and the suppression of the order parameter becomes small. Decreasing the coupling jJ below the critical value jJ ≈ j0 / the phase core expands beyond the magnetic screening length  and the transformation to the Josephson vortex with only one length scale J describing the phase, the currents, and the field across the junction has been completed. Upon a further decrease in jJ the order parameter suppression then is reduced to the value ı¦ ∼ (jJ /j0 )/. The analogous discussion for a layered superconductor produces quite a different result: In a layered material, the screening currents flowing perpendicular to the planes have to cross not only one single junction but have to overcome the large number of /d junctions. Upon decreasing the coupling strength between the layers the screening current density along the c-axis is also reduced such that the current pattern of the original Abrikosov vortex immediately starts to expand along the junction. As a result, the ratio between the magnetic extent of the vortex and the size of its phase core remains always the same,  /c = d/ = const, independent of the coupling strength, and we always keep the two length scales   and c describing the core and the magnetic size of the Josephson vortex. From the above discussion we can understand that a Josephson vortex is very similar to an Abrikosov

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vortex, however, with a different core size and structure. This conclusion is also confirmed by the calculation of the Josephson vortex line energy [61,62] which is obtained through an integration of the kinetic energy of the currents    J ≈ ""0 ln + 1.55 , (12.108) d where the inner and outer cutoff lengths in the logarithm are provided by the magnetic lengths (c , ) and the dimensions of the phase core ( , d) of the Josephson vortex (the constant 1.55 is found in a more accurate analysis, see [198]). The corresponding result for an Abrikosov vortex in an anisotropic superconductor and directed along the planes is [125] l = ""0 [ln(/") + 0.497]; note the different temperature dependence for these two results arising from the substitution of c (T) = "(T) by the layer distance d under the logarithm. The line energy (12.108) defines the lower critical field along the planes Hc1 (# = 0) = 4 J /¥0 and the (in-plane) elastic tension of a Josephson vortex, "l = J . Finally, the viscous drag coefficient depends on the core structure and differs only by a numerical factor from the corresponding (Bardeen–Stephen) expression for an Abrikosov vortex: substituting the appropriate dimensions for the phase core the result reads [199] (see also [194]) J ≈

¥02 1.8 2cnc 2 " 2

,

(12.109)

with cn the normal-state resistivity along the c-axis n and where we have assumed the relation ab = " 2 cn between the in-plane and c-axis resistivity (the corresponding Bardeen–Stephen result for the Abrikosov  vortex is l = ¥02/2cnc 2 " 2 ). In high-Tc superconductors the resistivity–anisotropy relation is vin olated below Tc and ab /cn < " 2 ; in this case the in-plane dissipation is enhanced and we should replace cn in (12.109) by the expression cn /(0.79 + n n 0.21" 2 cn /ab ) ≈ 4.76ab /" 2 . 12.5.3

Pancake Vortices

We consider a magnetic field directed along the caxis and discuss the structure of a vortex and its

planar constituents, the pancake vortices [65]. In this context it is interesting to understand the similarities and differences in the vortex structure for the three cases of an isolated thin film [200], an uncoupled stack of superconducting layers with jJ = 0 [62, 64, 65, 201, 202], and a layered superconductor with jJ > 0 [64,195,201,203]. We start with the simplest case of a 2D thin film of thickness d and choose  to denote the penetration depth of the corresponding bulk material; the effective penetration depth of the film is eff = 22 /d. The solution for a vortex positioned at the origin of the coordinate system has been given by Pearl [200,204]: The London equation (cf. (12.99); we write A(2) = A) 2 A = dı(z)(a + A) , with the source term ¥0 zˆ ∧ R ¥0 ∇' = − a= , 2 2 R2

(12.110)

zˆ ∧ K , K2  (12.111) is solved with the Fourier Ansatz A(r) = d 3 k/(2)3 × Ak exp(ik · r), aK = i¥0

d(aK + AK ) Ak = − , with 2 k2  AK = d2 R A(R, z = 0) e−iK ·R , (12.112) and an integration over kz provides the result AK = aK /(1+eff K).Inserting this result back into (12.112) we obtain zˆ ∧ K 1 Ak = −2i¥0 (12.113) 2 K k (1 + eff K) and transforming back to real space making use of the cylindrical symmetry we find the vector potential ∞ A' (R, z) = ¥0 0

dK J1 (KR)e−K|z| , 2 1 + eff K

(12.114)

with J1 (x) the Bessel function of integer order. The associated magnetic field asymptotically resembles that of a positively/negatively charged magnetic monopole generating a magnetic flux ¥0 in the upper/lower half space, see Fig. 12.6, B(r  eff ) ≈

¥0 z r . 2 |z| r 3

(12.115)

12 Vortex Matter

523

vector potential on the scale d we rewrite (12.99) in the form (we use that A(2) = A) d¥0  ı(z − nd)∇'n . (12.117) [2  − 1]A = 2 n

Fig. 12.6. Pearl solution for a vortex in a superconducting film with a magnetic field taking asymptotically the form of a positively/negatively charged magnetic monopole in the upper/lower half space

The magnetic flux crossing the film within a circle of radius R is ⎧ R ⎪ ⎪ , R  eff , ⎨ eff ¥ (R) ≈ ¥0 (12.116) ⎪  eff ⎪ ⎩1 − , eff  R , R and approaches the unit flux quantum ¥0 at large distances. The circular current density J' (R) (per unit length) decays algebraically with distance and generates (via the Lorentz force) a screened logarithmic interaction Vint (R) between a pair of Pearl vortices, ⎧  ⎪ ⎪ eff , ¥0 c ⎨ R J' (R) ≈ 2 ⎪ 2 4 2 eff ⎪ ⎩ eff , R2 ⎧ eff ⎪ ⎪ , ⎨ ln R Vint (R) ≈ 2"0 d ⎪ eff ⎪ ⎩ , R

An individual vortex line threading the stack is characterized by the positions Sn of the vortex cores within each layer. The gradients of the phase fields ∇'n (R) = −ˆz ∧ (R − Sn )/|R − Sn |2 drive the screening currents in the individual layers, which in turn are coupled electromagnetically through (12.117); hence, screening currents in response to a pancake vortex in the n-th layer will be set up within the other layers of the stack. The linearity of (12.117) allows us to reduce the problem to the analysis of the elementary building block of the vortex line, the pancake vortex [62,64,65,201,202]. We consider a single pancake vortex with its driving field ∇'n (R) = −ˆz ∧R/R2 placed at the origin of the n-th layer (positioned at z = 0) and with all the other driving terms vanishing. Transforming (12.117) to Fourier space we find the vector potential Ak = −id¥0

eff  R .

Next, we discuss the case of an array of uncoupled (i.e., jJ = 0) parallel superconducting layers where the interaction is of electromagnetic origin [62,64,65,201,202]. Ignoring small variations in the

(12.118)

and transforming back to real space using cylindrical symmetry we obtain the solution d A' (R, z) = ¥0 2

∞

dK 2

(12.119)

0 √

J1 (KR)e− 1+ K |z|/ × √ . 1 + 2 K 2 2

2

The associated magnetic field (see Fig. 12.7) Bz (r) =

R  eff ,

zˆ ∧ K + 2 k2 )

K 2 (1

B(2) (r) =

¥0 d −r/ e , 42 r ¥0 d R signz (12.120) 42 R2   |z| −r/ −|z|/ e , × e − r

differs quite drastically from the monopole-like Pearl solution: The screening effect due to the other superconducting planes in the stack squeezes the field

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G. Blatter and V. B. Geshkenbein

into a layer of width  along the z-axis and the magnetic field has to escape parallel to the layers rather than spread out uniformly over the entire solid angle. This field redistribution due to screening has important consequences: First, the squeezing of the field reduces the flux threading the central layer n to ¥ (R) = ¥0(d/2)[1 − exp(−R/)] hence the current flow is not able to screen the driving phase-field in the central layer. Indeed, the current density of the central layer decays like 1/R to all length scales,   d  ¥0 c eff −R/ J' (R, z = 0) = 1 − 1 − e , 2 R 2 4 2 eff (12.121) and therefore the logarithmic dependence of the interaction potential Vint between two vortices placed in the same layer persists to infinity, R em Vint (R, z = 0) = ±2"0 d ln + O(d/) , L 
(12.122)

with L the system-size cutoff (the − (+) sign refers to equally (oppositely) charged vortices; note that the total energy includes an additional self-energy term 2"0 d ln(L/), with the bulk planar coherence length  serving as a short distance cutoff).

Fig. 12.7. Pancake vortex in a layered superconductor with vanishing interlayer Josephson coupling. The screening currents present in the neighboring layers squeeze the magnetic field into the planar direction. The screening current density in the central layer decays like j ∝ 1/R to all scales

In order to find the general expression for the interaction energy between pancake vortices (positioned at Sn in layer zn = nd) we express the current and magnetic field terms in the free energy (12.97) through  the current density j = −(c/42 )[(¥0/2)d n ı(z − nd)∇'n + A] and the vector potential A, in Fourier space, c id¥0zˆ ∧ K 2 k2 42 K2 1 + 2 k2  × e−iK ·Sn e−ikz nd , (12.123)

jk = −

n,

id¥0zˆ ∧ K 1 2 K 1 + 2 k2  × e−iK ·Sn e−ikz nd ;

Ak = −

(12.124)

n,

combining terms we arrive at the free energy  ¥ 2d 2 1 d3 k k 2 F= 0 (12.125) 8 (2)3 K 2 1 + 2 k2  × e−iK ·(Sn −Sm ) e−ikz (n−m)d . n, ,m,

In order to make contact with the expressions (12.62) and (12.80) describing interacting vortex segments in the continuous anisotropic situation one has to take the appropriate limit " → 0 and account for the continuity of vortex lines via the relation ∇ · s = 0 (we remind the definition s = r  + u of the vortex positions), in Fourier space, K · Sk = −kz szk , and hence S˛,k K⊥˛ K⊥ˇ Sˇ,−k = K 2 |Sk |2 − |K · Sk |2 = K 2 |Sk |2 −|kz szk |2 .The terms ∝ |szk |2 in the free energy 2 2 (12.62) then combine into a term |szk k /K 2 (1 + 2 k2 ) 2 (cf.(12.126)),while those ∝ (|sxk | +|syk |2 ) come with a prefactor 1/(1 + c2 K 2 + 2 kz2 ) which vanishes in the limit " → 0. An individual pancake vortex involves the energy  d3 k ¥ 2d 2 k2 Epc = 0 3 2 8 (2) K (1 + 2 k2)   d L L ln , (12.126) = "0 d ln −  2  where the first term is due to the driving term ∝ ∇' and involves the core size  as a short scale cutoff,

12 Vortex Matter

while the second term is due to screening. A vortex– anti-vortex pair in the same layer (n) and separated a distance R costs an energy  ¥02d 2 d3 k k2 [1 − cos(K · R)] E+− (R) = 4 (2)3 K 2 (1 + 2 k2 )  1 − J0 (KR) = 2"0d dK K   d/2 × 1− √ 1 + 2 K 2 R (12.127) ≈ 2"0d ln , 

reproducing the result (12.122) above. Placing a second pancake vortex (with equal phase turn) in layer m we obtain the expression E++ (R, z) = "0 d

d 

 dK

J0 (KR) K

√ − 1+2 K 2 |z|/

e , × √ 1 + 2 K 2

(12.128)

with R = |Rm − Rn | and z = zm − zn = (m − n)d. In order to handle the (logarithmic) divergence at small K (cf. the correction term in (12.126)) we determine the (attractive) force f (R, z) = −@R E++ (R, z) between the two pancake vortices, d f = "0 d 



√

e− 1+ K |z|/ dK J1 (KR) √ . 1 + 2 K 2 2

2

(12.129)

At very small distances R < |z| <  the integral is cut by the exponential and we have f ≈ "0 (d 2 /22)(R/|z|); at intermediate scales |z| < R <  we find the constant force f ≈ "0 d 2/2 ; for large distances  < R the integral is cut by the J1 factor and f ≈ "0 (d 2/R). In the end we obtain an attractive interaction between vortices which is small (by a factor d/), but which is long ranged (extending over /d layers),

525

⎧ 2 R ⎪ ⎪ , R  |z|   , ⎪ ⎪ 4|z| ⎪ ⎪ ⎨ d R em Vint (R, z = nd =  0) ≈ "0 d , |z|  R   , ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ ln R ,   R . 

(12.130) For separations |z| >  the interaction decays exponentially ∝ exp(−|z|/) (the factor 1/2 quoted in [150] is incorrect). Finally, constructing a full vortex line with a phase singularity present in each layer, the calculation of the field distribution, the line energy, and the lower critical field Hcc1 [65] gives results in agreement with the continuous anisotropic description,   ¥0 R Bz = , K 0 22  ¥0 c l = H = "0 [ln  + const.] . (12.131) 4 c1 Similarly, the line energy l (# ) for a vortex tilted by an angle  = /2 − # away from the c-axis agrees (to logarithmic accuracy) with the continuous anisotropic result (12.25) in the limit " → 0,    1 + sin # l (# ) = "0 sin # ln (12.132) sin # 2 (note that "# = sin # for the present case of perfectly decoupled layers with " = 0). A finite Josephson coupling between the superconducting layers renders the introduction of a single pancake vortex into the layered superconductor impossible [64,195,201,203]: in addition to the electromagnetic interaction between the layers, the interlayer Josephson coupling contributes the energy  2   dR

d"0 FJ = 1 − cos ¥n+1,n(R ) . (12.133) 2    The phase difference ¥n+1,n = 0 between the plane containing the vortex and its two neighboring planes and the coupling energy FJ to these two layers becomes infinite: ignoring a possible relaxation of the phase pattern, equation (12.133) predicts an energy of an individual pancake vortex which grows with

526

G. Blatter and V. B. Geshkenbein

the sample area. However, the phase pattern due to a single pancake vortex can relax to a state with an energy growing only linearly in the sample size, the linear energy arising from the Josephson strings which take the flux to and away from the pancake vortex. Introducing a pair of oppositely “charged” pancake vortices at a distance R into the same plane creates two Josephson strings which contribute the energy (12.133) with the phase differences ¥n+1,n (R ) determined by the coupled set of equations (12.101). In order to truncate this system of equations we assume the phases 'n±1 to be undisturbed, while the phase 'n is distorted due to the presence of the pair in layer n. The phase difference ¥ = 'n+1 − 'n = 'n−1 − 'n then is a solution of  2 (2) ¥ = −2 sin ¥ ,

(12.134)

where we have neglected the additional screening term due to the vector potential which becomes relevant on large scales R > /". For small phase differences ¥ we can expand the sine in (12.134) and we obtain the natural length scale (screening length)   = d/" of the problem. On scales R <   the Josephson currents cannot build up and the system essentially behaves like an uncoupled layered material with its ideal 2D behavior. On the other hand, for distances R >   the currents along z are no longer hampered by the discreteness of the material and a continuous anisotropic 3D behavior results. At small separation R    the solution of (12.134) behaves as ¥ (R  R   ) ≈ R·∇' = (R/R) cos ' , with ' = arctan(y/x) the azimuthal angle in the plane,and vanishes ∝ exp(−R / ) at larger distances R   .For small phase differences ¥ we can expand the cosine in (12.133) and a trivial integration supplies us with the Josephson interaction energy of the pair, J Vint (R) ≈ "0 d

 2   R ln ,   R

R    . (12.135)

For large separations R   , two Josephson strings connecting the pair are created between the layers n and n ± 1. The interaction between the two pancake vortices then is dominated by the energies of the two strings and takes the form (see (12.108))

    R J Vint (R) ≈ 2"0 d ˛ − ,   4R

   R  /" ,

(12.136) with ˛ a numerical of order unity arising from cutting off the logarithm in (12.108) at a distance of the order of the separation d between the Josephson strings [70,205]. The last term in (12.136) is a higher order correction to the leading string term (arising again from the Josephson coupling [203]) and decays exponentially at distances R > /". The comparison of the Josephson interaction energies (12.135) and (12.136) with the electromagnetic energy (12.122) of the pair shows that the electromagnetic contribution is larger at small distances R <   and the Josephson term dominates for distances R >  . In summary, we arrive at the following picture for the interaction between two pancake vortices placed in the same layer of a Josephson coupled system: At small distances R <   the Josephson currents have not yet built up, the electromagnetic energy is dominant and we obtain a logarithmic interaction. As R grows beyond  , the Josephson currents have been established, the magnetic field between the two vortices has been redirected into two Josephson strings and the interaction grows linearly, i.e., the pair becomes confined. Two further pancake-vortex configurations are the kink–anti-kink excitation along one Josephson vortex [203] with the energy ⎧ R ⎪ ⎪ ln ,  < R <   , ⎪ ⎪  ⎪ ⎪ ⎪ ⎨     K,−K ,   < R < /" , Vint (R) ≈ 2"0 d ln − ⎪  4R ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ ln , /" < R ,  (12.137) and the double-kink configuration providing the building block of a tilted vortex with an energy [201] (we assume   < ; otherwise, the linear term ∝ dR/2 saturates at R ∼ , cf. (12.130)) K,K (R) ≈ Vint

⎧   dR 1 R 2   ⎪ ⎪ + ln ,  < R <   <  , ⎨ R "0 d   2   ⎪   ⎪ ⎩ ,   < R < /" . 2R

(12.138)

12 Vortex Matter

These basic pancake-vortex configurations allow to study more complicated vortex configurations and their energies in terms of point-like interacting particles. An example is the calculation of the line energy for a tilted vortex from which we can find the elastic line tensions for the in-plane and for the outof-plane tilt modes: Summing up the energies of the individual pancake vortices including their interaction as given by (12.138) and the contributions from the Josephson strings (12.135) and (12.136) we find the line energy of a vortex tilted by an angle # with respect to the layers. For large angles # > " the separation between neighboring pancake vortices is less than   and the line energy becomes [195,206] l (# > ") = "0 sin # (12.139)   "2 ˛   tan # ˛ + , ln × ln  sin # 2 tan2 # d where ˛ and ˛  are numericals of order unity. For small angles 0 < # < ", the Josephson string contributes a large energy (12.136) and the result reads l (0 < # < ") = "0 sin # (12.140)   "  1   ln + tan # . × ln +  tan # d 2" Using the definitions (12.37) for "l⊥ and for "l above we easily find the expressions for the out-of-plane and the in-plane line tensions in a layered material [150] (we ignore logarithmic corrections and the divergence of "l⊥ due to the lock-in transition [207–209] at # = 0): ⎧ 1 ⎪ , 0<# <" , ⎪ "0 ⎨ cos3 # ⊥ "l (# ) = 3 "⎪  ⎪ ⎩ " , , "<# < 3 2 sin # "l (# ) = ""0

⎧ ⎪ ⎨ cos # , 0 < # < " ,

 " ⎪ ⎩ , "<# < . sin # 2 These equations strongly resemble the results for the anisotropic superconductor, a consequence of the similarity in the current flow set up by a vortex in an anisotropic and in a layered material.

12.5.4

527

Renormalized Superfluid Stiffness

The presence of pancake vortices leads to a downward renormalization of the superfluid density in the superconducting layers [76, 210]; this effect has various consequences for the structural and statistical properties of the vortex system in layered superconductors, such as a reduction in the effective anisotropy, a modification of the characteristics of the Josephson vortices, and implications for the layer decoupling transition. Strictly speaking, in the absence of pinning the presence of pancake vortices removes the (homogeneous) superfluid stiffness altogether. However, a finite superfluid stiffness survives at finite distances: we can define a short-wavelength (kz ≈ /d) superfluid stiffness J in an individual superconducting layer. This stiffness J(K) is dispersive, where the (in-plane) long-wavelength limit J(0) is determined by the pinning of the pancake vortices within the pancake-vortex stacks produced by the vortices in the other layers; its dispersive component ∝ K 2 is due to the finite shear energy associated with the deformation of the pancake-vortex lattice in the plane. In order to gain a better understanding of this renormalized in-plane dispersive phase stiffness we search for a free energy functional ıF[ıA, u] describing the stiffness of the gauge-invariant vector potential ıA in the presence of mobile vortices described by their displacement field u (we first concentrate on a 2D (translation invariant) situation with the field B  zˆ ); note that ıF and ıA describe the deviations in the free energy and in the gauge field with respect to the equilibrium vortex lattice state. In particular, the gauge field ıA contains a term −j/s, s ≡ c/42 = e2 ns /mc, originating from the (macroscopic) current density j and a second term u ∧ B due to the vortex displacement u, originating, e.g., in response to an applied external current density jext , ıA = −

1 j+u∧B . s

(12.141)

Here, the field B  zˆ generates the vortices reducing the superfluid stiffness. The vortex-induced gauge potential is easily understood in the dynamical situation: taking the derivative of ıAu = u ∧ B

528

G. Blatter and V. B. Geshkenbein

with respect to time we find the Josephson relation ıE = −@t ıAu /c = B ∧ v/c with the vortex velocity v = @t u. Alternatively, the relation ıAu = u ∧ B follows from explicit calculation of the distortion in the vortex phase field ı'v = 'v (R−u)−'v (R) ≈ −u·∇'v and suitable averageover areas ıA > a20 : using the definition ∇'v = −  zˆ ∧ (R − S )/|R − S |2 , where S denote the vortex positions, and transforming to Fourier space one finds the useful relations uK = ı' vK =

a20 (ˆz ∧ iK) ı' vK , 2 2 (uK ∧ iK) · zˆ . K2 a20

(12.142)

Substituting iK ı' vK → ∇ı' vK → (2/¥0)ıAu one easily reproduces the relation ıAu = u ∧ B. The free energy functional we are searching for then should contain terms describing the kinetic and magnetic energies associated with ıA and terms accounting for the elastic energy following from the displacement field u; indeed, the free energy density   2 1 1 ıf [ıA, u] = (∇ ∧ ıA)2 + 2 ıA − u ∧ B 8  c66 (12.143) + [∇ ⊗ u]2 2 exhibits all the correct features (the tensor product in (12.143) stands for [∇ ⊗ u]2 ≡ @x˛ uˇ @x˛ uˇ ): i) Variation with respect to the vector potential ıA reproduces the relation (12.141). ii) “Integrating” over the gauge field ıA via minimizing with respect to ıA, u ∧ B + 2 K [K · (u ∧ B)] , (12.144) 1 + 2 K 2 and inserting the result back into (12.143), we find the elastic energy density ıfel = (1/2){[c11(K) − c66 ] |K · uK |2 + c66K 2 |uK |2} with the correct modulus c11(K) − c66 = (B2 /4)/(1 + 2 K 2 ) (note that we should identify (B2 /4)/(1 + 2 K 2 ) with the combination c11(K) − c66, cf. [13] and the remark below (12.68); the tensor product in (12.143) then transforms to c66K 2 |uK |2 /2 and the combination c66 [K 2 |uK |2 − |K · uK |2] = c66|K ∧ uK |2 ). And iii), variation with respect to u produces the force equation ıAK =

1 j ∧ B = −c66∇ 2 u , c

(12.145)

where j accounts both for current densities applied externally as well as those induced internally via compression. Note that the kinetic energy of the smooth macroscopic currents (external and from vortex lattice compression) enter the free energy density (12.143) via the term ∝ −2 , whereas the small microscopic currents due to the distortion u of the vortex system are accounted for in the shear term. Further insight into the nature of the superfluid stiffness in the presence of vortices is gained when the force equation (12.145) is modified to account for vortex pinning: using the Labusch [6] parameter ˛L we introduce an effective pinning force density f pin = −˛L u balancing the Lorentz force density jext ∧ B/c (note that the elastic forces due to shear and compression are taken care of in the effective pinning parameter ˛L ). Inserting the displacement field u = jext ∧ B/c˛L back into (12.141) we find that the response jext = −seff ıA involves the renormalized superfluid density seff =

c 1 ; 4 2 + B2 /4˛L

(12.146)

as expected, in the absence of pinning, ˛L → 0, the effective superfluid density seff → 0 vanishes. The renormalized superfluid density (12.146)  defines the Campbell penetration depth C = 2 + B2 /4˛L describing the penetration of a static external signal into the superconductor in the presence of (pinned) vortex lines [211]. On the other hand, accounting for the finite shear force in (12.145) the same calculation provides the dispersive response c c a20 K 2 1 . ≈ 2 42 1 + 16/a0K 2 42 16 (12.147) Next, we modify (12.143) to describe our pancake vortex system in a layered superconductor (we assume that B  zˆ ). We separate the term describing the smooth macroscopic current flow in the planes and complete with the elastic and Josephson coupling terms, seff (K) =

12 Vortex Matter

ıF =

 1 d3 r 8   2 1  (2) 2 × (∇ ∧ ıA) + 2 ıA − u ∧ B    + d2 R  ×

n

vem 2 c66 [∇ ⊗ un ]2 + u 2 2 n   " 2 "0

1 − cos ¥n+1,n , + d

2"0 d a40 2   (2) A K − uK ∧ B  × B2

(12.149)

 a20 K 2 2 2 |uK | + g |uK | . + 16

The first term describes the kinetic energy density of the smooth macroscopic current flow. The second term describes the shear stiffness and the third contributionoriginates from the electromagnetic tilt energy which “binds” the pancake vortices into stacks; here, we have rewritten the electromagnetic tilt energy in the form g (2"0 d/a40 )|uK |2 with the reduction factor

a20 a20 ln ; 82 d 2 + u20

(12.150)

the displacement u0 in (12.150) is determined by the pancake displacement typical for the actual physical situation (e.g., see (12.159) below). Note that it is this “pinning” via the pancake vortex stack which will establish a finite short-wavelength superfluid density in the individual layers. Finally, we “integrate” over the displacement field u: minimizing (12.150) we find

(12.148)

where the shear energy is determined by the density 2 of pancake vortices npc = B/¥0 = a−2 0 , c66 = "0 d/4a0 . The electromagnetic component of the tilt energy (12.85) is strongly dispersive ∝ kz−2 and turns the elastic tilt energy into a potential term with a prefactor vem ≈ ("0 d/2a202 ) ln[a20 /(d 2 + u2 )], where we have assumed kz ∼ 1/d in the logarithm. Finally, a finite Josephson coupling between layers is accounted for within a Lawrence–Doniach type description (cf. (12.97)) and involves the gauge invariant phase difference ¥n+1,n . In order to find the effective free energy describing the gauge invariant vector potential ıA we integrate over the displacement field u in (12.148); the relevant terms involving the displacement field define the partial free energy density ıfn =

g=

529

uK =

B ∧ ıA(2) 1 2 2 2 B 1 + a0 K /16 + g

(12.151)

and inserting this solution back into (12.150) we obtain the free energy density d g + a20 K 2 /16 2 |ıA(2) K | , 82 1 + g + a20 K 2 /16 (12.152) from which we find the renormalized superfluid stiffness ıfn [ıA] =

seff (K) =

c g + a20 K 2 /16 . 2 4 1 + g + a20 K 2 /16

(12.153)

Comparing with (12.146) we can trace back the finite long-wavelength superfluid stiffness ∝ g/(1 + g) to the out-of-plane pinning action of the pancake vortex stacks with the effective Labusch parameter ˛L = 4g"0 /a40 derived from the “pinning” potential Vpin = ˛L u2/2 due to the electromagnetic interaction defining the stacks. Similarly, comparison with (12.147) shows that the dispersive contribution ∝ a20 K 2 /16 is due to the in-plane vortex shear energy. The full result (12.153) then follows from the combination of the force equation j ∧ B d/c = (vem + c66K 2 ) u with (12.141). Finally,we derive an expression for the free energy in terms of the phase variable ı' through the substitution ıA(2) → (¥0/2)∇ı' (the overline reminds us that ıA(2) and hence also ∇ı' are smooth fields averaged over distances larger than the vortex lattice spacing); expanding (12.152) in a20 K 2 /16 and combining with (12.148) we arrive at

530

G. Blatter and V. B. Geshkenbein

 ıF[ı' ] ≈

+

d2 R

2  "0 d  g  ∇ (2) ı' n 2 1 + g n

a20 16(1 + g)2

(ı' n )2

  2" 2

+ 2 1 − cos ¥n+1,n d  1 d3 r (∇ ∧ ıA)2 . + 8

(12.154)

Comparing with (12.97) we note the factor g/(1 + g) which accounts for the suppression of the superfluid density due to the presence of the pancake vortices. This suppression is substantial for fields B  Bcc1 where g ∼ Bcc1 /B  1, while no renormalization appears at low fields B  Bcc1 where g  1 and hence g/(1 + g) ≈ 1. Note that at low fields we can drop the shear term ∝ (ı' n )2 in comparison with the electromagnetic and Josephson tilt energies. On the other hand, at high fields where the shear term dominates g is small and we can drop the factor (1 + g)−2 in the shear term. The renormalization of the (short wavelength) superfluid stiffness due to the presence of pancake vortices affects various properties of the vortex system in layered superconductors: important modifications appear in the context of the vortex (lattice) structure associated with a finite in-plane field component By , see Sect. 12.5.5, and in the analysis of the fluctuationinduced decoupling transition, see Sect. 12.7.3.

12.5.5

Vortex Lattice

In a continuous anisotropic superconductor the tilted vortex lattice usually preserves its triangular structure up to a trivial distortion,see (12.78).Deviations occur at low fields H < Hcc1 /"# where the attractive interaction between tilted vortices [179,182,183] leads to the chain state [179–181]). Increasing the anisotropy, an individual tilted vortex turns unstable [212] within some angular regime, suggesting that in layered material the vortex lattice may split into two separate families of pancake-vortex stacks and Josephson vortices, each defining one of the field components Bz and By , respectively [195, 213, 214]. Indeed, for weak Josephson coupling (we assume   > ) the pancake vortices mainly interact electromagnetically and their homogeneous tilt costs a large amount of magnetic energy; at the same time, the field component By parallel to the layers can penetrate easily into the superconductor in the form of Josephson vortices. In this situation a new structure with a lattice of Josephson vortices coexisting with a lattice of pancake vortex stacks, the so-called crossing lattice, may be energetically more favorable, see Fig. 12.8. Comparing the energies of the tilted vortex lattice and the crossing vortex lattices within the Lawrence–Doniach description,Benkraouda and Ledvij [214] indeed find the crossing-lattices state to be stable within an intermediate angular regime away from the extremal angles  = 0, /2 if the magnetic field is low enough; on the other hand, the tilted lattice remains stable in the vicinity of the highsymmetry directions perpendicular and parallel to

Fig. 12.8. Schematic drawing of tilted and crossed vortex lines/lattices where individual vortex lines can be identified. In the crossed vortex lattice Josephson vortices and pancake vortices define two interpenetrating lattices

12 Vortex Matter

the superconducting planes. Below we compare the energies of the two states and determine the transition angle × separating the (nearly c-axis directed) tilted vortex lattice ( < × ) from the crossed (or combined) lattice (× < ). In doing so, we follow [210] and account for the renormalization of the superfluid density due to the pancake-vortex lattice as derived in Sect. 12.5.4 above; the suppression of the superfluid density then favors the crossed lattice phase. A direct consequence of the crossed-lattices vortex state is the appearance of a (mixed) vortex-chain state in layered superconductors, where either a fraction or even all pancake-vortex stacks are pinned onto stacks of Josephson vortices [210, 215]. This unconventional vortex state has been observed in Bitter decoration experiments on layered BiSCCO superconductors [216]. Note that this vortex-chain state is different from the chain state appearing in anisotropic superconductors at small tilted fields, cf. the discussion in Sect. 12.4.1: whereas the chain state in continuous anisotropic superconductors is a consequence of the attractive interaction between tilted vortices [179,182,183],the chain state in layered superconductors is due to an attractive force appearing between stacks of Josephson and pancake vortices. Crossed Lattices In order to find the transition angle × near the caxis separating tilted and crossing vortex lattices we compare their free energies [210]: Tilting the field by the (small) angle  ≈ By /Bz away from the c-axis we have to pay for the tilt energy density ftilt =

c44 (k = 0) 2  , 2

(12.155)

with the homogeneous tilt modulus c44 (0) = B2z /4 + 3.7 ¥02 /(4)4 ; while the first term derives from the 0 limit k → 0 of c44 (cf. (12.83)), the second term is the contribution from individual pancake-vortex c stacks in the limit kz → 0, c44 ≈ ("0 /22 kz2 a20z ) ln(1 + 2 2 2 4 a0z kz /4) ≈ 2¥0 /(4) (see [217] for a precise evaluation of the numerical prefactor). On the other hand, leaving the pancake-vortex lattice unchanged and adding a Josephson vortex lattice

531

in the ab-plane we have to account for the free energy density associated with the latter. However, the Josephson vortices themselves are modified by the presence of the pancake vortices through the renormalization of the superfluid stiffness, cf. Sect. 12.5.4: For fields Bz > Bcc1 the suppression of the in-plane superfluid density by g leads to a reduction in the ef√ fective anisotropy, " → "˜ = "/ g.As a consequence, the Josephson length   describing the 2D core of a pancake vortex and the phase core of the Josephson ˜ = √g , as is easily seen vortex is reduced,   →   by comparing in-plane and out-of-plane energies in (12.154). Similarly, the line energy of the Josephson √ vortex is modified, J → ˜J = g J; again, this follows trivially from (12.154) after isotropizing the kinetic energy terms along the x and z directions and ˜ /"] integration over the region [ , ˜ away from the phase core where the discrete structure is unimportant. Finally, the Josephson-vortex lattice structure is modified too, taking a less anisotropic form with √ √ short and long basis vectors ∼ "a ˜ 0y and ∼ a0y / ". ˜ Note that we ignore here an additional reduction factor c(T) = cos(¥n+1,n )th in the Josephson coupling due to thermal fluctuations of pancake vortices, see [210]. Taking the above modifications into account, the Josephson vortex lattice contributes with the energy density, cf. (12.54),   B2y By √ "¥0 fJ = + . (12.156) g""0 ln √ 8 2¥0 gBy d 2 Comparing the two energy densities the terms B2y /8 cancel and the remaining corrections determine the critical angle (we choose a cutoff u0 ∼ a0z /  in g, cf. (12.159) below) "    1/2 a0z × ≈ 24.1 . (12.157) ln ln a0z  d In the most interesting low-field part of the phase diagram (with B of order B = 0 /2 ) the crossed vortex lattice is favored as soon as the field is tilted beyond the angle × of order 1–10 degrees. Chain State The second interesting peculiarity of the low-field vortex lattice in layered superconductors is the

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vortex-chain state, where a finite fraction of the pancake-vortex stacks are pinned onto the stacks of Josephson vortices [210,215,216]. This rearrangement is a consequence of the attractive interaction between a pancake-vortex stack and a Josephson vortex. Consider a pancake-vortex stack in the current field set up by a Josephson vortex directed along the y-axis, see Fig. 12.9: the Lorentz force acting on the pancake vortices deforms the stack and allows the system to gain energy. In order to estimate the displacement uy of the pancake-vortex stack we start from the screening current density jx set up by the Josephson vortex, √ c g¥0 egns √ jx ∼ @y ' ∼ ∼ j0 g , (12.158) 2 m     where we have assumed large fields B > Bcc1 such that g/(1 + g) ≈ g. This current drives the distortion of the pancake-vortex stack, shifting pancake vortices residing above and below the Josephson vortex core in opposite directions along the axis of the Josephson vortex,see Fig.12.9.The distortion is counteracted by the electromagnetic force ("0 d/2 )uy confining the pancake-vortex stacks; comparing the two forces we obtain the estimate uy ∼

2 √ a0z . g∼    

(12.159)

In low fields Bz < Bcc1 , g/(1 + g) ≈ 1 and the distortion uy ∼ 2 /  (note that uy also determines the displacement u0 entering the suppression factor g, cf. (12.150)). Returning to the chain state we consider a weak caxis field Bz < ¥0 / 2 = B  < B , then each Josephson vortex core contains only one row of pancake vortices (we assume non-overlapping cores, By < B  /"; note that Bz < B implies g/(1 + g) ≈ 1). These pancake vortices are pinned to the Josephson vortex core with an energy ∼ ("0 d/2 )u2y ∼ "0 d2 / 2 , a consequence of the distortion of the pancake-vortex stack due to the current of the Josephson vortex (see Fig. 12.9; with g/(1 + g) ≈ 1 the distortion is of √ order uy ∼ 2 / ). With a distance " a0y (a20y = ¥0 /By ) between Josephson vortices along the c-axis, a pancake-vortex stack pinned onto the Josephson√ vortex stack gains the line energy ∼ "0 d2 / " a0y  2 .

Fig. 12.9. Deformation of a pancake-vortex stack in the current field of Josephson vortices. The displacement u away from the straight line configuration (dotted line) is large, of order u ∼ a0z / 

On the other hand, moving a pancake-vortex stack from the bulk to the  Josephson-vortex stack costs a shear energy ∼ "0 a0z / exp(−a0z /); for fields Bz below Bchain ∼

B   2 ln   a0z a20y /"5 2

(12.160)

the chain state is energetically favorable. Recent experiments on BiSCCO single crystals have unraveled a complex Bz -By phase diagram with various vortex lattice phases (tilted, crossed, chain-state, mixed chain-lattice state) [218–221].

12.6

Anisotropic Scaling Theory

The discovery of high temperature superconductivity has enforced the rapid development of the phenomenology of anisotropic type II superconductors. Anisotropy has been traditionally incorporated into

12 Vortex Matter

533

Fig. 12.10. Schematic comparison of the traditional and scaling approach for obtaining physical results in anisotropic superconductors. The traditional approach starts from an anisotropic Ginzburg–Landau or London free energy functional and determines the desired quantity Q by performing all the steps done previously for the isotropic case. In the scaling approach the desired quantity Q˜ is obtained by a rescaling of the isotropic result Q. The scaling rules are determined only once by a rescaling of the anisotropic functional to an isotropic one

the theory via introduction of an anisotropic effective mass tensor into the Ginzburg–Landau or London equations. In the conventional approach one then repeats all the calculations which have been done for the isotropic case before. Due to the appearance of additional parameters and the breaking of spherical symmetry, the analysis becomes very tedious for the anisotropic problem.As a consequence, new results become available first for the isotropic case and in a subsequent step the case of cylindrical symmetry, where the magnetic field is aligned with the c-axis, is treated. Typically, only few results are known for the general case of arbitrary field direction. In this section, we discuss a scaling approach to the problem of anisotropy [176] which allows to generalize known results for isotropic superconductors to anisotropic materials in a simple way, including also the case of an arbitrary direction of the magnetic field with respect to the anisotropy axes; this new scheme provides a simple and direct access to the most general anisotropic results. In the following, we first rescale the anisotropic problem to a corresponding isotropic one on the level of Ginzburg–Landau or London equations and extract the scaling rules. In a second step, these scaling rules are used to generalize the isotropic results to the anisotropic situation; the two approaches are schematically illustrated in Fig. 12.10. Consider the Gibbs free energy (12.1) for an anisotropic superconductor with masses mx = my = m, mz = M > m. As usual we denote the mass anisotropy by " 2 = m/M < 1 and choose a coordinate system with the external field H in the yz-plane, enclosing an angle # with the y-axis, see Fig. 12.4.

We will make frequent use of the angle-dependent anisotropy parameter "# , see (12.26). A first transformation mapping (12.1) to an isotropic form has been introduced by Klemm and Clem [126] and scaling ideas have been used in calculations of the reversible magnetization and of the torque [222,223]. The important step in the construction of a general scaling theory with a wide regime of applications is the realization that for strong type-II superconductors with   1 fluctuations in the magnetic field can often be neglected [176]. In (12.1) the anisotropy enters only in the gauge invariant gradient term and a simple rescaling of the coordinate axes (we denote a quantity q in the rescaled isotropic system by q˜ ) x = x˜ ,

y = y˜ ,

z = " z˜ ,

(12.161)

together with a scaling of the vector potential, ˜z A , (12.162) " will render this term isotropic. As a consequence, we obtain the scaling rules for the magnetic field, Ax = A˜ x ,

Ay = A˜ y ,

Az =

B˜ y B˜ x , By = , Bz = B˜ z . (12.163) " " Inserting this result back into the free energy functional (12.1), the last two terms describing the magnetic field energy are transformed to   2  1 B˜ ⊥ 3 2 ˜ Gm = dr + Bz 8 "2   B˜ ⊥ · H ⊥ ˜ + Bz Hz . (12.164) −2 " Bx =

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Hence, we have removed the anisotropy from the gradient term but have reintroduced it in the magnetic energy term: in general it is not possible to isotropize both terms in the Gibbs energy simultaneously. However, depending on the physical question addressed, we can neglect fluctuations in the magnetic field. Problems such as vortex pinning or vortex lattice melting involve the coherence length  or the intervortex distance a0 = (¥0 /B)1/2 as their natural length scales; in strongly type II superconductors (with   ) or for large magnetic fields (with a0 < ) the length scales  and a0 are small compared to the scale  of variations in the magnetic field. In such situations the magnetic field is uniform on the natural length scale of the problem and we can adopt a mean-field decoupling scheme, where we first minimize the magnetic field energy Gm with respect to B˜ and then insert the resulting uniform field back into the free energy. More rigorously, let us consider the case  → ∞ or, equivalently, charge e → 0. The coupling between the order parameter ¦ and the gauge field A is given by the gradient term |[∇/i+(2e/c)A]¦ |2 and vanishes in the limit e → 0. The external magnetic field then merely fixes the average density of vortices.Hence the approach is exact for the case of an uncharged superfluid. Minimizing the magnetic field energy Gm we obtain B˜ = ("Hx , "Hy , Hz ), corresponding to B = H in the original system. Thus in the rescaled system the magnetic field is reduced to B˜ = "# B

(12.165)

as compared to the field in the original system. Formally, the deviation between the direction of B and H (cf. (12.28)) is beyond the scaling approach, as the uniformity of the magnetic induction requires H  Hcc1 /"# . On the other hand, proper transformation of the free energy (12.54) to the anisotropic situation allows for a correct derivation of the results (12.58) and (12.59), see [224]); hence, depending on the situation, the scaling rules might be used even beyond their formal regime of applicability. Next, let us transform energy and temperature: Since the volume scales as V = " V˜

(12.166)

all energies E and temperatures T scale as E = " E˜ ,

T = " T˜ .

(12.167)

The latter can be easily understood from the invariance condition on the Boltzmann factor, ˜ = exp(−G/T). An interesting subtlety ˜ T) exp(−G/ is the distinction between the microscopic temperature entering the Ginzburg–Landau functional via the parameter ˛ and the fluctuation temperature T entering in the Boltzmann factor: The appearance of an additional temperature dependence in the “effective Hamiltonian” (= Ginzburg–Landau functional) is a consequence of the partial summation over microscopic degrees of freedom in the partition function when going over from the microscopic formulation in terms of electronic degrees of freedom to the phenomenological description in terms of the order parameter ¦ . According to the above derivation it is only the fluctuation temperature T determining the statistical mechanics of the macroscopic wave function ¦ which is rescaled, whereas the microscopic temperature determining the size of ˛ and hence of ¦0 , , and  remains unchanged. Finally, when discussing vortex pinning in anisotropic superconductors we will need the scaling rule for the disorder: the latter is introduced into the model via spatial disorder in the GL coefficient ˛(r) describing disorder in the transition temperature Tc , and/or by spatial variation of the effective mass m (r) describing disorder in the mean free path l. Usually, the disorder landscape is characterized by a Gaussian distribution , e.g., ˛(r) = ˛0 + ı˛(r) with ı˛ = 0 and ı˛(r)ı˛(r  ) = ˛ ı(r − r  ), and similarly for m (r) = m0 + ım (r) with ım  = 0 and ım (r)ım (r  ) = m ı  ı(r − r  ). We first concentrate on the scaling of Tc -disorder: In the isotropized system the correlator reads ı ˛(˜ ˜ r )ı ˛(˜ ˜ r  ) = ı˛[r(˜r )]ı˛[r(˜r  )] = ( ˛ /") ı(˜r −  r˜ ), thus the disorder strength ˛ scales as ˛ = " ˜˛ . The second type of disorder is generated by the spatial variation of the mean free path [10,18] and can be described by a variation of the effective masses m(r) and M(r). For a layered superconductor, the disorder in m and M is due to disorder within the conducting plane and between adjacent planes, respectively, and thus in general the two need not to be the same af-

12 Vortex Matter

ter rescaling. The difference between the disorder in m and M is only relevant in the small angle regime |# | < ", since for angles larger than " the vortices are redirected mainly along the c-axis after rescaling and thus disorder in M can be neglected. Except for this small angle regime, the disorder in the mean free path can be treated as a scalar field and therefore transforms in the same manner as the disorder in Tc . We then obtain the more general rule

= " ˜

(12.168)

which is valid as long as the discreteness of the layered structure is not important. Note, that defining the scale transformation according to (12.161), the parameters  and  are not rescaled as the planar coordinates are not affected by the transformation,  = ˜

and

 = ˜ .

(12.169)

We are now in a position to set up the general scaling rule for transferring results from isotropic superconductors to anisotropic materials: Consider a uniaxially anisotropic superconductor (axis  z) characterized by the planar coherence length  and London penetration depth , the anisotropy ", and the scalar disorder strength , in an applied magnetic field H enclosing an angle # with the xy-plane, at a temperature T. Let Q be the desired quantity for which the isotropic result Q˜ is known. Then we obtain Q for the anisotropic superconductor by the scaling rule: ˜ # H, T/", , , /") . Q(# , H, T, , , ", ) = sQ Q(" (12.170) Typical scaling factors are sV = sE = sS = sT = " for volumes, energies, actions, and temperatures, and sB = sH = 1/"# for magnetic fields. We wish to point out that the scaling rule (12.170) is not unique: Defined via a mathematical construction rendering the gauge invariant gradient term in the Ginzburg–Landau functional isotropic. It is clear that there exist alternative transformation rules achieving the same goal. In particular, here we have chosen a transformation which leaves the main planar parameters  and  of the superconductor invariant, with the consequence that volume, energy, temperature, and disorder are rescaled. Another transformation (e.g., the transformation used by Klemm

535

and Clem [126]) leaves the volume (and hence also energy, temperature, and disorder) invariant. However, the planar parameters  and  are rescaled. The important point to notice is that all these consistent sets of scaling rules are equivalent to the originalrule (12.170) (see also the discussion in [225] and [226]). The scaling rule (12.170) can be used in various ways: The most straightforward application is the transformation of an expression Q˜ known for isotropic superconductors to the result Q describing an anisotropic material and we give an example below. But even in the absence of a formal expres˜ the scaling rule (12.170) provides useful sion for Q, insight into the behavior of anisotropic materials; e.g., assume a quantity Q(H) has been measured as a function of field H  c, then the scaling rule (12.170) predicts the complete angle and field dependence Q(# , H),or conversely,allows for a consistency check of a measured data set Q(# , H) through an appropriate data collapse. We begin with an example of a straightforward application of (12.170), illustrating how to transform a known isotropic expression Q˜ to the anisotropic situation. For this demonstration we choose the singlevortex elasticity "l ; other examples will be discussed below in the context of vortex lattice melting (angular scaling of the melting line, see Sect. 12.7.2), vortex pinning (Sect. 12.9), and vortex creep (see Sect. 12.9.4). The elastic energy of a single vortex is given by (see Fig. 12.4)    1   2 ⊥ 2 F[u] = dz "l (# )(@z  ux ) + "l (# )(@z  uy  ) . 2 (12.171) Transforming the in-plane tilt energy to the isotropic system we obtain the relation 

"l (# )

(ıux )2 (ı u˜ x )2 = " "˜l .  ız ı z˜ 

(12.172)

Since  is invariant, "˜l ≈ (¥0 /4)2 = "0 , where we again concentrate on the short wavelength limit of the elastic modulus. Furthermore, the amplitude ıux is not affected by the rescaling and therefore ı u˜ x /ıux = 1. On the other hand, the length ız is affected by the transformation (12.161): Let us consider a segment of length ll directed along the z -axis

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and let us transform this “longitudinal length” to the rescaled system. There, we have ˜ll2 = ˜lx2 + ˜ly2 + ˜lz2 and using (12.161) we obtain ˜ll2 = ll2 (cos2 # + sin2 # /" 2 ). With the definition (12.26) for the angle-dependent anisotropy parameter "# , we find that longitudinal lengths ll scale according to the rule ll =

" ˜ ll , "#

ll  B ,

(12.173)

and therefore also ız /ı z˜  = "/"# . The final expression for the in-plane elasticity then is "l = "0 " 2 /"# ,in agreement with the result (12.38) of the conventional approach. When transforming the out-of-plane tilt energy we should take care about the change of angles due to the scale transformation: The transformed vectors ı u˜ = ıuy  (0, − sin # , cos # /") and ı z˜ = ız (0, cos # , sin # /") are no longer orthogonal. Orthogonalizing, we obtain "l⊥ (# )

(ıuy  )2 (ı u˜ ∧ ı z˜ )2 = " " ˜ , l ız ı z˜  3

(12.174)

and the final expression for the out-of-plane line tension then is "l⊥ (# ) = "0 " 2 /"#3 , in agreement with (12.38).Here,we have used that ı z˜ = ı z˜  = ("# /")ız . As an additional result, we obtain the scaling rule for “transverse lengths”, l t , ˜l t ⊥ ex . (12.175) Next, let us use the scaling relation (12.170) to predict the angular dependence of a quantity for which we do not know an explicit expression but only its field dependence along some particular direction, usually along the c-axis. As a first simple example we consider the scaling behavior of the inplane resistivity: the scaling factor for the in-plane resistivity is s = 1 and using (12.170) we obtain (# , H) = ("# H) ≈ (sin # H). In addition, for # > ", after rescaling, the magnetic field is mainly directed along the c-axis and thus the Lorentz force is essentially independent of the direction of the current in the plane. Going back to the original system, one then expects the in-plane resistivity to be independent of the angle between the magnetic field and the current; these predictions are in good agreement lt = "# ˜lt ,

lt ⊥ B ,

˜l t ⊥ B˜ ,

with the experimental findings of Iye et al. [227] (see also [228]). In particular, assuming a slight misalignment of the field away from the ab-plane direction, the above arguments offer a natural explanation for the angular independence of the dissipation for the case where both the current and the magnetic field are (nominally) aligned with the superconducting planes [227]. Similarly, the anisotropy of the critical current density as measured by Roas et al. [66] in YBCO and by Schmitt et al. [67] in BiSCCO thin films exhibits all the features predicted by the scaling rule (12.170): The latter predicts a dependence of the in-plane critical current density jc on the angle # in the form jc (# , H) = jcc ("# H), where jcc (H) denotes the critical current density measured in a field H  c. In large magnetic fields the critical current density jcc (H) decreases with increasing field H. Hence changing the direction of the magnetic field leads to a dependence of the in-plane critical current  density jc (# , H) on the angle # through the combi nation "# H, resulting in sharp maxima of jc (# , H) when the field is aligned with the superconducting planes and a sharpening of these maxima with increasing field amplitude H. A detailed discussion of the angular dependencies of thermodynamic and electromagnetic properties in anisotropic superconductors as predicted by (12.170) and observed in various experiments can be found in [229]. We proceed with the more complex analysis of the general resistivity tensor describing the electric transport in the vortex state [230], Ei = iks (# , H)jk + ika (# , H)jk ,

(12.176)

where iks (# , H) and ika (# , H) denote the symmetric and antisymmetric (in H) response tensors. The basic idea then is to write this equation for the isotropized system and then replace all quantities by their anisotropic counterparts.In an isotropic system ˜ H ˜ of the symmetry is broken by the direction n˜ = H/ the magnetic field alone. The most general symmetric and antisymmetric tensors then take the form   ˜ = ( ˜ ıik + ˜ (H) ˜ n˜ i n˜ k , ˜iks (H) ˜ H) ˜ = ˜ H (H)" ˜ ijk n˜ j . ˜ika (H)

(12.177)

12 Vortex Matter

In order to generalize the relations (12.177) to the (uniaxially) anisotropic situation we have to rescale current densities, electric, and magnetic fields using the relations ˜j = j⊥ , ⊥

E˜ ⊥ = E ⊥ ,

H˜ ⊥ = "H ⊥ ,

˜jz = jz , "

E˜ z = "Ez ,

H˜ z = Hz ,

(12.178)

which follow from the scaling laws for A, B, and the London formula for the current density. We start from (12.176) rewritten for the isotropized system ˜ and H˜ with the help of (12.178) by and replace ˜j, E, their values in the original anisotropic system. The two equations for the symmetric part then read ˜ ⊥ + ( ˜ ˜ (H) ˜ E ⊥ = ( ˜ H)j ˜ H) ˜ ˜ (H) ˜ H˜ ⊥ H ˜ ⊥ ·j⊥ ) ( ˜ ⊥ (H ˜z ˜ H) H × + jz , 2 2 " H˜ H˜ (12.179)  2 ˜ ˜ ( ˜ H) H ˜ z jz Ez = 1 + ˜ (H) "2 H˜ 2 ˜ ˜ (H) ˜ H ˜ z (H˜ ⊥ ·j⊥ ) ( ˜ H) + , (12.180) " H˜ 2 and similar expressions are found for the antisymmetric part describing the Hall effect, ˜y ˜ H ˜z ˜ H (H) H jy + j , (12.181) ˜ ˜ z " H H ˜ H˜ x ˜ H (H) H˜ ˜ z jx − Ey = ˜ H (H) j , (12.182) ˜ z " H˜ H ˜y ˜ H˜ x ˜ H ˜ H (H) ˜ H (H) Ez = jy − j . (12.183) ˜ ˜ x " " H H

˜ Ex = −˜ (H) H

The relations (12.179)–(12.183) can be further processed in two ways: Expressing the resistivity tensor in terms of its isotropic counterpart we find (˛, ˇ ∈ {x, y}) s,iso s ˛ˇ (# , H) = ˛ˇ ("# H) , a,iso a (# , H) = ˛ˇ ("# H) , ˛ˇ

1 s,iso  ("# H) , " ˛z 1 a,iso a ˛z (# , H) = ˛z ("# H) , "

(12.184)

s ˛z (# , H) =

(12.185)

s (# , H) = zz

1 s,iso  ("# H) . " 2 zz

537

(12.186)

These results emphasize the modifications occurring in going from an isotropic system to the uniaxially anisotropic material characterized by the parameter ". In fact, the above derivation can be short-cut by proper application of the scaling laws (12.178) for current densities and electric fields, telling us that each appearance of z is associated with one factor 1/". In reality, however, the “equivalent” isotropic system is usually not available. A more practical way to express the above results then is in terms of quantities measurable in the anisotropic system. This is particularly convenient for the antisymmetric (Hall) response where all coefficients can be expressed H through xy (H) (= −˜ H (H)), the Hall coefficient measured in the geometry with j  yˆ , E  xˆ , and H  zˆ , H ika (# , H) = −xy ("# H)"ijk

Hj . "# H

(12.187)

In particular sin # , "# Hy a H . (# , H) = xy ("# H) zx "# H

a H (# , H) = xy ("# H) xy

(12.188)

Similarly, the symmetric response can be expressed z through the longitudinal resistivity xx (H) (= (H)), ˜ measured with the magnetic field H aligned with x the material’s zˆ -axis, and the response xx (H) (= (H) ˜

˜ (H)) measured in the force-free geometry with j  H  xˆ . Regarding the regime of applicability of the scaling rules (12.170) we point out that, in spite of starting from a GL-type description, the scaling approach is not limited to the regime near Tc . In fact, the above scaling rules can be obtained as well by starting from the London equations which are valid throughout the entire temperature regime. The scaling rules for point-like disorder will not be changed as long as the anisotropies in the penetration depths and in the vortex core size remain the same. On the other hand, extended pinning centers will be elongated (by the

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factor 1/") along the c-axis and this might give rise to enhanced pinning for magnetic fields aligned with the c-axis, cf. Sect. 12.10.1 where pinning by columnar tracks is discussed (the critical defect size along the c-axis beyond which correlated pinning appears is given by c = ").Also,the scaling approach can be used for the case of layered superconductors as long as the discreteness of the structure is not important. The crossover between quasi-2D and 3D anisotropic behavior depends on the physical quantity of interest, however, the regime where the anisotropic description is valid is usually large. Finally, the scaling rule (12.170) clarifies the following features of anisotropic superconductivity: First, the effect of anisotropy is to reduce the field component in the superconducting planes and, second, to enhance the effective strength of the pinning, both favorable effects in view of technological applications of the new materials. On the other hand, the anisotropy increases the temperature of thermal fluctuations, favoring phenomena such as thermal depinning and melting of the vortex lattice, effects which are scientifically very interesting, but rather undesired in view of applications.

12.7

Statistical Mechanics

Thermal fluctuations can have a dramatic effect on the vortex system in type II superconductors: strong fluctuations produce vortex lattice melting [11–13, 231] and trigger a decoupling transition in layered superconductors [64, 71–76]. Also, they change the constitutive B(H) relation near the lower critical field Hc1 [11]. In the presence of pinning, thermal fluctuations smear the pinning landscape and lead to the depinning of the vortex lattice [23]. Furthermore, thermal activation of vortex segments or vortex bundles produces an exponentially suppressed but finite creep motion [14–17]. The strength of thermal fluctuations is conveniently quantified through the dimensionless Ginzburg number Gi which derives from a comparison of Tc with the condensation energy in the coherence volume. For a bulk superconductor [232] (we set the Boltzmann constant to unity, kB = 1),

Gi =

1  Tc 2 1  Tc 2 = , 2 2 Hc0 8 ""00 0 "03

(12.189)

while for a 2D thin film [233] Gi2D =

Tc . 2"00 d

(12.190)

Here, Hc0 and "00 = 2"0(0)d are the Ginzburg– Landau expressions for Hc and "0 extrapolated to T → 0, cf. (12.12). The condition 1 − T/Tc = Gi determines the width of the critical region around the transition temperature Tc ; note that this width increases with field, Gi(H) = Gi1/3 [H/Hc20 ]2/3. In conventional bulk superconductors the Ginzburg number is small, Gi < 10−8 typically. Since Gi ∝ Tc4 /" 2 fluctuations gain dramatically in importance in highTc superconductors; for YBa2 Cu3 O7 we have Gi ≈ 0.01 (0 ≈ 1000 Å, 0 = 12 Å, " ≈ 1/8, Tc ≈ 90 K), while in layered Bi2Sr2 CaCu2 O8 thermal fluctuations are predominantly of 2D character with Gi2D ∼ 0.03 (0 ≈ 1400 Å, d = 15 Å, Tc ≈ 90 K). The phenomena of vortex lattice melting, layer decoupling, thermal depinning, and thermal creep then are particularly prominent in the copper-oxide based high-Tc superconductors and to a somewhat lesser degree in other anisotropic superconductors such as NbSe2 and the organic BEDT superconductors. Below, we concentrate on the thermodynamic phase transitions of vortex lattice melting and layer decoupling. An early suggestion that the vortex lattice may undergo a melting transition is due to Eilenberger [234]. Later, vortex lattice melting in 2D thin films has been proposed and analyzed by Huberman and Doniach and by Fisher [235, 236]; early work on bulk systems is due to Br´ezin, Nelson, and Thiaville [237]. Attention that this transition should be prominent in high temperature superconductors has been drawn by Nelson, by Houghton, Pelcovits, and Sudbø, and by Brandt [11–13]. The first-order nature of this transition has been conjectured in the work of Br´ezin, Nelson, and Thiaville [237] and numerical evidence for a sharp transition has been derived from Monte-Carlo simulations [238–242]. Similarly, experimental work on bulk vortex lattice melting has proceeded in steps, with early work by Gammel et al. [243]. First experimental results hinting towards a discontinuous transition have been reported

12 Vortex Matter

by Charalambous et al., Safar et al., and Kwok et al. [36–38] and clear thermodynamic evidence through the measurement of jumps in the magnetization and in the entropy (latent heat) has been provided by Pastoriza et al., Zeldov et al., Welp et al., Schilling et al., and Junod et al. [39–43,244]. While melting in systems made from point-like objects seems obvious, melting of a line crystal appears to be a more difficult concept — what is the elementary segment of the line which should be identified with an independently fluctuating degree of freedom? A simple answer is provided by the cage model: consider a line trapped in the potential produced by the other vortices in the lattice.The thermal displacement u of the line involves tilt modes with energy "l (u/L)2 L where L is the length of the segment. On the other hand, this displacement produces a shear interaction with the vortices defining the cage (going beyond the nearest neighbors) with an energy c66 (u/a0)2 a20 L and comparing the two energies we √ find the length L ≈ "l /c66 ∼ a0 of the fluctuating segment. In an anisotropic material with H  zˆ the line elasticity is reduced by the factor " 2 and hence L ∼ "a0 , in agreement with anisotropic scaling. Finally,we can expect the crystal to melt when the mean 1/2 thermal displacement u2 th becomes of the order of the lattice constant a0 ; the equipartition theorem √ tells us that u2 th ∼ TL/"l ∼ T/ c66 "l ∼ Ta0 /"0 , and hence the melting line follows from the relation T/"0 a0 = const.; in an anisotropic material this is modified to read T/""0 a0 = const. A more rigorous argument starts from the London free energy functional of the vortex system, see (12.62). In the dense vortex system of a strongly type II superconductor there is a large field regime (  a0  ) where we can neglect screening ( → ∞) and ignore the cutoff ; expressing √ all lengths in terms of the lattice constant a = (2/ 3)1/2a0 (e.g., x  = s /a ) and including anisotropy via scaling (cf. Sect. 12.6) we arrive at theform

 ""0 a  1 . dx  · dx FL {x  } = 2 , |x  − x | (12.191) The statistical mechanics of the vortex system (as given by the partition function over the free energy (12.191)) then is completely determined by the pa-

539

rameter ratio ""0 a /T containing all the dependencies on the field B and on the temperature T; the melting line will be determined by some universal constant derived from the dimensionless integral in (12.191). Using the numerical results in [242] we obtain a melting line of the form Tm = 0.098 ""0 a = 0.105 ""0 a0 ,   ""0  2 . (12.192) Bm = 0.069 Hc2 (T) T Below we will further elucidate the various aspects of vortex lattice melting in anisotropic and layered superconductors. Thermally induced layer decoupling is due to phase fluctuations in the superconducting layers and results in the destruction of phase coherence in the direction perpendicular to the layers. Such a transition has first been proposed for the zero-field situation by Friedel [245], however, subsequent work has shown that this transition is unfavorable [246]. Later,the concept of a decoupling transition has been extended to the finite field situation (with B  zˆ ) by Feigel’man et al. and by Glazman and Koshelev [64,71,72] with the idea that fluctuations of pancakevortices will help the transition. More refined discussions have been given by Daemen et al., by Frey, Nelson, and Fisher, by Horovitz and Goldin, and by Dodgson et al. [73–76]. Work related to layer decoupling has been carried out on BiSCCO crystals [39, 247–249] and on MoGe/Ge multi-layers [250, 251]; however, while detailed and consistent experimental results on vortex lattice melting are available today, much less is known experimentally about the decoupling transition in layered superconductors. Below we develop various ideas related to vortexlattice melting and layer decoupling. After a brief review of fluctuations in thin-film and layered superconductors, we start with a discussion of the Lindemann approach to melting and decoupling, providing us with a quick overview over the various melting lines in 2D superconducting films, layered, and anisotropic superconductors. We proceed with a more in-depth discussion of dislocation-mediated melting in 2D thin-film superconductors. Further progress beyond the Lindemann analysis can be

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made in magnetically coupled layered superconductors where we describe melting within a selfconsistent substrate model. In this system, decoupling is replaced by a defect transition, the finitefield generalization of the Berezinskii–Kosterlitz– Thouless transition in an individual superconducting layer, and we discuss this topic in detail. Further aspects such as the angular dependence of the melting line, Tc − T-scaling, the characterization of the first-order transition, the entanglement of the vortex lines in the liquid phase, and the constitutive relation B(H), are discussed as well. 12.7.1

Fluctuations in 2D Thin Film and Layered Superconductors

Superfluids and superconductors break the continuous U (1) gauge symmetry and hence cannot develop an order parameter describing true long-range order in two dimensions [252–254]. Still, we can define a local order parameter ¦ = |¦ | exp(i' ) and a superfluid stiffness J describing the low energy excitations in the film,   2 J F= d2 R ∇' . (12.193) 2 In a thin film superconductor (of thickness d) we have "0 d , (12.194) J=  with a well defined limit e → 0; screening is relevant on the large scale eff = 22 /d which we ignore for the time being, allowing us to replace the gauge invariant phase difference '˜ by the phase ' of the superconductor. Besides the Gaussian phase fluctuations described by (12.193), the compactness of the phase variable ' allows for vortex excitations which cost an energy Evortex = J ln(L/R0 ) ,

(12.195)

where L denotes the system size and R0 is a short scale cutoff (for a superconductor, R0 = ). The low temperature phase is characterized by the absence of free vortices; vortex-pair excitations merely renormalize the stiffness J → s (T)J with s < 1 [255],

while the Gaussian phase fluctuations produce an algebraic decay of the order parameter correlator,  R  2 0 , ¦ ∗ (R)¦ (0)th = e[' (R)−' (0)] th/2 ∼ R T . (12.196) = 2s J At high temperatures the entropy T ln L2 can compensate for the vortex energy and vortex pairs unbind at the Berezinskii–Kosterlitz–Thouless temperature [256,257]  TBKT = s∞ J , s∞ = s (TBKT ) > 0 . (12.197) 2 The coefficient s∞ renormalizing the strength J of the interaction is not universal and depends on the short scale properties of the system; for the 2D XY-model on a square lattice Monte Carlo simulations [258] provide the result s∞ ≈ 0.569 while typical values s∞ ∼ 0.5 − 0.9 are found in experiments [259, 260]. At high temperatures T > TBKT the order parameter correlations (12.196) decay exponentially with a correlation length   BKT (T) ∝ exp b TBKT /(T − TBKT ) , (12.198) diverging upon approaching TBKT from above. The parameter b is again non-universal: Monte Carlo simulations on the 2D square lattice XY-model provide the value b ≈ 3.17 [261], while typical values derived from measurements on thin superconducting films are in the range 2–16 [259, 260]). On scales R < BKT (T) the vortex pairs are still bound, whereas the individual vortices become free on scales R > BKT (T), hence the density of free vortices is −2 (T) . nv ∼ BKT

(12.199)

The presence of free vortices leads to the screening of the transverse stiffness s J on the scale BKT , hence s J(K) ∝ [1 + (BKT K)−2 ]−1 and s J(K = 0) = 0. At the transition the exponent  takes the universal value (TBKT ) = 1/4 and the superfluid density vanishes with a universal jump [262]. For a bosonic superfluid with superfluid density n2D s and boson mass m this jump is given by the expres2 sion n2D s (TBKT )/TBKT = 2m/ . In a superconducting film the superfluid density relates to the order parameter via |¦ |2 d = ns d/4 = n2D s /4 and

12 Vortex Matter 2 the jump is larger, n2D s (TBKT )/TBKT = 8m/ ; on the other hand, as the superconductor is charged, the superfluid density is conveniently defined via s2D = −js d/A (with js and A the supercurrent density and the gauge potential) and we find the jump in s2D at TBKT ,

s2D(TBKT ) 8c = 2 . TBKT ¥0

(12.200)

The transition itself is continuous with a singular + part in the free energy density fsing (T → TBKT ) ∝ −2 BKT and thus all derivatives are continuous. In a superconducting film we have to account for screening as well as the underlying mean-field type temperature dependence of the parameters: making use of the universal jump relation (12.200) we find that at TBKT screening becomes important on the large scale eff (TBKT ) = ¥02 /16 2TBKT ≈ 2 cm/TBKT [K]; free vortices only appear beyond this large distance and hence the rounding of the transition is weak. The mean-field type temperature dependence mainly enters the discussion via the parameter "0 d ≈ "00 d(1 − T/Tc0) ≈ 2"0(0)d(1 − T/Tc0), with Tc0 the mean-field transition temperature describing the onset of pairing and the appearance of a local order parameter |¦ | > 0 (here, "00 is the Ginzburg–Landau expression for "0 extrapolated to T → 0, cf. (12.12)). The BKT transition temperature then follows from a self-consistent solution of the relation s∞ "0 (TBKT )d/2 = TBKT , cf. (12.197), Tc0 Tc0 = . 1 + 2Tc0/s∞ "00 d 1 + (4/s∞ )Gi2D (12.201) In a weakly fluctuating film the Tc -reduction is small and so is the jump in the superfluid density at the transition, s2D(TBKT )  s2D (0); on the other hand, in the strongly fluctuating case the Tc -reduction is large and s2D (TBKT ) ∼ s2D (0). The second place where the underlying meanfield dependence shows up is the correlation length BKT ; replacing TBKT in (12.198) by the temperature dependent expression (s∞ "00 d/2)(1 − T/Tc0) ≡ xTc0(1 − t), cf. (12.197), we find TBKT =

541

x x(1 − t) TBKT 1−t = → T − TBKT t − x(1 − t) 1 + x t − x/(1 + x) Tc0 − T ≈ , (12.202) T − TBKT where we have used that TBKT /Tc0 = x/(1 + x) remains close to unity in the last expression.As a result we find the following modified form for the coherence length,   BKT (T) ≈ (TBKT ) exp b(Tc0 − T)/(T − TBKT ) , (12.203) with (T) the Ginzburg–Landau coherence length [263]. In layered superconductors the Josephson coupling between the superconducting planes introduces a finite anisotropy parameter " 2 = m/M and defines the new length scale   = d/". We then have to compare the crossover parameter cr = 2 2(0)/ 2 defining the 3D fluctuation regime around Tc with the 2D fluctuation parameter cr2D ≈ Gi2D . For a strongly coupled material with cr  cr2D the crossover from 3D-bulk to 2D-layered behavior can be discussed within a mean-field description. Here, we are interested in weakly coupled material with cr  cr2D where the transition is strongly influenced by the 2D-fluctuations [70, 264]. Whereas the conventional superconductors as well as the YBCO compound belong to the first class of materials, the layered Bi- and Tl-based materials are members of the second group with cr ∼ 10−3 . Starting from decoupled layers where fluctuations are governed by vortex pairs with logarithmic interaction, the coupling between the layers introduces a linear confinement potential on scales R >  , see (12.136), leading to a cutoff in the BKT-behavior of the system. The 3D-bulk transition at Tc occurs when the BKT-coherence length becomes of the order of  , BKT (T) ≈ ˛ , hence Tc ≈ TBKT +

b(Tc0 − TBKT ) , {ln[˛ /(TBKT)]}2

(12.204)

and the width ıTf3D of the 3D-fluctuational regime can be estimated from the condition [265] ıTf3D | @T BKT | ≈ ˛ ,

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2b(Tc0 − TBKT ) 2(Tc − TBKT ) = . ln[˛ /(TBKT )] {ln[˛ /(TBKT)]}3 (12.205) Below the bulk transition at Tc and outside the 3D-fluctuation regime the system still develops strong 2D fluctuations on length scales R < ˛ . With the 2D phase correlator [' (R) − ' (0)]2 th ≈ (T/s∞ "0 d) ln(R/) we find that strong phase fluctuations ([' (˛ ) − ' (0)]2 th ∼ 1) appear within a temperature regime ıTf3D ≈

ıTf2D ≈ Tc0

Tc0 s∞ "00 d

ln(˛ /(TBKT ))

(12.206)

below Tc0. This result for the 2D phase fluctuations is larger than the usual 2D Ginzburg criterion [233], cf. (12.190). The numerical ˛ determining the length scale ˛  where the interlayer coupling becomes important can be estimated from Monte Carlo simulations of the 3D anisotropic XY-model [266]: Comparing the relative Tc -shift (Tc − TBKT )/TBKT ≈ 0.28 obtained numerically for a system with anisotropy " 2 = J⊥ /J = 0.02 in the couplings J⊥, with the theoretical value b/[ln(˛/")]2 expected from the condition BKT (Tc ) ≈ ˛a/" one arrives at the estimate ˛ ≈ 4 (here, a denotes the lattice constant and we use a value b ≈ 3.2). Numerical estimates for the various temperatures in the layered BiSCCO superconductor then take the values (we choose parameters Tc0 ≈ 100 K, d ≈ 15 Å, 0 ≈ 18 Å, 0 ≈ 1400 Å, " ≈ 1/100, resulting in "00 d ≈ 1500 K, and we choose s∞ = 0.57, b = 3.2, ˛ = 4): Tc0 −TBKT ≈ 20 K, Tc − TBKT ≈ 2.5 K, ıTf3D ≈ 1 K, ıTf2D ≈ 60 K. We see that fluctuations strongly (by 20 K) reduce the mean field transition temperature Tc0 as we go to the thin film geometry, while the small coupling between the layers induces only a small upward shift (by 2.5 K) of Tc , hence BiSCCO indeed is a “strongly 2D-like” material. An interesting proposal regarding the transition of a layered material into its bulk superconducting state goes back to Friedel [245], who suggested that the proliferation of low-energy in-plane Josephson vortex-loop excitations should lead to a decoupling of the layers above some critical temperature Tc = Tloop < TBKT , producing a system of electro-

magnetically coupled 2D superconducting layers. However, as shown by Korshunov [246] this interesting scenario is never realized as the loop transition temperature always appears above the Berezinskii– Kosterlitz–Thouless transition temperature, Tloop = 4"0 d = 8TBKT (see also [82]). We will return to this issue below, see Sect. 12.7.5. 12.7.2

Lindemann Analysis of Vortex Lattice Melting

The theoretical analysis of melting is a difficult subject and consistent melting scenarios are known in few special situations only, notably the dislocationmediated melting scheme in two dimensions discussed in Sect. 12.7.4 below, or the self-consistent stability analysis applicable to the pancake-vortex system in layered superconductors presented in Sect. 12.7.5. On the other hand, a simple and practical scheme to locate a first-order melting transition is given by the Lindemann analysis providing a stability criterion for the lattice phase: Making use of the continuum elastic description of the vortex lattice, one calculates the mean-squared thermal displacement u2 th . The Lindemann criterion [267] then states that the crystal will undergo a melting 1/2 transition once u2 th becomes comparable to the lattice spacing a0 , u2 th |Tm ,Bm ≈ cL2 a20 . The Lindemann number cL is usually chosen to be a constant of order cL ≈ 0.1 − 0.3. Though not rigorous, the Lindemann melting scenario has proven very useful and reasonably accurate in predicting the positions of first-order melting transitions in general [268] and the line shape of the vortex-lattice melting transition in particular. Below, we first apply the Lindemann analysis to the vortex lattice melting in 2D thin superconducting films and proceed with the discussion of magnetically coupled layered systems, before going over to Josephson coupled [231] and continuous anisotropic systems [12]. Our main task is the calculation of the meansquared thermal displacement   d3 k T 2 u th ≈ (12.207) 3 2 (2) c66K + c44 (k)kz2  T , + c11 (k)K 2 + c44 (k)kz2

12 Vortex Matter

with the shear modulus c66 (cf. (12.73)) and the dispersive tilt and compression moduli c44(k) and c11 (k), cf. (12.68) and (12.75) (the appearance of two modes in (12.208) accounts for the vectorial character of the displacement field u involving two components ux and uy ). Here, we concentrate on the first term in (12.208) involving transverse shear/tilt modes and neglect the second term adding a contribution from compression modes which is smaller than the shear/tilt term (although comparable in size). In anisotropic material, the tilt modulus involves the bulk term (12.83) and the single vortex contribution (12.84). The second term in the single c vortex tilt c44 (cf. (12.84)) is due to the electromagnetic coupling between the layers and is the only term in c44 surviving the limit " → 0 (decoupled layers). The electromagnetic contributionto the tilt modulus is strongly dispersive and produces the large stiffness "l ≈ "0 /2 of the vortex lines in the long-wave-length limit kz < 1/. With increasing kz the electromagnetic stiffness decays ∝ 1/2 kz2 and the line tension crosses over to the well known result "l ≈ " 2 "0 for the anisotropic superconductor as kz increases beyond 1/" (note that this residual line tension is due to the Josephson coupling and is relevant only for " > d, where d denotes the layer separation). The expression given in (12.84) is valid for small displacements, in the elastic regime. For large displacements ukz > 1 the logarithm in the second term of (12.84) should be cut on 2/u rather than kz [65]. In our analysis below we replace the logarithm by the factor 2 kz2 /(1 + ˇ2 kz2 ) with ˇ = 1/ln(1 + 42 /cL2 a20 ) producing a smooth interpolation between the hard and soft tilt modes at large and small wave-lengths, respectively. 2D Superconducting Films We start with the Lindemann analysis of the vortex lattice melting in an individual superconducting layer. In calculating the mean squared displacement (12.208) we can drop the tilt energy ∝ c44 and the integral over kz provides a factor 2/d. The integration over K is log-divergent — as first-order melting is expected to involve short scales we cut the integration on a few lattice spacings ∼ ˛a0 and obtain the

ratio

u2 th ln ˛ T ≈ . a20 2 c66da20

543

(12.208)

The resulting melting temperature is independent of the magnetic field, 2D ≈ Tm

1 Tc cL2  "0 d "0 d ≈ ≈ , 2 ln ˛ 70 140 2Gi2D

(12.209)

where we have chosen a cutoff parameter ˛ ≈ 2 and a Lindemann number cL ≈ 0.08 in the last two expressions. These values are compatible with results from Monte Carlo simulations on the classical onecomponent Coulomb plasma with logarithmic interactions V (R) = −e2 ln(R/L) in two dimensions [269]. The latter is characterized by the dimensionless coupling parameter  = e2/T measuring the ratio of the potential energy to the temperature T in the lowtemperature solid. Simulation of the vortex liquid phase and subsequent comparison of the free energies of both solid and liquid phases disclose a firstorder melting transition at m ≈ 140 characterized by an entropy jump S ≈ 0.4 kB (we go over from the charge plasma to vortices via the replacement e2 → 2"0 d). Similar results have been found in a direct simulation of the transition [270], m ≈ 142 and S ≈ 0.17 kB . The (weak) first-order nature of this melting phenomenon found through numerical simulations substantiates the use of the Lindemann criterion in describing the transition. On the other hand, in two dimensions the above first-order melting scenario competes with the dislocation-mediated melting scheme for which an analytic description is available, see Sect. 12.7.4. This dislocation-mediated melting scenario predicts a smooth two-step melting transition with an intermediate hexatic liquid phase between the quasi-ordered solid and the fully disordered √ liquid. The melting 2D transition at Tm ≈ Aa20 dc66/2 3 ≈ 0.023 A "0d has the same parametric dependence as the Lindemann result (12.209) (the parameter A describes the 2D renormalization of the shear stiffness at Tm ). The dislocation-mediated melting scenario is realized in the limit of a small density of dislocations, while a first-order melting transition (involving the collapse of the two smooth transitions into a single sharp one) is expected for the situation where the dislocation

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density is large at the transition,cf.[271].The numerical simulations on the 2D Coulomb plasma [269,272] as well as further results based on the lowest-Landaulevel approximation describing the situation in high magnetic fields [273–275] indicate that the 2D vortex crystal melts in a first-order transition. However, the numerical simulations accurately reproduce various characteristics of the melting transition as predicted by the dislocation-mediated melting scenario, such 2D as the position of Tm , the jump in the shear modulus c66 at the transition, and the exponent  describing the decay of the quasi-long range crystalline order. In layered superconductors,the field-independent result (12.209) describes well the situation in high magnetic fields where the in-plane shear interaction outweighs the tilt interaction, see Fig. 12.11; using parameters typical for the layered high-Tc ’s we find 2D Tm ≈ 10–15 K.

interaction dominates and we recover the 2D melting result (12.209) (using (12.211) the logarithmic divergence is cut on large distances by the tilt stiffness providing an effective substrate potential; as melting involves short distances we should cut the divergence at a small distance of the order of a lattice constant a0 , equivalent to calculating the quantity [u(R ≈ a0 ) − u(0)]2th ). For low fields B < B the short wave-length tilt modes dominate and push the melting line to high temperatures,see Fig.12.11; expanding the logarithm in (12.211) we find the melting line in the form Bem m (T) ≈

cL2 "0 d B 2ˇ T

(12.211)

  cL2 T2 2 Tc ≈ 1− 2 , B(0) 8ˇGi2D T Tc where we make use of the interpolation formula 2 (T) = (0)2 /(1 − T 2 /Tc2) in the last relation.

Electromagnetically Coupled Layered Superconductors Next we include the electromagnetic coupling between the layers while keeping " = 0 (no Josephson coupling yet). In the high-field regime (a0 < ) the shear term in (12.208) dominates over the tilt energy and we recover the field-independent 2D-result (12.209). For small fields with a0 >  the tilt energy becomes relevant; here, the electromagnetic contribution in (12.84) is the dominant one and we obtain the Lindemann criterion in the form u2th 2T 2 2 ≈ c ≈ (12.210) L "0 d a20 a20   d 1 ln(1 + 4ıˇ) + , × 4ı (4ı)1/2 with ı = 2c662 /"0 = 2 /2a20; the first term originates from the soft tilt modes with kz  > 1, while the second term involves the long wave-length modes hardened by the electromagnetic coupling and becomes relevant only at very small fields a0 / > 2 ln(/d), where the shear modulus is exponentially small, cf. (12.72). Here, we concentrate on the first term, see below for a discussion of the second term producing a reentrant melting line at very low magnetic fields. At high fields B  B = ¥0 /2 the shear

Josephson Coupled Layered Superconductors We now include the Josephson coupling between the layers which produces a finite anisotropy parameter " > 0 and a new length scale   = d/". This additional coupling becomes relevant when a0 ,  >   and favors the solid phase, thus pushing the melting line further towards high temperatures and fields. Accounting for the additional single vortex elasticity " 2 "0 in the tilt modulus c44 (k) we evaluate the Lindemann criterion in the low-field regime (a0 > ) and recover the previous result (12.211) withthe modification that soft tilt modes are cut on 1/" ˇ instead of /d,  leading to the replacement of ln(. . . )/4ı by (d ˇ/")[ln(. . . )/4ˇı + 1]. The melting line then takes the new form cL2 ""0   B Bem,J (T) ≈ m T 4 ˇ

(12.212)

  T 2 3/2 cL2 (0) Tc B(0) 1− 2 ≈  . T Tc 16 ˇGi2D  

For (0) <   the line Bem,J m goes over into the electromagnetic melting line Bem m as the temperature drops below T em ≈ Tc [1 − ˇ((0)/ )2 ]1/2

(12.213)

12 Vortex Matter

(we make use of the interpolation formula 2 (T) = (0)2 /(1 − T 2/Tc2 )). For the opposite case with strongly coupled layers, i.e.   < (0), the electromagnetic melting line Bem m is completely hidden. Continuous Anisotropic Superconductors As the melting line increases beyond B the tilt energies are dominated by the dispersive bulk term 0 c44 ≈ 4" 2 "0 /a40K 2 (see (12.83)) and we arrive at the result for the melting line in the continuous anisotropic superconductor, Fig. 12.11, " 2 "02 2 (12.214) T2   T2 2  2 cL4 T2 Hc2 (0) c2 1 − 2 . = Gi T Tc

BJm (T) ≈ 4cL4 B

At large fields (a0 <   < (0)) the 2D result (12.209) is recovered. Comparing (12.192) and (12.215) we find a Lindemann number cL ≈ 0.17; note, however, that several approximations (e.g., we have dropped the compression term in (12.208) and have simplified integrals) have been made in the calculation of the result (12.215) and hence the numerical value of the above Lindemann number should not be overinterpreted. With a temperature dependence ∝ (1 − T/Tc )˛ and ˛ > 1 the melting line is located far below the upper critical field line Hc2 ∝ (1 − T/Tc ) for T close to Tc . For strongly coupled layers, e.g., in YBCO, the melting line rises steeply at low temperatures and the suppression of the order parameter close to Hc2 has to be taken into account (in BiSCCO the melting line remains far below Hc2 and there is no suppression of the order parameter). The melting transition of the vortex lattice under high field conditions has been studied within the lowest-Landau-level approximation [276,277]: Using a high-temperature expansion (up to 9th order) and a subsequent analysis based on Pad´e and Borel–Pad´e summation techniques,the free energy of the liquid phase has been calculated and compared to the result for the Abrikosov lattice phase [278].The two lines cross (with a finite angle,indicating a first-order transition) at a reduced temperature y ≡ −[1 − T/Tc − B/Hc2 (0)]/Gi(B) ≈ −7.0, where

545

Gi(B) is the field dependent width of the fluctuation region,Gi(B) = (Gi/2)1/3[TB/Tc Hc2 (0)]2/3  Gi (B  GiHc2 (0) within the field regime considered here). The resulting melting line takes the shape  3/2 T Hc2 (0) Tc B 1− ≈ 0.08 √ − Tc Hc2 (0) Gi T (12.215) and is located at an appreciable distance away from the critical fluctuation regime. Assuming that BJm (T) is close to Hc2 (T) we obtain the scaling result 1−bJm ∼ (Gi t 2 /(1 − t)]1/3 , where b = B/Hc2 (T) and t = T/Tc denote the reduced field and temperature. This result is easily reproduced within the Lindemann approach by noting that BJm ∝ "l c66 ; taking into account the renormalization of the elastic constants due to the suppression of the order parameter close to Hc2 (see (12.4.1), "l ∝ (1 − b)(1 − t) and c66 ∝ (1 − b)2 (1 − t)) we find that BJm ∼ (Hc2 (0)/Gi)[(1 − t)/t]2 (1 − b)3 and assuming BJm (T) ≈ Hc2 (T) we recover the above scaling formula. In addition, we can obtain an estimate for the Lindemann number cL ≈ (2/73 2 )1/4 ≈ 0.15. BJm (T)

Discussion The bulk melting line (12.215) cuts the B line at 1 − T/Tc ≈ Gi/cL4  2 ; for YBCO this parameter is of order 10−4 and all the melting line is given by the bulk result (12.215). The situation is quite different in weakly coupled layered material such as BiSCCO with (0) <  : at high temperatures the melting line is strongly suppressed to fields below B and is given by the expression (12.213) with its characteristic (1−T 2/Tc2 )3/2 power law behavior;  this 3/2 powerlaw is valid provided that  / ˇ <  < a0 and sensitively depends on the values of the various parameters (e.g., assuming (0) ≈ 2000 Å, " ∼ 1/300, and ˇ ≈ 0.2 we find that T em ≈ 0.8 Tc ). Note that the results (12.212), (12.213), and (12.215) not only differ in their temperature scaling but also in their de0 pendence on the anisotropy parameter ", Bem m ∝ " , em,J 1 J 2 Bm ∝ " , and Bm ∝ " . This implies that the extraction of a numerical value for the anisotropy parameter " from experimental data on the melting line [162,279] is problematic as it is a priori unclear to which shape of the melting line the data should

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be compared to. The analysis of the melting line can provide a reliable estimate for the anisotropy parameter only if " is large enough, such that either the bulk result (12.215) is valid or the mixed electromagnetic/Josephson result (12.213) can be identified via its particular line shape. On the other hand, if the anisotropy parameter is very small, say " < 1/500, the melting line is dominated by the electromagnetic coupling and no anisotropy parameter can be extracted.

maximum Bmax mr of the order of a few Gauss realized max at a temperature Tmr ≈ Tc /3. We conclude that the low-field reentrance is a rather delicate phenomenon appearing at very low magnetic fields of the order of a few Gauss. Traces of a dilute liquid phase have been observed in Bitter decoration experiments on BiSCCO single crystals [280]. The high and low-field branches of the melting line cut in the vicinity of Tc . For the extreme case of an electromagnetically coupled system the two branches merge at

Reentrant Melting

   Tx ˇGi2D (2ˇ)1/2 cL (0) −2 ln ≈ √ , Tc (3)1/4 Gi2D d cL2 (12.217) and no solid phase can exist at high temperatures beyond Tx . Using typical parameters for the layered high-Tc materials and adopting a value cL ≈ 0.17 for the Lindemann number (cf. Fig. 12.11) we find Tx close to Tc , 1 − Tx /Tc ≈ 0.05 (note that even a small but finite parameter " pushes Tx up close to Tc , see below and Fig. 12.11). The reentrance at Tx ends up in the critical region close to Tc ; while our approach does account for the fluctuations of the phase-field via the thermal motion of vortices, it neglects fluctuations in the amplitude of the order parameter and hence our analysis breaks down in this regime. Including a finite Josephson coupling " > 0 the melting line Bem,J (see (12.213)) and the crossing m point of the lower and upper branch of the melting line are shifted towards higher temperatures, cf. Fig. 12.11,

Up to now we have considered melting driven by increasing thermal fluctuations. However, the vortex crystal is not a generic crystal as obtained from competing attractive and repulsive interactions producing a potential minimum and hence a well defined lattice constant. Rather, the interaction between the vortices (so far) is purely repulsive and the lattice constant is determined by the pressure of the external H field (but cf.Sect.12.7.8).Decreasing H,the distance between vortices increases and screening leads to an exponentially weak interaction ∝ exp(−a0 /); correspondingly the shear modulus vanishes exponentially fast, cf. (12.72), and any finite temperature melts the crystal at sufficiently low fields. As a result, the vortex crystal can melt upon decreasing the magnetic field and we obtain a reentrant melting line, cf. Fig. 12.11. Using the result for the mean-squared displacement (12.208) and the Lindemann criterion we obtain the melting condition (12.211). At low fields the parameter ı ∝ exp(−a0 /) is exponentially small, the second term in (12.211) dominates, and we obtain the reentrant branch of the melting line,    B 4cL2 "0  −2 Bmr (T) ≈ ln . (12.216) 4 (3)1/4 T Note that this result involves the long wave length tilt modulus "l ∼ "0 and hence is independent of the anisotropy parameter ". Choosing typical values for the Lindemann number and for the penetration depth we have 4cL2 /(3)1/4 ≈ 0.2 and "0 (0)(0) ∼ 105 K, resulting in a reentrant melting at Bmr (T) ≈ B /[2 ln(2. 104 K 1 − T 2 /Tc2/T)]2 with a

1−

    2 2ˇGi2D     4 ˇ −2 2 Tx 1− ln ≈ . Tc cL2 (0) (3)1/4 " (12.218) Cutting the lower branch of the melting line with the bulk melting temperature (12.215), we find a result deep in the fluctuation regime, 1 − Tx /Tc ≈ (Gi/8cL4  2 )/[ln(. . . )]2 . A similar reentrance also appears in a 2D vortex system: in this case the low-field limit (a0 > eff = 22 /d) of the shear modulus decays algebraically [281] rather than exponentially, c66 ≈ 0.46 "0 eff /a30 ∝ B3/2, and we obtain the low-field branch of the 2D melting line in the form

12 Vortex Matter

Fig. 12.11. Low-field phase diagram of anisotropic/layered superconductors. Reduced units b = B/(¥0 /(0)2 ) and t = T/Tc have been used (cL = 0.17 and parameters (0) ≈ 2000 Å, d ≈ 15 Å, Tc ≈ 100 K). The dashed line shows the result for the isolated 2D layer. The solid lines illustrate the 3D bulk results for anisotropy parameters " = 0 (only electromagnetic coupling), " = 1/1000, 1/300, 1/100, and 1/8 (data derived from numerical evaluation of (12.208) and application of the Lindemann criterion u2 th = cL2 a20 ). The dotted line traces b (t) = B/[¥0 / 2 (t)]

B2D mr

¥◦ ≈ 2 eff



37 T "0 d

2 .

(12.219)

The result for the melting line of an isolated layer is illustrated in Fig. 12.11 (dashed line). The reentrant melting lines defined by (12.212), (12.213), (12.215), and (12.216) are illustrated in Fig. 12.11. Combining the various elements in the above discussion, we can easily understand the characteristic “nose”-like shape of the melting line of a 3D vortex crystal: The melting lines of most conventional 3D crystals come with a positive slope @T p > 0, with the notable exception of the ice–water transition which involves a retrograded melting line with @T p < 0 (in this comparison between vortex and conventional crystals we have to identify the induction H with the pressure p). A soft 2D vortex crystal with c66 ∝ a−2 0 exhibits an upright melting line with 2D Tm independent of H over a large portion of the phase diagram. Going over to a 3D vortex crystal via electromagnetic coupling of superconducting layers, the presence of the vortex crystals in the other layers

547

stabilizes the solid phase; considering one 2D vortex crystal in a particular layer, the vortex lattices in the other layers provide a stabilizing substrate potential which pushes the melting transition to higher temperatures.Finally,the Josephson coupling further enhances the interaction between the layers and the melting line is pushed up towards the mean-field Hc2 line as the pancake-vortex stacks transform into increasingly stiffer vortex lines. While the upward shift of the melting line with decreasing anisotropy is a consequence of an increase in the layer coupling and the establishment of vortex lines, the reentrant part of the low-field melting line is a consequence of the collapse of the vortex-vortex interaction due to screening, rendering the vortex crystal super-soft at low fields with c66 ∝ exp(−a0 /). In summary, the coupling of the soft 2D pancake-vortex crystals into a 3D crystal of vortex lines leads to an increasingly steeper retrograded melting line and the 3D vortex crystal melts like ice, resulting in a vortex liquid phase which is denser than the crystal. Angular Dependence of Melting In anisotropic superconductors an additional degree of freedom in setting up the system is the angle  between the magnetic field and the c-axis of the material (alternatively we may use the angle # = /2− with respect to the planes).In continuous anisotropic material the upper part of the melting line involves fields B > B and the characteristic scale of the melting process is small, a0 < . In this situation we can use the scaling rule (12.170) and obtain the angle dependence of the melting line, BJm (T, # ) =

1 J B (T, /2) . "# m

(12.220)

The prediction (12.220) is in excellent agreement with experiments on YBCO crystals [282,283]. In weakly coupled layered superconductors the most interesting part of the melting line appears at low fields B ∼ B and the vortex lattice is expected to rearrange into a crossed lattice, see Sect. 12.5.5, hence simple anisotropic scaling may fail. Indeed, experiments [218–221] show that the c-axis component Hz

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of the melting field decreases linearly with the inplane component Hy . Such a behavior can be understood on the basis of the discussion in Sect. 12.5.5, see [210]: We start from the thermodynamic condition for melting (at fixed magnetic field H), gcr (Hmz ) + ıgcr (Hmz , Hy ) = gl (Hmz ) + ıgl (Hmz , Hy ) ,

(12.221)

with gcr (Hz ) and gl (Hz ) the Gibbs free energy densities of the crystal and liquid states for a field arranged along the c-axis; the corrections ıgcr (Hmz , Hy ) and ıgl (Hmz , Hy ) account for the effect of the parallel field Hy . Expanding Hmz (Hy ) = Hmz (0) + ıHmz and the Gibbs free energies (we use @H g = −B/4) gcr (Hmz ) − gl (Hmz ) = gcr (Hmz (0) + ıHmz ) − gl (Hmz (0) + ıHmz ) ≈ ıHmz ıHmz − gl (Hmz (0)) + Bl gcr (Hmz (0)) − Bcr 4 4 we obtain the shift (note that gcr (Hmz (0))−gl (Hmz (0)) = 0) ıHmz ≈ −4g(Hmz (0), Hy )/B ,

(12.222)

with g(Hmz (0), Hy ) = ıgcr (Hmz (0), Hy )−ıgl (Hmz (0), Hy ) and the jump B = Bl − Bcr > 0 in the induction. In the vortex crystal the correction ıgcr (Hmz , Hy ) ≈ fJ is given by the energy density (12.156) of the Josephson vortex lattice; assuming an efficient layer decoupling upon melting the Josephson term in (12.156) vanishes and the correction ıgl (Hmz , Hy ) involves the field energy alone, hence g ≈ fJ − Hy2/8 ≈ √ √ (Hy /2¥0) g ""0 ln("¥0 / gHy d 2 ) and we arrive at the linear (in Hy ) shift   Hy az c(T)B   1/2 ln ln . ıHmz ≈ "B 4B d 4Hz  (12.223) The factor c(T) ≡ cos(¥n,n+1 )th accounts for the suppression of the Josephson coupling due to the thermal fluctuations of the pancake vortices, cf. [210]; at the (H  zˆ ) melting transition c ≈ 0.64 [25,284].

Comparison Theory–Experiment How well do the above theoretical considerations on vortex lattice melting agree with experiments? A direct comparison can be carried out for continuous anisotropic superconductors such as YBCO — in this situation the scaling analysis combined with results from numerical simulations provide us with the theoretical prediction (12.192) which contains no unknown parameters. On the experimental side, penetration depth measurements (surface impedance) by Kamal et al. [285] provide us with the result (T) ≈ .895 (0)(1−T/Tc )−1/3 valid at high temperatures T > 80 K, with (0) = 1400 Å and Tc = 92.5 K. The accurate position of the melting line is known from experiments by Schilling et al. [42]. The comparison between the theoretical expression (12.192) and the experimental data points is given in Fig.12.12 [286]: choosing a standard value " ≈ 1/8 for the anisotropy close to Tc we find excellent agreement between theory and experiment for fields below 5 Tesla; unfortunately, the measurement of (T) and of the melting line have not been carried out on the same sample, which may be responsible for the deviations observed at higher field values. This comparison illustrates the consistency we have reached in our understanding of vortex lattice melting: once the material anisotropy and the temperature dependence of the superfluid density are known we can accurately predict the location of the melting line. The penetration depth measurements of Kamal et al. [285] on YBCO single crystals report to find a −2 ∝ (1 − T/Tc )2/3 scaling over a region of order Tc itself, while in the discussion above we have usually assumed a mean-field type scaling of the superfluid density with −2 ∝ ns ∝ (1 − T/Tc ). Indeed, close to Tc one expects that fluctuations change character √ and exhibit critical scaling: with ¦ (r) = ns exp(i' ) a two-component order parameter field, the critical fluctuations belong to the 3D–XY universality class [16] and the superfluid density is expected to scale as a power law with an exponent close to 2/3, hence −2 ∝ ns ∝ (1 − T/Tc )2/3 (note that  ∝ 2 , hence  ∝ 1/ → 0, the charge e → ∞ becomes relevant, and the system seems to approach a type I behavior very close to Tc ). Such a critical scaling then

12 Vortex Matter

implies a bulk melting line which scales with a 4/3 power law,   T 4/3  2 cL4 T2 BJm ≈ Hc2 (0) c2 1 − . (12.224) Gi T Tc The dispute then is about the range where critical fluctuations should be expected: the usual Ginzburg criterion which estimates the width of the specific heat transition predicts a rather narrow critical region of the order of 1 Kelvin.On the other hand,other quantities may exhibit a different critical range, cf. equation (12.206) which predicts a larger critical region in terms of 2D phase fluctuations than expected from the conventional Ginzburg criterion. Note that the mean-field based power law (12.215) predicts a melting line which is too steep when compared to typical experiments on YBCO [42]; taking into account the suppression of the order parameter at high fields close to Hc2 , cf. (12.215), produces a flatter melting line also within the mean-field scaling scheme, see Fig. 12.12.

Fig. 12.12. Melting line in YBCO. The experimental data points (circles) are taken from [42]. The full line shows the theoretical result (12.192) with a penetration depth (T) taken from experimental data [285] and an anisotropy parameter " = 1/8 for YBCO close to Tc . The dashed line is a fit to the melting line using the result from a London model (with (Tc −T)1/2 -scaling and additional fitting parameters) taking into account the suppression of the order parameter at high fields close to Hc2

12.7.3

549

Lindemann Analysis of Layer Decoupling

In layered systems an additional thermodynamic transition takes the 3D bulk system into a system of decoupled 2D layers [72]. The loss of inter-layer coherence is due to strong thermal fluctuations of the pancake vortices within the individual layers and the transition line can be estimated within a Lindemanntype approach [72,231].A convenient starting point is the free energy (12.154) describing the phase degrees of freedom in the presence of pancake vortices — the mutual locking of the phase between layers is due to the electromagnetic and Josephson coupling terms. Using the equipartition theorem we determine the mean squared thermal fluctuation amplitude of the phase ı' within one layer, T ı' th ≈ 2"0 d 2

×

KBZ dK

(12.225)

0

K a20 K4 16(1+g)2

+

g K2 1+g

+

2  2

,

and find the line in the B-T diagram where this quantity becomes of order unity. The calculation of the integral in (12.226) is trivial but involves many cases: below we distinguish between the two cases of weak (2 /  <  <  ) and strong (  <  < 2 / ) interlayer coupling. We will use the estimate g ≈ a20 ln(a0 /d)/42 ≈ Bc1 /B for the parameter g, cf. (12.150). At high fields, the shear term ∝ K 4 is dominant and g is small, 1 + g ≈ 1: the shear term dominates over√the Josephson and electromagnetic terms if K > 1/ a0   and K > 1/, respectively (we drop numericals in this qualitative discussion). Also, we can disregard the electromagnetic energy as compared to the Josephson energy if K < /a0  . As a result, we can ignore the electromagnetic contribution to the integral (12.226) if the high-field condition a0 < 2 /  is satisfied and we obtain 4T ı' th ≈ 2 a0 "0 d 2

2 4/a  0

√ 4 2/a0  

dK 2 ≈ K4

7

 T   , 2 "0 d a0 (12.226)

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where we have assumed that a0 <   in the last equation. For a strong interlayer coupling the above conditions require a0 <   <  < 2 /  while a weak coupling requires a0 < 2 /  <  <  . The highfield decoupling line then takes the form 

"0 d T B > B× ≈ B  ,

2

BJdc (T) ∼ B 

B > B× ≈ B

  , 2 2

,  <, < ,

(12.227)

with the usual definitions B ≡ ¥0 /2 and B  ≡ ¥0 / 2 ; note that with decreasing coupling strength (increasing  ) the crossover field B× first decreases to approach B at   ≈  and then increases again as   increases further; since B > B we are allowed to approximate 1 + g ≈ 1 in this regime, as assumed in the beginning. For strong coupling   <  and intermediate fields   < a0 < 2 /  the shear term ∝ K 4 and the electromagnetic energy ∝ K 2 lose the competition against the Josephson coupling and the decoupling line takes the form "0 d , T   <  or

BJdc (T) ∼ B  B < B  ,

 <   . (12.228)

The same result is obtained in the low-field regime 2 /  < a0 whenever the Josephson interaction outplays the electromagnetic coupling, which is the case whenever B drops below B  . This completes the analysis for the strong coupling situation. For weak coupling ( <  ) and intermediate fields B  < B < B× (i.e., 2 /  < a0 <  ) the phase fluctuations are limited by the electromagnetic interaction. The ∝ K 2 term in (12.226) is dominant 2 between the lower cutoff Kmin ∼ (1 + g)/g 2 set by the Josephson term and the upper cutoff determined by either the Brillouin zone boundary with 2 Kmax ∼ 1/a20 or by the shear term, in which case 2 Kmax ∼ g/(1 + g)a20 ; correspondingly we find the mean squared fluctuation

⎧ B× ⎪ ⎪ ln , ⎨ T 1+g 2 B ı' th ≈ B 4"0d g ⎪ ⎪ ⎩ ln , B 

B < B < B× , B  < B < B .

(12.229) Comparing to more sophisticated calculations (cf. Sect. 12.7.5 below) we find that the logarithm in (12.229) is a peculiarity of the Lindemann analysis and actually should be replaced by a number ˛,hence the low-field decoupling line assumes the form,   ˛T B "0 d a0 em Bdc (T) ≈ ln 1− , ˛ T d 4"0d B  < B < B× ,  <   , (12.230) where the factor [1 − ˛T/4"0d] appears as a consequence of the renormalization of the superfluid density through the factor g/(1 + g). Note that the above weak-coupling result Bem dc does not depend on the anisotropy parameter ". The Lindemann criterion cannot tell us about the underlying mechanism triggering the transition (in fact, the Lindemann criterion assumes that the transition follows from a trivial increase in the fluctuation amplitude). We will see later (cf. Sect. 12.7.5 below) that in the low coupling limit " → 0 the decoupling transition assumes a topological character and is triggered by the appearance of vortex loops, either in the form of interstitial/vacancy pancake-vortex pairs (these correspond to vortex loops in planes perpendicular to the layers) or in the form of Josephson vortex loops (i.e., loops parallel to the layers). The defect transition is of the Kosterlitz–Thouless type (cf. Sect. 12.7.5) and appears at Bem dc (T) as given by (12.230) with ˛def = 8. The position of the loop transition has been determined via a self-consistent calculation of the inter-layer coupling [73] and later via a renormalization group approach [75], "0 d a0 ln , T d  2 B ≤ B× ≈ B 2 ,  <   .  Bem dc (T) = B

(12.231)

Again,this result follows the Lindemann line (12.230) if we set ˛L = 1 (the additional factor (1 − T/4"0d) in (12.230) is due to the renormalization of the superfluid density which has not been taken into account

12 Vortex Matter

551

Fig. 12.13. Schematic low-field phase diagrams of layered superconductors with melting and decoupling lines. At high fields/low temperatures the phase fluctuations triggering decoupling are limited by shear modes, while tilt modes are relevant in the low-field/high-temperature region close to Tc . Left: For weak coupling  <   the decoupling and melting lines em merge at intermediate temperatures with a joint transition at the electromagnetic melting/decoupling line Bem m = Bdc . With increasing coupling, T em → 0 and the common melting/decoupling transition at Bem disappears as the transition m,dc rises above the B line for   <  (right). It is unclear whether the decoupling line in the vortex liquid corresponds to a real phase transition or a mere crossover

in [73, 75]). Since the defect transition involves an eight times larger value of ˛, it preempts the Josephson loop transition and the true decoupling is triggered by the proliferation of free interstitials and vacancies in the small " limit. Finally, let us combine the above results for the decoupling transition with those obtained previously for the melting transition and construct the corresponding B–T phase diagram, cf. Fig. 12.13. Starting from the small coupling limit " → 0,  <  , we note that the low-field expression (12.230) for the decoupling line Bem dc coincides (up to a factor of order unity) with the form of the electromagnetic melting line Bem m , see (12.212), suggesting that the two transitions may join up within the temperature regime 2D Tm < T < T em [249]. Indeed, we will see below in Sect. 12.7.5 that the melting line undercuts the decoupling line and the two transitions merge into a single one. As the temperature increases beyond T em the Josephson interaction becomes relevant and

decoupling occurs in the liquid phase following BJdc as given by the low-field result (12.228) above, see Fig. 12.13. Note that the interlayer phase fluctuations are limited by tilt interactions in the low-field regime where the decoupling occurs in the liquid phase, giving some justification to the application of (12.226) in the liquid regime. On the other hand, the primary expression (12.226) has been derived with the crystal as a starting point (i.e., a well defined displacement field u) and it remains unclear how much the result of a second transition in the liquid can be trusted. In fact, many suggestions have been made regarding the existence of intermediate vortex liquid phases [287,288] but a final conclusion has not been reached (note that the various proposed intermediate liquid phases involve layered and anisotropic superconductors as different starting points; e.g., the decoupling line (12.228) involves the layer separation d and does not scale in the parameter T/").

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Increasing the layer coupling the branching point at T em moves down in temperature and joins up with the point where decoupling and melting merge at low temperatures. The intermediate line Bem m,dc is squeezed as the melting/decoupling transition rises above the B line. The phase diagram then simplifies in the regime   <  with a simple cutting of the melting and decoupling lines in the vicinity of the 2D point (Tm , B  ) [71], see Fig. 12.13. 12.7.4

Vortex Lattice Melting in 2D Thin Films

An attractive scenario for the melting of 2D crystals has been put forward by Nelson and Halperin [289] who base their analysis on the unbinding of dislocation pairs in the framework of a BKT transition. This concept has later been applied to the 2D vortex system in thin superconducting films [235, 236] (however, we remind that numerical simulations [273–275] suggest 2D vortex lattice melting to be first-order). The role of the logarithmically interacting topological excitations is played by the edge dislocations. Using a continuum description, the displacement field of an edge dislocation with a Burgers vector b = (b, 0) pointing along the x-axis is given by [171,290]   x2 b xy y udis = , − (12.232) + arctan 2 R2 x R2 and the energy associated with this defect is R  d edis (R) = d2 R c66 ∇⊥ · udis )2 2 R0

=

db2 R c66 ln . 2 R0

(12.233)

The lower cutoff R0 is provided by the lattice constant a0 of the vortex lattice, whereas the upper cutoff R is given either by the sample dimension L (→ selfenergy of the topological defect) or by the distance R to the nearest dislocation with an opposite Burgers vector (→ logarithmic interaction between two oppositely “charged” topological excitations). Note that the distortion (12.232) does not involve any compression (∇ · udis = 0) but only shear forces, a consequence of the incompressibility of the vortex lattice, c11  c66 . For an elementary dislocation the Burgers

vector (the misfit-vector when encircling the dislocation) is equal to a unit lattice vector and the energy edis (R) becomes "0 d R , (12.234) ln dis(R) = √ a 4 3 0 √ which is a factor 4 3 ≈ 7 smaller than the corresponding energy (12.195) for the vortex excitation itself. The BKT-type dislocation-mediated melting transition of the 2D vortex lattice then takes place at a temperature [235,236] Aa2 c66 d A"0 d "0 d 2D Tm ≈ √  TBKT . (12.235) = ≈ 4 70 8 3 The numerical A ≈ 0.62 accounts for the renormalization of the shear modulus c66 due to bound dislocation pairs [269]. Vortex lattice melting in thin Nb3 Ge films following the dislocation-mediated melting scenario has been observed by Berghuis, van der Slot, and Kes [291]. The usual dislocation-mediated melting scenario predicts a two-step transition [289, 292]: The lowtemperature phase is characterized by quasi-longrange translational order with a density–density correlator

K  (R)−K  (0)th = exp −K2 [u(R) − u(0)]2th /4 (12.236) ∝ R− ,

TK2 ,  (T) = 4c66 d

(12.237)

(here, K  (R) = exp{iK  · [R + u(R)]} and we have used [u(R) − u(0)]2th ∼ 2u2 (R)th = (T/c66d) ln(R/a0)) decaying algebraically with distance and producing algebraic Bragg peaks in the structure factor, SK  (K) ∝ |K − K  |−2+ . (12.238) In addition, the 2D crystal exhibits orientational long-range order as quantified by the angle correlator ¦6 (R)¦6∗ (0)th , ¦n = exp[in(R)] with the bondangle field (R) = (@x uy − @y ux )/2. In the first step 2D of the melting process, dislocations unbind at Tm and destroy the translational quasi-long-range order, while orientational long-range order transforms into quasi-long-range order, ¦6 (R)¦6∗ (0)th ∝ R−6 with 6 = 18T/KA ,KA the Franck constant describing the stiffness of the bond-angle field (R). This

12 Vortex Matter

intermediate phase is the hexatic liquid. The melting is completed in a second BKT type transition (at Th2D = KA /72) where disclinations unbind (a dislocation can be viewed as a tightly bound pair of disclinations), destroying the hexatic quasi-long-range order in the bond-angle field. This “classic” topological melting scenario is modified for high fugacities of the dislocations, implying a large density of dislocation pairs at the transition. The two continuous transitions described above then collapse into a single first-order transition [271–275] and the melting transition is more conventional (a similar collapse is familiar from the merging of the two critical fields Hc1 and Hc2 into the first-order transition at Hc as √  = / decreases below the value 1/ 2). 12.7.5

Electromagnetically Coupled Layered Superconductors

The extreme case of a layered superconductor is a stack of 2D superconducting films without any Josephson coupling between the layers. This limit is difficult to realize in a material as even a weak coupling " ∼ 10−3 already has an appreciable effect on the phase diagram, cf. Fig. 12.11; materials approaching this limit are the layered BiSCCO compound and the artificially grown multi-layer structures such as the (Y/Pr)Ba2 Cu3O7−y [188] and the Mox Ge1−x /Ge [189] systems. The interaction between pancake vortices is of electromagnetic origin (see (12.126)): the logarithmic intra-plane repulsion between pancake vortices extends to infinity, V0(R) = −2"0 ln(R/L) (cf. (12.122), L denotes the system-size cutoff) and the attractive inter-layer interaction is small by the small factor d/ but long-ranged,extending over /d layers (cf. (12.130)). The resulting line elasticity is strongly dispersive,   "0 d 2 kz2 "l ≈ 2 2 ln 1 + . (12.239) 2 2 kz 1 + 2 KBZ Note that short wavelength distortions with a large kz ∼ 1/d produce a potential energy "0 d ln(/d)(u/)2 (this energy results from adding the pairwise interaction energy (12.130) over /d pairs),while long wavelength (kz < 1/) fluctuations of a vortex line involve the elastic energy ∼ "0 (kz u)2 .

553

An individual stack of pancake vortices already defines an interesting thermodynamic system: thermal fluctuations blur the vortex line and even trigger its evaporation at high temperatures [65]: pulling a pancake-vortex out of its stack costs the logarithmic energy 2"0 d ln(u/) [65] and the usual energyentropy argument predicts a vortex-line evaporation at Tevap = TBKT = "0 d(TBKT )/2. Note that with the appearance of free vortices through vortex-pair unbinding at TBKT the concept of a pancake vortex stack looses its meaning; this is consistent with its evaporation at Tevap = TBKT . This stack evaporation is readily rederived through a self-consistency argument [293]; the latter provides the basis for the self-consistent substrate model [294] used below in the derivation of the pancake-vortex lattice melting transition. Each pancake vortex of the stack is subject to a quadratic potential Vs (u) =

1 ˛ s u2 2

(12.240)

produced by the other members of the stack. The parameter ˛s of this “substrate potential” is given by the thermal average (cf. (12.148) where we have introduced the elastic energy vem ≈ ˛s /a20 ; here, we account for the thermal renormalization of this quantity)  em ˛s = @u2x0 Vint (un − u0 , nd)th . (12.241) n=0

Given the long-range nature of the pancake-vortex interaction, we can ignore correlations in the pancake-vortex fluctuations and hence ˛s (u2 th ) =   d2 K 2 2 K 2 V em (K, nd)e−K u th /2 ≈ − 2 x int (2) n=0  d2 K exp(−K 2 u2 th /2) ≈ 2"0 d (2)2 1 + 2 K 2 ⎧ 2 u th 22 ⎪ ⎪ ⎪ ln , u2 th  22 , ⎨ "0 d 22 u2 th ⎪ u2 th ⎪ 22 ⎪ 22  u2 th , ⎩1− 2 , u th (12.242)

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G. Blatter and V. B. Geshkenbein

with the pancake-vortex interaction energy  em Vint (K, nd) = ¥02d 2 (dkz /8 2 ) k2 /[K 2 (1 + 2 k2 )] exp(ikz nd) deriving from (12.126). On the other hand, the equipartition theorem for a harmonic potential tells us that u2 th = 2T/˛s. The two expressions for ˛s (u2 th ) cut at a finite value u2 th < ∞ if T < "0 d/2; approaching T → "0 d/2, u2 th > 22 becomes large and we can find the precise solution using the asymptotic expression in (12.242). The mean square displacement u2 th =

22 1 − 2T/"0d

(12.243)

diverges at TBKT = "0 d(TBKT )/2 and the pancake vortex stack evaporates. The discussion above and in Sect. 12.7.4 provides us with two important results [294]: First, we know the position of the melting line for the pancakevortex lattice in the two limits of high and low fields (ignoring reentrance in the latter case). For high fields B  ¥0/2 = B the in-plane repulsion is the dominant interaction between pancake vortices and the melting process exhibits a 2D character: the lattice melts as the temperature is increased beyond 2D Tm ≈ "0 d/70. In BiSCCO the penetration depth is of order (0) ≈ 2000 Å and d ≈ 15 Å, "0d ≈ 750 K, and the high-field limit of the melting line approaches 2D Tm ≈ 11 K. At low fields, the electromagnetic interaction between pancake vortices in different layers is dominant. In the limit B → 0 we can ignore the interaction between pancake-vortex stacks and these stacks melt or evaporate at TBKT (note that the true melting line will reenter before reaching the stack evaporation temperature); for BiSCCO we have TBKT ≈ 0.9 Tc (note, however, that Josephson coupling cannot be ignored in BiSCCO, hence one has to expect that the low-field melting line is pushed closer to Tc ). Second, we have learnt how to use a self-consistent substrate model for an accurate description of the evaporation transition of an individual pancake-vortex stack — below we will generalize this idea to describe the melting (evaporation) of the pancake-vortex lattice.

Pancake-Vortex Lattice Melting Knowing the position of the melting line in the two limits of high and low fields (up to reentrance), the question is how to obtain the line in between. Here, we can make use of the above analysis: The selfconsistent substrate model accurately describes the melting/evaporation of a single pancake-vortex stack (a self-consistent harmonic approximation has been used in the analysis of the 2D XY-model [295]). The idea then is to extend this analysis to the 3D pancakevortex system at finite fields. Let us choose one of the layers (say n = 0) as our reference, then the pancake vortices in the other layers form stacks which provide a periodic substrate potential in which the pancake vortices of our reference layer are trapped. Without this substrate potential, our 2D pancake-vortex sys2D tem would melt at Tm . The substrate potential stabilizes the crystal and pushes the melting temperature to higher values — the 3D crystal turns unstable when the substrate potential collapses. Given the long-range nature of the pancake-vortex interaction one expects that the mean-field type analysis as formulated through the self-consistent substrate model provides accurate results for the stability range of the vortex lattice.Note that we ignore here the reentrance of the melting line at low magnetic fields. The accurate determination of the stability region of the pancake-vortex crystal requires numerical calculation of the thermal renormalization of the shear modulus c66 (T) and the substrate potential ˛s (T): the simplest analysis is based on a self-consistent harmonic approximation (SCHA) with a Hamiltonian  1 d2 K Hh = u0˛ (K) ¥˛ˇ (K) u0ˇ (−K) (12.244) 2 (2)2 BZ

and requires calculation of the thermally averaged elastic matrix  1  1 − eiKR (12.245) ¥˛ˇ (K, u2 th ) = 2 a0 R 

em × @˛ @ˇ Vint (R , 0)th   em + @˛ @ˇ Vint (R , nd)th n=0

12 Vortex Matter

(here, the first term corresponds to the expression (12.69) while the second accounts for the substrate potential (12.241)) together with the mean squared displacement amplitude  d2 K u2 th = T [¥ −1 ]˛˛ (K, u2th ) . (12.246) (2)2 BZ

After Fourier transformation the thermal average in (12.246) can be expressed through the Debye–Waller factor exp(−K2 u2 th /2); the 3D crystal becomes unstable when this system no longer exhibits a solution for u2 th . However, the proper inclusion of anharmonic effects require a more accurate analysis going beyond the SCHA scheme, e.g., the two-vertex SCHA which includes cubic anharmonicities [294,296]. The numerical results [294] exhibit a smooth decrease of c66 (T) and ˛s (T) with increasing temperature up to the instability field Binst (T), with reduction factors ∼ 0.5 and ∼ 0.1 near the collapse, cf. Fig. 12.14. In order to find the melting line we have to go one step further: for a first-order transition the lowtemperature phase can be overheated to the upper spinodal and the instability temperature Tinst only provides an upper estimate for the melting temperature Tm < Tinst . The determination of the melting temperature requires calculation of the free energies (here at fixed B) of the solid and liquid phases; their intersection defines Tm . The free energy of the solid can be calculated within a variational scheme: Given

555

the Hamiltonian H = H2D + Hs with    2 N  em d R em V (R , 0) − V (R, 0) H2D = 2 R =0 int a20 int 

(12.247) and N Hs = 2

  R ,n=0

 em Vint (R , nd) +

 d2 R em V (R, 0) , a20 int

(12.248) the variational free energy F h + H − H h h is minimized through the SCHA Hamiltonian H h , cf. (12.244), and involves the harmonic free energy of 2D fluctuations     NTa20 d2 K (/e)T/a40 h F =− ln 2 (2)2 c66K 2 + ˛s /a20 BZ   (/e)T/a40 + ln , c11 K 2 + ˛s /a20 h the ground state energy H2D − H2D h = −0.749N"0 d of a 2D lattice of N log-interacting particles [297], and the substrate energy

Hs − Hsh h =    exp[−K 2 u2 th /2] ˛s 2  2 + u −N 2 "0 d  th . K2 (1 + 2 K2 ) 2 a0  =0 The free energy Fliq of the pancake-vortex liquid is calculated from the internal energy U ( ) of a 2D log-interacting system via integration,  Fliq ( ) =

Fig. 12.14. Instability line Binst for the pancakevortex lattice (PVL) in the B–T plane calculated with the two-vertex-SCHA. The line goes asymptotically to Tm2D at high fields and ends at TBKT at zero field. The left inset shows the twovertex-SCHA results for the shear modulus c66 , the substrate strength ˛s , and the fluctuation width u2 th with increasing temperature at the field B = B . The right inset is a zoom of the instability line at low fields, and shows the result (dashed line) of the simple SCHA scheme

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Fig. 12.15. Instability (dashed) and melting (solid ) lines in an electromagnetically coupled layered superconductor. The melting line derives from cutting the free energies of the solid and liquid phases, see inset (data for B = B ). The lower graph gives the entropy jump per pancake-vortex Spc . Also shown are the real temperature and field scales assuming (T) = (0)/[1 − (T/Tc0 )2 ]1/2 , with (0) ≈ 2000 Å, d ≈ 15 Å, and Tc0 ≈ 100 K

0 Fliq (0 ) +



d  U (  ), with the internal energy (we

0

remind that  = 2"0 d/T) U ( ) ≈ N"0 d [−0.751 + 0.880 −0.74 − 0.209 −1.7 ] known from numerical simulations [269] and the integration constant deriving from an exact solution at 0 = 2 where Fliq (2) = 0.081 N"0 d [297] (we assume that the 2D liquids in neighboring layers have lost all correlations). The result of this analysis is shown in Fig. 12.15 and produces a melting line Bm (T) very close to the instability line Binst (T) (similar results have been found for the stability region of the magnetically coupled pancake-vortex lattice within the framework of density functional theory [298, 299]). The intersection of this melting line with the lowfield reentrant line (12.216) is located at a field value B/B ≈ 0.005 and a temperature Tx /"0d ≈ 0.35 or 1 − Tx /Tc ≈ 0.15 (we have used "00 d/Tc ≈ 15; note that the expression for Bmr /B involves the parameter /d when expressed through the combination

Fig. 12.16. Comparison of melting (solid) and instability (dashed) lines obtained within the self-consistent substrate model with the electromagnetic melting line (dotted) from a Lindemann analysis with cL = 0.065. The discrepancy between these results at high temperatures is traced back to the thermal renormalization of elastic coefficients, accounted for in the substrate model but not in the Lindemann analysis. The true melting line is expected to reenter near Tx /"0 d ≈ 0.35 where our line is terminated

T/"0 d and hence the point of reentrance is not universal; in BiSCCO, the Josephson coupling cannot be ignored and the point of reentrance is pushed close to Tc ). The jump in the slope S = −@T F gives the latent heat TS = U ; at high fields the entropy jump per pancake-vortex takes a value Spc ≈ 0.4 kB , consistent with experiments [42] and simple estimates [300]. The increase of the entropy jump with temperature follows naturally from the analysis of the magnetization and entropy jumps in layered superconductors, see (12.268) in Sect. 12.7.6 below. In Fig. 12.16 we compare the results obtained within the substrate model with those of the Lindemann analysis presented in Sect.12.7.2 above.Choosing a Lindemann number cL = 0.065 the results agree well at low temperatures and high fields, close to the 2D-melting limit (note that a similarly small Lindemann number has been found in the discussion of vortex lattice melting in 2D films, cf. (12.209)). However, for high temperatures T/"0 d → 0.5 the Lindemann result largely overshoots the result from

12 Vortex Matter

the substrate model (also, the Lindemann result involves the dimensionless temperature T/Tc while the substrate model scales in T/TBKT ). We can trace back this discrepancy to the thermal renormalization of the elastic constants at high temperatures, which is accounted for within the substrate model, cf. Fig. 12.14, butnot in the Lindemann analysis. Replacing the tilt modulus c44(k) ≈ "0 /2a202 kz2ˇ in (12.208) by the renormalized coefficient ("0 /a20 u2 th kz2 )(1 − 22 /u2 th ) (cf. 12.242) we can properly account for the thermal softening of the pancake vortex stacks and the renormalized Lindemann result comes to lie close to the instability and melting lines of the substrate model. Defect–Unbinding Transition The pancake-vortex crystal in an electromagnetically coupled superconductor exhibits a second phase transition besides the melting/sublimation transition discussed above. This second transition is due to the unbinding of interstitial-vacancy pairs (of the Kosterlitz–Thouless type) and transforms the lowtemperature, low-field crystal into a 3D defected crystal at high temperatures or fields [76]. The basis for this transition is given by an interesting screening effect which exactly (partially) compensates the logarithmic defect energy in a 2D vortex crystal (3D pancake-vortex crystal) [301,302]. Consider a rigid 3D pancake-vortex lattice in a layered superconductor and move one pancake-vortex by the distance R¯ within the same layer to produce an interstitial-vacancy defect pair. The excitation energy associated with this manipulation contains a self, an interaction, and a core-energy part and reads 

 L L ¯ Ei,v (R, L) = 2"0 d ln − ln + (i + v )"0 d . a0 R¯ (12.249) The numericals i,v quantify the core energies of the interstitial and the vacancy-defect and are of order 0.15–0.2 [303,304] and L denotes the system size. Next, we let the lattice relax and determine the screened self-energy of the defect — the interaction between the defects is screened in the same manner. We first ignore the electromagnetic coupling to the

557

pancake vortices in the other layers and take them into account in a second step later. The interaction energy of the pancake-vortex configuration reads 1 Eint = Vpc (R − R ) (12.250) 2 ,  d2K 1 n(K)Vpc (K) n(−K) = 2 (2)2 BZ

(Vpc (R) = −2"0 d ln(R/a0) is the pancake-vortex interaction energy, see  (12.122), and n(K) =  eiK·R denotes the Fourier transform of the pancake-vortex density), from which one can derive the dispersive compression modulus 4"0 d c11(K) = n2 Vpc (K) = 4 2 , (12.251) a0 K with n = B/¥0 the vortex density. The change in energy due to the presence of the defect can be written in the form  d2 K ıEint = (12.252) (2)2 BZ   c11 |K · u(K)|2 + in[K · u(K)] Vpc (K) . × 2 While the first term in (12.253) describes the elastic energy of the relaxed lattice, the second term originates from the forcethe interstitial exerts on the pancake vortices, Esource =  Vpc (R −u ) (we place the interstitial at the origin). Minimizing (12.253) with respect to u we obtain the distortion u(K) = −

inKVpc (K) i K =− ; 2 c11(K)K n K2

(12.253)

the displacement u(R) = R/2nR2 removes exactly one pancake-vortex from the neighborhood of the interstitial and thus produces perfect screening [301]. The log-divergent self-energy in (12.249) then is compensated by the gain in elastic energy under the relaxation of the lattice: inserting the solution (12.253) back into (12.253), we obtain  1 L d2 K ıEint = − Vpc (K) = −"0 d ln . 2 2 (2) a0 BZ

(12.254)

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This perfect screening is spoiled by the interaction of the pancake vortices in the layer (n = 0) with those in the other layers (n = 0); adding the substrate potential (12.240) with ˛s = ("0 d/2 ) ln(a0 /d), cf. (12.148) where vem ≈ ˛s /a20, to the energy (12.253) and minimizing, the displacement field (12.253) is reduced by the factor 1 + g with [301] n˛s Hc1 a20 a0 g= ≈ . (12.255) = ln 2 c11 (K)K 42 d B The screening charge is reduced to the fraction 1/(1 + g) and the divergent self-energy in (12.249) is only partially canceled by the lattice relaxation; inserting u(K) back into the expression (12.253) for the change in the interaction energy we obtain ıEint = −"0 d ln(L/a0 )/(1 + g). The relaxation of the pancake-vortex lattice then reduces the self-energy in (12.249) by the factor g/(1 + g). At large fields, B > Hc1 ,where the vortex–vortex interaction is dominated by intraplanar forces, we have g < 1 and a large downward renormalization of the energy of the defects.When g > 1 at small fields B < Hc1 ,the interaction between pancake vortices is dominated by the interlayer forces and the effect of screening is small. The above analysis extends trivially to the case of a pancake-vortex vacancy as well as to the interaction between the defects. The energy for an interstitial-vacancy pancake-vortex pair of extent R placed into one plane of a decoupled layered superconductor then involves the screened self and interaction-energies 2g"0 d R¯ sc ¯ Ei,v ln (R, L) = + (i + v )"0 d . (12.256) 1+g a0 The result (12.256) with its logarithmic interaction between interstitial-vacancy defect pairs immediately implies a Berezinskii–Kosterlitz–Thouless transition [257] where these defects unpair, producing a crystal with free defects. The field dependent screening of the interaction pushes this defect-unbinding transition to low temperatures, Tdef (B) =

g "0 d , 1+g 2

(12.257)

and solving for B we obtain the defect-unbinding line   2T B "0 d  a0  ln 1− . (12.258) Bdef (T) = 4 2T d "0 d

Thermal fluctuations soften the substrate potential and modify the result (12.258): Using (12.242) with (12.243) we obtain ˛s (T) = (T/2 )(1 − 2T/"0d). The defect-unbinding line then has to be determined from (12.257) using the renormalized suppression factor g(T) = (a20 /82 )(2T/"0d)(1 − 2T/"0d) for the superfluid density and we obtain the final result near TBKT     2T 2 B 1 − T/TBKT 2 B r 1− Bdef (T) = = . 8 "0 d 8 1 − T/Tc (12.259) Here, we have used that (1 − 2T/"0d) ≈ [(1 − T/TBKT )/(1 − T/Tc )] close to Tc . The last factor in (12.259) replaces the Ginzburg–Landau temperature dependence in −2 ∝ 1 − T/Tc by the temperature factor 1 − T/TBKT accounting for 2D fluctuations and pushes the defect-unbinding line to zero at TBKT.Note that the second factor (1 −T/Tc )−1 ≈ (1 −TBKT /Tc )−1 enhances the prefactor in (12.259). The position of the defect line Bdef (T) should be compared with the melting line as found in the previous paragraph. The result of such a comparison is shown in Fig. 12.17, where we plot the full melt-

Fig. 12.17. Phase diagram for an electromagnetically coupled layered superconductor. The solid line marks the defect-unbinding transition Bdef , which transforms into the decoupling transition Bdec in the case of a finite weak Josephson coupling with " < d/. The dashed line shows the melting line Bm calculated within a self-consistent analysis (with parameters (0) ≈ 2000 Å, d ≈ 15 Å, Tc ≈ 100 K)

12 Vortex Matter

ing line Bm (T) (dashed) together with the defectunbinding line Bdef (T) (solid); we find that the melting line undercuts the defect-unbinding line, leaving only one (sublimation) transition Bm (T) at high temperatures, see also Fig. 12.13. Note that the renormalized defect line Brdef (T) vanishes rapidly as T approaches TBKT ; in principle this allows for a reappearance of the defect line below the melting line as T → TBKT . The analysis of the structure of the phase diagram near TBKT then requires precise knowledge of the shape of the defect line and of the melting line including its reentrance; in fact, the reentrance of the melting line may well preempt the reappearance of the defect line at high temperatures. At low temperatures, Bdef (T) remains below the melting line, allowing for two transitions separating a defect-free solid, a defected solid, and a pancake-vortex gas. The low-temperature defect transition at Bdef ≈ B ("0 d/8T) ln(a0 /d) in electromagnetically coupled systems should be compared with the decoupling transition Bem dc ≈ B ("0 d/T) ln(a0 /d) (cf. (12.231)) in weakly coupled layered superconductors. Indeed, with a small but finite Josephson coupling present, the lattice defects trigger the decoupling of the layers and the defect-unbinding line Bdef (T) turns into a decoupling line Bdef dc (T). For fields above Bdef (T) the system develops a finite cdc axis resistivity with c proportional to the density of free mobile defects nd , see [101]. Close to the unbinding transition, free defects appear in the solid  with a density nd ∼ a−2 exp[−2 b/(T/T def − 1)] 0 (with b a non-universal constant). At higher temperatures the core energy Ecore = "0 d determines nd a20 ∼ exp(−Ecore /T). Such a topological decoupling transition based on a quartet-unbinding has first been proposed by Feigel’man et al. [64] (see also [74] for a similar supersolid transition). Whereas the defect-unbinding transition Bdef is the finite-field generalization of the zero-field vortex-unbinding transition at TBKT , the decoupling transition Bem dc [73, 75] is the finite-field generalization of Friedel’s loop transition at Tloop = 8TBKT , cf. Sect. 12.7.1. This pair of transitions is easily derived from the Lawrence–Doniach model (12.97): for zero coupling " the individual layer develops a vortexunbinding transition at TBKT . On the other hand,

559

restricting ourselves to a pair of (coupled, " > 0) layers and going over to sum (¥+ = (¥1 + ¥2 )/2) and difference (¥− = ¥2 − ¥1 ) phase variables we arrive at the Sine–Gordon model with a “roughening” temperature at 8TBKT , independent of the coupling ". Obviously, the zero-field vortex-unbinding transition at TBKT preempts the zero-field loop transition in the limit of small interlayer coupling. Increasing the coupling " the transition is shifted upward, cf. (12.204). The same pair of transitions can be continued to finite magnetic fields.The presence of pancake vortices then is easily accounted for via the renomalization of the superfluid density, cf. (12.97) and (12.154), and we obtain the finite-field versions of the two zero-field transitions where vortices/defects unbind and where Josephson loops are created spontaneously. Again, in the limit of small coupling the vortex-induced decoupling transition Bdef dc is eight times lower than the “loop” transition at Bem dc , cf. (12.231). At finite coupling the true transition is exem pected at a value between Bdef dc and Bdc : comparing 2 the Josephson energy def EJ with EJ = "0 d/ 2 , see 2 (12.97), within the coherence area def ≈ n−1 d with the temperature we estimate the upward shift of the decoupling transition due to the Josephson interaction Ecore J Tdc (B) ≈ . (12.260) ln [(Ecore /"0d)(B/B  )] While the result (12.258) is valid at low fields, the steep upward shift described by (12.260) becomes effective when g ≈ Hc1 /B < / ln(d/"), with  = Ecore/"0 d the numerical quantifying the core energy of the defects (see [303, 304] where numerical values  ≈ 0.15 − 0.2 are reported). The quantitative condition for the applicability of the result (12.258) thus depends both on the anisotropy parameter " as well as on the defect core-energy Ecore . At higher fields B > (8/) ln(d/")Hc1 the decoupling line crosses over into the “loop” transition at Bem dc as given by (12.231) [73,75]; an accurate description of the crossover between the defect and “loop”transitions requires a generalization of the analysis in [82] to finite fields. The pair of transitions discussed above is not unique to the vortex system discussed here — it appears in the context of the XY -model (with coupling

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J) subject to a symmetry breaking field ∝ cos[m#R ] (here, #R denotes the angle of the spin at the position R, m is an integer), where the BKT transition at TBKT = J/2 competes with the unlocking or“roughening” transition at TR = 8J/m2 [255]. A similar scenario shows up in the context of the adsorption of a 2D crystal (elasticity C, lattice constant a) on a commensurate substrate (lattice constant a/m), where the dislocation-mediated melting transition at Tm = Ca2 /4 competes with the roughening transition at TR = 4Ca2 /m2 [305] (in both cases the ratio between the two transition temperatures involves the factor 16/m2; for m = 1 this is twice larger than in the present layered situation). Quite interestingly, proper modification of the Lindemann estimate (12.230) accurately reproduces the two transition lines: replacing the logarithm in (12.229) by the numerical ˛L = 1 produces the loop transition at   T "0 d a0 Bem ln 1 − ; (12.261) = B  dc T d 4"0 d accounting for the renormalization of the superfluid stiffness, we find that Bem dc correctly extrapolates to vanish at T = 8TBKT .On the other hand,replacing the logarithm by ˛def = 8 reproduces the result (12.258). 12.7.6

Characteristics of First-Order Melting

The mean-field transition at the upper critical field Hc2 (T) is of second order: the U (1) symmetry of the complex superconducting order parameter ¦ forbids the cubic term in the Landau expansion and hence |¦ | is coming up smoothly from zero upon cooling. However, thermal fluctuations turn this transition into a crossover where Cooper pairs form, establishing a finite local order parameter and a finite density |¦ |2 but without establishing any long-range order; no symmetry is broken in this vortex liquid phase which is no different from the normal metallic one. The true transition into the superconducting state then changes its nature and is shifted to the vortex melting transition. The broken symmetry appears as a periodic density modulation in |¦ |2 and the standard symmetry considerations for a melting transition apply [306]: the cubic term in |¦ |2 enters the

usual Landau expansion for a liquid–solid transition and a sharp first-order transition is expected where translational (not U (1)) symmetry is broken (note that no long range phase order is expected in the unpinned vortex lattice state [71,72,307]). This scenario is confirmed by experiments through the observation of jumps in the magnetization in BiSCCO [40] and in YBCO [41] as well as the measurement of a latent heat in YBCO [42,43] at the vortex lattice melting transition; the thermodynamic consistency between the magnetic and calorimetric measurements has been demonstrated via the Clapeyron relation [42]. A simple estimate for the jump B in the induction is provided by the following scaling analysis ( [300]): we start from the thermodynamic relation B = −(4/V )@H G|T , with G the Legendre transform of the free energy F, G(T, H) = F(T, B) − BHV /4, and V the system volume.Attributing the thermal energy kB Tm and the volume Vedf to each elementary degree of freedom we can estimate G ∼ kB Tm V /Vedf . The volume Vedf is determined by the dominant modes leading to melting, which are√located at the Brillouin zone boundary with KBZ ≈ 4/a0 . Hence, we can define the volume per degree of freedom in the form Vedf ≈ a20 L.Making use of the power-law dependence of Tm/Vedf on H we obtain the jump  4 @G  kB Tm kB Tm B = − ; ≈

=

 V @H Tm HVedf ¥0 L (12.262) the numerical ≈ 1 follows from comparison with results from numerical simulations [240,300]. In an anisotropic material the important fluctuations involve the wavevector kz ∼ 1/"a0 along the field and thus L ≈ "a0 . We then arrive at the final result for the jump in B as it applies to continuous anisotropic superconductors, B ≈

kB Tm ¥◦ , ≈ 6. 10−4 2 ¥◦ "a0  (Tm )

(12.263)

approaching zero as Tm → Tc . Note that in an incompressible (uncharged, e → 0 and  → ∞) system, we correctly find B → 0. Rewriting (12.263) in the form B[G] ≈ (1.5·10−6/")Tm [K](Bm [G])1/2 and choosing " = 1/8 we arrive at a good agreement with the magnetization data of Schilling et al. [42, 308]

12 Vortex Matter

Fig. 12.18. Top: the jump B in the induction versus temperature T as measured in an YBCO single crystal [42] and calculated from the expression (12.263). The deviations close to the transition are possibly due to the increased relevance of disorder. Bottom: the same for a BiSCCO single crystal [40] using the result (12.266). The drop in B on approaching Tc is explained in terms of a temperature dependent cutoff in the electromagnetic fluctuations through the Josephson coupling at temperatures T > T em

on an YBCO single crystal, see Fig. 12.18 (we have made use of the experimentally measured melting line Bm (T)). The magnetization jump describes a transition from a “dilute” vortex solid into a “dense” vortex liquid, hence a vortex crystal melts like ice. The jump s in the entropy density is easily derived via the Clapeyron relation 1 dHm s = − B . (12.264) 4 dT This is converted into the entropy jump Sd = a20 ds per vortex per layer (we approximate Hm ≈ Bm away from Hc1 ) Sd ≈ −kB

d Tm dBm

˛d Tm ≈ kB , 4L Bm dT 4L Tc − Tm (12.265)

561

where we have assumed that the melting line vanishes Bm ∝ (Tc − T)˛ as T approaches Tc in the last equation. In the continuous anisotropic situation the singular factor 1/(Tc − Tm ) is compensated by the divergence of L ≈ "a0 ; on the melting line, the product a0 (1 − Tm/Tc ) is (roughly) temperature independent and we find a constant but material dependent entropy jump per vortex per layer. Using parameters for YBCO ((0) ≈ 1400 Å and d = 12 Å) we obtain the value Sd ≈ 0.4 kB , in good agreement with experiment [42]. Note that the entropy jump Sd > 0 naturally describes a transition from the low-entropy solid into a high-entropy liquid; combining the positive entropy jump Sd > 0 with the retrograded melting line dBm /dT < 0 the negative jump in the magnetization appears as a direct consequence of the Clausius–Clapeyron relation (12.264). Note that the result (12.265) differs from the simple estimate Sd ∼ kB (d/L) by the important factor 1/(Tc /Tm − 1) which becomes large near Tc ; in our derivation this factor stems from the shape of the melting line which involves the dependence on 1 − T/Tc through the temperature dependence of the Landau parameter ˛(T) (without this “internal” temperature dependence the melting line would decrease according to Bm ∝ 1/T 2). Deriving the entropy jump directly from the free energy via the relation S = −dF/dT care must be taken to take this additional temperature dependence in the free energy into account, cf. [300]. In layered BiSCCO the melting line is pushed down to low fields Bm (T) < B (T) over a large portion of the phase diagram. The dominant interaction in the vortex system is given by the electromagnetic one. The loosely bound pancake vortices undergo large thermal fluctuations and dominate the melting process, hence L = d; the shape of the melting line is given by (12.212). Close to Tc the Josephson interaction becomes relevant as "(T) > d: for T > T em ≈ Tc [1 − ("(0)/d)2]1/2 the dominant fluctuations at melting are cut off on the scale L ∼ " and the melting line is given by (12.213). In a layered material the jump B in the induction then takes the particular temperature dependence

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B ≈

⎧ ⎪ ⎪ ⎨

kB Tm , ¥0 d

⎪ k T ⎪ 2 /T 2 , ⎩ B m 1 − Tm c ¥0 "(0)

Tm < T em , T em < Tm ,

(12.266) where we have assumed that ≈ 1. Note that it is the temperature dependence of L, which goes from "a0 (Tm ) in a continuous anisotropic superconductor to L ≈ d and L ≈ "(Tm ) in a layered material, that leads to the different dependencies in the jump B(T). The result (12.266) is in good agreement with the experimental observations of Zeldov et al. [40] (see Fig. 12.18; for BiSCCO we use parameters d = 15 Å, (0) = 2000 Å, " = 1/400): At low temperatures Tm < Tc − 7 K the jump B increases linearly with temperature.About 7 K before reaching the transition, B drops sharply and vanishes at Tc , a behavior which can be explained in terms of the crossover at T em , where the Josephson coupling between the layers becomes relevant and cuts off the further growth of fluctuations. The jump in the entropy again follows from the general expression (12.265). In a layered material we have to insert the length L ≈ d or L ≈ " and the singular factor Tm/(Tc − Tm) remains uncompensated; as a result, the entropy jump per vortex per layer diverges on approaching Tc , ⎧

kB ⎪ ⎪ , Tm < T em , ⎪ ⎨  1 − (Tm /Tc )2 Sd ≈ 3 d kB ⎪ ⎪ ⎪ , T em < Tm . ⎩ 4 "(0)  1 − (Tm /Tc )2 (12.267) This divergence in the entropy jump upon approaching Tc is in agreement with experimental observations [40] and with the result of our melting analysis in Sect. 12.7.5 above, cf. Fig. 12.15. Note that in this latter case the divergence arises from 2D-like fluctuations pushing the melting line to zero at TBKT ; assuming a melting line in the form B ∝ (1 − 2T/"0 d) we obtain a divergent entropy Spc

 2T 1 . ≈ kB 4 "0 d 1 − 2T/"0d

(12.268)

12.7.7

Vortex Liquid

While the structure of a liquid (plasma, gas) of particles is intuitively understood, the structure of a line liquid is less obvious. In particular, one could think about a liquid state where the vortex lines still remain ordered with respect to their neighbors, see the disentangled liquid in Fig. 12.19, or alternatively, convey an entangled liquid where the vortex lines wind around each other, effectively producing large vortex loops, see Fig. 12.19. A very fruitful concept allowing to quantify this idea has been put forward by Nelson [11,305] who proposed a mapping of the 3D system of classical vortex lines to a system of 2D quantum bosons: Consider the simplified free energy 1 F = T T

L dz

  "l dR 2 2 dz

 

0

(12.269)

   1 2"0K0 |R − R |/ , + 2 = where the 2D coordinates R (z) define the (zdependent) positions of the individual lines. Here, the first term describes the elastic energies of the individual lines (of length L) while the second term describes the interaction with K0() the MacDonald √ function,K0( < 1) ∼ − ln ; K0 ( > 1) ∼ e− / . Here,we approximate the non-local (in z) interaction between vortex segments, see (12.62), by a local one — this approximation is valid as long as the angular distortions @z R remain small, i.e., not to deep into the vortex liquid phase. The expression (12.270) should be compared with the imaginary time () action of massive (m), twodimensional, interacting (q2 ) Yukawa bosons, B

1 S = B B  

T B

d 0

 

  m dR 2 2 d

(12.270)

  1 2  q K0 |R − R |/ , + 2 = with B and T B the parameters describing quantum and thermal fluctuations. The bosonic statistics

12 Vortex Matter

563

Fig. 12.19. Possible equilibrium phases for the vortex-liquid: disentangled vortex-liquid (equivalent to the normal ground state of 2D quantum bosons) and entangled vortexliquid (superfluid groundstate of 2D quantum bosons). The entangled liquid is characterized by vortex-loop excitations (cooperative ringexchange processes) and is thermodynamically equivalent to the normal metal phase

is realized through periodic boundary conditions R ( = 0) = RP( ) ( = B /T B ) with P a permutation of the particles. The classical statistical mechanics of the vortex system then can be mapped to the quantum statistical mechanics of 2D bosons through the identification T ↔ B , z ↔ , L ↔ B /T B, "l ↔ m, 2"0 ↔ q2 . Rewriting (12.270) and (12.271) in dimensionless form (we measure energies in units of q2 , times in units of B /q2 , and lengths in units of a , a2 = √ 2/n 3 with the density n = 1/a20 ), S = B

  dR 2 1 d 2 2dB d  0   1  + K0 |R − R |/ , (12.271) 2 = ˇ



we identify two relevant parameters, the inverse temperature ˇ and the de Boer parameter  dB ,

2"0 L q2 , = TB T T B = . √ = √ a g m a 2"l "0

ˇ=  dB

(12.272)

This mapping then brings together the physics of 3D vortices with that of 2D quantum bosons, of which a number of results are known. The phase diagram for the Bose model, as shown schematically in Fig. 12.20, contains three phases: A high temperature normal liquid, a low temperature (Wigner) crystal, and a superfluid due to large quantum fluctuations. This can be understood heuristically by considering the three energy scales involved: The transition from a normal liquid to a lattice is determined by the competition between the thermal energy 1/ˇ and the potential energy 1. In the limit  → ∞ and  dB = 0 the (classical) transition takes place at ˇm ≈ 140 [269], cf. the discussion on the one-component Coulomb plasma in Sect. 12.7.2. Increasing quantum effects, a further

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G. Blatter and V. B. Geshkenbein

Fig. 12.20. Schematic phase diagram for a system of 2D charged bosons in terms of the de Boer parameter  dB and inverse temperature ˇ. The solid lines represent phase transitions; quantum fluctuations are relevant in the shaded region. In the vortex system the parameter  2dB = T 2 /2"l "0 a2 and ˇ = 2"0 Lz /T. The constant field line for a thin sample (Lz < 70T/"0 ) is shown (dashed-dotted line) as it runs through all three phases. For thicker samples the line is pushed upward

transition takes the normal liquid to a superfluid when the thermal energy matches the kinetic energy,  2dB ˇ ≈ 1. Finally, at low temperatures, the competition between potential and kinetic energy determines whether the system is a crystal ( dB <  dBm ) or a superfluid ( dB >  dBm ). For ˇ,  = ∞, it is known that  dBm = 0.062 [309]. The boson phase diagram can be reinterpreted in terms of the vortex system, where ˇ is proportional to the sample thickness and  dB measures the strength of thermal fluctuations. For thin samples, ˇ < ˇm , 1/ 2dB, we find a disentangled vortex-liquid. In thicker samples, the system is either a lattice or an entangled vortex-liquid depending on temperature and magnetic field. Note the non-trivial mapping between the H–T and  dB –ˇ phase diagrams: In a thin sample, the constant field line (dashed-dotted in Fig. 12.20) passes through the crystalline phase (low T), the disentangled liquid phase (intermediate

T), and the entangled liquid phase (high-T). With increasing Lz , this line moves to higher values of ˇ, the vortex lattice melting line is determined solely by the value of  dBm , and we recover the bulk melting line in the form Tm ≈ 0.09 ""0 a (when compared to (12.215) this corresponds to a Lindemann number cL ≈ 0.17; we use "l = " 2 "0 for an anisotropic superconductor, while the 3D-XY simulations producing the result (12.192) involve a slightly different effective elasticity [242]). Rigorously, the bosonic phase transition into the superfluid state maps to a corresponding phase transition in the vortex system with a toroidal geometry. In a real vortex system the boundaries at z = 0 and z = L are free and the transition transforms into a crossover, except for the thermodynamic limit L → ∞. The phase diagram proposed in Fig. 12.20 is the simplest possible one, but more complicated versions are possible in principle. In fact, the transition from the Wigner crystal to the superfluid involves the change of two symmetries: the transverse translational symmetry of the lattice and the gauge symmetry of the superfluid. If these two symmetries do not change simultaneously an intermediate phase will appear,either a supersolid with both symmetries broken [74] or a normal liquid with none of the symmetries broken. The latter phase would correspond to a disentangled vortex liquid (cf. Fig. 12.19) and various arguments have been put forward in support of such an intermediate phase [287, 310]. However, numerical simulations of the 2D quantum bosons via a path-integral Monte Carlo technique [240] (see also [311]) show that no such complication of the phase diagram is realized, at least in the local approximation (12.271). To this end, the two symmetries in the problem are traced via a measurement of the structure factor S(K, ) =

: 1 9 K ()−K (0) N

(12.273)

and of the winding number [312] W 2  ns = . n 2 2 ˇN

(12.274)

Here, K () is the partial Fourier  transform of the density operator, (R, ) =  ı[R − R ()] and

12 Vortex Matter

 ˇ W =  0 d @ R is the winding vector measuring the diffusion of the center-of-mass of the system in imaginary time (periodic boundary conditions R (0) = RP( ) (ˇ) are used, with P a permutation of the N bosons; ns , n denote the superfluid and total density, respectively, and N is the number of bosons, N = 36–64 in the simulations of [240]; for larger systems, the fraction of entangled lines ne /n is measured instead of ns /n). Figure 12.21 shows the first Bragg peak S(K 1 ) and the superfluid/entanglement fractions ns /n and ne /n as a function of the de Boer parameter  dB ; as the Bragg peak collapses at  dBm = 0.062 the superfluid density rises steeply, showing that the system undergoes a sharp (first-order) transition from a crystal to a superfluid (entangled) liquid. As an additional result these simulations provide quantitative results for the jumps in the energy and in the density at the transition which compare well with the jumps in entropy and magnetization observed in experiments on YBCO single crystals [240], cf. Sect. 12.7.6 above.

12.7.8

565

Constitutive Relation

Thermal fluctuations of the vortex lines also change the constitutive B(H) relation and modify the meanfield results (12.56) and (12.57) for the magnetization [151,313,314], particularly in 2D thin films and in layered superconductors. Their effect is most dramatic near the onset of the vortex phase where fluctuations change the functional behavior (12.55) at the Hc1 transition and even render the transition first order at low temperatures. While these effects are very interesting from a theoretical point of view, their experimental observation appears to be difficult. In determining the new constitutive relation B(H) we have to include additional terms in the free energy density f which arise from thermal fluctuations of the vortex lines. One obvious term is the entropy contribution −Tsl of a fluctuating line [11]. The line entropy of an isolated line sl0 (T) is reduced by the potential cage set up by the surrounding vortices, sl (T, a0) = sl0(T) − (1/L) ln ˛ L/LT with the thermal collision length LT ≈ "l a20 /T and the numerical constant ˛ > 1. The free energy density

Fig. 12.21. The first Bragg peak and the entanglement parameter ne /n for a system with 64 lines and ˇ = 300. A sharp transition from a crystal to an entangled liquid is found at  dBm = 0.062. The structure factors just before and after the melting transition are displayed. The inset shows the superfluid density ns /n for a system with 36 lines

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G. Blatter and V. B. Geshkenbein

g(B) =

   Tsl0 B H l 1 − (12.275) − ¥0 l Hc 1   T2 + 6K0 ( ¥0 /B)) + ln ˛ "l a20

then picks up an additional steric repulsion ∝ T 2/a20 due to the above entropy reduction (here, l denotes the line energy). Minimizing g(B) with respect to B we obtain the new constitutive relation B(H): Close to the transition the vortex density 1/a20 = B/¥0 is small and the steric repulsion dominates over the usual repulsive interaction due to the vortex currents; as a result we obtain the new constitutive relation B ∼ ("l l 2 /T 2 )[H − Hcr1 ], changing the logarithmic mean-field result (12.55) into an algebraic form. In addition, the field-independent line entropy sl0 renormalizes the lower critical field Hcr1 = Hc1 (1 − Tsl0/ l ). Note that the long-wavelength elasticity "l relevant at low fields is equal to the line energy, "l ≈ l ∼ "0 . A similar renormalization of Hc1 due to disorder is discussed in [315,316]. In layered superconductors additional short wavelength distortions of the vortex lines have to be accounted for.These short wave-length distortions produce an attractive vortex–vortex interaction of the van der Waals type: Consider the case of a magnetically coupled layered system with two straight vortices 1 and 2 at a distance R. Displacing an individual pancake vortex in stack 1 by u1 is equivalent to placing a pancake-vortex–anti-pancake-vortex pair, i.e., a pancake-vortex dipole d1 , onto the vortex line. The pancake-vortex dipole d1 induces a second dipole d2 in vortex 2 within the same layer. With two pancake vortices interacting logarithmically within the same layer, the force from dipole d1 acting on vortex 2 follows a 1/R2 law, hence d2 ∼ d1 2 /R2. Finally, the interaction potential between two vortex dipoles follows a 1/R2 behavior as well and we obtain a longrange attractive potential VvdW ∼ (d1 /)2 4 /R4 between the vortex segments.For thermally driven fluctuations we have d12 ∝ u21 and the van der Waals attraction takes the form VvdW ∼ (T/d)(/R)4 [317] (alternatively, the fluctuations can be driven by disorder [318] or, at low temperatures, quantum mechanically). This additional attractive vortex–vortex interaction adds a contribution −(T4 /d)(B/¥0)3 to

the Gibbs free energy density g which drives the transition first-order at low temperatures [317].With a finite Josephson coupling " > 0 the attraction decays ∝ (/R)5 , VvdW ∼ (T/")(/R)5 (the power-laws R−4 and R−5 correspond to the well known ∝ r −6 and ∝ r −7 behavior of the usual van der Waals attraction between neutral atoms). The relation between the van der Waals attraction of vortex lines and the Casimir effect in vortex systems is discussed in [319].

12.8

Quenched Disorder: Pinning and Creep

Defects produce an inhomogeneous material and pin the vortices at energetically favorable locations in the crystal. The transport and magnetic properties of a bulk superconductor then are described by the Bean model [320]: vortex pinning leads to the formation of a vortex density gradient which is equivalent to a current flow j in the bulk via Maxwell’s equation j = (c/4)∇ ∧ B with B = nv ¥0 zˆ , nv the vortex density. This current flows free of dissipation and hence the optimization of the defect structure in a material is an important goal regarding applications. Science-wise, the problem of pinning confronts us with the physics of instabilities (strong pinning) or with the statistical summation of competing forces, a challenging problem in classical statistical physics (collective pinning). The maximal vortex density gradient is obtained when the pinning force density F pin acting on a vortex becomes equal to the driving Lorentz force density, resulting in a critical or depinning current density jc , 1 jc ∧ B = F pin . (12.276) c Usually, this critical current density is considerably reduced with respect to the depairing current density j0 , which can be estimated from Maxwell’s equation ∇ ∧ B = (4/c)j to be of the order of j0 ≈ cHc /4 ∼ c"0 /¥0, see (12.7) for the exact result from GL theory. The critical current density depends strongly on the strength of the pinning force. Since the Lorentz force is a force per unit volume, pinning has to pro-

12 Vortex Matter

567

Fig. 12.22. Left: Strong pinning centers due to individual defects produce large displacements (plastic deformations) in the vortex lattice. The vector r denotes the position of the unperturbed lattice far away relative to the impurity; the displacement u points in the opposite direction. Right: Weak pinning due to point-like disorder. The forces f originating from different impurities compete due to the elastic properties of the vortex lattice, hence forces add randomly within the collective pinning volume

duce a force per unit volume, too. In the strong pinning situation [6, 10, 134] defects act individually and produce plastic deformations (see, Fig. 12.22) [171,321,322]);the pinning force then is given by the product of the active defect density and the average pinning force, see Sect. 12.8.1 below. In the copperoxides a strong pinning situation naturally appears due to the layered structure, resulting in individual pancake-vortex pinning at low magnetic fields, cf. Sect. 12.9.7 [64,323,324]. When pinning is weak, the elastic forces dominate over the pinning forces and the defects compete, see Fig. 12.22; we then are faced with the problem of the statistical summation of individual pinning forces.Weak point-like pinning centers pose a generic problem relevant in a number of physical situations and the corresponding summation problem has been solved by Larkin and Ovchinnikov within their weak collective pinning theory,see Sects. 12.8.2 and 12.9 [8,10] (similar concepts have been applied to the problem of charge density wave pinning [325–327]). Such point-like pins produce an uncorrelated disorder landscape; other types of defects, such as columnar tracks produced by heavy ion irradiation [29, 30, 55, 89–91] or twin bound-

aries [27, 28, 328, 329] generically present in some of the copper-oxides, produce a correlated pinning potential and will be discussed in Sect. 12.10. A third distinction in the type of pinning, besides “strong” versus “weak” and “correlated” versus “uncorrelated”, is between “bulk” and “non-bulk”. The classic example of a “non-bulk” type pinning is that due to surface barriers [330]; furthermore, the platelet shape of layered copper-oxide superconductors (e.g., BiSCCO) produces the even stronger geometrical barriers [331–333], see Sect. 12.11. 12.8.1

Strong Pinning

The classic arguments characterizing strong pinning go back to Labusch [6], see also [10, 134]: In order to calculate the mean pinning force acting on the vortex lattice one has to determine the interaction of the latter with an individual strong pinning center and then average the force over randomly positioned impurities or defects. At low impurity concentration nimp , the displacement fields of different defects do not interfere and the critical current density jc can be obtained within a linear approximation in nimp ,

568

G. Blatter and V. B. Geshkenbein

jc ∧ B/c = Fpin = nimp fpin  .

(12.277)

The pinning force of an individual impurity fpin derives from the free energy epin (r) of the vortex system in the presence of the impurity, f pin = −∇r epin(r) ,

(12.278)

where r denotes the position of the unperturbed lattice relative to the impurity, see Fig. 12.22. If pinning is weak the energy landscape epin (r) is a single valued periodic function (see Fig. 12.23, usually with a period a˜ of the order of the lattice spacing a0 ) and averaging the pinning force over impurity positions the resulting force is zero — weak pinning is due to fluctuations in the defect structure and thus collective, see Sects. 12.8.2 and 12.9 below. Strong pins induce plastic deformations in the vortex lattice and the energy landscape epin (r) becomes a multi-valued function in the displacement r [6], see Fig. 12.23; as a result the averaging process produces a non-zero pinning force determined by the jump epin connecting different metastable states: Choosing r along the x-direction, we first express the average along the x-axis over the force −@x epin(x, y) through the jump epin (y) in the pinning energy, Lx fpin  = − 0 a0

= 0

dx Lx

Ly 0

dy @x epin (x, y) Ly

dy epin (y) . a0 a˜ (y)

(12.279)

The jump epin (y) depends on the “impact parameter” y, the asymptotic transverse distance between the pin and the trapped vortex line.Assuming a maximal transverse trapping distance t⊥ along the y-axis and approximating a˜ ≈ a0 we obtain the mean pinning force Strap t⊥ t⊥ t epin (0) ≈ 2 fimp ≈ 2 fimp , a20 a0 a0 (12.280) where we have expressed the jump epin (0) ≈ t fimp in terms of the typical impurity force fimp acting over the bistability range t of the function epin (x, 0); fpin  ≈

the product t⊥ t then defines the trapping area Strap associated with the strong pin. Note that this trapping area approaches zero at the crossover to weak pinning where the bistable region vanishes. The independent action of these strong pins finally determines the critical current density c nimp fimp  B c c t⊥ ≈ nimp 2 epin (0) ≈ neff fimp B B a0 fimp ∼ j0 nimp Strap  , (12.281) "0

jc =

with the effective impurity density neff = nimp (Strap /a20) and the depairing critical current j0 ∼ c"0 /¥0. In order to derive a quantitative criterion for strong pinning we consider a single defect center at r pin . Such a point-like defect exerting a force f pin (r) on the vortex produces a deformation at the impurity  u˛ (r pin ) = d3 r G˛ˇ (r pin , r)fpin,ˇ (r) = C¯ −1 f˛ , (12.282) with (cf. (12.92))  1 d3 k G˛˛ (k) . = (2)3 C¯

(12.283)

In (12.282) we have assumed an impurity with a force profile of width smaller than the lattice constant a0 . As the vortex lattice deforms (by u) in response to the force exerted by the defect we should evaluate the force in (12.282) at the position r + u and the displacement u(r) has to be determined selfconsistently from u(r) =

1 f pin [r + u(r)] . C¯

(12.284)

For weak pinning the displacement u is small and the solution u(r) ≈ f pin (r)/C¯ is a single-valued function.Strong pinning,however,produces multi-valued functions u(r) and f pin (r), cf. Fig. 12.23. The solution of (12.284) turns multi-valued whenever the displacement collapses, i.e., when @x u → ∞ (we move the vortex lattice in the x-direction). Deriving (12.284) with respect to x we find that u =

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569

Fig. 12.23. Energy landscape epin and pinning force fpin versus displacement r of the vortex lattice relative to the defect. In the weak pinning case the energy landscape epin is a single-valued function in r (dashed lines), while strong pinning produces plastic deformations and renders epin multi-valued (solid lines; dotted lines indicate unstable branches). The average pinning force fpin  is determined by the jump epin connecting different metastable branches. Left: large defects. Right: small defects with rpin < a0   fpin (x + u)/[C¯ − fpin (x + u)] (note that x > 0 implies u < 0) and we arrive at the (Labusch) criterion

@x fpin = −@x2 epin = C¯ ;

(12.285)

hence, in order to produce strong pinning the (negative) curvature of the pinning energy epin has to overcompensate the elasticity of the lattice (note that the curvature in the potential minima is positive; the Labusch criterion involves the maximal negative curvature above the inflection point). A strong pinning situation then occurs whenever the curvature in the pinning energy becomes large or the elastic moduli take on small values. Note that the Labusch criterion tests an individual isolated pinning center and classifies it as a weak or strong one. 12.8.2

Weak Collective Pinning

When the displacement fields u produced by individual impurities remain single valued the pinning centers are weak; trivial averaging over these individual pinning forces results in a zero average pinning force density, cf. (12.280). In this situation, pinning is due

only to local fluctuations in the pinning force density. The competition between elastic and pinning forces then produces a sublinear growth of the pin2 ning energy Epin (V )1/2 with volume V ∼ Ld : Let us characterize the disorder potential by the density of pins nimp of size rpin ∼  and the individual pinning force fimp . Due to the competition between the pins, the individual pinning forces add up only randomly within the volume V ∼ Ld and the fluctuations in 2 the pinning energy Epin (V )1/2 can be written as  2 1/2 2 Epin (V )1/2 ≈ fimp nimp (/a0)2 V rpin , (12.286) where the factor (/a0)2 accounts for the fact that only the vortex cores are pinned by the disorder. The result (12.286) implies that the pinning force acting on an individual stiff vortex grows √ only sublinearly 2 in the volume V , Epin (V )1/2 ∝ V . Since the driving Lorentz force increases linearly with volume we have to conclude that a stiff vortex lattice is never pinned and hence the critical current density vanishes in this limit. On the other hand, a real vortex lattice is characterized by a finite elasticity allowing its accommodation to the pinning potential on some

570

G. Blatter and V. B. Geshkenbein

length scale. Within weak collective pinning theory 2 the sublinear growth of Epin (V )1/2 goes over into a linear law when the displacement field u increases beyond the characteristic size rpin of fluctuations in the pinning potential Epin . Each volume of size Vc then is pinned independently with a pinning energy 2 Uc = Epin (Vc )1/2 . As the pinning energy grows linearly with volume beyond Vc we obtain a proper pinning force density Fpin ∼

Uc ∼ Vc rpin



2 nimp (/a0)2 fimp

Vc

1/2 . (12.287)

Balancing this pinning force density against the Lorentz force density jB/c we arrive at a finite critical current density jc ∼ cFpin /B. Note the opposite scal√ ing with Vc of the pinning energy U ∝ V and the c c √ critical current density jc ∝ 1/ Vc , a characteristic feature of the weak collective pinning theory. The remaining task is the determination of the collective pinning volume Vc ; its calculation is complicated by the dispersion and anisotropy of the vortex lattice and will be carried out in Sect.12.9.Assuming an isotropic non-dispersive model system with an elasticity C,the comparison of elastic (C(rpin/Lc )2 Ldc ) and pinning (Uc ) energies provides the collective pinning length (we choose d = 3) Lc ∼

C 2 a20 . 2 nimp fimp

(12.288)

Combining (12.287) and (12.288) we obtain the critical current density jc ∼ j0 [nimp a20 ]2

4 fimp

"04

,

(12.289)

where we have reexpressed the bulk elasticity C = "0 /a20 through the corresponding line elasticity "0 . 12.8.3

Weak versus Strong Pinning

It is instructive to compare the weak and strong pinning schemes and their dependence on dimensionality, particularly in the limit of a small defect density. We analyze strong pinning and the Labusch criterion (12.285) using a simple (isotropic) Ansatz

G(k) = 1/Ck2 for the elastic response function (G(k) = 1/"0k2 in the 1D problem, G(k) = a20 /"0dk2 in a 2D film of thickness d, and G(k) = a20 /"0k2 for the 3D situation).We will find that in one and two dimensions the effective elastic coefficient C¯ drops to zero as the integral in (12.283) diverges and we have to modify our analysis. The original Labusch criterion (12.285) identifying individual pinning centers as strong ones is applicable in 3D only where the integral in (12.283) converges. One Dimension: Single Vortex Pinning In 1D, individual pins are always strong as the effective elastic coefficient C¯ vanishes due to the diverging integral in (12.283). At the same time, the deformation of the line due to the action of these pinning centers is large and we cannot ignore their mutual competition. The collective pinning analysis in 1D then provides us with the collective pinning length 2 1/3 Lc ∼ ("02 /nimp fimp ) and a critical current density  jc ∼ j0

2 nimp  3fimp

"02

2/3 .

(12.290)

This result is valid as long as many pins compete within the volume  2 Lc ; the condition nimp  2 Lc > 1 puts a lower limit n¯ 1D ∼ fimp /"0  3 < nimp on the defect density, where the critical current density assumes the value j¯c ∼ j0 (fimp /"0 )2 . For small densities nimp < n¯ 1D each pin acts individually and we can determine jc from the force balance (¥0 /c)jc lu ∼ epin ∼ fimp u, with u ∼ t the displacement directed along the force acting on the vortex.The length l between two subsequent pins fixing the vortex derives from an analysis of the pinned vortex geometry, see Fig. 12.24: Integrating the force equation @2u "0 2 = f (z) (12.291) @z (here, f (z) denotes the force per unit length acting on the line) over one pinning center we find the angle  = @z u ∼ fimp /"0 distorting the vortex line at the location of the defect; alternatively,we cut the diverging integral in (12.283) on the scale l, C¯ ∼ "0 /l, and the relation u ∼ fimp /C¯ provides us with the deformation angle  ∼ u/l ∼ fimp /"0. A vortex segment

12 Vortex Matter

571

that the collective pinning expression (12.290) dominates over the strong pinning expression (12.293) at large densities nimp > n¯ 1D . Two Dimensions: Pinning in a Film Again, individual pinning centers are strong, however, the effective elastic constant C¯ diverges merely logarithmically, C¯ −1 ∼ (a20 /"0d) ln(L/). The cutoff length L is given by the distance to the next effective pin as determined from  the condition nimp (Strap /a20)dL2 ∼ 1, L ∼ a0 / nimp t t⊥ d, cf. (12.280). Assuming a transverse trapping length t⊥ ∼  we solve for the longitudinal displacement u ∼ t ∼ fimp /C¯ and find u ∼ a0 (fimp a0 /"0d) × ln("0 /fimp nimp  3). Given the trapping area Strap we easily find the mean force density Fpin = nimp fpin  ∼ 2 [nimp fimp /"0 ] ln("0 /fimp nimp  3 ) and the corresponding critical current density 2

jc ∼ j0 nimp a0  2 Fig. 12.24. Strong pinning geometry for a vortex line. The deformed vortex probes the volume l2  of width  in the direction transverse to the line and the motion

of length l deformed by the angle  in the direction of the driving force encounters l2 nimp defects (assuming a transverse trapping length t⊥ ∼ ) and we find the distance l when this number is unity, hence l ∼ "0 /nimp fimp . (12.292) Combining this result with the expression for the critical current density we find  jc ∼ j0

3 nimp  3 fimp

"03

1/2 .

(12.293)

At the crossover density n¯ 1D ∼ fimp /"0 3 the critical current density matches up with the weak pinning result, j¯c ∼ j0 (fimp /"0 )2 . Also, the displacement u ∼ lfimp /"0 ∼  (fimp /"0 )/(nimp  3 ) is of order  at the strong to collective-pinning crossover density n¯ 1D and hence smoothly matches the displacement field relevant in the collective pinning scenario. Note

"0 a0 fimp ln . d "02 fimp nimp  3

(12.294)

This result is valid for large impurity forces fimp and at small enough pinning densities nimp where neighboring pinning centers do not compete. The competition between neighboring centers appears when the displacement u ∼ a0 (fimp a0 /"0d) ln("0 /fimp nimp  3) drops below the width  of the individual pinning centers — in this situation, the competing elastic forces inhibit the optimal use of all the pins. Hence pinning turns collective below the critical force d 1 fLab ∼ "0 2 ; (12.295) 2 a0 ln(a0 /nimp d 4) the same expression derives from the Labusch criterion (12.285). Note that this “Labusch force” involves the distance to the next pin as a cutoff in the logarithm, in contrast to the original concept characterizing individual centers. Below fLab the collective pinning scheme provides the results d "0 1 Lc ∼  , a nimp d 0 fimp jc ∼ j0 nimp a0  2

2 a0 fimp ; d "02

Fpin ∼ nimp 

2 fimp

"0

,

(12.296)

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G. Blatter and V. B. Geshkenbein

at crossover, the collective pinning volume contains L2c d nimp  2 /a20 ∼ ln2 (a20 / nimp d 4) > 1 active pinning centers. Comparing (12.294) and (12.296) we note that the strong pinning result is logarithmically larger. Three Dimensions: Bulk Pinning The Labusch criterion (12.285) offering a distinction between weak and strong individual pinning centers becomes applicable in 3D, where the integral in (12.283) converges and the elastic coefficient C¯ remains finite, C¯ ∼ "0 /a0 . According to (12.285) an individual pinning center becomes strong when fimp > fLab ≡ (/a0)"0 . We begin with the analysis of the weak collective pinning situation assuming fimp < fLab (where necessary, we code quantities in this regime with a superscript “< ”) and find the following results for the 3D vortex system   f4 1 "0 2 2 2 imp Lc ∼ , F ∼ n a  , pin imp 0 nimp a20 fimp "03 jc ∼ j0 (nimp a20 )2

4 fimp

"04

∼ j0 (nimp  3 )2

4 fimp

. 4 fLab (12.297)

These results are valid as long as many (competing) pins are active in the volume Lc 3 , nimp ( 2 /a20 )Lc 3 > 1. For densities larger than n¯ 3D ∼ (fLab /fimp )3 (a0  2 )−1 each pin would act individually (note that pinning is collective at low densities nimp < n¯ 3D in 3D, but at high densities nimp > n¯ 1D in 1D). However, before reaching the critical density n¯ 3D the collective pinning length Lc drops below the distance a0 between the vortex lines: the condition Lc ∼ a0 provides us 2 2 with the crossover density n¯ < a0 ∼ (fLab /fimp ) /a0  < n¯ 3D where 3D weak collective pinning smoothly crosses over into the 1D weak collective pinning result (12.290), see Fig. 12.25. Turning to strong pinning fimp > fLab (coded with a superscript “> ”) we first consider high densities; as the Labusch criterion is not effective in 1D the system remains collectively pinned as long as nimp > n¯ 1D and then smoothly crosses over to the 1D strong pinning as nimp is decreased below

Fig. 12.25. Pinning diagram delineating various pinning regimes involving collective versus individual pinning and 1D-line versus 3D-bulk pinning: 3D wcp – bulk weak collective pinning, 1D cp – collective line pinning, 1D sp – strong line pinning, 3D sp – bulk strong pinning

n¯ 1D . With decreasing density nimp the pinning dis tance l ∼ a0 (flab /fimp )/nimp a0  2, cf. (12.292), increases beyond a0 as nimp decreases below n¯ > a0 ∼ (fLab /fimp )/a0  2 and the system enters the 3D strong pinning regime, see Fig. 12.25. The calculation of the mean pinning force density Fpin ∼ nimp fpin  proceeds along the lines discussed above in Sect. 12.8.1 and involves the trapping area Strap ∼ t (we assume a transverse trapping length t⊥ ∼ ) with t ∼ u ∼ fimp /C¯ determined by the effective elas¯ we obtain a mean pinning force dentic constant C; 2 sity Fpin ∼ nimp (/a0)fimp /"0 and a critical current density jc ∼ j0 nimp a0  2

2 fimp

"02

2

∼ j0 nimp  3

 fimp . (12.298) 2 a0 fLab

The bulk strong pinning result (12.298) smoothly transforms into the 1D expression (12.293) at n¯ > a0 where l ∼ a0 . On the contrary, the strong pinning expression (12.298) apparently does not match up with the bulk weak collective pinning result (12.297) upon crossing from strong centers fimp > fLab to

12 Vortex Matter

weak impurities with fimp < fLab (here, we concentrate on low impurity densities nimp < 1/a0  2, cf. Fig.12.25).However,we have to keep in mind that our rough derivation of the strong pinning result (12.298) breaks down on approaching the critical force fLab . Indeed, since the displacement field u(r) turns single valued below fLab , strong pinning vanishes altogether (with a power [fimp − fLab ]˛ ) and pinning survives only in the form of weak collective pinning due to fluctuations in the impurity density. Within the approximative scheme adopted here the sharp rise of the critical current density at fimp > fLab is encoded in a jump jc |sp /jc |wcp ∼ 1/nimp a0  2 > 1. The above discussion sheds more light on the concept of pinning in general. Starting from the rigid limit we understand that such a stiff lattice cannot be pinned. Going over to a finite but still large elasticity (such that fimp < fLab ) the weak deformations produced by individual pins do not allow individual pins to hold the lattice; indeed, following the discussion in Sect. 12.8.1 the averaging of individual pinning forces produces a null result, cf. (12.280). In this situation pinning appears only as a consequence of fluctuations in the pinning forces,i.e.,the second moment rather than the first moment in the force distribution function establishes a finite average pinning force density. Reducing the elasticity, the individual centers turn into strong ones when fLab drops below fimp . These strong pinning centers now start to pin the lattice individually and strong pinning, linear in the defect density nimp , outperforms collective pinning; hence, the first moment of the force distribution function takes over the leading role in the pinning process. This crossover between weak collective and strong pinning can in fact be realized in an experiment: increasing the magnetic field to approach the upper critical field Hc2 leads to a marked softening of the elastic moduli, cf. the discussion in Sect. 12.4.1. The reduction in the elastic moduli entails a decrease of the Labusch force fLab and triggers the crossover from weak to strong pinning, producing the famous peak effect in the critical current density [10].

12.8.4

573

Vortex Creep

Thermal fluctuations affect the vortex system in many respects; while their implications on the thermodynamic equilibrium state have been discussed in Sect. 12.7 above, thermal fluctuations interfere with pinning in two ways: the rapid thermal motion of the flux lines within their pinning valleys produces a smearing of the energy landscape and leads to a downward renormalization of the pinning strength, see Sect. 12.9.5 below. On the other hand, the current carrying state of a hard type II superconductor as given by the Bean state is only metastable and thus bound to decay: large thermal fluctuations help the flux lines to overcome their pinning barriers and move to the next favorable pinning valley where they are retrapped. This leads to a thermally activated motion of vortices commonly known as creep [14, 334] (see Kramers’ pioneering work on the problem of thermally activated processes and metastability [335] and [336] for a review on this topic). The basic equations determining the decay of the current density j are given by Maxwell’s equation, @t B = −c @x E, where we use a geometrical arrangement with the field B  zˆ and the current density j and electric field E = B ∧ v/c pointing along the yaxis. The velocity of the vortices is parallel to the Lorentz force, v = (v, 0, 0). Inserting the expression for the electric field into Faraday’s law we obtain the equation of continuity for the vortex lines, @t B = −@x (vB), and expressing j through B using Amp`ere’s law we arrive at the dynamic equation for the current density j, @t j =

c 2  @ vB . 4 x

(12.299)

Here, the most important factor is the velocity v of the vortices which is due to thermal activation over the pinning barrier U (j), v = vc exp[−U (j)/T] ,

vc ∼ jc B/c ,

(12.300)

leading to a non-linear diffusion equation [337] for the current density j   U (j) jc @t j ≈ − exp − . (12.301)  T

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The relaxation time  ∼ (Hc2 /H) nw2 /c 2 depends on the sample dimension w and hence is a macroscopic rather than a microscopic time scale [38,337]. The above equation can be solved with logarithmic accuracy [338] and we obtain   t U (j) = T ln 1 + , (12.302) t0

and the vortices depin. The numerical value of the exponent ˛ depends on the physical situation: e.g., the activation of a particle over a smooth barrier involves the exponent ˛ = 3/2,while creep of an elastic string trapped along its length in a smooth potential well is described by the exponent ˛ = 5/4 (this corresponds to the situation of intrinsic pinning with vortex lines arranged parallel to the ab planes or to vortex pinning by columnar tracks, see Sect. 12.10.1). In his phenomenological discussion, Anderson [14] used the value ˛ = 1 and this value is also found in an analysis of pinning by random point-like defects within the framework of weak collective pinning theory [339]. Combining the dynamics (12.302) with the disappearance of barriers near jc we obtain the famous logarithmic time decay of the (diamagnetic) current (we choose ˛ = 1 here),   T  t j(t) = jc 1 − . (12.304) ln 1 + Uc t0

typical experimental results for the activation energy Uc in high-Tc superconductors measured at low temperatures are in the range Uc ∼ 100–1000 K [44–46,340,341]. Since pinning in these materials is usually weak, such measurements of the activation energy Uc provide a useful test of the validity of weak collective pinning theory. So far we have concentrated on the case where j is near to its critical value jc , a restriction reasonably justified in conventional hard type II superconductors with typical decay coefficients T/Uc of the order of 10−3 [342,343]. In the oxide superconductors pinning is weak and the corresponding decay coefficients are much larger, reaching values of order 5% at temperatures T ≈ 20 K [46,344]. These large logarithmic decay rates are a result of various factors such as the high temperatures available in an experiment and the small pinning energies Uc (small coherence length , large anisotropy). Combining the large decay coefficients with a typical logarithmic time factor ln(t/t0 ) of the order of 20 we conclude that the current density j can decay to a (small) fraction of the critical current density jc during an experiment. The determination of the critical current density in the oxides then is always affected by the presence of “giant” creep [44–46] and the condition jc − j  jc is not valid in general. From a scientific point of view the case j  jc and in particular the limit j → 0 is very interesting: If we wish to probe the thermodynamic state of the vortex structure we should perturb the system only infinitesimally and record its response. For a truly superconducting state we expect to observe a vanishing resistivity  in the limit j → 0 or, in a somewhat different language, to see a sublinear “glassy” response of the vortex structure.Assuming weak collective pinning, we will see below (cf. 12.9.4) that the barriers U (j) against creep diverge algebraically with vanishing current density j,  j 

c U (j) ≈ Uc , (12.305) j

The temporal decay of the transport current is thus determined by the ratio T/Uc which can be found experimentally by measuring the relaxation of the diamagnetic moment of a sample in the critical state;

implying a strongly subohmic current-voltage characteristic of the form    Uc  jc

V ∝ exp − . (12.306) T j

with t0 = T/jc |@j U |. Using (12.302) the time evolution of the screening current density j(t) is obtained by simple inversion. The important quantity which we need to know is the activation energy U (j), in particular, its functional dependence on the current density j. The energy scale for the pinning barrier U is determined by the pinning energy Uc . The dependence of U on the transport current density j is due to the Lorentz force; as j approaches jc from below the barriers vanish,   j ˛ U (j → jc ) ≈ Uc 1 − , (12.303) jc

12 Vortex Matter

The divergence of the creep barriers described through the collective creep theory (cf. Sect. 12.9.4) is based on the elastic properties of the vortex system. If these elastic properties are lost, the barriers are expected to saturate at a value Upl (the so-called plastic barrier), resulting in thermally activated flux flow response (TAFF [56]). This scenario is realized in a vortex liquid phase where the shear modulus collapses, resulting in a plastic barrier Upl ∼ ""0 a0 and an ohmic current-voltage characteristic. The above ideas are amenable to experimental verification if the sample can be prepared in a state characterized by a small transport current flow (Maley plot) [17,345]: warming a sample in the critical state, the decay rate of the shielding current can be increased by several orders of magnitude such that the current density j drops far below its critical value jc within experimentally accessible time scales, allowing to probe the small current regime j  jc .

12.9

Uncorrelated Disorder: Collective Pinning and Creep

The theory of weak collective pinning produced by an uncorrelated disorder potential due to a high density of weak point-like defects has been studied in great detail [10, 150]; below we first concentrate on weak collective pinning of vortex lines and vortex bundles and then proceed with a discussion of collective creep, both thermal and quantum. Thermal fluctuations not only produce vortex motion; they also smear the pinning potential and thus lead to a downward renormalization of the critical current density.We end this section with a discussion of pinning in layered superconductors where the layered structure introduces elements of strong pinning. The classic example of weak collective pinning is based on a short-ranged (uncorrelated) disorder potential as produced by point-like defects. On a technical level such a pinning landscape is introduced into the Ginzburg–Landau model via spatial variation in the GL coefficient ˛(r) describing changes in the (local) transition temperature Tc , and/or by spatial variation of the effective mass m (r) describing changes in the mean free path l. For the generic case of weak point-like pins the disorder

575

landscape is usually characterized by a Gaussian distribution, e.g., ˛(r) = ˛0 + ı˛(r) with ı˛ = 0 and ı˛(r)ı˛(r  ) = ˛ ı(r − r  ), and similarly for m (r) = m0 + ım (r) with ım  = 0 and ım (r)ım (r  ) = m ı  ı(r − r  ). The pinning energy acting on a vortex line then is obtained by a convolution of the disorder potential Upin (r) and the form factor p(R) of the vortex line [we concentrate on the isotropic case here; the anisotropic situation is handled with the help of anisotropic scaling theory, see (12.170)],  pin(u, z) = d2 R Upin(r) p(R − u) , (12.307) with

⎧ ⎪ ⎨ Upin(r) = 1 p(R) =

ı˛(r)|¦0|2 ,

ım(r) ⎪ |¦0|2 , ⎩ ˛0 m0 1 − |¦v (R)|2 ,

ıTc − pins ,

 |∇¦v (R)| ,

ıl − pins ;

2

2

a convenient choice for the (normalized) vortex solution is [123, 124] ¦v ≈ [R/(R2 + 2 2)1/2] exp(i' ). The pinning energy correlator Upin(r)Upin(r  ) = ı(r − r  )

(12.308)

has a weight ([ ] = energy 2 /volume) ⎧  2

˛ Hc2 ⎪ ⎪ , ıTc − pinning , ⎪ ⎨ ˛ 2 4 0

= (12.309) ⎪ m  H 2 2 ⎪ ⎪ c ⎩ , ıl − pinning . m20 4 The correlator for the vortex pinning energy takes the form  pin (u, z) pin(u , z )  = × d2 R d2 R Upin(r)Upin (r  )p(R − u)p(R − u ) = ı(z − z )K(|u − u |) , with the disorder correlator  K(u) = d2 R p(R − u)p(R) .

(12.310)

(12.311)

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G. Blatter and V. B. Geshkenbein

The detailed shape of the correlator K(u) depends on the vortex form factor p(R); making use of the above vortex solution ¦v we find a sharply peaked function of width u ∼  and height K(0) ∼  2 (we find K(0) = 2  2 for ıTc -pinning and K(0) = (12/15)  2 for ıl-pinning). The slow (algebraic) decay of the vortex currents at distances R <  produces a correlator with slowly decaying tails, K(u → ∞) ∼ ( 4 /u2) ln(u/). Quite often, the simple form function p(R) = exp(−R2/2 2) is used, producing a Gaussian correlator K(u) =  2 exp(−u2 /4 2) [346,347]. In calculating the pinning energy correlator of the vortex lattice the structure factor has to be replaced by that of the lattice which occupies only a fraction  2 /a20 of the volume, cf. (12.286). For small displacements u < a0 the disorder landscape is short-scale correlated with 1 K(|u − u |) . a20 (12.312) For displacements u > a0 the vortex lattice probes the same disorder landscape and the correlator K(u) has to be replaced by a periodic one [19–21]. The generic abstract problem we are confronted with in this section is that of a d-dimensional elastic manifold moving in n transverse dimensions and subject to a disorder potential; the free energy of this object takes the form Epin (u, r)Epin (u , r  ) = ı(r − r  )



 F=

dd r

 C (∇u)2 + Epin (u, r) , 2

(12.313)

with the correlator Epin (u, r)Epin (u , r  ) = ı d (r − r  )K(|u − u |) . (12.314) For a single vortex line d + n = 1 + 2, r → z, C → "l , and Epin → pin with a correlator (12.310), K(u) = K(u); for the vortex lattice d + n = 3 + 2, the elastic part (C/2)(∇u)2 of the free energy has to be replaced by the dispersive expression (12.95) and the disorder correlator is given by (12.312), K(u) = K(u)/a20 .

12.9.1

Dirty Elastic Manifolds

To acquaint ourselves with the dirty elastic manifold problem we briefly outline a few pertinent results. The interesting behavior of this system results from the competition between elasticity, thermal fluctuations, and quenched disorder. Let us study the behavior of the manifold on the scale |r| = L and first ignore the disorder energy Epin ; then the competition between the elastic energy ∼ Cu2 Ld−2 and the tem1/2 perature T results in distortions [u(L)−u(0)]2th ≡ √ 1/2 u2 (L)th ∼ T/C Lth with the thermal exponent dth =

2−d , 2

(12.315)

independent of the number of transverse dimensions n. An elastic string (d = 1) then exhibits thermal wandering u2 (L)th ∝ TL, while the distortions grow only logarithmically for an elastic membrane, u2 (L)th ∝ T ln L for d = 2. Above two dimensions elasticity wins over thermal fluctuations and the manifold remains undistorted at large scales. As an illustration consider a scattering experiment probing the locations of atoms in a crystal: the long-range order in the three-dimensional crystal produces sharp ı−function peaks which are only reduced in magnitude by thermal fluctuations (via the Debye–Waller factor). On the other hand, in a two-dimensional crystal thermal fluctuations turn the reflections into algebraic peaks, cf. Sect. 12.7.4. Next, let us ignore thermal fluctuations and investigate the effect of quenched disorder. Assuming weak disorder we can study its effect in perturbation theory [18] using the Green’s function G ∼ 1/Ck2 of the elastic free energy (the “free” problem). Then u(k) = −G(k)∇u Epin (u, k) and we find (to lowest order in u) a disorder-induced roughening  [1 − cos(k · r)] 2  u (L)dis ∼ K (0) dd k . (Ck2 )2 (12.316) For d ≤ 4 the integral in (12.316) is determined by the lower cutoff ∼ 1/L and we obtain the result u2 (L)dis ∼ [K (0)/C 2]L4−d . Hence disorder is relevant in dimensions d < 4 and marginally relevant in d = 4 with u2(L)dis ∼ [K (0)/C 2] ln L. Within perturbation theory the effect of the disorder potential

12 Vortex Matter

is encoded in the force–force correlator K  evaluated at u = 0. The above perturbative approach breaks down in low dimensions d ≤ 2 as the integral in (12.316) diverges at small k; however, the result 4−d (12.317) = 2 for the wandering exponent remains valid, as scaling arguments show [10]: balancing the elastic energy ∼ Cu2 Ld−2 against the disorder energy [K (0) u2 Ld ]1/2 we recover the Larkin exponent dL independent of the number of transverse dimensions n (this scaling result is confirmed by other, more rigorous, approaches [348,349]). Perturbation theory also breaks down at large 1/2 distances L > Lc when u2 (L)dis > rpin increases beyond the scale rpin defined by the disorder po2 tential Epin ; the condition u2 (Lc )dis = rpin defines 2 the crossover length Lc = (rpin C 2 /K (0))1/(4−d) [10]. While the manifold probes only one pinning valley at small scales L < Lc , the availability of many (metastable) minima allows for further optimization of the manifold’s state at large scales L > Lc [350,351]. The determination of the large-scale wandering exponent d,n is a difficult problem in the statistical physics of disordered systems [111, 112]; the simplest approach producing an acceptable estimate is based on simple Flory- [352]/Imry-Ma [353] type scaling arguments. The crucial element in such a scaling estimate is to find an appropriate Ansatz for the disorder energy. At short distances, where only one pinning valley is relevant, the potential can be expanded, Epin (u) ∼ −F pin · u, resulting in 2 the above estimate Epin  ∼ [K (0)u2 Ld ]. On large distances we cannot expand any longer and have 2 to estimate the disorder energy Epin  directly from the potential Epin . Assuming a disorder landscape with Gaussian disorder described by the correlator (12.314) we adopt the asymptotic power-law Ansatz K(u) ∼ u−ˇ ; the value ˇ = n describes a shortrange correlated random potential (K(u) ∼ ı n (u)), for ˇ = −2 we recover the Larkin Ansatz for a random force field, and ˇ = −n corresponds to the (longranged) random-field situation [354]. This scaling Ansatz for the disorder correlator then produces a 2 pinning energy scaling as Epin [V = Ld , u(L)]1/2 ∼

577

(Ld u−ˇ )1/2 ∼ 1/2 L(d−ˇ)/2 , and comparing to the elastic energy C(u/L)2 Ld ∼ CLd−2+2 we obtain the Flory type result F d,n =

dL

4−d . 4+ˇ

(12.318)

Again, we find that disorder disrupts the manifold in dimensions d < 4 [18] and choosing ˇ = −2 we recover the Larkin exponent dL = (4 − d)/2. The “Flory” expression is believed to provide the exact result [354] for the random-field problem (ˇ = −n) where a mean-field analysis is valid. Also, a more sophisticated approach based on the introduction of replicas and use of a variational harmonic Ansatz for the free energy confirms the Flory type formula (12.318) when an appropriate replica symmetry breaking scheme is applied [355]. On the other hand, it is clear that the “Flory” result is only approximate for the short-range case ˇ = n as it misses to reproduce the exact result 1,1 = 2/3 known for the directed polymer problem (d = 1) with one transverse dimension (n = 1) [356]. The functional renormalization group (FRG) theory [357] allows to elaborate on the problem in a more controlled manner. Within an = 4 − d expansion the 1-loop renormalization group equation for the (scale dependent) disorder correlator Kl (u) involves the usual scaling terms plus quadratic corrections arising from integration over short scale disorder [357,358],

@l Kl (u) = ( − 4)Kl (u) + u Kl (u) +



 I[Kl (u)Kl (u)/2 −

(12.319)



 Kl (u)Kl (0)]

,

with I = Sd−1  d−4 /(2)d C 2 , l = ln R , and   a short scale cutoff; Sd−1 is the surface area of the unit sphere in d dimensions and we have introduced

.. the notation Kl ≡ @u .. Kl to denote derivatives (the last term in (12.320) can be rewritten in terms of derivatives with respect to u and takes the form I[Kl2 (u)/2 − Kl(u)Kl(0) + (n − 1)Kl2 (u)/2u2 − (n − 1)Kl(u)Kl(0)/u]). A straightforward numerical solution of the fixed-point equation provides the results d,1 ≈ 0.2083 [357] and d,2 ≈ 0.1766 [359]. An alternative analytic scheme to determine the wandering exponent  puts the emphasis on the asymptotic behavior of the fixed-point function [360]: looking

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G. Blatter and V. B. Geshkenbein

for a fixed-point solution with an algebraic asymp∗ ∗ totic KLR ∝ u−ˇ , the scaling terms define the exponent  = /(4+ˇ ∗ ) and thus reproduce the“Flory”result (12.318) with ˇ = ˇ ∗ . In addition, the full 1-loop fixed-point equation also allows for a short-ranged fixed-point function with a rapidly decaying Gaus∗ sian form, KSR (u) ∝ u4+n− / exp[−u2 /2 ] [357]. The following general picture capturing the behavior of both long-range random-field (ˇ < 0) and short-range random-bond (ˇ > 0) problems then has been proposed [360]: For large negative ˇ the FRG flow does not modify the asymptotic behavior of the correlator, ˇ ∗ = ˇ and the “Flory” result holds. Increasing ˇ from below (i.e., decreasing the correlator’s range) we should stop the further increase of ˇ ∗ when we reach the critical value ˇc∗ associated with the short-ranged random-bond problem. Following the proposal in [360] a possible criterion determining ˇc∗ derives from the asymptotic form of ∗ the fixed-point function KSR (u): setting ˇc∗ equal to the exponent 4 + n − / in the algebraic prefactor ∗ of KSR (u) and using the simultaneous validity of the “Flory” result (12.318) at ˇc∗ one obtains ˇc∗ = n/2, resulting in the short-range exponent SR d,n =

4−d . 4 + n/2

(12.320)

Further decrease of the correlator’s range does not change this result, hence all short-range problems are supposed to flow to a fixed-point solution characterized by ˇc∗ . Although the use of the power-law ∗ correction to KSR (u) as the relevant criterion to de∗ termine ˇc is somewhat questionable, this scheme does reproduce the correct value 1,1 = 2/3 for the random polymer problem in 2D space. The scheme adopted in [357] can be extended to follow the same philosophy regarding the evolution of the long-range F SR exponent d,n to the short-range result d,n ; the in∗ crease in ˇ should be stopped at a different value ˇc∗ = 0.801 for n = 1 and ˇc∗ = 1.663 for n = 2. These values deriving from the numerical solution of the fixed-point function (and hence involving all scales) are larger than the result ˇc∗ = n/2 emphasizing the tail of the correlator. The same result (12.320) has been found via a different route based on physical arguments and scaling

estimates [17]: The argument starts with the observation that the elastic and pinning energies competing for the manifold’s state involve separate parameters, elasticity but no disorder on the one hand, and disorder but no elasticity in the disorder term. Second, the disorder energy is expressed with an Ansatz considering separately the scaling arising from the manifold’s volume (∼ (L/Lc )dı ) and its transverse degrees of freedom (∼ (/u)˛ ), Epin ∼ Uc (L/Lc )dı (/u)˛ . The volume exponent ı = 1/2 then derives from the requirement that Epin does not depend on the elasticity C; this is the usual volume scaling as it is also assumed in the Flory-type analysis. Balancing disorder and elastic energies, this Ansatz provides the SR exponent d,n = (4 − d)/(4 + 2˛). Second, we have to find the exponent ˛ describing the influence of the transverse degrees of freedom. The Flory-type analysis valid for a long-range correlator leads to a value ˛ F = ˇ(n)/2; on the other hand, we expect the correlator to be renormalized at large distances in the short-range random-bond problem and hence ˛ SR is not a priori known. For a rigid manifold, the dependence of the disorder energy on the length L is entirely due to the random addition of forces producing the simple square root dependence on volume, hence a stiff manifold is characterized by an exponent ˛ = 0. The finite elasticity allows the medium to wander in the transverse dimensions and to explore all the metastable states within the transverse volume V⊥ = un . The number of available low lying metastable states should thus scale with the transverse volume V⊥ , hence ˛ SR = ˛0SR n, with ˛0SR a fixed constant independent of d and n. Using the exact result 1,1 = 2/3 we obtain ˛0SR = 1/4, hence ˛ SR = n/4 and we recover the short-range exponent (12.320). The large numerical value of the wandering expoSR nent d,n , cf. (12.320), characterizes the strong effect that quenched disorder exerts on an elastic manifold. However, additional “microscopic” features of the elastic manifold may reduce the effect of disorder. Here, we are interested in the behavior of Vortex Matter where the elastic manifold is made from a periodic array (with periodicity a0 ) of interacting vortices; both, the long-range interaction between the vortices and their periodic arrangement have drastic consequences on the impact of quenched disorder

12 Vortex Matter

on the Abrikosov vortex lattice and reduce the wandering exponent to zero. Let us first concentrate on the effect of long-range interaction. Neglecting the effect of screening, the interaction between vortex segments decays only algebraically ∝ 1/r, cf. (12.62), and the elastic tilt and compression moduli become dispersive, cf. (12.68). This dispersion in the elastic response lifts the effective dimensionality of the vortex system to four dimensions where disorder is only marginally relevant [10, 346, 347]. As a consequence, the displacement field u2 (R, L) ∝ ln(1 + R2/a20 + L/a0 ) grows only logarithmically, cf. Sect. 12.9.3, and the longrange order of the Abrikosov vortex lattice is transformed into quasi-long-range order in the presence of disorder. The algebraic decay of the translational correlation function K  (R)K  (0) ∝ R− depends on the disorder parameter ,  ∝ , and algebraic Bragg peaks in the structure factor survive in the limit of weak disorder (cf. the discussion on quasilong-range order in Sect. 12.7.4). The above behavior is cut off due to screening on the scales R < , L < 2 /a0 , but is resurrected on larger scales due to the periodic nature of the vortex manifold: for small displacements u < a0 the lattice experiences a different disorder potential and, although taking place in the manifold’s space, the motion can be considered as a transverse one (note that a longitudinal motion, e.g., the motion of a string along itself, is not subject to pinning forces as the manifold glides along the pinning valley). However, when displaced a distance u > a0 the vortex lattice is exposed to the same pinning potential and the disorder is effectively reduced. This feature is captured by a periodic correlator K(u+a) = K(u) with a a lattice vector; hence, besides random bond (short-range) and random field (long-range) disorder a third type of random periodic disorder turns out to be relevant [19–21]. This type of random periodic disorder is realized in the context of charge density wave pinning and pinning of the periodic vortex lattice; the wandering exponent takes the value  CDW = 0 and the displacement field grows only logarithmically with distance, u2 (R > Ra)/a20 ∼ c ln(R/Ra) [19–21], with the crossover length Ra defined via the condition

579

u2 (Ra ) ∼ a20 . Note that within this random periodic pinning regime the prefactor of the logarithm is a number, [u(R) − u(0)]2/a2 ∼ 2c ln(R/Ra), c ≈ 0.0434, [21, 361]) independent of the disorder strength, producing a power-law decay R−(0,1) in the density–density correlator with an exponent (0,1) = [a K(0,1) ]2c/2 ≈ 1.14, cf. (12.236) and (12.237) (see [21, 361] for a detailed discussion of nonuniversal features). Hence the destruction of translational order in the vortex lattice is quite subtle, with logarithmic regimes arising due to the long-range interaction between vortex segments at intermediate scales and due to the periodic nature of the manifold at large scales. By virtue of this smooth algebraic decay of translational order the asymptotic logarithmic regime has been termed the Bragg-glass scaling regime [21] (characterized by the Bragg-glass exponent  BG =  CDW = 0). The problem of vortex pinning and creep in a weak, uncorrelated random potential is intimately related to the problem of elastic manifolds in disordered media. Vortex pinning is determined by the competition between elasticity and disorder on short scales, and hence the crossover length Lc is a crucial quantity in the determination of the critical current density. Vortex creep involves the barriers separating metastable states. Near jc these barriers again are determined by the short-scale properties of the elastic manifold. Decreasing the driving force, the vortex system probes increasingly larger length scales and the manifold’s large-scale properties become important; the details of pinning and creep in the vortex system and their relation with the above results will be discussed in the following sections. Finally, the interplay of thermal and quenched disorder will lead to the phenomenon of thermal depinning discussed in Sect. 12.9.5. 12.9.2

Single Vortex Pinning

An individual vortex line subject to a disorder potential is described by the free energy (12.313): while the elastic term tries to keep the vortex line straight, the disorder term allows the vortex to lower its energy as it aligns itself with the energetically favorable valleys in the pinning landscape. The energy gain due

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to the disorder potential arises from fluctuations in the pinning landscape and hence involves the square of the pinning potential,  2 Epin (L) = dz dz  pin(0, z) pin (0, z ) ∼ K(0) L  2 L .

(12.321)

"l u2 − f ∼ 2 L2

7

2 u L 

(12.323)

for the maximal displacement u ∼ rpin ≈  we obtain the collective pinning length Lc and the pinning energy Uc ,  1/3 , (12.324) Lc ∼ "l2  2 /  2 1/2   4 1/3 2 4 Uc ∼  Lc ∼ "l  ∼ Hc  /Lc .

The sublinear growth of the pinning energy 2 Epin (L)1/2 has its origin in the competition between the individual pinning centers: Modifying (12.286) to describe the pinning of an individual vortex line the 2 fluctuations in the pinning energy Epin (L)1/2 can be written as (note that only defects within a distance  away from the vortex core contribute to the pinning energy, hence V ∼  2L, and we can drop the factor (/a0)2 )

Each segment of length Lc of the vortex then is pinned by the collective action of all the defects within the collective pinning volume Vc ∼  2Lc , which then act to produce a finite pinning potential Uc . Within each pinning valley u <  the displacement u grows with distance following a 3/2-power law, cf. (12.317)

2 2 Epin (L)1/2 ≈ (fimp nimp  2L)1/2  ,

u ∼ (L/Lc )3/2 .

≈

2 fimp nimp

 . 2

(12.322)

The last equation relates the disorder parameter to the individual pinning force fimp and the impurity density nimp ; the disorder parameter then can be determined via the defect induced variation in the Ginzburg–Landau parameters ˛ and m following the above scheme [150] or via a direct calculation of the individual pinning force fimp [362,363]. Following the general ideas of weak collective pinning theory (cf. Sect. 12.8.2) we have to cut the sublinear growth of (12.321) on the collective pinning length Lc . The latter is found by minimizing the free energy of the vortex with respect to the length L over which the pinning forces add up randomly and we will do this via scaling estimates (in such estimates we will drop all numerical factors and we will remind the reader by the use of the symbol “∼”; a more accurate scheme providing estimates for these numerical prefactors is the dynamical approach introduced by Schmid and Hauger and by Larkin and Ovchinnikov [8,9]).We start from the free energy (12.313) of a vortex line subject to a disorder potential pin and estimate the energy involved in displacing a vortex segment of length L by u ≤ rpin. While the disorder  potential contributes an energy gain −  2L(u/), cf. (12.321), the elastic distortion of the line costs an energy "l u2 /L. Minimizing the free energy density

(12.325)

Finally, using the characteristic transverse scale rpin ≈  of the pinning landscape pin , we obtain a pinning force fpin ∼ ( Lc )1/2. The critical current density jc follows from comparing fpin with the Lorentz force jc ¥0 Lc /c and we obtain   2 c  1/2 ∼ j0 , (12.326) jc ∼ ¥0 Lc Lc with the depairing current density j0 ≈ "0 c/¥0 introduced above, see (12.7). The regime of weak collective pinning is characterized by a large suppression of the critical current density jc with respect to the depairing value j0 and hence the collective pinning length Lc should be much larger than the coherence length ,Lc  .The latter condition is consistent with our use of elasticity theory in the determination of the collective pinning length. On the other hand, for Lc <  the distortion of the vortex line is large, elasticity theory breaks down, and the pinning is strong. Experimentally, the collective pinning length Lc is difficult to determine, one possibility being to investigate the thickness dependence of the critical current density in a thin film [364–366]. On the other hand, the collective pinning energy Uc can be determined more directly in a magnetic relaxation experiment by measuring the creep rate, see Sect. 12.9.4.

12 Vortex Matter

The quantity most easily amenable to experimental measurement is the critical current density jc . A comparison of the critical current density jc with experimental results provides us with a check for the validity of collective pinning theory. We then are forced to study the microscopic origin of the disorder and determine the parameter . Alternatively, we can take a more phenomenological point of view and regard jc as the simplest experimentally accessible quantity which we can use for the characterization of the disorder potential. Taking the latter point of view we express the collective pinning length Lc and the collective pinning energy Uc by the critical current density jc , Lc ∼ 

 j 1/2 0

jc

,

Uc ∼ Hc2  3

 j 1/2 c

j0

.

(12.327)

The results (12.325) and (12.326) apply to the limit of an isolated vortex, i.e., for weak enough magnetic fields such that we can neglect the interaction between neighboring vortices. Comparing shear (c66 (u/a0)2 La20 ∼ "0 (u/a0)2 L) and tilt ("l u2 /L) energies produced in a deformation u on scale L we find that we can neglect the interaction with other vortices if a20 > ("0 /"l )L2 . Inserting the collective pinning length L ∼ Lc we find that the single vortex pinning regime extends up to magnetic fields of order jc Bsv ∼ Hc2 . (12.328) j0 Anisotropic Material In generalizing the above results to the situation in anisotropic materials we can follow the conventional route and rederive the various expressions while taking the material anisotropy into account [206]. However, since pinning involves the length scale    it is more convenient to use the scaling approach (see Sect. 12.6) and generalize all isotropic results directly to the anisotropic case. For isotropic superconductors we have found a collective pinning length Lc ∼ ("02  2/ )1/3, a pinning energy Uc ∼ (  2 Lc )1/2, a critical current density jc ∼ j0 (/Lc )2 , and the limiting field Bsv ≈ (jc /j0 )Hc2 . Using our scaling rule (12.170) we proceed

581

with the following arguments: The collective pinning length Lc is a longitudinal length, therefore Lc (# ) = ˜ c ∼ ("˜02 ˜2/ ˜ )1/3 we have ("/"# )L˜ c .In order to obtain L ˜ , ˜ ˜ ) to express all rescaled material parameters (, ˜ ˜ by the original parameters ( = ,  = , ˜ = /"), hence L˜ c ∼ ("02  2"/ )1/3 and we obtain the final result  2 2 4 1/3 Lcc "  " c 4/3 iso Lc (# ) = , Lc = " Lc ∼ . "#

(12.329) Here, Lcc is the collective pinning length for the case where the magnetic field is aligned with the c-axis and Liso c denotes the collective pinning length in an equivalent isotropic material (" = 1) with identical material parameters , , and . The pinning potential Uc is a scalar quantity; after rescaling, the isotropized system does not emphasize a particular direction and hence the pinning potential should not depend on the angle # . The scaling factor for Uc is sE = " and rewriting U˜ c in terms of the original material parameters we obtain the result  1/3 Ucc = " 2/3 Uciso ∼ " 2/3 "0  4 ,

(12.330)

independent of the angle # , as expected. In the anisotropic material we have to distinguish between two critical current densities, the in-plane critical current density jc  x involving a Lorentz force along y , and jc⊥  y , the out-of-plane critical current density, pushing the vortices along the direction of the xaxis. The in-plane critical current density is obtained from rescaling the force balance equation between the pinning and the Lorentz force, ˜jc¥0 L˜ c  ∼ U˜ c . The in-plane critical current density then scales like a  planar length and thus jc = ˜jc = " −2/3 jciso = jcc . On the other hand, for the out-of-plane critical current density we have to orthogonalize the Lorentz force after rescaling (cf. (12.174) and (12.175)) such that jc⊥ scales like a transverse length, therefore jc⊥ = "# ˜jc and we obtain the result jc⊥ (# ) = "# jcc ; in summary, jc = " −2/3 jciso = jcc ,

jc⊥ (# ) = "# jcc .

(12.331)

Finally, the field Bsv limiting the regime of single vortex collective pinning becomes Bsv (# ) ∼ (jcc /j0)Hc2 (# ), where we have used the scaling rule for

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G. Blatter and V. B. Geshkenbein

magnetic fields and the above result for the transformation of the in-plane critical current density jc . 12.9.3

Vortex Lattice Pinning

When studying the weak collective pinning of a bulk vortex lattice we have to account for both shear and tilt deformations; the additional shear energy takes care of the interaction energy produced in the relaxation to the disorder landscape (the large compression modulus inhibits density modulations and we can ignore this term in the elastic energy). In comparing the tilt and shear energies of a lattice deformation u we have to account for the dispersion of the tilt modulus c44, see (12.74), B2 /4 u2 u2 Etilt ∼ c44 2 ∼ , V L 1 + 2 /R2 L2 ¥0 B u2 u2 Eshear ∼ c66 2 ∼ . (12.332) V R (8)2 R2 Here, R and L are the transverse (perpendicular to the field) and longitudinal (along the field) lengths associated with the deformation u and V ≈ R2 L is the bundle volume. Comparing these tilt and shear energies we find the geometry of the pinned vortex bundles, ⎧ R  ⎪ ⎪ R, a0 < R <  , ⎨ c44 (R) a0 L∼ R∼ ⎪  c66 ⎪ ⎩ R, 
Balancing the pinning energy against the elastic (shear) energy within the volume V we obtain the second relation between the transverse and longitudinal bundle lengths R2 ∼

 L 3 2 2 c66  2 c a0 L ∼ a0 L .

a0

(12.335)

Here, we have made use of (12.325) and the expression "l ≈ "0 for the line elasticity in an isotropic material (note that we use the single vortex pinning length Lc as a disorder parameter). Making use of the relation L ∼ (/a0 )R valid in the nondispersive regime we find the expressions for the bundle dimensions Rc and Lbc , the pinning energy Ucb ∼ c66 (/Rc )2 Vc and the critical current density jcb ∼ cUcb /BVc ,  L 3 c Rc ∼  >, a0   2  L 2  c Lbc ∼ Lc >  , (12.336) a0 a0 a0   2  L 4 c Ucb ∼ Uc , a0 a0  a 2  a 4 0 0 , jcb ∼ jc  Lc

(12.337)

where we have expressed the results for the vortex bundle through the single vortex quantities Lc ∼ ("02  2 / )1/3, Uc ∼ "0  2 /Lc , and jc ∼ j0 Uc /"0Lc . 2 Comparing the pinning energy [Epin (u, V )]1/2 ∼ 2 2 1/2 [( u /a0 )V ] within one pinning valley with the shear energy we find the scaling of the displacement u with distance R,  a 3  R2   a 3 R 0 0 u2 ∼  2 ∼ 2 . (12.338) Lc a0 L Lc  When attempting to repeat this scaling analysis for the dispersive regime a0 < R <  we find that the transverse scale R drops out of the various relations (cf. equations (12.333) and (12.335), (12.338)), hence one may suspect that the displacement u grows only logarithmically with distance R, a behavior that cannot be obtained within a simple scaling approach. We then have to change our strategy and determine the mean squared displacement

12 Vortex Matter

field u2 (R, L) ≡ [u(R, L) − u(0, 0)]2  with the help of perturbation theory using the small disorder parameter [10]; the relation u2(Rc , Lc ) ≈  2 then directly determines the bundle geometry in terms of the disorder parameter and the results for the pinning energy and the critical current density are trivially obtained via scaling. We start from the static (! = 0) version of the force equation (12.90) with the force Fpin,ˇ (u, r) ≡ −@uˇ Epin (u, r) deriving from the disorder landscape, u˛ (k) = G˛ˇ (k, ! = 0)Fpin,ˇ (u, k) .

(12.339)

To lowest order we can set u = 0 on the right hand side of (12.339) and determine the correlator  3 3  d kd k u2 (k) = G˛ˇ (k)G˛ˇ  (k  ) (2)6 × Fpin,ˇ (0, k)Fpin,ˇ  (0, k ) , (12.340) with the force correlator deriving from the energy correlator (12.312), in Fourier space,

ıˇˇ  ı(k + k  ) . a20 (12.341) Combining (12.340) and (12.341) we obtain the real space mean squared correlator  2 d3 k u2(r) = 2 [1 − cos(k · r)] (2)3 a0 (12.342) × G˛ˇ (k)G˛ˇ (−k) . Fpin,ˇ (0, k)Fpin,ˇ  (0, k ) = −(2)3

The main term in (12.342) originates from the transverse part in G˛ˇ and the largest contribution to the integral  d3 k [1 − cos(k · r)] 2 2 u (r) = 2 (2)3 [c66 K 2 + c44(k)kz2 ]2 a0 (12.343) derives from long wavelengths k → 0. The evaluation is done separately for the dispersive (c44 ≈ 4"0 /a40 K 2 ) and the non-dispersive (c44 ≈ 4"0 2 /a40 ) regimes describing the situation at small (a0 < R < ) and large (R > ) distances [150] and interpolating between the two results we find (under the conditions that u2 ≤  2 and a0 < Lc )

u (R, L) ∼  2

2

583

 a 3  R2 0

Lc

 a20 L2 1/2 + 4 2    R2 L . + ln 1 + 2 + a0 a0 (12.344)

While the first term in (12.344) reproduces the results (12.333) and (12.338) found previously in the nondispersive regime, the second term dominates in the dispersive region on small scales R <  and provides us with the new result u2 (R) ∼  2(a0 /Lc )3 ln(R/a0) — indeed the mean squared displacement grows only logarithmically in the intermediate regime a0 < R < . The condition u2 (R) ∼  2 provides us with an expression for the pinning length Rc    Lc 3 ; (12.345) Rc ∼ a0 exp c a0 the constant c in the exponent depends on the numerical ˛ chosen in the pinning criterion u2 (R) ≈ ˛ 2 and cannot be obtained within the present perturbative analysis. The remaining characteristic quantities are obtained with the help of scaling estimates and we arrive at the following results valid in the dispersive region,    Lc 3 Rc ∼ a0 exp c <, a0     Lc 3 Lbc ∼ a0 exp 2c < (12.346) a0 a0    Lc 3 Lc , ∼ Uc exp 2c a0 a0   L 2  L 3  c c b jc ∼ jc . (12.347) exp −2c a0 a0

Ucb

Lacking a numerical value for the constant c in the exponent heavily reduces the degree of accuracy of these scaling results as compared to the results obtained for the single vortex pinning regime and the vortex bundle pinning regime in the non-dispersive case — while the latter are accurate up to a numerical prefactor, the indeterminacy of c lifts our ignorance to the exponent. This situation can be greatly

584

G. Blatter and V. B. Geshkenbein

improved with the help of the functional renormalization group theory [357], cf. (12.320): the dispersion in the elastic modulus c44(k) renders the problem effectively four-dimensional (where the disorder is marginally relevant) and the divergence in the IV fourth derivative Kl→l (0) → ∞ of the disorder corc relator at the origin allows for a precise definition of the Larkin length Rc ∝ exp(lc ). The crucial quantity relating the disorder potential to the pinning force then is the fourth derivative K IV (0) of the disorder correlator at the origin — this differs from the perturbative analysis [cf. (12.316)] where this role is played by the combination K  (0)/ 2 involving the second derivative K  (0) (due to the force–force correlator describing the disorder potential in one pinning valley) and the disorder scale  (from the condition u2 (Rc ) ≈  2); within a scaling analysis, K IV (0) ∼ K  (0)/ 2 ∼ K(0)/ 4. Integrating the RG flow through the Larkin regime 0 < l ≤ lc and identifying the divergence of KlIV (0) at l = lc the one-loop analysis provides the result [347]   3/2 "02 4 . (12.348) Rc ∼ a0 exp 3 a30 K IV (0) A similar analysis [367] for the single vortex pinning provides the result Lc ∼ ["02 /K IV (0)]1/3 but fails to provide an accurate numerical for the prefactor. It is then quite appropriate to redefine Lc as given by (12.325) to read Lc ≡ ["02 /K IV (0)]1/3; this allows us to cast the result (12.348) into the form (12.345) with c = 4 3/2/3,  3/2     L  7−5 ln(4/3) Lc 3 4 9 c . (12.349) exp Rc ∼ a0 a0 3 a0 The remaining formulae then follow from scaling. The prefactor ∼ (Lc /a0 )0.62 in (12.349) can be obtained within a two-loop analysis of the RG flow [346, 347]; the result (12.349) then provides us with an expression for Rc up to an unknown overall numerical prefactor and thus puts the dispersive regime on the same level of accuracy as the single vortex and large bundle pinning regimes. The dispersive regime with its “small” bundles of size a0 < Rc <  is limited to the intermediate field region

  2/3 jc 2 jc ln  Bsv < B < Bsb ∼ Hc 2 ; j0 j0

(12.350)

the field Bsb marking the onset of the non-dispersive regime with its characteristic “large” bundles differs from Bsv only by a logarithmic factor. The explicit field dependence of the results (12.346) and (12.347) follows from the substitution  j B 1/2 Lc 0 ∼ . (12.351) a0 jc Hc2 Anisotropic Material The discussion of pinning in anisotropic materials is quite involved and we restrict the analysis to the case of uniaxially anisotropic superconductors with the magnetic field applied along the high symmetry direction. We proceed along the traditional path using scaling estimates and calculate the bundle size, the pinning energy Ucb and the critical current density jcb . Repeating the derivation of (12.353) above with the appropriate expression (12.83) for the tilt modulus we obtain the displacement correlator  3  R2 a2 L2 1/2 2 2 a0 " u (R, L) ∼  " 2 + 04 (12.352) Lcc     L R2 . + ln 1 + 2 + a0 "a0 The bundle size Rc then follows from the definition u2 (Rc ) ∼  2,   c  ⎧ L 3 ⎪ ⎪ a , a0 < Rc < /" , exp c c ⎨ 0 "a0 Rc ∼ ⎪   Lcc 3 ⎪ ⎩ , /" < Rc , " "a0 (12.353) and the corresponding expressions for Lbc follow trivially. Replacing Lc /a0 → Lcc /"a0 and  → /" in (12.337) and (12.347) we obtain the pinning energy Ucb and the planar critical current density jcb in the anisotropic situation. The boundary Bsb limiting the small bundle pinning regime takes the form   2 c 2/3 jcc  j ln 2 c Bsb ∼ Hc 2 . (12.354) j0 " j0

12 Vortex Matter

12.9.4

Collective Creep

In the absence of thermal (and quantum) fluctuations the vortex system remains firmly pinned for driving forces f below critical, f < fc ; the depinning at fc then is an example of a dynamical phase transition [368, 369]. Thermal or quantum fluctuations allow the vortices to overcome the pinning barriers at small applied forces f < fc , either via thermal activation or quantum tunneling; the resulting vortex motion (cf. 12.8.4) smears the transition at fc and reintroduces dissipation. However, within the collective pinning scheme the barriers inhibiting vortex motion diverge as the driving force f is decreased, resulting in a“glassy”and hence“truly superconducting” response as f → 0. In the following, we analyze this collective creep type motion, concentrating first on thermal processes; we discuss the motion of individual vortex lines and proceed with classical creep of vortex bundles. Second, we extend the analysis to quantum motion of vortices. Single Vortex Creep The classical creep-type motion of an individual vortex line can be visualized as a thermal diffusion process where vortex segments move between metastable states. In the absence of an external current density j a vortex segment lowers its energy by finding the optimal low energy state. With an applied current density j a new metastable state becomes more favorable and the vortex moves. The new optimal states are determined by the condition that the energy gain due to the driving Lorentz force matches the energy of the newly deformed vortex configuration. For a current density j near criticality this condition is already fulfilled for the neighboring metastable state a distance ∼  away. However, upon decreasing the current density j the Lorentz force is reduced and the next favorable metastable state is further away. As a consequence, the thermal motion of the vortex will involve more extended segments hopping larger distances in order to reach the next optimal low energy state, see Fig. 12.26. For a quantitative analysis we need to know more about the low lying metastable states of the vortex in its random pinning environment. This touches upon

585

the general problem of elastic manifolds in quenched random media, of which our vortex line is a typical example [111, 112, 370], see Sect. 12.9.1. Consider a d = 1-dimensional elastic string moving in n transverse directions as described by the free energy (12.313). The statistical mechanics of this object is given by the partition function (u,L) 

Z= (0,0)



L 1 D[u ] exp − dz (12.355) T 0     "l @u 2   + (u , z ) . × pin 2 @z 

Disorder is always relevant for dimensions n ≤ 2 and the string is in a pinned phase characterized by a displacement correlator with a wandering exponent n > 1/2 (an exponent 1/2 corresponds to thermal wandering, "l u2 /L ∼ T)  L 2n . (12.356) u2 (L) ≡ [u(L) − u(0)]2 ∼  2 Lc For L < Lc (Larkin regime) the string probes only a single pinning valley; the wandering exponent 1L = 3/2 has been derived above using scaling arguments, see (12.325) and (12.317). The crossover length Lc limiting this regime also has been calculated above, Lc ≈ ["l2  2 / ]1/3 . At larger distances L > Lc the string can choose between many metastable minima [350,351].The determination of the wandering exponent 1,n is a nontrivial problem (cf. the discussion in Sect. 12.9.1); its numerical value depends on the dimensionality of the transverse space: 1,1 = 2/3 is an exact result [356], while for n = 2 numerical simulations [371,372] give a value 1,2 ≈ 0.620 ≈ 5/8. Metastable states extending over a distance L are separated by a typical distance u(L) ∼ (L/Lc )1,n and different minima vary in depth by an amount ı pin(L) ∼ Uc (L/Lc ) 1,n ; the exponent 1,n = 21,n − 1 follows from scaling, assuming that variations in the pinning energy ı pin(L) scale as the elastic energy "l u2 /L ∝ L21,n −1 . The situation is different for dimensions n > 2 [373], where a finite “roughening” temperature separates a low temperature disorder dominated phase from the high temperature thermal phase characterized by 1th = 1/2.The (1+2)-problem corresponds to

586

G. Blatter and V. B. Geshkenbein

Fig. 12.26. Effective tilted random potential acting on the flux line in the presence of quenched disorder and with an applied driving current j. The vortex line relaxes into a low energy metastable state. Close to the critical driving force, j ≤ jc , the next metastable state is near to the original state and separated from the latter only by a small barrier U (j) ∼ Uc (1 − j/jc ). At low driving currents j  jc the closest favorable metastable state is far away from the original one and separated from the latter by a large barrier, U (j) ∼ Uc (jc /j) . Hops to the closest valleys are not favorable and represent only an intermediate step in the diffusive motion of the vortex to its next optimal state

the lower critical dimension for this phase transition and thus exhibits marginal behavior, see Sect. 12.9.5 below. However, for the physically relevant case discussed here, n = 2 and a single vortex is always in a pinned or “glassy” state. Let us return to our original problem and apply the above results to the discussion of vortex creep. The free energy of a vortex segment of length L trapped in the disorder potential and driven by a small current density j  jc takes the form (see Fig. 12.27; below, we drop the indices “1, n” on the wandering exponent )  L 2−1  L 1+ ¥0 F(L) ∼ Uc − j Lc  , (12.357) Lc c Lc where we have assumed that the barriers inhibiting vortex motion scale the same way as the metastable minima [374]. Equation (12.357) reduces the present discussion to a simple nucleation problem [375] where the vortex segment plays the role of the nucleation bubble: whereas nuclei with lengths L smaller than some critical length Lopt will collapse back to zero length and thus are undercritical, activated segments with a length L > Lopt will expand and contribute to the vortex motion (see [376] for a similar analysis of domain wall motion in spin glasses and [377] for a description of dislocation motion in metals). The critical size of the nucleus derives from

the minimal barrier in (12.357) and defines a creep barrier which increases algebraically with decreasing current density j,

Lopt (j) ∼ Lc

 j 1/(2−) c

U (j) ∼ Uc

 j 

c

, j (12.358) with the exponent = (2 − 1)/(2 − ). Correspondingly, the current-voltage characteristic exhibits a highly non-linear “glassy” behavior at low drive j

,

  Uc  jc 

. V ∝ exp − T j

(12.359)

Using the above estimate 2 ≈ 5/8 for a single vortex in three dimensions we obtain Lopt (j) ∼ Lc u(j) ∼ 

 j 8/11 c

j  j 5/11

U (j) ∼ Uc

,

c

, j  j 2/11 c

j

.

(12.360)

A non-linear increase of the creep barrier with decreasing current density has been observed experimentally by various groups, see [345,378–381].

12 Vortex Matter

Combining (12.358) and (12.302) we obtain a nonlinear logarithmic time decay of the current density,  j(t) ∼ jc

t T ln Uc t0

−1/

.

(12.361)

For current densities near to critical the result (12.304) is more appropriate and interpolating between these two formulae we obtain the general behavior  

T  t  −1/

j(t) ∼ jc 1 + ln 1 + (12.362) Uc t0 valid close to jc as well as for j  jc [337]. A nonlinear time logarithmic decay of the trapped diamagnetic moment has been observed in several experiments [46–48,345,380–382]. Within the single vortex pinning regime the exponent 1/ ≈ 11/2 is large and rewriting the solution (12.362) for small temperatures and small times in exponential form,  T t j(t) ∼ jc exp − ln , Uc t0

t 11 Uc , < t0 2 (12.363) we find an algebraic time decay of the current and an exponential dependence on temperature,  t T/Uc

T ln

 T , j(T) ≈ jc exp − t T0 (12.364) with T0 ≈ Uc / ln(t/t0 ). A sample prepared in a critical state at t = 0 will carry a strongly reduced screening current j  jc after only a few seconds of waiting time: The quantity measured in a magnetization experiment is usually not the critical current density jc but the strongly reduced value j(T). Such an exponential decrease of the apparent critical current density has indeed been observed in the oxide superconductors and the characteristic temperature T0 has been measured to be of the order of 10 K [49]. Typical experimental values for the activation energy Uc are of order 102—103 K [45,46,345,383,384] while typical experimental waiting times are of order t/t0 ∼ 1010, hence T0 is related to Uc as expected. Returning to the general result (12.362) we determine the normalized creep rate S = −d ln j/d ln t j(t) ≈ jc

0

,

587

which differs from Anderson’s result T/Uc due to the nonlinear dependence of the activation barrier U (j) on the current density j, T S≈ . (12.365) Uc + T ln(1 + t/t0 ) Equation (12.365) shows two interesting features which are a direct consequence of the collective creep behavior with its characteristic strong increase of the pinning barrier (12.305) with decreasing current density: First,the decay rate S decreases with increasing time, producing an upward curvature in a plot of the logarithmic time decay of the diamagnetic current [48, 385]. Second, the decay rate S saturates for temperatures T > Uc / ln(t/t0 ), Ssat ≈ 1/ ln(t/t0 ). Both these effects are a consequence of the decaying current density j due to creep and the concomitant increase of the system barriers as time evolves [386]. However, we cannot expect this saturation to appear within the single vortex pinning regime: Near saturation the ratio T ln(t/t0 )/Uc in (12.362) has become larger than unity and the current density has decayed by a factor ∼ 10−2 . At such small current densities the interaction between neighboring vortices cannot be neglected any longer and the vortex motion proceeds by the diffusion of vortex bundles rather than independent single vortices.As discussed below, creep due to vortex bundles is characterized by an exponent of order unity and a saturation Ssat of order of a few percent is expected, in agreement with experiment. Also, experiments show a saturation at temperatures T > 10 K in agreement with a collective pinning energy of order 102 K as obtained above. As we have already mentioned above, the analysis of collective creep in terms of individually moving vortices applies only for a limited regime of current densities j, the reason being that the pinning energy grows only sublinearly in the length L, whereas the interaction energy between neighboring vortices grows linearly in L. Thus with decreasing current density j the relative importance of the interaction between the vortices grows. We identify the length scale where the intervortex interaction becomes dominant by comparing the shear energy c66(u/a0 )2 a20 L with the tilt energy "l (u/L)2 L; with c66 = "0 /4a20 and "l ≈ "0 we find a (longitudinal)

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along the superconducting planes. For this case we have to substitute the in-plane current density ratio jcc /j and the boundary jsv by their out-of-plane counterparts "# jcc /j and "# jsv . Creep of Vortex Bundles

Fig. 12.27. Free energy F versus length L of the hopping segment. The free energy involves the barrier energy Uc (L/Lc )2−1 growing slowly with L and the energy gain due to the Lorentz force, (j/jc ) Uc (L/Lc )1+ , with a small prefactor j/jc  1 but a more rapid growth in L. For a fixed driving current density j a minimal segment of length Lopt (j) has to overcome the barrier U (j) in order to reach the next metastable state. With decreasing j the barrier diverges with U (j) ∼ Uc (jc /j) and = (2 − 1)/(2 − )

crossover length L ∼ a0 ; single vortex creep thus is limited to currents such that Lopt (jsv ) < a0 and using (12.358) we obtain jsv ∼ jc

 L 2− c

a0

∼ jc

 B j (2−)/2 0 . Hc2 jc

(12.366)

For small current densities j < jsvc or large fields B > Bsv the interaction between vortices is important and the minimal barrier for flux motion involves a small vortex bundle instead of a single vortex line. We add a note on single vortex creep in anisotropic materials: The relevant length Lopt (j) (size of the critical nucleus) increases with decreasing (inplane) current density j according to Lopt (j, # ) ∼ Lc (# )(jcc /j)8/11, and similar results apply for the thermal activation barrier (classical creep), U (j) ∼ Ucc (jcc /j)2/11 (note that the creep barrier is a scalar quantity and hence one expects a result independent of the angle # ). The boundary of the single vortex pinning regime is reached when the current density √ j drops below jsvc (# , B) ∼ jcc ( "# Lcc /"a0 )11/8, a result obtained from rescaling the condition Lopt (j, # ) ∼ a0 . If the vortex lattice is subject to an out-of-plane current density j  y the creep motion is directed

Creep of vortex bundles involves an additional new feature not present in the discussion of single vortex creep: the jump of the vortex bundle to the next valley involves a finite compression of the vortex lattice. Indeed, in order to gain energy from the driving Lorentz force the lattice has to undergo a compression: Consider a displacement u(r) restricted to the bundle volume V , then the energy gain EL due to the Lorentz force is given by  1 dV (j ∧ B) · u (12.367) EL (V ) = c =−

1 c

 dV [(j ∧ B) · r] (∇ · u) .

With the compression modulus c11 much larger than the shear modulus c66 the compressed vortex bundle elongates along the direction of motion (we call the resulting elongated bundle a “superbundle”). A simple scaling analysis can be applied in the nondispersive regime: comparing the compression and shear energy densities c11(u/R )2 ∼ c66(u/R⊥ )2 we find the asymmetry ratio R /R⊥ ∼ /a0 between the parallel and perpendicular (to the bundle motion) dimensions of the superbundle. The superbundle is made from subbundles forming independently via the competition between the elastic tilt and shear energies and the disorder potential, hence R⊥ = Rc follows from optimizing the geometry of the subbundles. For the creep process, the large value of the compression modulus c11  c66 prohibits the hopping of individual subbundles and therefore the latter are coupled together, leading to the hop of the large superbundle.The activation energy Ucb for such a correlated hop then is given by the sum of the activation energies for the subbundles, Ucsb ∼ c66

R Lb u2 b R R L ∼ Uc . ⊥  Rc Lbc R2⊥

(12.368)

12 Vortex Matter

The analysis of the superbundle dimensions in the dispersive and mixed dispersive/non-dispersive regimes requires some care in the evaluation of the compression, tilt, and shear energies [38, 387] and produces the results R2⊥ R⊥ , a20  R  1/2 ⊥ R⊥ , R ∼ Lb ∼ a20  R ∼ Lb ∼ R⊥ , a0 R ∼ Lb ∼

R⊥ < R ∼ Lb <  , R⊥ <  < R ∼ Lb ,  < R⊥ < R ∼ Lb .

(12.369) Combining (12.369) and (12.368) we obtain the final result for the creep activation energy Ucsb of a vortex superbundle near jc ,

Ucsb

⎧  L 3 Lc c ⎪ ⎪ Uc exp 5c , a0 < Rc < (a20 )1/3 , ⎪ ⎪ a a ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎨ L  L 3 c c ∼ Uc 2 exp 2c , Rc <  < (R3c /a20 )1/2 , ⎪ a a 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪   3  L 4 ⎪ ⎪ c ⎪ ⎩ ,  < Rc . Uc a0 a0

(12.370) At small current densities the creep of vortex bundles involves hops beyond  and we have to extend the analysis of single vortex creep above using the general concepts from the theory of dirty elastic manifolds. In a first step we have to identify the exponents describing the bundle’s transverse fluctuations u2 (R, L) ≡ [u(R, L) − u(0, 0)]2  on length scales beyond the pinning lengths Rc and Lc , extending the analysis from the 1 + 2-dimensional vortex string to 3 + 2 dimensional vortex manifold. Here, we can make use of the discussion in Sect. 12.9.1 above, cf. (12.320), which provides us with the short-range SR SR wandering exponents 1,2 = 3/5 and 3,2 = 1/5 (and correctly reproduces the exact result 1,1 = 2/3). Furthermore, on large distances the periodic nature of the vortex manifold becomes relevant and the impact of disorder decreases as expressed through a crossover to the Bragg glass scaling regime characterized by the exponent  BG = 0. Note that the result SR for 1,2 = 0.600 used here differs by 3 % from the result 1,2 ≈ 0.620 derived from numerical simulations

589

[371,372]; the corresponding scaling results for single vortex creep then take the form Lopt (j) ∼ Lc (jc /j)5/7 , u(j) ∼ (jc /j)3/7, and U (j) ∼ Uc (jc /j)1/7. We will use these results in our interpolation scheme below. In generalizing the results for the isotropic manifold’s wandering to the vortex lattice with its anisotropic elastic moduli we interpolate between the manifold’s short scale fluctuations as given by (12.325) for the single vortex regime and (12.344) for the vortex bundle regime (the Larkin results) and the long distance random manifold and Bragg glass scaling,

u2 (R, L) ∼ 2

⎧  L 21,2 ⎪ ⎪ , Lc < L < a0 , ⎪ ⎪ Lc ⎪ ⎪ ⎪   ⎪ 2 ⎪  a0 3  ⎪ R2 L  3,2 ⎪ ⎪ , a0 < R <  , ⎪ ⎨ Lc ln 1 + a2 + a0 0    2  ⎪ a0 3 R a2 L2 1/2 23,2 ⎪ ⎪ + 04 ,  < R < Ra , ⎪ ⎪ 2 ⎪ L   c ⎪ ⎪ ⎪   ⎪  a2  ⎪ R2 a2 L2 ⎪ 0 ⎪ ln 1+ 2 + 20 2 ,  < Ra < R , ⎩ 2 Ra  Ra

(12.371) where the last three expressions refer to the weak pinning case with Lc > a0 . The crossover length Ra ∼ (Lc /a0 )3 (a0 /)1/3,2 ∼ Rc (a0 /)1/3,2 is obtained by matching the random manifold result in (12.371) to the Bragg glass scaling at u ∼ a0 , assuming the crossover takes place in the nondispersive regime with Ra > . Note that tracing the growth of u with distance requires some care; e.g., starting out in the “strong” pinning regime with Lc < a0 the single vortex scaling has to be matched to the dispersive regime using the proper random manifold scaling expression. Let us return to the problem of vortex creep at low drives. We first generalize the free energy (12.357) for the disordered string to the (isotropic) driven random manifold, F(L) ∼ Uc

 L  d,n Lc

−

F  L d+d,n , Fc Lc

(12.372)

where F and Fc denote the applied and the critical force density and d,n = d − 2 + 2d,n is the energy exponent. The critical nucleus is characterized by the size Lopt (F) and jumps a distance uopt (F),

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G. Blatter and V. B. Geshkenbein

 F 1/(2−d,n ) c Lopt (F) ∼ Lc , F  F d,n /(2−d,n ) c uopt (F) ∼  . F

(12.373)

Inserting the result for Lopt (F) back into (12.372) we obtain the creep barrier with its characteristic glassy exponent , U (F) = Uc

 F 

c

F

,

4 , 2− 1

→ + , 2−

=

2d,n + d − 2

= . (12.374) 2 − d,n

When applying this general result to the vortex lattice we have to account for the modifications due to the dispersive nature of the elastic moduli and the anisotropic geometry of the superbundles. Within the small bundle pinning regime, the bundle dimensions along the field as well as along the direction of the hop grow very fast, Lb ∼ R ∼ R3⊥ /a20 . Using the scaling behavior of R⊥ with current density j, R⊥ (j) ∝ (jc /j)1/(2−) (this relation follows from balancing the elastic (shear) energy against the energy gain from the driving Lorentz force), we obtain the scaling laws Lb ∼ R ∝ (jc /j)3/(2−) for the bundle dimensions in the field/force plane. For intermediate size bundles we have Lb ∼ R ∼ (R3⊥ /a20 )1/2 and hence Lb ∼ R ∝ (jc /j)3/2(2−) . Finally, for large bundles, Lb ∼ R ∼ (/a0 )R⊥ and no additional scaling factors arise. Summarizing, we have to modify the result (12.374) within the small (and intermediate) bundle pinning regime by changing the exponent

(the correction terms follow from the additional geometry factor Lb R /R2⊥ , cf. (12.368)),

→ +

the transverse length R⊥ due to the shear relaxation in the subbundles is still smaller than the screening length . We then arrive at the following exponents

characterizing creep of vortex bundles within the lattice pinning regime:

small bundles

(12.375)

intermediate bundles .

A further complication concerns the wandering exponent : Within the small bundle pinning regime, the generalized Larkin–Ovchinnikov result (12.371) for the displacement correlation function u2 (R, L) depends only logarithmically on the distances R and L and hence the wandering exponent changes to the new value  = 0 within the small bundle creep regime. This reduced exponent remains also valid within the intermediate bundle creep regime where

5 , 2

small bundles , a0 < R⊥ , R ∼ Lb <  ,

= 1,

≈

7 , 9

intermediate bundles , a0 < R⊥ <  < R ∼ Lb , large bundles ,

 < R⊥ , R ∼ Lb . (12.376)

The above results are applicable within the random manifold pinning regime where the typical hopping distance u is less than the lattice constant a0 of the flux line lattice. On the other hand, the relevant hopping distance u can grow well beyond this value, e.g., by decreasing the current density j or in the context of relaxation on large scales, implying a crossover to the Bragg-glass (BG) scaling regime. Within this regime, the wandering exponent  is again reduced to zero and the exponent takes the value

=

1 , 2

BG regime ,

Ra < R⊥ .

(12.377)

In calculating the above creep exponents we have assumed wandering exponents  = 0 (small bundles),  = 0 (intermediate bundles),  ≈ 1/5 (large bundles), and  = 0 in the BG creep regime. We now are ready to collect the results describing creep at small driving forces j  jc . Let us assume that the sample has been initially prepared in a critical state belonging to the single vortex pinning regime, i.e., Lc < a0 (note that we use the line SR wandering exponent 1,2 = 3/5 rather than the result 1,2 ≈ 5/8 derived from numerical simulations and used in Sect. 12.9.4 above). As the current j decays due to creep, the activation barrier U (j) evolves according to the following scheme:

12 Vortex Matter

⎧  j 1/7 c ⎪ ⎪ Uc ⎪ ⎪ ⎪ j ⎪ ⎪ ⎪  j 5/2 ⎪ ⎪ sv ⎪ ⎪ Usb ⎪ ⎪ j ⎪ ⎪ ⎪ ⎨ jsb Uib U (j) ≈ ⎪ j ⎪ ⎪ ⎪   ⎪ 7/9 jib ⎪ ⎪ ⎪ Ulb ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪  ⎪ jlb 1/2 ⎪ ⎪ ⎩ UBG j

,

jsv < j  jc ,

,

jsb < j < jsv ,

,

jib < j < jsb ,

,

jlb < j < jib ,

,

0 < j < jlb ,

591

(12.378)

with Usb ≈ Uc (a0 /Lc )1/5, Uib ≈ Usb (/a0 )5/3 , Ulb ≈ Uib (/a0 )4/3 , UBG ≈ Ulb (a20 L3c / 5)7/5 ; jsv ≈ jc (Lc /a0 )7/5, jsb ≈ jsv (a0 /)2/3, jib ≈ jsb (a0 /)4/3 , jlb ≈ jib ( 5/a20 L3c )9/5 . The time decay of the current density follows trivially from combining (12.302) and (12.378) and solving for j. The various creep regimes are illustrated in Fig. 12.28. Again we add a note on anisotropic materials making use of our scaling rules (12.170); basically, the scaling approach is limited to length scales R < , however, one can show [150] that the results can be extended to the larger regime R < /". We assume a current density j directed along the planes, j ⊥ H and have the vortex system start decaying from the single vortex pinning regime such that √ Lcc < "a0 / "# . As the current density j drops be√ low jsv (# , B) ≈ jcc ( "# Lcc /"a0 )7/5 (for 1,2 ≈ 5/8 the exponent is 11/8), we enter the vortex bundle creep regime where the activation barrier increases rapidly, U (# , j, B) ≈ Usb (# , B)[jsv (# , B)/j]5/2, with √ Usb (# , B) ≈ Ucc ("a0 / "# Lcc )1/5, obtained by requiring continuity across the boundary between the single vortex and the lattice pinning regimes. The boundary jsc (# , B), within which the scaling theory can be applied, is found by rescaling the condition R (jsc ) ≈ a0 (jsb /jsc )3/2 ∼ /", and we ob√ tain jsc (# , B) ∼ jsv (# , B)("a0 / "# )2/3 . Note that jsc is larger than the current density jib limiting the small/intermediate bundle pinning regime.For a current density j directed along the y axis (out-of-plane current) the creep motion is directed along the superconducting planes and we have to substitute the in-plane current density ratio jsv /j by its out-ofplane counterpart "# jsv /j. The extension of the creep

Fig. 12.28. Creep regimes and their relative position within the B–j plane. Disorder is irrelevant in the flux-flow regime at large fields and large current densities.Vortices move via creep in the low-field-low-current region below the critical current density jc (B). The flux creep regime is divided up into a single-vortex creep regime at large current densities, followed by the small to intermediate and large bundle creep regimes at lower current densities (larger fields). At very low current densities we enter the BG creep regime

analysis to anisotropic materials beyond the scaling regime is rather tedious and we refer the reader to the original literature [150]. Quantum Creep Above we have analyzed the decay of a vortex density gradient due to thermal activation of the vortices out of their metastable states. According to this classical picture the decay rate (12.365) should vanish in the zero temperature limit. However, one may expect that quantum fluctuations trigger a decay of the critical state via quantum tunneling at low temperatures, leading to a saturation of the decay rate S = d ln j/d ln t. Indeed, such a saturation has been observed in a number of experiments on oxide superconductors [50, 52, 388], as well as in Chevrel

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phase [389], organic [390] and heavy fermion superconductors [391]. Macroscopic quantum phenomena have always attracted much interest as they naturally touch the boundary between quantum and classical physics: macroscopic systems are coupled to their environment and thus are inherently dissipative. The classic and most carefully studied system exhibiting macroscopic quantum tunneling is the (rf) superconducting quantum interference device (SQUID) [149, 392–395]. In particular, Caldeira and Leggett [149] have discussed in detail how to extend the quantum description to macroscopic systems, the main effect of the dissipative environment being to suppress the probability for the tunneling process. Later, this early work has been extended to vortex motion by quantum tunneling in Josephson junction arrays [396], thin (disordered) films [397–399], and bulk superconductors [400,401]. The decay of a metastable state due to tunneling is similar to the classical decay process via thermal activation; the additional feature to be taken care of in a tunneling process is the time component of the motion. During tunneling the vortex moves under the barrier, hence the process is forbidden (virtual) and the longer the time the string spends under the barrier, the less chances there are for the process to happen (for the thermally activated situation time is irrelevant as the vortex hops over the energy barrier in a real process). While for an activated motion the probability is determined by the saddle-point solution of the free energy (12.313), one has to find the saddle-point solution of the Euclidean action in order to describe a tunneling event [149,402–406]. The quantum problem then is a d + 1-dimensional generalization of the d-dimensional classical problem, where d denotes again the dimension of the object, e.g., d = 1 for the vortex line. As discussed in Sect.12.3.2 above,the vortex equation of motion involves transverse (Hall) and dissipative forces, see (12.45), ml v˙ L + l v L + ˛l v L ∧ zˆ = f ext .

(12.379)

Here, we include an additional massive term ml v˙ L with the motivation that the quantum tunneling of a massive object is the most simple case to analyze.

Expanding the dynamical coefficients in (12.45) it is possible to derive a vortex mass (see also [407,408], where an electronic vortex mass mel ≈ (2/ 3)me kF has been found, and [409,410] reporting a mass enhanced by a factor (kF )2 in a superclean material; besides the electronic contribution, a second term mem = (¥0 /4c)2 of electromagnetic origin contributes to the vortex mass [407]; in a tunnel junction this result is modified to mem ∼ C¥02 /pc c 2 with C the junction capacitance and pc the size of the phase core, typically mel  mem ), however, it turns out that in a bulk superconductor the dissipative and Hall terms always dominate over the mass term, the dissipative contribution being the most relevant one in dirty and clean materials while the Hall term dominates in the superclean limit.In the following,we first discuss the simple and illustrative case of tunneling of a massive string and then proceed with the analysis of vortex tunneling,discussing the dissipative and Hall tunneling in sequence. The Lagrangian generating the classical equation of motion for a massive elastic string is 



 ml L u = dz (@t u)2 − F u , (12.380) 2 with the free energy functional F[u] given by (12.313). The displacement vector u(z, t) plays the role of the macroscopic variable. Substituting t → −it in (12.380) we obtain the (imaginary time) Euclidean action [405,406],   

 ml 2 SE = dt dz (@t u) + F u , (12.381) 2 for which we have to find the saddle-point solution. The saddle-point solution is given by the classical bounce trajectory where the string moves through the inverted potential and bounces back to the origin, see Fig. 12.29 for a sketch illustrating the tunneling trajectory of a particle. Here, we are interested in the scale for the action, i.e., we wish to determine the quantity corresponding to Uc , and thus we set the driving force equal to zero, j = 0. Equation (12.381) then has to be minimized with respect to the form (length L and displacement u) and the duration t of the bounce and we will do this on the level of scaling estimates. At this point we should note that the

12 Vortex Matter

Fig. 12.29. The tunneling trajectory (bounce, dotted line, massive dynamics) is found as the classical solution of the equation of motion in the inverted potential. In the dissipative situation energy is transferred from the system to the reservoir (dashed line)

tunneling process involves the same initial and final states as encountered above in the determination of the classical creep rate; the additional new feature in the quantum problem is the relevance of the time evolution of the vortex motion. In fact, the determination of the optimal geometry for the bounce solution involves only the free energy part of (12.381) and the result in the form of the collective pinning length Lc has been found above (12.325); the displacement u ∼  is given by the disorder landscape. The estimate for the tunneling time tc is obtained by equating the elastic and the kinetic energy densities, ml (/tc )2 ∼ "l (/Lc )2 , √ "l ml 2  , Lc , tc ∼ "l  (12.382) where we have inserted the result tc into (12.381) in order to find the minimal action for the tunneling process in the last equation. In the limit of vanishing dissipation the action does not depend on the collec m 1/2 l

tc SE ∼ Uc ∼  

593

tive pinning length Lc and hence is independent of the pinning potential. Let us then apply these ideas to vortex tunneling. In accounting for the dissipative environment we restrict ourselves to the simplest case of ohmic dissipation with the viscous drag coefficient (12.45). The corresponding dynamical term in the vortex action can be obtained by coupling the system to a bath of harmonic oscillators and subsequent integration over the environmental (bath) variables [149],      l u(z, t) − u(z, t  ) 2 dt dt  dz . 4 t − t (12.383) The expression (12.383) is nonlocal in time and we transform to Fourier space to obtain the effective action,   dq d! SEeff = (12.384) 2 2    1

2 2 l |!| + "l q |u(q, !)| + pin(q, u) . 2 Equating the dynamical and elastic energy densities, l !c u2 ∼ "l q2c u2 , with qc ∼ 1/Lc the inverse length of the tunneling segment, we find the tunneling time tc ∼ 1/!c and the corresponding action, l 2 L , (12.385) tc ∼ "l c l  2Lc    j0 1/2 1  j0 1/2 SEeff ∼ ∼ 2 n ∼ ,   e  jc Qu jc with Lc given by (12.327). Here, we have introduced the dimensionless parameter Qu = (e2/)( n /) quantifying the strength of quantum fluctuations and playing the role analogous to the Ginzburg number in the context of thermal fluctuations. For an anisotropic material we have to substitute Lc → Lcc and hence SEeff = " 4/3 SEeff ,iso . The normalized creep rate S near jc takes the form, S=

 d ln j ≈ − eff . d ln t SE

(12.386)

The main parameter determining the action and the tunneling rate is the ratio  n /.A small action favoring tunneling is obtained in materials characterized by a large normal state resistivity  n and a small coherence length . The quantum unit of resistance is

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G. Blatter and V. B. Geshkenbein

/e2 ≈ 4.1 k§, thus the phenomenon of quantum creep should become experimentally observable for ratios  n / ≥ 1 k§. The result (12.386) depends only weakly on the disorder potential via a square root dependence on the critical current density jc . Finally, let us discuss the case of a Hall-type vortex dynamics as it is relevant in a superclean superconductor (cf. Sect. 12.3.2; the Hall dynamics is also relevant for the tunneling of vortices in superfluid He [411]). Substituting the dissipative dynamical term l |!|u2 by the Hall term ˛l !u2 in the action for the tunneling process and proceeding with scaling estimates we immediately reproduce the result (12.386) above but with l replaced by ˛l ,  j 1/2 ˛l  2 Lc SEH n ∼ ∼ n 3 . (12.387)   jc However, there is a lot of additional interesting physics associated with the Hall tunneling of vortices [412, 413]. In particular, the geometry of the bounce solution is very different from that found in the massive or in the dissipative situation. In the following, we discuss a model calculation which highlights the important differences and which also is relevant for the tunneling motion of a (collectively) pinned vortex. We concentrate on the 2D situation, describing a vortex in a thin film (of thickness d) or a pancake vortex in a strongly layered superconductor, see Sect. 12.5.3. Rewriting the external force f ext in (12.379) in terms of a potential U (R) the equation of motion reads ˛d v L ∧ zˆ = −∇U (R) ,

(12.388)

with ˛d = ˛l d = ¥0nd/c (see (12.45)). Equation (12.388) is the equation of motion of a charged (e) particle with zero mass in a magnetic field B = (0, 0, ˛d c/e). Such a particle always follows the equipotential lines defined by the potential U , v = ∇U ∧ zˆ /˛d . Expressing (12.388) in components we obtain the set of equations @U (x, y) dy =− . dt @x (12.389) The action producing these equations of motion is 

 SH = dt ˛d yx˙ − U (x, y) , (12.390) ˛d

dx @U (x, y) = , dt @y

˛d

where the first term is just the Lagrangian j · A/c of a charged particle moving in a magnetic field B produced by the vector potential A = (By, 0, 0). The set of dynamical equations (12.389) is of the Hamiltonian form (see also [411,414] where the same approach has been used in the calculation of vortex tunneling in a superfluid and for particle tunneling in the presence of a strong magnetic field): if we define √ √ the coordinate q = ˛d x, the momentum p = ˛d y, and the Hamiltonian H(q, p) = U (x, y) we find that the set (12.389) is equivalent to the canonical equations dq @H(q, p) = , dt @p

dp @H(q, p) =− . (12.391) dt @q

The Hamilton equations (12.391) describe a particle moving in one dimension with a dynamics defined by the y dependence of the potential U (x, y). The coordinates x and y play the role of canonical variables and we can immediately quantize the theory by imposing the commutation relation [x, y] = (1/˛d )[q, p] = i/˛d = i/nd. An alternative way to go over to a quantum description starts from the action (12.390): In order to describe the tunneling motion of the vortex we have to go over to an imaginary time formalism. However, a mere substitution t → −it in (12.390) as done previously leads us to an action describing a particle moving in an imaginary magnetic field (only one time derivative appears in the dynamical part of (12.390)). On the other hand, if we perform a simultaneous rotation of both the time t and the y-axis, t → −it and y → iy, the dynamical term in (12.390) becomes real again; in addition,if the potential U (x, y) is even in the coordinate y we recover a real Lagrangian. Finding a saddle-point of the original action describing the decay of the metastable state then becomes equivalent to solving a real classical problem. In fact, this scheme for obtaining the desired saddle-point solution can be put on more firm grounds by exploiting the equivalence between the original 2D Hall problem and the 1D Hamiltonian problem: we assume that the original potential U (x, y) can be rewritten in a form U (x, y) = (f /2)y2 +Ux (x),then the resulting 1D-problem is that

12 Vortex Matter

of a massive particle (with mass m = ˛d /f ) moving √ in a potential Ux (q/ ˛d ) ≡ U1 (q) (note that close to a (local) minimum we can expand any potential U (x, y) into this form if we choose the appropriate axes). Writing down the action for the 1D-particle problem and going over to the imaginary time formalism we obtain the Euclidean action (t → −it)   1  SE = −iS(t → −it) = dt q˙ 2 + U1 (q) 2m  

= dt p q˙ − HE (q, p) , (12.392) with the definition HE (q, p) = p2 /2m − U1. The usual Wick rotation within the Lagrangian formalism is equivalent to the combined rotation t → −it and p → ip in the Hamiltonian formalism as the substitution t → −it implies p → ip; equivalently, we can invert the potential U1(q) → −U1 (q). Returning to the original Hall-tunneling problem in 2D we can set up the appropriate rule to go from the action (12.390) to the imaginary time expression SEH . Remembering that the y coordinate plays the role of the momentum p in the 1D Hamiltonian problem we obtain the Euclidean action (assuming a potential U (x, y) = (f /2)y2 + Ux (x)) SEH = −iS(t → −it, y → iy)    f = dt ˛d y x˙ − y2 + Ux (x) . (12.393) 2 According to the above rule we should only invert the potential along the x-axis, Ux → −Ux ; this partial inversion is crucial for obtaining the appropriate “bounce” solution for the tunneling action. A potential U (x, y) describing a tunneling situation should exhibit a local minimum (say at the origin) and a connection via a saddle to free space along one direction (the x-axis); we choose the form U (x, y) = Uc [y2 / 2 + x 2 / 2 − x 3 / 3] with the parameters Uc and  describing a weak pinning situation, cf. Fig. 12.30. The Hall-tunneling problem maps to the 1D-particle problem with a particle mass m = ˛d  2 /2Uc in the cubic potential U (q) = √ √ Uc [(q/ ˛d )2 − (q/ ˛d )3]. Using the exact solution for the tunneling action of a massive particle in a cubic potential [149] and transforming this result back to the Hall problem we obtain the final result

SEH 8 ˛d 2 8 =  = nd 2 .  15  15

595

(12.394)

The result (12.394) does not depend on the depth of the pinning potential but only on its geometry. This can be readily understood if we consider the shape of the “bounce” solution. Inverting the x-component in U (x, y) we obtain an inverted potential with the minimum and the saddle interchanged, cf. Fig. 12.30. The trajectory describing the escape of the particle out of its metastable state follows the equipotential lines in the inverted potential and describes a circle-shaped ∼ . The resulting  curve of radius  action S = 2 dq p → ˛d dx y is a consequence of the “geometrical” quantization and is solely determined by the encircled area without any further dependence on the depth of the potential, in agreement with the result (12.394). The generalization of this result to three dimensions (tunneling of a vortex line) is straightforward: The area enclosed by the trajectory simply has to be replaced by the enclosed volume, SEH / = (˛l /)V = nV . Within the framework of collective pinning theory (where a vortex line segment of length Lc tunnels by a distance ∼ ) we immediately reproduce the result (12.387). The general situation including both dissipative and Hall terms is discussed in [150]. A second illustrative example of a Hall tunneling event is the depinning of a pancake vortex from a trap, which is equivalent to the tunneling of an electron in the“classic”Hall problem (e.g.,out of a defectinduced trap within the 2D electron gas to the edge of the Hall bar).  We choose a potential of the form U (x, y) = U0( x 2 + y2 ) − Fx with a narrow symmetric trap potential U0() increasing monotonously from U0(0) = 0 to U0(∞) = U0 within a distance  = a. The applied force F is assumed to be small, Fa  U0 . The inversion y → iy of the trap potential turns circular equipotential lines U0 (x, y) = const. into hyperbolas x 2 − y2 = const., cf. Fig. 12.31. The tunneling trajectory then forms a triangle CAA’ with the distances CB ≈ U0/F and AA’ ≈ 2U0/F determined by the depth U0 of the trapping potential and the applied force F. The area A ≈ (U0/F)2 encircled by the tunneling trajectory then determines the Euclidean action SEH / ≈ ˛d (U0 /F)2 [415].

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Fig. 12.30. Metastable potential U = (27/4)[x 2 − x 3 + y2 ] (left) and its inversion Ui = (27/4)[−x 2 + x 3 + y2 ] (right). A particle trapped in the minimum at the origin escapes via Hall tunneling out of its metastable state along an equipotential line, ˛d v = ∇⊥ Ui , with ∇⊥ = (@y , −@x ). The Hall tunneling trajectory starts from the saddle point (obtained from the minimum after inversion) and follows the Ui = 0 equipotential line shown in the plot

an isotropic medium from classical to (dissipative) quantum creep one easily finds the relevant tunneling time  ∼ u2 /U and the creep action S(j) ∼ Sc (jc /j) s with the exponent s = (d + 2)/(2 − ) and Sc ∼  2 Ldc (the same result is found within a quantum dynamical FRG scheme (taken from unpublished work by D.A. Gorokhov, D. S. Fisher and G. Blatter); note the exponent s differs from the incorrect expression quoted in [150]). 12.9.5

Fig. 12.31. Quasiclassical trajectory describing the Hall tunneling of a particle out of a metastable state

As with classical creep, the discussion of quantum creep can be extended to vortex bundles and to low current densities: extending the scaling analysis for

Thermal Depinning

The high-Tc superconductors can be operated at high temperatures where thermal fluctuations of the vortex lines are important. We distinguish between two types of thermal motion: The large intervalley fluctuations producing flux creep have been discussed in Sects. 12.8.4 and 12.9.4 above. Here, we concentrate on (phonon-like) small-amplitude fluctuations restricted to the individual pinning valleys (intravalley fluctuations) which lead to a smoothing of the pinning potential, reducing the pinning strength and the critical current density while increasing the creep barrier. Early studies of this phenomenon have been based on the dynamical approach [23]; here, we follow the simpler analysis using scaling estimates and providing the same results [38, 417]. We first discuss the interplay between thermal fluctuations and quenched disorder in the single vortex pinning

12 Vortex Matter

regime and then extend the discussion to pinned vortex bundles. The key quantity entering the discussion of thermal depinning is the mean squared thermal displacement u2(L, t)th = [u(L, t) − u(0, 0)]2th . A simple scaling analysis tells us that an individual vortex line exhibits diffusive motion under the action of (long-wavelength) thermal fluctuations: the mean squared displacement u2 (L, t = 0)th ∼ (T/"l ) L diverges at large distances L → ∞ and, relating length to time for a dissipative dynamics (cf.(12.51)), t ∼ (l /"l )L2 , we find the time-domain diffusion in the form u2 (L = 0, t)th ∼ T(t/l "l )1/2 . More rigorously, these results follow from the fluctuationdissipation theorem [416] relating the mean squared displacement u2 (L, t)th to the single vortex Green’s function G(q, !) (see (12.51)),  d! dq u (L, t)th = 2 (12.395) 2 2 × [1 − cos(qL − !t)] ! × ctgh Im G(q, !) . 2T 2

In the presence of disorder we have to cut the qintegration in (12.396) on the inverse (temperature dependent) pinning length 1/Lc (T), leading to the finite result u2(Lc (T), t → ∞)th ∼

TLc (T) . "l

(12.396)

Thermal fluctuations lead to a smoothing of the quenched disorder potential: Due to thermal motion of the vortex line the vortex core will sample the disorder potential over an extended spatial region ∼ u2 (Lc (T))th .As the amplitude of the thermal fluctuations increases beyond the vortex core radius, u2 (Lc (T))th >  2 , the vortex will experience an averaged disorder potential and thereby pinning will be reduced. Let us calculate the mean squared pinning potential in the presence of thermal fluctuations. Before taking the average over the disorder potential (see (12.310)) we perform a time average over vortex positions,

597



 dt dt   dz dz t0 t0 ×  pin [z, u(z, t)] pin [z , u(z , t  )]   dt dt  dn R K =L t0 t0 (12.397) × [|u(0, t) − u(0, t  )|] .

Epin (L)2t0  ≡

Transforming to Fourier space and assuming Gaussian fluctuations we obtain  dn K 2 Epin (Lc (T))t0  = Lc (T) (12.398) (2)n × K(K) e−K u (Lc (T),t→∞)th /2 ⎧ 2 2 ⎪ ⎨  K(u = 0) , u th   , ∼ Lc (T) dn uK(u) ⎪ ,  2  u2 th , ⎩ 2 n/2 u (Lc (T))th 2

2

and using K(u = 0) ∼  n ,  ≡ we can interpolate



dn uK(u) ∼  2n

(12.399) Epin (Lc (T))2t0  ∼ Lc (T) n n/2  2  × 2  + u2 (Lc (T))th (at high temperatures where u2 th >  2 the cutoff on the K-integration is due to the “Debye–Waller” factor exp[−K 2 u2 (Lc (T))th /2] while it is due to the correlator K(K) = |pK |2 at low temperatures). Finally, we have to determine the temperature dependent collective pinning length Lc (T). For n = 1 (e.g., a vortex line trapped in between the superconducting planes of a layered superconductor or confined to the plane of a thin crystal [418]) our scaling estimates work fine and equating the pinning energy (12.400) to the elastic and thermal energies "l u2 (Lc (T))th /Lc (T) ∼ T we obtain the collective pinning length  T 5 sv , Tdp = ( "l  3)1/3 ∼ Uc , (n = 1) . sv Tdp (12.400) sv Note that the depinning temperature Tdp is itself temperature dependent through the dependence of the parameters "l , , and  on 1 − T/Tc . The boundary Lc (T) ∼ Lc

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sv Tdp,0 beyond which thermal smoothing becomes relevant is determined from a solution of the equation sv sv sv = Tdp (Tdp,0 ) Tdp,0

,

(12.401)

sv sv (T) = Tdp (0)(1 − T/Tc )˛ ; for ıTc -pinning with Tdp in high-Tc superconductors, ˛ = 1/3 [150]. The temperature dependent activation energy becomes sv Uc (T > Tdp ) ∼ T and the depinning critical current density decays with increasing temperature accordsv ing to jc (T) ∼ j0 (/Lc )2 (Tdp /T)7 [377,419]. For n = 2 the length Lc (T) drops out of the scaling analysis due to the marginal relevance of the disorder for the 1 + 2-dimensional single vortex problem — the situation is similar to that encountered in the dispersive pinning regime discussed in Sect. 12.9.3 above. Again, we can resort to perturbation theory, starting from the thermal result and expanding in powers of the disorder parameter . To lowest order the result reads [367]  1 nTL u2 (L) = (12.402) 1+ "l (4)n/2(4 − n)    "l n/2 1−n/2 × 2 , L T T  with  ≡ dn uK(u) ≈  2n and where . . .  denotes the average over thermal and quenched disorder [note the modified dimensionality [ ] = Energy 2 L−(1+n) , cf. (12.309)]. For n < 2 the disorderinduced fluctuations grow faster and dominate for L > T 5 /"l 2  4 , in agreement with (12.400). For n = 2 we can extract the depinning temperature sv Tdp = ("l )1/3 , however, in order to find the pinning length Lc (T) we have to push the expansion one term further; the result takes the form (  denotes a longitudinal short scale cutoff)

  1 "l  2TL 1 "l2 2 1+ + ln( L) . "l 8 T 3 16 2 T 6 (12.403) Equating the two correction terms we find an expression for the temperature dependent collective sv 3 pinning length, Lc (T) ≈ Lc exp[C (T/Tdp ) ], with C a constant of order unity which cannot be determined within the present perturbative approach, a u2 (L) =

consequence of the marginality of the problem. Using functional renormalization group theory the collective pinning length is defined as the length scale where the amplitude of the renormalized disorder correlator Kl appears to diverge, a scheme which allows for the determination of the constant C in the exponent [367]. For a short-range correlated potential the final result reads    32 T 3 Lc (T) ∼ Lc exp , (n = 2) , sv  Tdp sv Tdp = ("l )1/3 ∼ ( "l  4 )1/3 (12.404)  1 j 1/2 c ∼ Uc ∼ Tc . Gi j0

Note, that the temperature dependence of the prefactor remains undetermined. For temperatures T > sv Tdp thermal fluctuations are large, the pinning energy follows the temperature, and the critical current density decays rapidly   48  T 3 Uc (T) ∼ T , jc (T) ∼ jc exp − . sv  Tdp (12.405) These results are modified when the correlations in the disorder potential are not short-ranged. Indeed, the slowly decaying form factor of the vortex core produces a non-integrable correlator (12.311) K(u) ∼ K(0)(/u)2 ln(u/) and the result (12.405) sv is only reached at higher temperatures ln() Tdp ; in sv sv the intermediate regime Tdp < T < ln() Tdp the pinning length Lc (T) grows with a simple exponensv tial law Lc (T) ∼ Lc exp[C T/Tdp ] [420]. Note, that the large scale wandering exponent does not change with this modestly longer range of the correlator. sv At low temperatures T < Tdp,0 the above results smoothly cross over to the corresponding zero temperature expressions derived in 12.9.2. However, note that the rapid increase of the pinning length Lc (T) with temperature sharply limits the single vortex pinning regime and hence the applicability of the results (12.405): using the condition Lc (T) < a0 we find that the single vortex pinning regime extends up to temperatures (cf. (12.328)) sv T < Tdp,0 ln

Bsv . B

(12.406)

12 Vortex Matter

The thermal depinning of vortex bundles is analyzed in close analogy to the case of single vortex pinning. Again we have to determine the mean squared thermal displacement amplitude and use the result in the determination of the renormalized pinning energy. Here, the discussion is considerably simplified due to the finiteness of the mean squared thermal displacement u2 (t → ∞)th as the amplitude u2 (r, t)th saturates on scales r ≈ a0 , t ≈ a20 /c66 (such a separation of scales is absent in d = 1, 2 where thermal fluctuations produce a divergent u2 th , which then has to be cut off by the pinning length introduced via the quenched disorder potential, cf. (12.396)). The mean squared thermal displacement ampli2 tude has √ been calculated in Sect. 12.7.2, u th ≈ Ta0 /2 "0 . Thermal smoothing is relevant as soon 1/2 as the amplitude u2 th grows beyond the core radius , and using the definition u2 th (Tdp ) ≡  2 we find the depinning temperature  B 1/2 √ Tdp = 2 "0  2 ¥0  1 j 1/2 B 1/2 c ∼ Tc . Gi j0 Bsv

(12.407)

Note that the right hand side of (12.407) does not depend on temperature; in YBCO the parameter Gi j0 /jc |T=0 is of order unity and we find that the depinning line in the H–T phase diagram, Bdp ∼

 T 2 j0 Gi Bsv , jc Tc

(12.408)

increases much steeper with temperature than in conventional superconductors, emphasizing the important role of thermal fluctuations in the high temperature superconductors. Also, the expression (12.407) for the depinning temperature of the vortex lattice crosses over to the result (12.405) for the single vortex depinning temperature as the field B drops below Bsv , see Fig. 12.32. The depinning line Bdp in Fig. 12.32 marks the crossover to the thermally smoothed pinning regime. In order to have the first-order melting line survive in the presence of disorder, pinning should be weak, i.e., the melting line should stay away from the 1D

599

Fig. 12.32. Pinning regimes and their relative position within the B–T plane. The single-vortex pinning regime is bounded by Bsv and by the single-vortex depinning tempersv ature Tdp ; all quantities within this regime are independent of field. The small bundle pinning regime is limited by Bsb b and Tdp ; dependencies on field and temperature are exponential. The large bundle pinning regime is characterized by algebraic dependences on T and on B. The crossover lines Bsv (T) and Bsb(T) drop rapidly beyond the depinning line Bdp (T) (extrapolated to zero with the dotted line)

pinning regime. In anisotropic material the depinning line crosses the melting line at H ∗ ≈ 8GiHc2 and the single vortex pinning regime remains below sv the melting line if Bm (Tdp ) > H ∗ . In estimating the position of the single vortex depinning temperature sv Tdp we may distinguish between a weak pinning masv terial with Tdp ∼ [jc /j0Gi]1/2 Tc  Tc (cf. (12.405)) sv apand a more strongly defected material where Tdp proaches Tc [150]. In the latter case we have to solve sv self-consistently for Tdp,0 ; for ıTc pinning in high-Tc material we find

 sv (12.409) ≈ Tc 1 − (j0 (0)Gi/jc (0))3/2 . Tdp,0 Typical values for YBCO are Bsv ≈ 6 T, Bsb ≈ 10 sv sv T, Tdp,0 ≈ 60 K, H ∗ ≈ 10T, and Bm (Tdp ) ≈ 25 T, telling us that a specific sample can belong to either class of weakly or strongly pinning material (we have

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G. Blatter and V. B. Geshkenbein

used expressions valid for anisotropic superconductors here). Note that conventional superconductors with a small value for Gi belong to the class of strong pinning material. At high temperatures beyond the depinning line T > Tdp the disorder potential is smoothed due to intravalley thermal fluctuations; we determine the time averaged pinning energy following the same scheme as for the single vortex pinning problem above (cf. (12.397)) and find the expression

E(V )2t  ≈

lat V,  2 + u2 th

(12.410)

 with lat = /a20 = d2 uK(u)/a20 ≈  4 /a20. Comparing the results at low and high temperature we 1/2 find that the thermal displacement u2 th takes over the role of the pinning scale  if we express the disorder strength by the parameter lat (note that the disorder parameter lat describes fluctuations in energy, see (12.312), and therefore is not rescaled due to thermal smoothing; the latter affects only the pinning forces via the change of the length scale 1/2 rpin ∼  → rpin ∼ u2 th of the disorder potential but leaves its energy scale unchanged). Further1/2 more, we have to replace u2 th for  in the expression for the energy gain due to the Lorentz force. These substitutions allow us to generalize the results of Sect. 12.9.3 in a simple way, such as to include the effects of thermal fluctuations. In particular, we can combine the low and high temperature results into one expression by defining the temperature dependent effective pinning scale rpin(T) of the disorder potential,   u2th  T  2 rpin = 2 1 + . (12.411) (T) ∼  2 1 + 2  Tdp The expressions for the collective pinning length Rc (T), the pinning energy Ucb (T) ∼ c66 [rpin/Rc (T)]2 Vc (T), and the critical current density jc (T) ∼ [rpin/R2c (T)]j0 then take the form

Rc (B, T) ∼ (12.412) a0 ⎧     B  T 2 3/2 ⎪ ⎪ exp c , a0 < Rc (T) <  , 1 + ⎪ ⎨ Bsv Tdp   ⎪ ⎪ T 2 3/2  B  ⎪ ⎩ 1+ ,  < Rc (T) , a0 Bsv Tdp Ucb (B, T) ∼ (12.413) Uc ⎧     B  T 2 3/2 ⎪ ⎪ exp 3c , a0 < Rc (T) <  , 1 + ⎪ ⎨ Bsv Tdp  ⎪   2 B 3/2 B  T 2 2 ⎪ ⎪ ⎩ 1+ ,  < Rc (T) , Hc 2 Bsv Tdp jc (B, T) ∼ (12.414) jc ⎧     T 2 3/2 B  ⎪ ⎪ ⎪ exp −2c , a0 < Rc (T) <  , 1 + ⎨ Bsv Tdp ⎪ T −11/2 j0  B −3  ⎪ ⎪ 1+ ,  < Rc (T) . ⎩ 2 jc  Bsv Tdp In order to obtain the explicit dependence on the magnetic field B and on temperature, we have substituted   2 Lc rpin T 2 1/2 B  1+ → (12.415) a0  2 Bsv Tdp in the above formulae. Finally, the temperature dependent crossover fields Bsv (T) and Bsb (T) are given by the expressions    T 3 Bsv (T) ∼ Bsv exp −2c 1 + sv (12.416) Tdp and

 T −2 Bsb (T) ∼ Bsv 1 + Tdp     T −2 2/3 2 jc 1+ × ln  j0 Tdp (12.417)

and decrease rapidly with increasing temperature T upon crossing the depinning line Bdp (T), see Fig. 12.32.

12 Vortex Matter

12.9.6

601

Anisotropic Material

The generalization of the above results (12.413), (12.414), and (12.415) to the anisotropic situation using the scaling rules (12.170) is straightforward and involves the substitutions jc → jcc , Uc → Ucc , Hc2 → Hc2 (# ) , Bsv → Bsv (# ) , (12.418) Tdp → Tdp (# ) , with the field rescalings Hc2 (# ) = Hcc2 /"# , Bsv (# ) = Bcsv /"# and the rescaled depinning temperature √ Tdp (# ) = 2 ""0  2("# B/¥0 )1/2. However, the scaling approach is limited to phenomena involving length scales smaller than , i.e., the regime Rc <  when discussing pinning; rescaling the condition Rc <  we obtain the constraint   c 2/3 jcc j . B < Bsc ≈ Hc2 (# ) ln  2 c j0 j0 12.9.7

(12.419)

Layered Superconductors

Pinning and creep in layered superconductors require a special discussion as the discreteness of the structure can modify the results obtained for a continuous anisotropic description [64, 324]. We assume that pinning is due to the collective action of weak point-defects and consider the situation with the magnetic field B aligned with the material axis. Depending on the strength of the pinning and on the magnitude of the magnetic field B, the collectively pinned object as well as the nucleus for the creep-type motion can take any form between a zero-dimensional pancake vortex and a three-dimensional vortex bundle. Let us start with the usual continuous anisotropic description and consider an isolated vortex line. The length of the collectively pinned segment is given by (12.329), Lcc ∼ " 4/3 ("02  2/ )1/3. For weak enough pinning ( small) we will have Lcc > d and the collectively pinned object is a line segment. Second, in order to neglect the interaction between the vortex lines the magnetic field has to be weak enough such that

Fig. 12.33. Schematic pinning diagram for a layered superconductor: strong pinning leads to 0D pancake-vortex pinning and 2D pancake-vortex bundle pinning. Decreasing the materialanisotropy or the disorder such that " > "ca ∼ (d/) Up /Eem the 0D and 2D strong pinning regimes vanish and we remain with the 1D line and 3D bundle-pinning regimes characteristic for the anisotropic material, cf. Fig. 12.32. The elastic approach used in the determination of the various pinning regimes is limited to the low-temperature low-field regime below the melting line

Lcc < "a0 . Together, we find the requirements for single vortex line pinning to be d < Lcc < "a0 ,

1D pinning .

(12.420)

Increasing the magnetic field B such that "a0 < Lcc we enter the regime of 3D bulk pinning, d, "a0 < Lcc ,

3D pinning .

(12.421)

We now increase the pinning parameter (or decrease the anisotropy parameter "); as the collective pinning length Lcc drops below the interlayer distance d the line elasticity cannot compete any longer against the pinning force and we enter the strong pinning regime where pancake vortices are pinned individually. In addition, we have to make sure that the intraplanar interaction energy between

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the pancake vortices is smaller than the pinning en ergy Upc = d 2 of an individual pancake vortex, c66  2d < Upc . Combining these results the requirements for single pancake-vortex pinning are (cf. Fig. 12.33) Lcc < d ,

B < B2D ∼

Upc Hc , "0 d 2

0D pinning .

(12.422) Note that within the weak collective pinning scheme  the pinning energy Upc = d 2 of an individual pancake vortex still derives from many competing point-like pinning centers. Increasing the magnetic field B such that a0 <  d"0 /Upc the possibility of 2D collective pinning arises. In this case all the pancake vortices within a radius Rc > a0 relax collectively to the underlying pinning potential. The collective pinning radius Rc is obtained from the elastic-disorder energy balance, c66d (/Rc )2 R2c ∼ (Rc /a0)Upc , hence Rc ∼ a0 ("0 d/Upc )(/a0)2 . A second requirement for the existence of a 2D collective pinning regime is given J by the smallness of the interlayer coupling Uint ∼ 2 2 2 c44 (K ∼ /Rc )(/d) dRc ∼ "0 d(/ ) (Rc /a0 )4 as 2D compared to the intraplanar energy Uint ∼ c66 d 2. The two conditions then define the boundaries of the 2D collective pinning regime, Lcc < d ,

a0 < Rc <

 a0   ,

2D pinning . (12.423) When the anisotropy is large (i.e., for a small anisotropy parameter " < d/ where the Josephson tilt energy " 2 "0 u2 looses the competition against the electromagnetic tilt energy "0 d 2u2 /2 ) we have to account for the electromagnetic coupling between the pancake vortices; comparing the electromagnetic tilt energy ("0 d 2/a202 )R2c with the intraplanar shear 2D energy Uint we obtain the modified condition Rc < . Note that pinning is strong when compared to tilt deformations but weak compared to the intraplanar shear interactions. The 0D and 2D strong pinning regimes vanish (simultaneously) as the material anisotropy (or the  disorder potential) is pushed beyond "ca ∼ (d/) Up /Eem . √ Finally, for large fields such that Rc > a0   we enter the 3D collective pinning regime,

 Lcc < d , a0   < Rc ,   2 U 1/3 pc B2D 2 < B 3D pinning.  "0 d

(12.424)

The latter condition is equivalent to the previous condition "a0 < Lcc ; the boundary to the 3D collective pinning regime does not depend on the interlayer distance d and is the same whether it is reached from the 1D or from the 2D-pinning regime. Note that an additional 1D-pinning regime appears at low fields when pinning is weakened by thermal fluctuations [323]. The complete description of pinning and creep in layered superconductors follows the same scheme as discussed in Sects. 12.9.1–12.9.6 above for the case of isotropic and continuous anisotropic materials; here, we wish to concentrate on some additional aspects which are peculiar to the layered situation. One of these aspects is the strong dispersion in the electromagnetic tilt interaction manifesting itself in the limit of large anisotropy (small anisotropy parameter "  d/). This dispersion is responsible for the appearance of a first-order like jump in the pinning length Lc versus disorder strength in the small induction regime [323] (a similar jump between different pinning regimes appears at high fields [421]): while a large disorder potential produces 0D pancake-vortex pinning with Lc = d, decreasing the disorder leads to a jump in Lc with a large value Lc > , placing the pancake-vortex system into the 1D pinning regime. A second aspect we wish to address here is a consequence of the weak interaction between pancake vortices at small magnetic fields: the elastic potential between pancake vortices reaches a lower limit in the electromagnetic coupling between layers. On the other hand, no such limit applies to the disorder strength which then can outplay the elastic coupling between pancake vortices. In a strong disorder potential the 0D pancake vortices are quite free to optimize their pinning energy by exploring large regions of the disorder landscape; this probing of the distribution of low-lying states then leads to a further enhancement of strong pinning. In the following, we present a refined analysis of single pancake-vortex pinning where we combine the two new aspects discussed above.

12 Vortex Matter

In order to simplify the discussion we consider the decoupled limit with a purely electromagnetic interaction (in a real material this corresponds to the limit "  d/ where the electromagnetic coupling dominates over the Josephson coupling). The presence of point disorder leads to a distortion of the vortex line with a typical relative displacement u between neighboring pancake vortices. The optimal pinning state is determined by the competition between the elastic and the pinning energies. The deformation u of the vortex line on a length L costs an energy Eel (u, L) ∼ "l (u, kz ∼ 1/L) u2/L with the strongly dispersive elasticity "l (u < , kz ) = ("0 /22 kz2 ) ln[1+2 kz2 /(1+u2kz2 )]. On the other hand, when adjusting to the disorder potential, a vortex segment of length  L gains the pinning energy Epin (u, L) ∼ |E0(u)| L/d, where d is the layer separation and E0 (u) is the deepest minimum a pancake vortex can settle in within the area u2 . This energy E is determined by the condition u2 0 dEg(E) ≈ 1, where g(E) is the distribution of pinning energies, which for a large number of defects we assume to be Gaussian, %  1 E2 g(E) = √ (12.425) exp − 2 .  Up  2 Up Here, Up quantifies the disorder strength, in the col lective pinning scenario, Up ≈  2d. For u   (strong pinning) each pancake vortex can explore many minima and settle in the lowest available state    u2 1/2 E0 (u) ∼ −Up ln 1 + 2 . (12.426)  Introducing the energy scale Eem = "0 d 2/2 , we arrive at the vortex free energy f per unit length (cf. (12.323)), 7 Eem  2  u2 E0 (u) d f (u, L) ∼ ln 1 + 2 . + d L + u2  2 d L (12.427) Minimizing f with respect to u and L provides us with the optimal pinning state. For strong pinning the minimum is realized by the 0D pancake-vortex configuration (L = d) and minimizing (12.427) with respect to u we obtain the optimal search area [422]

603

  −1/2  2  Up Up  Eem −1 ln ln 2 ≈ . Eem Eem  Up (12.428) The activation barrier for pancake-vortex motion is    Up  1/2 Upc = −E0 (ug ) ≈ Up ln . (12.429) Eem u2g

2

This value is larger than the naive estimate in (12.422) above as the pancake vortex optimizes its energy by exploring the various minima in the transverse dimension. Comparing the Lorentz force j(¥0/c)d with the pinning force Upc /, we find the pancake-vortex critical current density,    Up  1/2 jpc ≈ jp ln , (12.430) Eem with jp = j0 (Up /"0 d). In the weak-pinning situation Up < Eem we have ug ∼ , E0 ≈ −Up , and it is energetically more favorable for the system to settle in the 1D pinning regime: Minimizing f (u = , L) with respect to L, we find   4/3  E 2/3 em Lc ∼  , (12.431) d Up and the corresponding expressions for the pinning energy Uc and the critical current density jc follow from the usual scaling arguments,   2/3  E 1/3 em Uc ∼ Up , d Up  d 2/3 U 1/3 p . (12.432) jc ∼ jp  Eem Hence decreasing the pinning strength Up below the electromagnetic energy scale Eem triggers jumps in the pinning length Lc , the pinning energy Uc , and in the critical current density jc involving the large factor (/d)2/3(Eem /Up)1/3 . Such jumps in the characteristic quantities Lc , Uc , and jc can be triggered as well via an increase in temperature: thermal fluctuations smear the pinning potential and thus effectively reduce the disorder strength, with the corresponding first-order like thermal depinning transition taking place at a temperature [323]    Up  −1/2 Tg ∼ Up ln . (12.433) Eem

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Let us apply these ideas to the layered BiSCCO superconductor: We can ignore the Josephson coupling in our rough estimates if " < d/; with typical values d = 15 Å, (0) ≈ 2000 Å, and " < 1/150 this requirement is marginally satisfied. The parameter Up can be estimated from experiments measuring the critical current density at low B and T; in layered high-Tc superconductors (BiSCCO) Up ≈ 10 K, typically.Comparing with the electromagnetic elastic energy Eem ≈ 0.2 K one concludes that strongly layered high-Tc superconductors quite naturally generate a strong-pinning situation with Up  Eem . As usual, increasing the magnetic field or temperature couples the pancake vortices into a 2D collectively pinned structure (above B2D ∼ (Upc /"0d)) or into a 1D collectively pinned line (above Tg ∼ Up [ln(Up /Eem )]−1/2 ). Similarly, decreasing the current density via creep couples the pancake vortices along the c-axis below jg ∼

−1/4 jp (Eem /Up)1/2 [ln(2 Eem / 2Up )]1/2 ln(Up /Eem ) . The details of this temperature and creep induced coupling are quite non-trivial, involving variablerange hopping arguments [323] known from transport in doped semiconductors [423]. As discussed above, the 2D pinning regime bridges the field regime B2D < B < B2D [( 2 / 2) (Upc /"0d)]1/3 between the 0D strong and 3D weak pinning regimes. The pinning energy Ucb ∼ Upc Rc /a0 ∼ "0 d(B/Hc2 ) does not depend on the disorder parameter and the critical current density jcb ∼ jp a0 /Rc ∼ jp (B2D /B) smoothly extrapolates the pancake-vortex pinning result to higher fields. The analysis of creep at low current densities and the inclusion of thermal smoothing at high temperatures follows the usual scheme [150] discussed in Sects. 12.9.4 and 12.9.5 above. 12.9.8

Pinning and Creep Diagrams

Besides the thermodynamic phase diagrams discussed in Sects. 12.1 and 12.7, the pinning and creep diagrams introduced above provide a further useful characterization of disordered Vortex Matter, cf. Figs. 12.28, 12.32, 12.33. These diagrams describe the crossover between the various pinning regimes and are not of thermodynamic origin — rather they de-

scribe the situation under strong drive close to jc . The richest diagram appears in layered material (see Fig. 12.33) with 0D and 2D strong pinning regimes involving individual and in-plane coupled pancake vortices. Weakening the pinning forces by thermal smoothing at high temperatures or increasing the elastic forces by increasing the magnetic field produces a crossover to the intermediate and weakly pinned 1D lines and 3D bundles. Decreasing the material anisotropy (i.e., increasing ") the 0D and 2D strong pinning regimes shrink and  vanish completely at the critical value "ca = (d/) Up /Eem — the pinning diagram evolves into that describing the simpler continuous anisotropic case involving only 1D and 3D regimes, see Fig. 12.32. The depinning line Tdp triggers a rapid collapse of the crossover fields Bsv and Bsb . The sliver of small bundle pinning between Bsv and Bsb is narrow. Finally, we can join the pinning diagram at jc (Fig. 12.32) and the thermodynamic phase diagram Fig.12.3 at j = 0 through the creep diagram Fig.12.28, see Fig. 12.34. Moving away from the critical current density jc towards thermodynamic equilibrium (where j = 0) we cross the various regimes of single vortex creep, small bundle creep, large bundle creep, and finally, at very small current densities, the system ends up in the Bragg-glass regime. The three dimensional diagram Fig. 12.34 is very useful for the understanding of the time evolution of the system. Consider for example a typical magnetic relaxation experiment where a sample is first cooled in zero field to a point A = (j = 0, H = 0, T < Tc ) within the phase diagram. Upon switching on a field Hext a critical state is established within the sample and we jump to the point B = (j = jc , H = Hext , T).In the following,the critical state decays slowly in time and the system evolves gradually through the various creep regimes as indicated by the line until it reaches equilibrium (C). While the system should end up in the Bragg-glass regime, the time scales involved in reaching this asymptotic regimes are large; within an experimental waiting time t the system is able to overcome barriers of height T ln(t/t0 ). With typical (in the copper-oxides) barriers Uc ∼ 100 − 1000 K and a logarithmic enhancement ln(t/t0 ) ∼ 25 the currents can relax by a factor 25T/Uc ∼ 2 − 25 (we

12 Vortex Matter

605

Fig. 12.34. Three-dimensional phenomenological phase diagram B–T–j for an anisotropic high temperature superconductor. The different regimes of pinning, creep, and flow divide the phase space into separate regions. The various pinning regimes relevant at j ≈ jc are shown in the back. Upon decreasing the current density j the system probes the various creep regimes on approaching equilibrium. Under equilibrium conditions (front) the currents in the system vanish. The current axis also can be understood as a time axis via the relation j(t) ∝ [ln(t/t0 )]−1/ . The path of a typical magnetic relaxation experiment is illustrated where a sample is cooled under zero field conditions to the point A. Switching on a magnetic field a critical state is formed (B) which subsequently decays via creep ( ) towards the equilibrium Bragg-glass state (C)

assume a glassy exponent ∼ 1) and it appears difficult to establish glassy order throughout the sample.

12.10

Correlated Disorder

The most prominent type of correlated disorder is produced by columnar defects introduced artificially into the material (typically YBCO or BiSCCO) by irradiation with heavy ions. The columnar defects produce a marked increase in the critical current density and a shift in the irreversibility line towards higher temperatures and fields [29,30,89,90,424] (note that proton-irradiation producing either point-like defects or defect clusters leads to an increase in the critical current density but leaves the irreversibility line unchanged [425]). The angular sensitivity of these results with respect to the field-track orientation is much weaker for the BiSCCO compound [424]

as compared to YBCO [29]. This different behavior is attributed to the large difference in anisotropy between these materials [90,426]. Theoretically, the interplay between the vortex system and a columnar defect structure has been studied in much detail by Nelson and Vinokur [55] and the competition between point disorder and columnar defects has been analyzed in [427]. In the following, we discuss some selected aspects of strong pinning by columnar defects and then add a few remarks concerning strong pinning due to twin boundaries.We adopt a continuous anisotropic description for the material and assume both the magnetic field and the tracks to be directed along the c-axis. 12.10.1

Columnar Defects

Pinning in a type-II superconductor is optimized by introducing defects which trap the individual

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vortex lines all along while simultaneously destroying a minimal volume fraction of the superconducting material itself. A close to optimal defect structure then is obtained by introducing columnar defects into the material with cylinders of nonsuperconducting material of diameter ∼ , the vortex core size (note that an ordered array of tracks further optimizes pinning for a particular field value, see [428]). The resulting pinning properties will be highly anisotropic, with optimal pinning obtained for a configuration where the magnetic field is aligned with the linear defect structure. For this situation each trapped vortex gains an energy Ur = ˛Hc2  2L, where L is the size of the system along the direction of the magnetic field and ˛ a geometry factor (the index “r” refers to the properties of the cylindrical rods). For weak enough fields the defects outnumber the vortices and the interaction between the vortices is small compared to Ur ; within this single vortex pinning regime we can expect to obtain a critical current density jc ≈ ˛j0 , ˛ ∼ 0.1 — 1, of the order of the depairing current density j0 . Second,since the thermal softening of the linear pinning potential is much more gradual than for the pointlike pins one expects a reduction in the decrease of the critical current density with increasing temperature. Both effects have been observed on samples of YBCO irradiated with high energy (∼ GeV) Sn and Pb ions [29,30]. The fast heavy ions produce linear tracks of damaged material due to their large ionization energy-loss rate exceeding a few keV/Å. High-resolution electron microscopy confirms the formation of linear tracks of highly defected material aligned with the beam direction. The resulting defect structure can be modeled as a random array of parallel normal cylinders of diameter 2rr ≈ 50–70 Å embedded in a matrix of superconducting material. The density nr = dr−2 of these columns is conveniently measured in terms of the matching field B¥ = ¥0 /dr2 producing an equivalent density of vortex lines in the superconductor. Typical irradiation doses as used in the experiments produce values for B¥ between 1–5 T. The pinning potential produced by a (large) cylindrical cavity of radius rr with  < rr <  has been determined by Mkrtchyan and Schmidt [429],



R2 r (R) ≈ −"0 ln 2 R − rr2

 ,

rr +  < R < 

(12.434) (the above result is easily obtained using the concept of image charges from electrostatics and summing the ln-interaction potential from three aligned charges, two positive ones placed at 0 and at R and a negative charge at rr2 /R). Cutting the above expression at the distance  away from the cavity we obtain the depth   rr r (0) ≈ −"0 ln ,  < rr , (12.435)  of the pinning potential (this result derives from cutting the vortex currents on the defect radius rr instead of the core radius ). In order to reach an optimal pinning state the vortex lattice has to deform locally at a cost of c66(dr /a0)2 a20 , where dr is the mean separation between the tracks. A comparison of the shear and pinning energies allows us to determine the field regime B < Br where the individual vortices can accommodate to the defect structure, Br ≈ 4B¥ .

(12.436)

Increasing temperature, the vortex core becomes larger than the diameter of the cavity and the pinning energy can be estimated from the gain in condensation energy when the vortex and the (small) cavity overlap, r (0) ≈ −"0

rr2 , 2

rr <  ;

(12.437)

for a small cavity the pinning energy is reduced by a factor rr2 / 2. A further characteristic of the pinning potential is the extended tail r (R < ) ≈ −"0 rr2 /R2. The critical current density jc can be obtained from the force balance equation and is close to the depairing value  r 2 r jc (T < Tr ) ≈ j0 , jc (T > Tr ) ≈ j0 ,  (12.438) with the crossover temperature Tr /Tc = 1 − (02 /rr2) determined by the condition (Tr ) = rr . Using parameters appropriate for YBCO (0 ≈ 12 Å, rr ≈

12 Vortex Matter

30 Å) we obtain the estimate Tr ≈ 0.8 Tc . Above Tr the pinning potential decreases with increasing temperature and hence also the crossover field Br limiting the accommodation of the individual vortex lines to the columnar pins decreases,  r 2  T r . (12.439) Br (T) ≈ B¥ ∝ 1 −  Tc At large temperatures T the pinning potential is strongly renormalized by thermal fluctuations. The mean squared fluctuation amplitude u2  increases, first still limited to a region around the track, then expanding to cover many √ tracks, until it reaches the value u2  ≈ Ta0 /2  ""0 set by the neighboring vortices in the lattice. In the course of this expansion the vortex line averages the disorder potential over an increasingly larger area and the critical current density decreases accordingly. In the following, we discuss this thermal smoothing through the various regimes — we will see that correlated disorder helps in keeping the critical current density high as compared to uncorrelated disorder due to point defects. A useful tool in the discussion of thermal fluctuations is the vortex–boson analogy, mapping the classical statistical mechanics of the vortex system to the quantum statistical mechanics of 2D bosons through the identification T ↔ B , z ↔ , L ↔ B /T B, and "l ↔ m, cf. Sect. 12.7.7 above; note the columnar tracks produce a static disorder potential within the quantum formulation. The problem of vortex pinning onto a columnar track maps to the problem of binding a particle in the potential "r (R) and we have to solve the Schr¨odinger problem  2  T 2 ∇R + r (R) ¦ (R) = −eB ¦ (R) . (12.440) 2"l Assuming a rectangular potential of depth r and extent rr ≈ max(rr , ) the binding energy is 2 given by eB = r [1 − cT 2/2 r "l rr ] with c a constant of order unity. If the potential is shallow (or T is large) the binding is exponentially weak 2 [430], eB ∼ (T 2/"l rr ) exp(−T 2 /"l A) with A = ∞ 2 − 0 dR R"r (R) = − r rr /2. As a result, we obtain the renormalized pinning energy  T  rr √ r , Tdp "r (T) = r f = √ "l r , (12.441) r Tdp 2

607

with f (x  1) = 1 − x 2 and f (x > 1) ∼ x 2 exp(−x 2 ). Taking account of the slow decay of the pinning potential − r (R) ≈ −"0 rr2 /R2, we have to solve the Schr¨odinger equation for a particle in a shallow longrange potential and we obtain a√binding energy of √ r the form (12.441) with Tdp = ( 2rr /) "l r and f (x > 1) ≈ exp(−x) [150]; the slow decay of the pinning potential strongly reduces its thermally induced smoothing. The renormalization of the pinning energy due to thermal fluctuations implies a corresponding reduction in the crossover field, Br (T) ≈

rr2 r B¥ exp[−(T/Tdp )] . 2

(12.442)

Similarly, thermal fluctuations lead to a reduction in the critical current density: Increasing the temperar ture beyond Tdp the vortex line wanders away from its column. In order to find the mean squared thermal amplitude u2 th we consider again the 2D quantum problem of a particle trapped in a shallow potential. The binding energy eB is easily converted to a localization length for the wave function (l2 ∼ 2 /meB) and using our mapping rules bridging between the 2D-Bose system and the vortex system we obtain 1/2

u2 th ≈ rr

T r r exp[T/2Tdp] . Tdp

(12.443)

Combining (12.441) and (12.443) we obtain the fluctuation corrected critical current density (we assume r Tr > Tdp ), r

jc ≈

rr2 Tdp r j0 exp[−3T/2Tdp ], 2 T

r Tdp < T < Tdl .

(12.444) 1/2 As the thermal amplitude u2 th grows beyond the mean rod spacing dr the vortex line samples many tracks and becomes collectively pinned by fluctuations in the track density; the condition u2 th ∼ dr2 determines the delocalization temperature Tdl , r Tdl ∼ Tdp ln

dr2 . 2 rr

(12.445)

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r Note that both expressions Tdp and Tdl depend on r and Tdl,0) temperature and their onset values (Tdp,0 have to be determined self-consistently, cf. (12.401). Next we determine the fluctuation amplitude 1/2 u2 th in the high temperature regime T > Tdl . We then have to find the localization length for a particle subject to a random potential U (R) characterized by a correlator Ur (R)Ur (R ) = r ı(R − R ) 4 with r = r2rr /dr2 (here, r denotes the unrenormalized pinning potential). Analyzing the Schr¨odinger equation (12.440) with r (R) replaced by Ur (R) we can estimate the localization length by comparing the kinetic (T 2/2"l u2 th ) and potential energies 1/2 2 /dr u2th ), ( r rr 1/2

u2th ∼ dr

 T 2 , Tdl

T > Tdl .

(12.446)

Above Tdl both the pinning energy and the critical current density decay more slowly with temperature,

pin ∼ r

 r 2 T 2 r

dl

, dr T    r  rr 2 Tdl 4 j0 . jc ∼ "0 dr dr T

(12.447)

At high temperatures the single vortex pinning regime extends up to fields where the thermal amplitude u2 th of the individual lines remains smaller 2 than the √ thermal amplitude of fluctuations u th ≈ Ta0 /2 ""0 of the vortex lattice,  r 2  T 6 r r dl B < Br ∼ B¥ , dr "0 T

T > Tdl .

(12.448) Accounting for the temperature dependence of Tdl the crossover field Br ∝ B¥ (1−T/Tc )6 decays rapidly on approaching Tc . The results (12.438), (12.444), (12.447) describe the decrease of the single vortex critical current density with increasing temperature: At low temperatures T < Tr the critical current density is of the order of the depairing current density, (12.438). Above Tr the critical current density decreases, first due r to the reduction in r , (12.438), then, above Tdp , due to thermal fluctuations, (12.444), and finally, above

Tdl , due to delocalization from the individual rods and crossover to collective pinning by many rods, (12.447). Comparing the efficiency of vortex pinning by correlated columnar disorder with that of uncorrelated disorder due to point-like defects we note the advantage offered by the irradiation technique: for low fields and temperatures the vortices can take advantage of strong pinning centers. At elevated temperatures, thermal smoothing efficiently wipes out the effect of point-like disorder and strongly reduces the pinning strength of an individual columnar defect; on the other hand, the collective pinning due to many columnar tracks is less susceptible to thermal smearing, resulting in a slower decay of jc with increasing temperature. Various extensions complete the pinning diagram: At high temperatures vortex bundles are pinned collectively by fluctuations in the track distribution. Similarly, pinning involves vortex bundles at high magnetic fields. However, plastic pinning intervenes in an intermediate field range just above the matching field B¥ , see [150] for details. A further topic of interest is vortex creep. The generic geometry of the creep nucleus is a half loop with an aspect ratio following from the competition between the line elasticity and pinning, uhl ∼ ( r (T)/"l )1/2Lhl , and a size determined by the energy gain due to the Lorentz force j¥0 Lhl uhl /c, r (T) j0 , "l j √ "l r (T) j0 . Uhl (j) ∼ r (T) "0 j uhl (j) ∼ 

(12.449)

As j decreases these half-loops expand, transform into double-kinks as they link up to the neighboring columns (u ∼ dr ) and finally hop large distances u  dr to search for an optimal track; the careful analysis again involves variable range hopping arguments [55] known from transport in doped semiconductors [423]. A particular characteristic of pinning by columnar tracks is the locking of vortices into the track direction [38, 431]: Tilting the field away from the track direction √ produces kinks with an energy cost Ekink ≈ dr 2"l r (T). Multiplying with the number of kinks L||/dr and dividing by the vortex

12 Vortex Matter 2 volume La trapping energy density √ 0 we obtain the E() ≈ 2"l r (T)||/a20 . Rewriting this term as an energy gain of vortex lines in the presence of columnar defects, ıE ≈ −("l /2a20 )( − t )2 Ÿ(t − ) we obtain the trapping angle  2 1/2 r t = . (12.450) "l

Adding this energy gain to the Gibbs free energy density and minimizing with respect to the angle  at a given angle H of the external H-field we obtain the locking angle 4"l L = t . (12.451) ¥0 H For angles H < L the vortices remain fully locked to the columnar tracks. The trapping of vortices onto columnar tracks has been tested in crystals irradiated at a finite angle r (measured from the c-axis). The magnetization loop is measured twice, once with the field aligned parallel to the tracks (B = r ) and a second time with the vortices aligned in opposition to the tracks (B = −r , i.e., the vortices run within the plane defined by the tracks and the c-axis but point in the direction opposite to the tracks); the angular anisotropy then acts equally in both geometries. Comparing the two magnetization curves allows to estimate the pinning efficiency of the tracks and its angular sensitivity. With a track angle r = 30◦ , experiments on YBCO [29] show very different magnetization curves while the same experiment carried out on an irradiated BiSCCO sample [424] shows little difference between the two geometries. This result can be easily understood within the scaling approach: after rescaling to an equivalent isotropic system the magnetic field and the tracks are redirected towards the c-axis with the new isotropized angles ˜r,B = arctan(" tan r,B ). The trapping cone  is given by the isotropized version of (12.450), ˜t = 2 ˜r /"˜l , where the isotropized pinning energy and line tension take the values ˜r = "0 and "˜l = "0 and  denotes the pinning efficiency. The latter does not depend on the anisotropy and hence therescaled trapping cones have a similar width ˜t ≈ 2. While the track-aligned vortices obviously reside in the trapping cone (and hence are strongly pinned), the vor-

609

tices in the opposite geometry may or may not end up in the trapping cone: in YBCO the anisotropy is moderate and the redirected vortices end up outside the trapping cone, while the large anisotropy in BiSCCO redirects the vortices (and the tracks) very close to the c-axis allowing for an efficient vortex trapping and pinning.The width of the trapping cone depends on temperature, mainly via the temperature dependence of the pinning energy "r . Indeed, increasing the temperature reduces the pinning strength "r and hence narrows the trapping cone. This explains the reappearance of the angular sensitivity in BiSCCO observed at high temperatures [432]. The same scheme can be used to analyze the pinning efficiency of randomly splayed tracks [433]. 12.10.2

Twin Boundaries

Another type of correlated disorder is due to twinning planes (TPs) which naturally appear in orthorhombic YBCO crystals. In general, such planar perturbations scatter electrons and thus define a finite transparency for motion of carriers across — the boundary acts as a tunnel junction. In high temperature superconductors the combination of d-wave symmetry and scattering suppresses superconductivity: hence twinning planes in YBCO, in addition to introducing a finite transparency (corresponding to a local perturbation ım in the GL functional) also produce a suppression of the order parameter (a local perturbation ı˛ [434]); which of the two mechanisms is more efficient in pinning the vortices is difficult to tell and furthermore temperature dependent. The attraction of (parallel aligned) vortices to twinning planes leads to an enhanced vortex density along the twin boundaries, as clearly observed in Bitter decoration experiments [169].The relevance of the twin boundaries for the pinning problem has been indicated early on in resistivity [28, 328, 435], torque [436], and magnetization [437–439] experiments and more recently [440] via the angle dependence of the c-axis critical current density jc (, B) when the magnetic field B = B(cos , sin , 0) is rotated within the ab-plane.Pushing vortices out of this correlated pinning potential (with a driving current

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parallel to the twins) is hard and the critical current enhancement in the geometry j  TPs is quite substantial, with values of order 10 [440] up to 25 [441] measured at 77 K. Such a rather strong pinning is also suggested by low-temperature scanning tunneling experiments resolving the vortex structure near a twin boundary, where the current density required to push a flux line out of an individual twin boundary potential well is of the order of the depairing current density [442]. Furthermore, thermal depinning out of the well is strongly suppressed: following arguments similar to those used in the context of pinning by columnar tracks (see Sect. 12.10.1) one derives [150] a critical current density decaying ∝ T −4 as compared to the exponential decay produced in the competition of thermal fluctuations with uncorrelated disorder (again,the mapping to quantum mechanics is conveniently used). Vortex pinning along the twin boundary direction (as probed with a current j ⊥ TBs) is a more subtle issue. Various competing effects seem to be at work which enhance/reduce the pinning within the twin boundaries as compared to the bulk: while the elongation of the vortex core (implying a reduced suppression of the order parameter in the vortex center, cf. the discussion in Sect. 12.5.2 and [196]) tends to decrease the pinning of the vortex within the twinning plane, one expects an enhanced defect density within the twin boundary region generating the opposite effect. A generic mechanism producing an enhanced pinning within the plane (as compared to pinning in the bulk) is again due to thermal smoothing at high temperatures: while pinning of a vortex line decays exponentially with temperature in the bulk (cf. (12.405) and (12.405)), the thermal motion of a vortex line confined to a plane is suppressed, see (12.400), and hence thermal smoothing of the disorder potential within the twin boundary plane is reduced with a critical current density decaying ∝ T −7 (note the difference between thermal depinning within the well due to smoothing of uncorrelated disorder as compared to thermal depinning out of the well described above). Experiments report conflicting results, on the one hand providing evidence for easy-flow channels [443,444], while others are favorably explained in terms of enhanced

pinning within the twinning planes [28,445]. These contradicting results may find a consistent interpretation in terms of the above thermal smoothing scenario: experiments on the same sample carried out at low and high temperatures exhibit vortex channeling along the twinning planes at low temperatures, while large (relative to the bulk value) barriers to flux motion in the twin planes are observed at higher temperatures [446]. A further interesting aspect of twin boundary pinning is vortex trapping and vortex locking, cf. (12.450) and (12.451) for the analogous phenomenon in the context of pinning by columnar tracks. Indeed, the enhanced pinning described above is effective as long as the vortices are sufficiently aligned with the twinning planes such that finite vortex segments remain trapped [38, 431]. Typical trapping angles are of order 10◦ , while locking angles are found in the sub-1◦ regime [28,328,440].

12.11

Surface and Geometrical Barriers

While vortices find a thermodynamically stable state in the superconductor at fields H > Hc1 they cannot form spontaneously within the bulk sample but have to penetrate from the sample edge. This edge represents a perturbation which produces a barrier against vortex entry. For an ideal surface the barrier vanishes only at the critical field Hc > Hc1 , while inhomogeneities in the surface facilitate vortex entry. The vortex penetration into a bulk sample then is delayed to a penetration field Hp with Hc1 < Hp < Hc . Early studies of surface barriers are due to Bean and Livingston [330] and Clem [447]. The appearance of pronounced surface pinning in layered BiSCCO has strongly revived interest in surface pinning and creep [448]. Furthermore, the platelet type form of BiSCCO single crystals produces geometrical barriers generating a dome like flux profile within the sample [331–333].Below we summarize the main features of surface pinning and creep and then discuss the nature and consequences of geometrical barriers.

12 Vortex Matter

611

surf (x, H) = −"0 K0(2x/) (12.453) ¥0 [Hc1 + H exp(−x/)] . + 4

surf (, Hc1 ) ≈ surf (∞, Hc1 ),hence the vortex is thermodynamically stable for fields H > Hc1 but cannot penetrate until the barrier vanishes at Hp . On the other hand, real samples will usually exhibit inhomogeneities producing local field enhancements and reducing the first penetration field to a value below Hc . Above Hp vortices rush into the sample until their mutual repulsion matches their compression due to the Meissner surface current. Following Clem [447] we average the magnetic field over the intervortex distance a0 and split the sample into two regions, a vortex-free surface region 0 < x < xf where Meissner currents flow and a current-free vortex region xf < x with an induction B. The field profile h(x) follows from the London equation and decays exponentially from h(0) = H > Hp at the surface to h(x > xf ) = B < H in the vortex region. Furthermore, the Meissner current j(x) = (c/4)@x h(x) should vanish at xf . Solving the London equation in the vortex free regime with the Ansatz h(x) = a exp(−x/) + b exp(x/) and enforcing the above boundary conditions at x = 0, xf we find the field profile h(x) = Bcosh[(x − xf )/], in particular H = Bcosh(xf /).The position xf of the front is fixed by requiring that the “last” vortex penetrating does not encounter a surface barrier, hence j(x = 0) ≈ j0 and we obtain the second relation Hp ≈ Bsinh(xf /). Given the external field H,the field profile in the sample then is characterized by the induction B and the front position xf in the form  H + Hp . (12.454) B = H 2 − Hp2 , xf (H) = ln 2 H − Hp

This surface barrier has a maximum up (H) ≈ "0 [ln(Hp /H) − 1] at a distance xp ≈ (Hp /H) from the surface and vanishes only at the first penetration field Hp ≈ ¥0 /4 ≈ Hc (at fields of order Hc the order parameter is strongly suppressed at the surface; using a Ginzburg–Landau description in the high- limit one can show that Hp = Hc is an exact result [449, 450]). At first penetration a current j(x = 0) ≈ cHc /4 of the order of the depairing current pushes the vortex into the sample. Note that at Hc1 the vortex energy at the sample edge becomes equal to its energy in the bulk,

For H  Hp we have xf ≈ (Hp /H) ≈ a20 /4  a0 for H  Hc2 and the width xf of the vortex free region is indeed larger than the intervortex distance a0 up to large fields of order Hc2 . Finally, we calculate the magnetization −4M = H − B = H − (H 2 − Hp2 )1/2 ≈ Hp2/2H, where the last equation is an expansion for H  Hp . This magnetization reaches the scale of the equilibrium magnetization −4Meq ≈ Hc1 only at a field H ≈ Hp2 /Hc1 ≈ Hc2 , close to the upper critical field. Note that the magnetization loop is strongly asymmetric: on decreasing H (while keeping B constant) the front xf

12.11.1

Surface Barriers

Consider a flux line running parallel to the superconductor–vacuum interface at a distance x (e.g., with the vortex directed along zˆ and the surface parallel to the yz-plane with a surface normal xˆ ). The boundary condition xˆ · [∇' + (2/¥0)A] = 0 tells that no current crosses the superconductor–vacuum interface and the vortex current is squeezed towards the surface; following Bernoulli’s law the vortex is attracted to the surface. For a quantitative analysis we can adopt the idea of image charges from electrostatics: placing an image vortex pointing along −ˆz at a distance −x from the interface the boundary condition is satisfied. The interaction energy between the vortex and its ghost image produces the attractive Bean–Livingston potential 1 ¥0 [hv (2x) − hv ()] (12.452) 2 4 = "0 [K0(/) − K0 (2x/)] ,

im (x) =

where hv (x) = −(¥0 /22 )K0(x/) is the magnetic field associated with a vortex (the factor 1/2 restricts the energy to the half space). A vortex penetrating into the superconductor has to overcome the potential im(x). In the presence of an external field H directed parallel to the vortex the Meissner current j(x) = (c/4)@x [H exp(−x/)] pushes the vortex into the sample and the total energy takes the form

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moves towards the surface until B = H and the magnetization M = 0 vanishes until H changes sign (the image force pulls the vortices out of the sample keeping B = H). At finite temperature vortices can enter the sample for fields H < Hp via thermally activated creep through the formation of half-loops (of height l and extending a distance u into the sample). Balancing the elastic energy ("l u2 /l) against the pinning ("0 u ln(u/)) and Lorentz force (¥0 ju2 /2c) energies we obtain (cf. (12.449) and [448]) u≈

j0 j0 ln , j j

U≈

 j0 j0 "0  ln2 . (12.455) 2 j j

In an anisotropic material the length l and energy U are rescaled by the anisotropy parameter ". In layered material such as BiSCCO the loop height l ≈ "(j0 /j) might be smaller than the layer separation d and penetration proceeds via single pancakevortex creep. Balancing again the surface barrier "0 d ln(u/) against the Lorentz force energy ¥0 jdu/c we obtain the creep characteristics j0 Hc ≈ , j H

j0 Hc ≈ "0 d ln , j H (12.456) where we have expressed the Meissner current j driving the creep in terms of the applied magnetic field, j ≈ cH/4. Combining this result with the time relaxation (12.302) we obtain the relation U (Hp ) = T ln(t/t0 ), from which we conclude that the effective penetration field Hp decays with temperature or time, u≈

U ≈ "0 d ln

Hp (T, t) ≈ Hc exp[−T/T0(t)] ,

T0 = "0 d/ ln(t/t0 ) . (12.457) Repeating the calculation of the induction (12.454) with the new boundary condition for the current density j(x = 0) ∼ j0 exp[−T/T0(t)] we recover the original expression (12.454) with Hp replaced by Hp (T, t). The tail (H  Hp (T, t)) of the magnetization loop then decays with temperature or time according to −4M(T) ≈

 2T  ∝ exp − , (12.458) 2H T0  t −2T/"0 d −4M(t) ∝ . t0 Hp2

Note, that the tail decays faster than the penetration field itself. As the hopping distance u increases beyond the front xf (H, Hp (T, t)) the creep barrier saturates and the magnetization becomes reversible; the condition u ∼ Hc /Hp = xf ≈ Hp /Hirr defines the irreversibility line 2

Hirr (T, t) ∼

 Hp ∼ Hc2 exp(−2T/T0) , (12.459)  Hc

where the magnetization loop M(H) closes when measured on a time scale t and at a temperature T. Inserting this result into (12.459) we note that the magnetization M has dropped to a value of order of the equilibrium magnetization, −4M ∼ Hc1 . Also,  inserting Hp ∼ Hc Hirr / back into the expression for the front position xf ∼ Hp /Hirr one observes that the front has approached the boundary √ up to a lattice constant, xf ∼ ¥0 /Hirr . Note that, depending on the field and time scale of the experiment, the irreversibility temperature Tirr (H, t) ≈ "0 d ln(Hc2 /H)/2 ln(t/t0 ) may end up on either side 2D of the 2D melting temperature Tm ≈ "0 d/70. Finally, although the system behaves reversibly at T > Tirr (H, t), the resistivity (as probed via a small transport current) is still activated,  ∝ exp(−U /T). The corresponding surface barrier U ≈ ("0 d/2) ln(Hc2 /H) follows from (12.456) with the driving current density j ∼ j0 /u obtained from the saturated hopping distance u ∼ xf ∼ a0 . 12.11.2

Geometrical Barriers

Consider a flat superconducting sample of ellipsoidal shape in the magnetic field Ha arranged parallel to the axis; then the induction B in the sample and the applied field Ha are related via the magnetization M and the demagnetization factor n, B = Ha + 4(n − 1)M .

(12.460)

In the Meissner state the field inside the superconductor vanishes and the magnetization loop rises more steeply, −4M = Ha/(1 − n). For a flat (rotational) ellipsoid of height d and diameter 2w the demagnetization factor is n ≈ 1 − (/2)(d/2w), hence −4M = (2/)(2w/d)Ha and the magnetic moment

12 Vortex Matter

is enhanced by a factor ∼ w/d  1 (for a cigar shape ellipsoid with d  w, n ≈ (d/2w)2 ln(2w/d), and n = 1/3 for a sphere). Quite often, flat platelet like samples are analyzed with the help of an idealized ellipsoidal approximation. However, the radius of curvature R at the perimeter is quite different for an ellipsoidal shape and for a platelet-shaped sample: the ellipsoid exhibits a “sharp” edge with Rell ≈ d 2/w, while the platelet’s edge is better approximated by a shelf of width d and the corresponding radius of curvature Rshelf ≈ d. This has quite severe consequences for the field enhancement at the sample edge: Consider the magnetostatic problem for a superconducting wedge with an opening angle 0 in the Meissner state; the external field H is arranged perpendicular to one of the wedge’s planes. With ∇ · B = 0 and ∇ ∧ B = 0 we can express B = ∇¦ through the scalar field ¦ and solve for the Laplace problem ¦ = 0 with Neumann boundary conditions ∇⊥ ¦ = 0 in cylindrical coordinates (, ' ) (we look for a solution in the form ¦ (, ' ) =  ˛ [a cos(˛' ) + b sin(˛' )] and impose the boundary conditions @' ¦ = 0 at ' = 0, 2 − 0 to obtain ˛ = /(2 − 0 )). Near the edge, the magnetic field diverges as B(, ' ) ∝  (0 −)/(2−0 ) and for a sharp edge with 0 = 0 we obtain the divergence √ B(, ' ) ∝ 1/ . Cutting this divergence at the radius of curvature we obtain the field enhancement Hedge ∼ Ha (w/d) for the ellipsoid (in agreement with the above analysis using the demagnetizing factor n), while for the platelet sample the fieldenhancement is parametrically smaller, Hedge ∼ Ha w/d. Let us determine the field and current configuration in a flat sample of strip-like geometry (extending along y) subject to a perpendicular field Ha directed along the z-axis. We can ignore the details of the sample’s thickness profile (d(x)) by integrating the current density along the z-axis, d(x)/2 

Jy (x) ≡

dzjy (x, z) ,

(12.461)

−d(x)/2

reducing the sample to a zero width strip extending from x = −w to x = w. The magnetostatics problem is solved via conformal mapping: We define the complex function §( = x + iz) ≡ Bz () + iBx () over

613

the complex -plane. Outside the sample the field equations ∇ · B = 0 and ∇ ∧ B = 0 are interpreted as the Cauchy–Riemann equations for the complex function §() which thus is analytic away from the cut |x| ≤ w. Amp`ere’s law then relates the current flow Jy (x) with the jump in Bx (x, 0) = Re §(x, 0) across the strip, Bx (x, 0) = (2/c)Jy (x), while the zcomponent of the induction is required to vanish in the Meissner state, Bz (|x| < w, 0) = 0. Therefore, § is purely imaginary near the strip while § ∼ Ha far away. The corresponding analytic function takes  the form §( = x + iz) = Ha  2/( 2 − w2 ) and we obtain the current and field profiles of the Meissner state cHa x √ , 2 w2 − x 2 |x| Bz (|x| > w, z = 0) = Ha √ . (12.462) x 2 − w2 Jy (|x| < w) = −

Given the current profile in the sample we now can determine the potential profile for a vortex penetrating the sample from the edge. For a flat strip of constant thickness d this potential takes the form (see Fig. 12.35) ¥0 geo(x) = l d + c

w

dx Jy (x ) ,

(12.463)

x

where the first term is due to the vortex line energy, while the second term is the contribution from the Lorentz force pushing the vortex into the sample. For Ha = 0 a sharp barrier of height l d rises over the distance ∼ d from the edge as the vortex line is being created (here, we ignore the presence of an additional surface barrier).A finite field produces an energy gain as the vortex glides into the sample and at Heq = (d/2w)Hc1 the vortex state has become thermodynamically stable. However, the vortex penetration is delayed by the geometrical barrier and  we need the “large” field Hp ≈ d/wHc1 in order to produce the edge current density jedge ≈ cHc1 /2d necessary to compensate the vortex creation energy l . The geometrical barrier l d is absent in a smooth sample with an ellipsoidal profile: here, the line energy rises continuously from zero with increasing

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free central vortex region with Bz (|x| < xf ) = 0 and a vortex free edge regime with a Meissner current Jy (xf < |x| < w) = (c/2)Bx (x, 0) = 0; correspondingly the complex function § is real and imaginary in the two regions and we find the solution §( = x + iz) = Ha [( 2 − xf2 )/( 2 − w2 )]1/2 producing the current and field profiles of the vortex penetrated state Jy (|x| < xf ) = 0 ,



Bz (|x| < xf , z = 0) = Ha

Fig. 12.35. Vortex potential at various applied fields 0 ≤ Ha ≤ Hp . At Ha = 0 the potential at the edges of the sample is assumed to drop linearly for simplicity. The inset shows the penetration of a vortex into an elliptical-shape sample (the vortex segment grows continuously and there is no geometrical barrier) and into a shelf-shape sample (the vortex segment grows rapidly over a distance ∼ d/2 resulting in a geometrical barrier)

sample thickness and the Lorentz force can match the cost at Hp = (d/w)Hc1 ; the penetration field Hp for a flat sample is enhanced  over that for an ellipsoidal sample by the factor w/d due to the presence of a geometric barrier. Accounting for the additional surface barrier the penetration field Hp is shifted to the value Hp ≈  d/wHc , where the square-root reduction factor is due to the field enhancement at the (flat) sample edge. Note that surface imperfections and creep over the surface barrier reduce Hp to smaller val  ues, d/wHc1 < Hp < d/wHc ; at the same time the macroscopic origin of the geometric  barrier prevents Hp from dropping below the value d/wHc1 . Extending the above calculation for the current and field profiles to the vortex penetrated sample we can determine the magnetization; even in the absence of bulk pinning we expect to find a hysteretic magnetization loop due to the presence of the geometrical barrier. Following the rules defined in Sect. 12.11.1 we split the sample into a current

xf2 − x 2

, w2 − x 2  2 2 x cHa x − xf , Jy (xf < |x| < w) = |x| 2 w2 − x 2 Bz (xf < |x| < w, z = 0) = 0 . (12.464) The position xf of the vortex front again follows from balancing forces at the edge: the edge current (at x ≈ w−d/2) should match the force required to overcome the geometrical barrier,Jy(w−d/2) ≈ cHc1 /2, and hence xf ≈ w 1 − Hp2 /Ha2 .

(12.465)

Upon increasing Ha beyond Hp vortices rush into the sample and set up a dome-like field profile. As the dome expands with field the magnetization decreases M ≈ Ha (w2 − xf2)/8wd ≈ Hc21 /8Ha . At Ha ≈ Hc1 we have −4M ∼ Hc1 ,xf ≈ w−d/2 and the system becomes reversible.Upon decreasing the field Ha from below Hc1 the front xf first moves towards the edge and vortices start leaving the sample when xf ≈ w − d/2. Decreasing Ha from above Hc1 vortices leave the sample immediately. Note that while vortices creep over surface barriers via half-loop formation (cf. Sect. 12.11.1 above), vortex creep over the macroscopic geometrical barriers would involve segments of length ∼ d moving over distances of order ∼ w and hence is strongly inhibited. Also, note that imperfections easily destroy the surface barrier, while the geometric barrier with its macroscopic origin is very robust.

12 Vortex Matter

12.12

Vortex Glasses

The presence of disorder turns the vortex lattice into a glassy phase which is most conveniently defined through its response: as the current j goes to zero the barriers inhibiting vortex motion diverge, U (j → 0) ∼ Uc (jc /j) , and the vortex velocity v ∝ exp[−U (j)/T] goes to zero in a singular manner [15–17, 19–21]. The glass exponent describing the divergence of the barriers is a characteristic quantity of the phase. This should be contrasted with the situation where the barriers saturate, producing a regular ohmic response with a finite resistivity TAFF ∼ FF exp[−Upl /T] (here, FF =  n (B/Hc2 ) denotes the flux flow resistivity). Such thermally activated flux flow (TAFF) occurs in a pinned liquid where the divergence of the activation barriers is cut off by plastic rearrangements in the vortex system [56]. A second characteristic of the glass phase is the loss of translational order. Here, we have to distinguish between weak and strong disorder and between uncorrelated (due to point defects) and correlated disorder (e.g., due to columnar defects or twin boundaries). Originally, the vortex-glass phase has been conceived as a strongly disordered phase where the lattice order has been largely destroyed through the appearance of dislocations [16]. Further analysis has shown, however, that the loss of translational order due to disorder is much more subtle: within the elastic description outlined in Sect. 12.9.4, the asymptotic loss of order is only logarithmic, u2 (R) ∼ a20 ln(R/Ra) [19–21]. Hence a scattering experiment would still produce algebraic peaks [21] (as for a 2D lattice with quasi-long-range order, cf. Sect. 12.7.4) and following this characteristic this phase has been termed a Bragg-glass. Also, it has been shown that for weak disorder and low magnetic fields this Bragg-glass phase is stable with respect to spontaneous creation of dislocations [21,451–453]. The other extreme is that of strong disorder due to columnar defects. Contrary to uncorrelated pointlike disorder which induces line wandering, correlated disorder due to columnar tracks favors localization of the lines onto the defects.Upon irradiation, the original vortex lattice is destroyed and the pinned

615

phase below the irreversibility line turns into a glassy phase which has been termed “Bose-glass” [55] in analogy with the Bose-glass phase appearing in the phase diagram of dirty 2D quantum bosons [454]. Besides these two “weak” and “strong” Bragg and Bose-glass phases a third type of glass, the “original” vortex-glass phase [15], is still much debated. Quite interestingly, the Hamiltonian originally proposed to describe the vortex-glass phase, when analyzed for the weak disorder case, has been shown to produce Bragg-glass order [20,21]. The present status of the (amorphous) vortex-glass phase is rather vaguely defined: again due to point like disorder, the latter is assumed to be strong enough to introduce line entanglement and a finite density of dislocations. With dislocations present, the question then arises whether this amorphous vortex-glass is a thermodynamically distinct phase as compared to the vortexliquid [455]. Indeed, isolated dislocations still can be described within an elastic theory and the competition between elasticity and disorder produces a glassy response for the individual dislocation line, similar to the glassy response of an individual vortex line [38,377,456]. However, going to a finite density of dislocations the elastic description becomes ill defined as the dislocations start interacting and it is unclear whether the resulting phase turns into a glass or a liquid phase. An interesting question in this context is about the relation between structural order and stiffness (if such relation exists at all): the analysis of the low-frequency response of a collectively pinned vortex manifold (a vortex-glass with a displacement field u2(R) ∝ R2 and diverging barriers described through the random manifold regime) gives a superfluid density s (!) which vanishes ∝ [ln(1/!)]−2/(2+d−2) in the limit ! → 0 [457]. The question then is whether the quasi-longrange-ordered Bragg-glass (with a displacement field u2 (R) ∝ ln(R/Ra)) behaves differently and develops a finite superfluid stiffness. Confronted with (possibly) three types of new glass phases a number of questions arise: what are their phase boundaries, what is the nature of the transitions, and what are the neighboring phases. Let us first concentrate on the Bragg-glass: Since the effect of disorder is weak the Bragg-glass melts

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via a first-order transition into a vortex-liquid upon increasing temperature, the melting line remaining essentially unchanged as compared to the disorder free case. Increasing the magnetic field the Braggglass is expected to become unstable as well [21], cf. Fig. 12.2. Magnetization experiments on anisotropic [458–460] and layered [461, 462] material exhibit a “second”pronounced peak suggesting a rapid change in the pinning and/or creep characteristic of the material as the magnetic field is increased. Muon spin resonance and neutron scattering experiments [162, 279, 463] provide evidence for a sudden loss of order at higher values of magnetic field. Hence increasing the magnetic field appears to produce a disorder-driven crossover or transition into a more strongly disordered phase, possibly characterized by a finite density of dislocations and/or vortex entanglement; numerical simulations [464–466] and theoretical considerations [22,422,453,467] support this expectation. The sharpness of the experimental signal as observed in layered BiSCCO and Nb2 Se [462, 468–470] suggests that indeed we deal with a first-order phase transition.The resulting disordered phase at high fields is either an amorphous vortexglass phase or a frozen liquid. In Sect. 12.12.1 below we discuss how to locate the position of this disorder induced transition via a Lindemann argument. While point disorder disfavors the crystal through forced line wandering, one expects that correlated disorder with its tendency to localize vortices disfavors the vortex liquid. Irradiation with heavy ions then not only transforms the (first-order) melting line into a (second-order) Bose-glass line but also induces an upward shift of the phase transition — we will estimate the size of this shift with the help of a modified Lindemann criterion in Sect. 12.12.1. At low fields the transition into the Bragg-glass phase upon cooling remains first-order: a small amount of disorder does not destroy the first-order melting transition. The jumps in magnetization and entropy disappear at high fields [37, 471] and the first-order line terminates, at least in continuous anisotropic material such as YBCO — the question then arises whether we deal with a critical endpoint (implying that the vortex-glass and liquid phases are no different) or possibly a multi-critical point with a

second-order transition continuing on and separating a vortex-glass phase from the vortex-liquid (cf. Fig. 12.2). In looking for such a second order transition a scaling analysis of the transition line and the analysis of critical scaling in the vortex-glass and liquid response are valuable tools in providing evidence for the existence of a generic phase transition; the resulting exponents characterize the transition. Below we derive the scaling rules for the vortex-glass transition, see Sect. 12.12.2. Based on such a scaling analysis the presence of a vortex-glass transition has been inferred from transport data measured on epitaxial thin films [35,472]. Early experiments on untwinned single crystals defected by electron [473] and proton irradiation [474] failed to identify vortex-glass scaling; however, increasing the (proton) irradiation dose above a critical value, vortex-glass scaling has been observed recently in an untwinned single crystal [475]. The same scaling analysis has been successfully carried out on twinned samples [476], thus providing evidence for a generic Bose–glass transition and a Bose-glass phase in the presence of correlated disorder. 12.12.1

Disorder-Induced Melting

We concentrate on low temperatures and consider a vortex system subject to uncorrelated point-like disorder; the magnetic field is assumed parallel to the axis of the anisotropic superconductor. At first sight collective pinning theory tells that disorder is more relevant at low fields: at low magnetic fields pinning involves individual vortex lines, while increasing the field the elastic forces gain in importance and at high fields pinning involves 3D vortex bundles. However, when asking about lattice order we should compare the disorder-induced mean squared displacement amplitude u2 (L) with a20 = ¥0 /B rather than pinning and elastic energies. Starting from short distances, the individual vortex line first proceeds along its pinning valley (u2 (L < Lcc ) ∼  2(L/Lcc )3 ) and then wanders between different such valleys with a superdiffusive behavior u2 (L > Lcc ) ∼  2 (L/Lcc )5/4 . This wandering is limited by the presence of other vortices — their repulsive interaction confining the vortex line to its lattice site becomes relevant at dis-

12 Vortex Matter

tances L > "a0 . However, if before reaching this length the vortex line has wandered a distance cL a0 away, the neighboring vortices cannot force the line back to its lattice site and the lattice melts [22]. The Lindemann condition for the disorder-induced melting then reads u2 ("a0 ) ≈ cL2 a20 and produces the melting field (cf. Fig. 12.2)  8/3  Hc2 5/3 Bm,dis ≈ 2cL2 Hc 2 , (12.466) Bsv where we have assumed that pinning is not very weak, Lcc < "a0 or equivalently 2cL2 Hc2 /Bsv < 1. Assuming a Lindemann number cL ≈ 0.15 this requires an extended single vortex pinning regime Bsv ≈ 0.15Hc2 ; assuming parameters for YBCO the disorder-induced melting field Bm,dis ends up in the few Tesla regime. A similar analysis can be carried out for layered material such as BiSCCO [422]. We consider the simplest case where the layers are coupled electromagnetically (" = 0); in this case the disorder spreads out each vortex line into a tube of area u2g , cf. (12.428). The Lindemann criterion for the disorder-induced melting transition takes the form u2g ≈ cL2 a20 and we obtain the melting field   1/2  2 Up  Eem  Eem Bm,dis ≈ 2cL2 . Hc2 ln ln 2 Up Eem  Up (12.467) Again,consistency requires that we can ignore the interaction between pancake vortices in the same layer: comparing tilt and shear energies, Etilt ∼ Eem

u2g 2

ln

u2g 2 2 , E ∼ c a d , (12.468) shear 66 u2g a20 0

we obtain the condition  2 E  em , Bm,dis < B ln 2  Up   1/2 Up Eem Hc2 ln <1. 2cL2 Up B Eem

(12.469)

The strong pinning condition (12.469) requires a pin2D ning energy Up > 102Eem ≈ Tm ; this condition is

617

roughly matched in typical crystals resulting in a disorder induced melting in the field range Bm,dis ∼ B . The above Lindemann estimates for the location of the disorder induced transition are easily done; a much harder question is to understand the nature of the resulting disordered phase, in particular its enhanced pinning and reduced creep properties. The most obvious interpretation of the sharp increase in magnetization at Bm,dis is in terms of a (roughly 10fold) increase in the critical current density [462]. However, the disorder-induced phase transition requires itself strong pinning and the transition ends up in the single pancake-vortex pinning regime; once each pancake vortex is optimally pinned it is hard to imagine how to enhance pinning further. Expanding the search area u2g ≈ cL2 a20 to a20 for the optimal accommodation of individual pancake vortices produces an increase in jc by a factor 1.5 at most. An alternative explanation of the large jump in the magnetization is based on vortex creep: what is measured in the magnetization experiment is not the critical current density jc but the relaxed current density j(t) after creep, cf. (12.302). Indeed, experiments on BiSCCO [477–479] show that the creep barriers diverge faster in the disordered high-field phase.The appearance of two vortex phases near the second magnetization peak, an ordered low-field and a disordered high-field phase, has recently been observed in fast magneto-optic imaging [478, 480]. The two phases are characterized by different current profiles separated by a front [477, 478]; the velocity of this front separating the ordered and disordered phases goes to zero with the magnetic field B approaching Bm,dis . Applying a field B below but close to Bm,dis the expulsion of the disordered phase from the sample can be traced on a time scale of a few seconds, while the dynamical establishing of the disordered phase throughout the sample is observed when ramping the field beyond Bm,dis . The same type of ordered and disordered phases have been discussed in the context of dynamical effects associated with the peak effect in NbSe2 [468,469]. Both, the sharpness of the transition [469] as well as the observation of supercooling of the disordered phase [477] suggest a first-order nature for this field-driven order-disorder transition.

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Melting in the Presence of Correlated Disorder Strong correlated disorder not only transforms the first-order melting line into a second-order Boseglass transition but also produces an upward shift in the melting/irreversibility line [481]. This shift can easily be derived with the help of a Lindemann analysis. In a first step we reformulate the usual Lindemann criterion u2 th (Tm ) ≈ cL2 a20 into an equivalent energy criterion: balancing the longitudinal ("l u2 /L) and transverse (c66(u/a0 )2a20 L) elastic energies we obtain the longitudinal length L ∼ "a0 , and equating the resulting elastic energy at the Lindemann displacement u ∼ cL a0 to the thermal energy, T ∼ "l cL2 a20 /"a0 , we obtain the criterion Tm ∼ cL2 "0 , "a0

(12.470)

telling us that the vortex system melts when the thermal energy per unit length (T/"a0 ) becomes a (Lindemann) fraction of the confining potential energy per unit length "0 . Let us use this energy criterion in order to estimate the shift in the melting line due to columnar disorder. Consider first the weak-field situation with 1/2 u2 th = cL a0 > dr (we assume a0 < ). We treat the effect of disorder perturbatively: requiring the temperature to be large, T > Tdl (cf. (12.445)), the vortex lines are pinned collectively by the action of many defects. Furthermore, the disorder energy is small compared to the elastic energies at melting if u2 th as given by (12.446) is larger than u2 th = cL2 a20 as determined by the elastic forces, requiring that T>

cL a0 Tdl > Tdl . dr

(12.471)

The presence of columnar defects within the potential well leads to an effective lowering of the thermal energy: The fluctuating vortex line probing the area cL2 a20 is exposed to Nr ∼ cL2 a20 /dr2 competing The latter produce the pinning energy √ defects. 2 r Nr (rr /cL2 a20 ), where the last factor gives the relative contribution of each trap (cf. the estimate of the pinning energy leading to (12.445)). The modified Lindemann criterion reads

2 r rr TBG − ∼ cL2 "0 , "a0 dr cL a0

(12.472)

resulting in a Bose-glass line TBG shifted to higher temperatures as compared to the melting temperature Tm in the unirradiated material, r 2

TBG − Tm ∼ r rr

rr cL a0 Tdp ∼ , cL dr dr Tm

(12.473)

where we have expressed the pinning energy r r through the depinning temperature Tdp (< Tdl ), see (12.441), in the last equation. The weak-pinning condition (12.471) then tells us that the relative shift (TBG − Tm )/Tm is small. The above analysis is valid as long as the vortex samples many traps within the potential well produced by its neighboring vortices. For high fields cL a0 < dr we have either none or of the order of one trap available in each vortex potential well. We then should expect the Bose-glass line to approach the original melting line in a smooth way. The weak pinning condition now requires comparison of u2 th as given by (12.443) with cL2 a20 and we obtain the weaker condition T > Tdl . The mean energy lowering over many vortices due to the presence of columnar traps 2 is given by r (cL2 a20 /dr2) (rr /cL2 a20 ), where the second factor gives the probability to have a defect available within the fluctuation area cL2 a20 , whereas the last factor gives the mean contribution of this defect. The result (12.473) then is modified to take the form "a0 rr TBG − Tm ∼ r rr ∼ dr2



cL a0 dr

2

r 2 Tdp

. Tm (12.474) √ The shift is reduced by a factor ∝ B¥ /B and TBG (B) approaches Tm (B) at large fields. While the Lindemann criterion quite successfully predicts the location of the melting line, one should appreciate that the determination of its shift due to disorder is a much more subtle question and a more rigorous analysis should account for the change in energy of both the solid and liquid phases. Nevertheless, the good agreement with experiments on irradiated YBCO single crystals [481] lends some support to the rough estimates given above.

12 Vortex Matter

12.12.2

Vortex-Glass Scaling

In the presence of strong enough disorder the firstorder melting transition is expected to transform into a second-order glass transition. The existence of such a generic phase transition can be traced with the help of a scaling analysis as suggested by Koch et al. [35], see also [16]. Assuming a continuous glass transition at Tg we can describe the emerging critical behavior in terms of the diverging length and time z scales VG ∝ |T − Tg |− and VG ∝ VG , with  and z the correlation length and dynamical exponents. Above the critical dimension dc = 6 of the glass transition these exponents take on the mean-field values  = 1/2 and z = 4 [482] (these are the same characteristics as for conventional spin glasses [483]), while critical fluctuations are expected to increase z in lower dimensions [484]. The characteristics of the glass transition can be extracted from the current-jelectric-field-E characteristic via the following scaling analysis: Since the vector potential A scales as an inverse length (this follows from gauge invariance), the electric field E ∝ @t A is expected to scale like 1+z 1/VG VG , hence EVG is an appropriate scaling combination. On the other hand, since j ∝ @A f , with f denoting the free energy density, the scaling comd−1 bination for the current density is jVG . Collecting results we can define the scaling Ansatz  d−1  −(z+1) E ∝ VG e± jVG .

(12.475)

The Ansatz (12.475) is consistent with the one obtained for the complex conductivity  (!) ≈ s /(−i! + "): Since the superfluid density s scales 2−d with length according to s ∝ VG [485], we expect a scaling law for the complex conductivity of the form 2+z−d z  (!) ∝ VG s± (!VG );

(12.476)

equation (12.476) then provides additional information on the frequency dependence of the response near the glass transition [16, 486]. Here, we concentrate on the dc-response encoded in the scaling Ansatz for the current-voltage characteristic (12.475). Above Tg , we expect the response to be ohmic at large scales L > VG . These large distances L > VG

619

are probed with small current densities j < jx+ ∝ 1−d VG , with the crossover current density jx+ vanishing on approaching the transition temperature according to jx+ ∝ (T − Tg ) (d−1) .

(12.477)

Within this ohmic regime the scaling function e+ (x) vanishes linearly in x, e+(x → 0) ∼ x, and we obtain the scaling law for the resistivity in the form (T) ∝ (T − Tg ) (z+2−d) .

(12.478)

Below Tg , the system shows glassy response at large scales L > VG , i.e., small current densities j < jx− ∝ 1−d VG , where again, the crossover current density jx− − scales according to jx− ∝ (Tg − T) (d−1) .

(12.479)

The scaling function e− takes the form e− (x → 0) ∼ exp(−a/x ) and we obtain the glassy characteristic     jc . E ∝ exp −c j

(12.480)

Approaching the transition at Tg , the diverging length scale VG in (12.475) has to cancel out and hence e± (x → ∞) ∼ x ˛ with ˛ = (z + 1)/(d − 1); we then obtain the power-law current-field characteristic E ∝ j(z+1)/(d−1) .

(12.481)

Since z > 4 is expected in d = 3 [482], an exponent ˛ > 2.5 is predicted. Right at the transition VG ∼ ∞ and the power-law characteristic (12.481) is valid on all scales, i.e., at arbitrary small currentdensities j. Moving away from the transition, VG and the crossover current-densities jx± are finite and the power-law (12.481) valid at small distances (L < VG ) and large current densities (j > jx± ) crosses over to the ohmic (12.478) and glassy (12.480) characteristics at large distances (L > VG ) and small currentdensities (j < jx± ) for high and low temperatures, respectively, see Fig. 12.36. The above scaling analysis in fact applies to an isotropic disordered superconductor, e.g., a gauge

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glass describing a disordered granular superconductor [487,488],  H = −J cos('i − 'j − Aij ) , i,j

2 Aij = ¥0

j A · dl ,

(12.482)

i

where the variables 'i denote the phases of the individual grains and A is the vector potential; the randomness and frustration is introduced through the potentials Aij , usually chosen as quenched random variables uniformly distributed between 0 and 2. Recent numerical simulations (via a parallel tempering Monte Carlo technique) of the three-dimensional gauge glass provide evidence for a glass transition at Tc ≈ (0.47 ± 0.03) J with characteristic exponents  ≈ 1.39 ± 0.20 and z ≈ 4.2 ± 0.6 [489]. However, including screening into the model (by adding a fluctuating  gauge field a, Aij → Aij + aij /, H → H + (1/2) [∇ ∧ a]2 , ∇a = 0) appears to destroy the finite temperature gauge glass transition [490,495]. On the other hand, the magnetic field penetrating the (weakly disordered) superconductor in the form of vortex lines naturally introduces an anisotropy into the system — it remains unclear what effect this change of symmetry has on the existence and nature of the vortex-glass transition in a 3D bulk disordered superconductor. The above scaling analysis can be extended to an anisotropic situation as naturally provided by the vortex system and definitely required for the Boseglass transition in the presence of correlated disorder [55]; in this case two diverging length scales l⊥ = lloc (T) ∝ (TBG − T)−⊥

2 l ∼ l⊥ /D0 (12.483) are introduced, with D0 a short distance “diffusion” constant. In the simplest scenario, D0 remains finite on approaching the phase transition, such that the scaling exponent  takes the value  = 2⊥ . The z dynamic scaling exponent z is defined via BG ∝ l⊥ . An (experimental) analysis of the current-voltage characteristic in the vicinity of the transition into the glass phase can then be used to extract the scaling exponents z (z ) and  (⊥ ) (see Fig. 12.36): The slope

and

Fig. 12.36. Current-voltage characteristic of a superconductor in the mixed state. The plot provides the vortex-glass scaling laws for E(j),(T),and jx± (T) and the glass exponent

. At the glass transition Tg the characteristic is algebraic, E ∝ j˛ , with ˛ = (z + 1)/(d − 1). Above Tg the system is a liquid on large distances with  ∝ (T − Tg ) (z+2−d) and critical at small distances. Below Tg we have a glassy response at large distances E ∝ exp[−c(jc /j) ] and a critical response at small scales. The crossover current densities jx± separating critical from liquid/glassy response scale as jx± ∝ |T − Tg |v(d−1)

of the power-law characteristic at Tg in a log–log plot provides a measure of the dynamic exponent z. Determining the crossover current densities jx+ and jx− allows for an estimate of the correlation length exponent  . Finally, a measurement of the resistivity (T) above Tg provides a consistency check for the two exponents z and  . The vortex-glass exponent

determining the response below Tg is usually difficult to measure. The first scaling analysis of this type providing experimental evidence for a continuous transition into a vortex-glass phase has been carried out on epitaxial thin films of YBa2 Cu3 O7−y by Koch et al. [35]. The large critical current densities jc (T = 77 K, H = 0) > 106 Acm−2 indicate the presence of

12 Vortex Matter

rather strong pinning (large disorder) in these samples. Magnetic field values ranging from 0.5 T to 4 T have been studied. The exponents z and  as determined from the power-law characteristic at Tg and the crossover current densities jx+ and jx− above and below the transition are z ≈ 4.8 ±0.2,  + ≈ 1.7 ±0.4, and  − ≈ 1.5–2.0. The consistency check on (T) provides z ≈ 4.8,  ≈ 1.7, in good agreement with the previous independent measurements of z and  . The vortex-glass exponent as obtained by fitting the characteristic to a form (12.480) takes the value

≈ 0.4 ± 0.2. One should remark that a large dynamic range is needed in order to distinguish the vortex-glass behavior (12.480) from a simple powerlaw characteristic. A careful scaling analysis of the resistive transition on a twinned YBCO single crystal provides the Bose-glass exponents z ≈ 6 ± 1 and ⊥ ≈ 1 ± 0.2 [476]. Originally, the scaling analysis of the irreversibility line in (both, electron and proton irradiated) untwinned YBCO single crystals at high fields has proven unsuccessful [473, 474]; however, increasing the proton irradiation dose above a critical value, good resistivity scaling with an exponent s =  (z + 2 − d) ≈ 5.1 ± 0.5 has been observed in a recent experiment [475] within a field range between 1 and 8 T. In the end, it appears still unclear what are the generic requirements for the existence of a vortex-glass transition in 3D bulk superconductors: Dirty samples exhibit a continuous transition/crossover from an irreversible low-temperature to a reversible high-temperature phase. Improving the sample quality the Bragg-glass to vortex liquid transformation proceeds via a sharp first-order transition and there is no second-order glass transition. While strong enough irradiation seems to bring the glass transition back, the generic requirements for its presence are still unclear at present. Finally,we note that no glass transition is expected in a 2D system as point-like vacancies and interstitials cutting the diverging barriers are present at any finite temperature; also, the absence of Braggglass order in 2D thin films due to the presence of a finite density of dislocations has been shown in [21, 492, 493] (this density, however, is extremely small when disorder is weak). Experiments on 2D YBCO films seem to confirm these predictions [494].

621

As a further consequence,the decoupled phase in layered superconductors exhibits normal-metallic response and true superconducting (glassy) response is only established in the 3D superconductor below the decoupling line.

12.13

Concluding Remarks

In the disorder-free situation the Vortex Matter in strongly fluctuating superconductors is characterized by the thermodynamic H–T phase diagram, see Figs. 12.2, 12.11, 12.13, 12.15, and 12.17. In an anisotropic material a first-order melting line separates the vortex lattice from an entangled vortexliquid phase, Figs. 12.2 and 12.11. The mean-field Hc2 line has turned into a crossover marking the binding of charged carriers into Cooper pairs as observable through a broad maximum in the specific heat. However, no phase stiffness has been established yet in the vortex-liquid below the crossover at Hc2 and the material is still in the metallic phase with its characteristic ohmic response. The true phase transition breaking the translational symmetry through creation of a vortex lattice and introducing a phase stiffness parallel to the magnetic field appears only at the first-order melting/solidification line. The phase transition √ line is given by the expression Tm = 0.105 ""0 ¥0/B (or, alternatively Bm = 0.069 Hc2 (T)(""0 /T)2, cf. (12.192)) and involves the superfluid density ns (T) via the penetration depth (T) — the knowledge of the material anisotropy " and the superfluid density allows for an accurate prediction of the location of the melting line, cf. Sect. 12.7.2. The characteristic jumps at this firstorder transition are well understood,too,with a typical entropy jump Sd ≈ 0.4 kB and a positive jump in the induction (few 0.1 Gauss,see Fig.12.18) following from the backward slope dHm /dT < 0 of the melting line and the Clapeyron equation B = −4s/@T Hm , hence the vortex liquid is denser than the vortex solid and the vortex lattice melts like ice (cf. Sect. 12.7.6). The melting line reenters at low fields and gives way to a narrow sliver of vortex-liquid (order of 10 G wide, see Figs. 12.2 and 12.11); the narrow extent of this phase and its susceptibility to disorder render the observation of this phase a difficult task.Thermal

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motion of the vortex lines also renormalizes the constitutive B(H) relation; in strongly anisotropic and layered superconductors large thermal fluctuations produce an appreciable attractive fluctuation force in the range  < R < /" leading to a van der Waals type low-field phase diagram. In layered superconductors the melting line is dramatically decreased as a consequence of the extreme softness of the Vortex Matter. In the absence of interlayer Josephson coupling the pancake vortex system can be viewed as an arrangement of 2D vortex lattices coupled by the residual electromagnetic interaction along the third dimension. The melting line then interpolates between the 2D shear-dominated melting at high fields/low temperatures and the evaporation of isolated pancake-vortex stacks at TBKT in the lowfield limit (before reentering near TBKT ). Away from these limits, the melting process then can be understood as the melting of a 2D pancake-vortex system subject to a substrate potential which itself is formed from the 2D pancake-vortex systems in the other layers; the corresponding self-consistent analysis provides us with a low-field instability at Binst (T) (cf. Fig. 12.14 and the comparison of the free energies of the solid and liquid phases shows that the real melting line is close by, cf. Fig. 12.15. Including a finite Josephson coupling into the analysis, the pancake vortices bind into line objects and the melting line is rapidly pushed upwards, crossing over into the continuous anisotropic result, cf. Fig. 12.11. In turn, thermal fluctuations can destroy this interlayer coupling and an additional decoupling transition appears in the low-temperaturehigh-field portion of the phase diagram and possibly also close to Tc , see Figs. 12.13 and 12.17. As the interlayer coupling goes to zero (implying " → 0) the decoupling transition turns into a topological defect transition (cf. Fig. 12.17) and the decoupling line close to Tc vanishes as T em → Tc , cf. (12.213) and Fig. 12.13. Adding weak point-like disorder destroys the vortex lattice; at low fields this destruction is “soft” as the ordered lattice is turned into a quasi-long-range ordered Bragg-glass phase. As a consequence, the first-order nature of the melting line is preserved, at least at low enough fields. In anisotropic super-

conductors (YBCO) the jumps in magnetization and entropy decrease with increasing magnetic field and the first-order line seems to terminate [37,244,459]. At present it is unclear how many phase lines emanate from this (multi) critical point; possible scenaria involve two ((frozen) vortex-liquid and Bragg-glass), three (vortex-liquid, Bragg-glass, and vortex-glass), see Fig. 12.3, or even more vortex phases appearing in the vicinity of this point. The persistence of a specific heat step above the (multi) critical point does provide thermodynamic evidence for the continuation of the phase line at higher fields, an obvious candidate being a second-order vortex-liquid to vortex-glass transition. However, in a recent experiment this specific heat step has been located away from the irreversibility line and an interpretation in terms a vortex-liquid to vortex-liquid transition has been suggested [496]; different types of vortex-liquid phases have been postulated in the past based on symmetry considerations and duality arguments [287,288] and on numerical simulations [497]. Although some agreement has been reached regarding the entangling of the vortex system upon melting, the final answer regarding the existence of additional vortex-liquid phases has not been given yet. An interesting recent development is the manipulation of the Vortex Matter phase diagram through deliberate introduction of additional pinning centers. Starting with a clean single-crystal YBCO sample exhibiting a first-order melting transition, disorder is enhanced through irradiation with electrons, protons, or heavy ions. With increasing proton irradiation dose the first-order transition retreats both from low and high fields [498], finally giving way to a second-order glass transition [475].Columnar tracks produced by irradiation with heavy ions shift the upper critical point up towards higher fields [499],while point-disorder (introduced via electron/proton irradiation) leads to a downward shift [498]. Columnar tracks produce localization, while electron irradiation leads to line wandering, hence the above shifts in the critical point are suggestive of an entangled high field “glassy” or frozen-liquid phase. Subjecting the pancake-vortex system in layered superconductors (e.g., BiSCCO) to disorder has a number of interesting consequences. Again, weak

12 Vortex Matter

disorder preserves the first-order nature of the melting line.Adding a few strong pins through irradiation with a low dose of heavy ions clearly establishes the lattice nature of the low-field phase: With a matching field B¥ = 5 G and an applied field of order 200–400 G each vortex pinned by a column is surrounded by 40–80 free vortices on average.The introduction of these pins has essentially no effect on the high-temperature phase which remains reversible. On the contrary, the low-temperature phase exhibits a marked increase in the width of the magnetic hysteresis and in the critical current density [500].These findings are consistent with the appearance of a finite shear modulus in the low-temperature phase; with a finite c66, pinning a few vortices is sufficient to pin all the vortex lattice. Hence a consistent interpretation of the first-order phase transition is in terms of a melting line [500]. Increasing the irradiation dose to matching fields above 40 G leads to the transformation of the sharp melting transition into a continuous (possibly Bose-glass) transition line shifted up to higher temperatures and fields. The addition of point-like disorder favors a disordered phase and the melting line flattens at low temperatures, apparently turning into a field-driven order–disorder transition [477], see Fig. 12.37, possibly capturing both decoupling and melting of the pancake-vortex structure. Recently, a smooth crossover has been observed between the thermal and disorder-induced melting lines, deferring a putative (multi) critical point to low temperatures [470, 477]: making use of a vortex-shaking technique [308], the first-order transition could be traced [470] far into the low-temperature regime where the disorder-induced melting is usually identified with the help of the “second peak” in the magnetization. The continuous evolution of the thermally induced melting line at high temperatures into a disorderinduced first-order melting/decoupling line at low temperatures then provides a natural interpretation of these findings. Note that all these observations are restricted to temperatures above ∼ 20–25 K. The absence of a “second peak” at lowest temperatures (below ∼ 20 K) casts doubts on an interpretation of the “second peak” line in terms of a jump in the critical current density, as such a jump should be vis-

623

Fig. 12.37. Schematic phase diagrams for a strongly fluctuating layered superconductor in the presence of disorder. The (first-order) melting line Hm at high temperatures smoothly crosses over into a (first-order) disorderinduced melting/decoupling transition (Hm,dis ). The nature of the low-temperature-high-field phase is presently unclear: Shown are two scenaria where the depinning line Hdep denotes a phase transition into a generic vortex-glass state different from the low-field Bragg-glass and the hightemperature vortex-liquid phases (top) and where the depinning line Hdep terminates in a critical point leaving the Bragg-glass phase as the only thermodynamic phase different from the vortex-liquid/normal-metal phase (bottom; note that the depinning line may be contracted to zero)

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Fig. 12.38. Schematic phase diagram for a low-Tc superconductor as inferred from experimental studies [469]. With thermal fluctuations remaining small, the melting line is expected to close up with the Hc2 line. On the other hand, quenched disorder still can induce a sharp (first-order) phase transition taking the vortex system between an ordered vortex solid (Bragg-glass) and a disordered (amorphous vortex-glass or frozen-liquid) vortex phase

ible down to the lowest temperatures. On the other hand, this observation does lend support to a scenario based on two states of Vortex Matter characterized by different creep barriers [477–479], departing from one another at low drives. Disorder also affects the decoupling line: indeed, in layered superconductors the solidification into a still decoupled array of (disordered) 2D lattices at high magnetic fields does not produce a glass phase as mobile pancake vortex defects can circumvent the diverging creep barriers and produce ohmic response. Glassiness is only established at the decoupling line: the coupling of the layers into a 3D coherent superconductor prohibits the appearance of point-like defects and all creep barriers diverge at small drive. It is unclear at present whether the decoupling in disordered layered superconductors joins up with melting into the disorder-induced first-

order line at low temperatures (with a joint melting and decoupling at the“second-peak”line) or whether the decoupling appears as a separate phase transition. Experimentally, an additional depinning line is observed at low temperatures and high fields, extending upward from the critical point and separating (thermal-fluctuation dominated) liquid and (disorder dominated) glassy/frozen phases [501]; at present it is unclear whether this line marks a continuous transition or a mere crossover.Note that in these layered systems pinning is strong at low temperatures and due to point defects in the bulk; at elevated temperatures strong thermal fluctuations efficiently smear the pinning potential and the irreversibility is due to surface and geometrical barriers [501]. Finally, these new theoretical and experimental concepts developed within the program to understand strongly fluctuating high temperature superconductors find their application to low-Tc materials as well. In a clean sample, the melting line is expected to close up with the Hc2 line and no further new vortex phases are expected. The presence of disorder, however, again allows for a disorder-induced sharp (first-order) phase transition taking the vortex system between an ordered vortex solid (Bragg glass) and a disordered (amorphous vortex glass or frozen liquid) vortex phase [468,469], see Fig. 12.38. The sharpness of this transition can be observed in experiments carried out in a Corbino disc geometry avoiding vortex entry from the sample boundary. Such vortex entry through an imperfect surface is expected to “contaminate” the Bragg-glass phase with a disordered vortex phase. The interplay between the ordered and the disordered vortex phase is at the origin of the peak effect [502–506] and numerous dynamical phenomena (such as history dependent response [507–509], memory effects [510,511], low-frequency noise [512–515], slow voltage oscillations [514,516]) observed in many low Tc superconductors. An interesting question is about the status of the Hc2 line — is the Hc2 line a phase-transition or crossover line? Combining our knowledge gained in the studies of anisotropic (YBCO), layered (BiSCCO), and low-Tc (NbSe2 ) superconductors we can summarize the situation as follows: experiments in-

12 Vortex Matter

dicate to us the existence of a depinning line, continuing upward from the (multi) critical point in YBCO and departing upward from the (disorder–induced) melting line in BiSCCO — the corresponding line in NbSe2 should be identified with the Hc2 line. The crucial question that future experiments may answer is, whether this depinning line is a phase transition line or a mere crossover into a “more metallic” phase. On the theoretical side we know about the existence of the Bragg-glass phase, a generic thermodynamic phase different from the normal metal and characterized by its specific structural (quasilong-range order) and dynamical (glassy response) properties. The Bragg-glass is a quite delicate phase, easily destroyed by thermal fluctuations (via a firstorder transition to the vortex liquid which is equivalent to the normal metal) or via strong disorder. The crucial question then is about the nature of the phase reached from the Bragg glass by increasing the disor-

der strength: is the resulting phase equivalent to the normal-metal — then the depinning line would be a mere crossover and type II superconductors develop only one generic phase, the Bragg-glass (we ignore here the possibility of multiple liquid phases). Or is the resulting phase different from the normal metal (a vortex glass, an amorphous vortex solid, etc.), in which case the depinning line should mark a thermodynamic phase transition and the Vortex Matter in type II superconductors involves more than one thermodynamic phase. We hope that this puzzle will be resolved in the near future. Acknowledgments We thank Matthew Dodgson, Thierry Giamarchi, J´erˆome Koopmann, Sergey Korshunov, Alexei Koshelev, Anatoly Larkin, Henrik Nordborg, and Manfred Sigrist for illuminating discussions.

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13

Unconventional Superconductivity in Novel Materials

M. Brian Maple Department of Physics and Institute for Pure and Applied Physical Sciences. University of California, San Diego La Jolla, California, USA Eric D. Bauer Department of Physics and Institute for Pure and Applied Physical Sciences. University of California, San Diego La Jolla, California, USA Vivien S. Zapf Department of Physics and Institute for Pure and Applied Physical Sciences. University of California, San Diego La Jolla, California, USA Jochen Wosnitza Hochfeld-Magnetlabor Dresden (HLD), Forschungszentrum Dresden-Rossendorf, Dresden, Germany

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 13.2 Conventional Superconductors Containing Localized Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . .641 13.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .641 13.2.2 Superconducting-Magnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 13.2.3 Paramagnetic Impurities in Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 13.2.4 Magnetic Field Induced Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 13.3 f-Electron Heavy Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 13.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .659 13.3.2 Normal-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 13.3.3 Superconducting-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 13.3.4 UPd2Al3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 13.3.5 Quantum Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 13.3.6 Non-Fermi Liquid Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 13.3.7 Magnetically Ordered Heavy-Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 13.3.8 Multiple Superconducting Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 13.3.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 13.4 Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .696 13.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .696 13.4.2 Normal-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 13.4.3 Superconducting State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 13.4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 13.5 Layered Cuprate and Ruthenate Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 13.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .717 13.5.2 The Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 13.5.3 Structure and Charge Carrier Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 13.5.4 Superconducting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 13.5.5 Normal-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 13.5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 13.6 Comparison of the Properties of Different Classes of Novel Superconductors . . . . . . . . . . . . . . . . . . . 741 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

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13.1

Introduction

The past three decades have witnessed a series of spectacular developments in the field of superconductivity, driven by advances in the synthesis of new and complex materials. This has spawned several classes of novel superconducting materials, including magnetically ordered superconductors, heavy fermion superconductors, organic superconductors, high Tc cuprate superconductors, ruthenate superconductors, as well as several other interesting superconducting materials such as Ba1−x Kx BiO3 , the fcc “buckeyball” compound Rb3 C60 , and MgB2 . The extraordinary physical properties of these materials are reviewed in this chapter. The focus of this chapter is on the unconventional superconductivity displayed by heavy fermion f -electron materials, organic compounds, and copper and ruthenium oxides (Sects. 13.3–13.6). In these novel materials, the superconducting phase often occurs in the vicinity of a magnetic phase in the hyperspace of temperature, chemical composition, pressure and magnetic field. This has been widely regarded as evidence that spin fluctuations play an important role in the pairing of superconducting electrons in these materials. In a few cases, this scenario has received additional support from experiments such as electron tunneling, although it has not been definitively established. The unconventional superconductivity observed in many of these materials is characterized by an anisotropic energy gap that appears to vanish at points or along lines on the Fermi surface. This suggests that the superconducting electrons are paired in states with angular momentum greater than zero (p or d-wave superconductivity). Multiple superconducting phases have been discovered in the heavy fermion compounds UPt3 and U1−x Thx Be13 and attributed to a multicomponent superconducting order parameter coupled to a second order parameter.In the case of UPt3,the second order parameter appears to be antiferromagnetic in nature. Recently, superconductivity has been observed under pressure in several antiferromagnetic Ce-based compounds (e.g., CeIn3 [1], CePd2 Si2 [2]) at low temperatures near the critical pressure where the N´eel temperature extrapolates to 0 K (quantum critical

point). An even more surprising development is the discovery of superconductivity under pressure in a ferromagnetic U-based compound (UGe2 [3]) within the ferromagnetic phase in the vicinity of the critical pressure where the Curie temperature drops abruptly to 0 K. These striking results all point toward an intimate relationship between superconductivity and magnetism in these novel materials. In order to provide a perspective for the role played by magnetism in the unconventional superconductivity of the heavy fermion f -electron, organic, and cuprate and ruthenate compounds described in Sects. 13.3–13.6, the extraordinary phenomena produced by the interplay between superconductivity and magnetism in conventional superconducting systems are briefly described in Sect.13.2. In conventional superconductors, the pairing of superconducting electrons is mediated by phonons, the superconducting electrons are paired in singlet states with zero angular momentum (s-wave superconductivity), and the superconducting energy gap is constant, or nearly so, over the Fermi surface. Conventional s-wave spin-singlet superconductivity is adversely affected by the presence of localized magnetic moments in both paramagnetic and magnetically ordered states. Nevertheless, the interaction between superconductivity and magnetism generates new and unusual physical phenomena that include: (i) the destruction of superconductivity at a second critical temperature Tc2 < Tc1,where Tc1 is the superconducting critical temperature, (referred to as reentrant superconductivity) in a superconducting material containing paramagnetic impurities that exhibits the Kondo effect; (ii) the coexistence of superconductivity and antiferromagnetic order; (iii) the destruction of superconductivity by the onset of ferromagnetic order at a second critical temperature Tc2  c < Tc1, where c is the Curie temperature and Tc1 is the superconducting critical temperature; (iv) the development of a new sinusoidally modulated magnetic state with a wavelength  ∼ 102 Å that coexists with superconductivity in a narrow temperature range above Tc2 due to superconductingferromagnetic interactions; and (v) magnetic field induced superconductivity (known as the Jaccarino– Peter effect [4]).

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The emphasis of this review is on experiment. In view of the enormous scope of the subject,this article is not comprehensive and the choice of examples discussed is very selective. The chapter is divided into the following sections: 1. Introduction 2. Conventional superconductors containing localized magnetic moments 3. f -electron heavy fermion superconductors 4. Organic superconductors 5. Layered cuprate and ruthenate superconductors 6. Comparison of the properties of different classes of novel superconductors It should be noted that the magnetic field is denoted by the symbol H throughout the text, and by both H and B in the figures.

13.2

Conventional Superconductors Containing Localized Magnetic Moments

13.2.1

Introduction

The focus of this article is on the unconventional superconductivity displayed by several classes of novel superconductors, including heavy fermion f electron compounds, organic compounds, and layered cuprates and ruthenates. As noted above, there is a preponderance of evidence for pairing of superconducting electrons with angular momentum L > 0 and a strong perception that magnetic interactions play a dominant role in mediating the electron pairing in these materials. However, in order to put these developments in perspective, we first describe the extraordinary effects that occur when magnetic moments are embedded in s-wave superconductors. We consider two situations: one in which magnetic ions dissolved in a superconducting host have a small concentration and do not exhibit spin glass or long-range magnetic order (paramagnetic impurities), and another in which a large concentration of magnetic ions occupy a sublattice in a superconducting intermetallic compound and exhibit long range magnetic order (magnetically ordered superconductors). Our

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discussion is brief; the interested reader is referred to several more extensive articles on the subject for further reference [5–13]. The interrelation between superconductivity and magnetism has a long history that spans nearly a half century. The first theoretical inquiry into the coexistence of superconductivity and magnetism was made by Ginzburg in 1957 [14] and experimental investigations by Matthias, Suhl, and Corenzwit soon followed in 1958 [15]. The early experiments were carried out on binary and pseudobinary systems in which a rare earth rare earth (R) impurity with a partially-filled 4f electron shell and corresponding magnetic moment was dissolved in a superconducting element or binary compound; e.g., La1−x Rx and Y1−x Rx Os2 . (Note that throughout this Chapter, rare earth R includes Y and the lanthanides La through Lu, upon successive filling of the f -electron shell.) These experiments were very provocative, but largely inconclusive with regard to questions concerning the coexistence of superconductivity and long-range magnetic order, primarily due to complications associated with chemical clustering and/or short-range or “glassy” types of magnetic order. However, the experiments stimulated some imaginative theoretical developments which, although largely inapplicable to the binary and pseudobinary systems then being investigated, anticipated several important phenomena that were later discovered in the 1970s. A significant “spin-off ” of the early experimental and work on the coexistence problem was the achievement of a rather good understanding of superconductivity in the presence of paramagnetic impurities, including the effects of the crystalline electric field, Kondo scattering, localized spin fluctuations, etc. [5,6]. 13.2.2

Superconducting-Magnetic Interactions

In a conventional superconductor, the superconductivity involves electron pairs (Cooper pairs) in which the two electrons have opposite momentum k and spin s - (k ↑, −k ↓). An applied magnetic field H or the magnetic moment  of an ion in a superconductor can interact with the superconducting electrons in two ways: via the Zeeman interaction of H or the exchange field generated by  with the spin s of a

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conduction electron (s·H), and via the electromagnetic interaction of the vector potential associated with H or  with the momentum k of a conduction electron (the k·A term in the one electron Hamiltonian). Both of these interactions raise the energy of one member of a Cooper pair and lower the energy of the other. Such “pair breaking” interactions are very destructive for superconductivity and generally lead to strong depressions of Tc . In a type II superconductor that does not contain ions carrying magnetic moments,superconducting electron pairs can be broken by both the Zeeman interaction of the applied magnetic field H with the spin of a superconducting electron, and the electromagnetic interaction of H with the momentum of a superconducting electron. The Zeeman interaction leads to the paramagnetic limiting field Hp = [N(EF )/( n - s )]1/2 , where N(EF ) is the density of states at the Fermi level EF , and n and s are the normal and superconducting state Pauli susceptibilities, respectively [16, 17], while the electromagnetic interaction yields the orbital critical field  Hc2 = ¥0/2 2, where ¥0 is the flux quantum and  is the superconducting coherence length [18,19]. According to the BCS theory, s (0) = 0, and Hp = Hp0 = 1.84 Tc [Tesla]. However, spin orbit scattering can increase s and, in turn, Hp . How does a superconductor overcome these magnetic pair breaking interactions? One way that has been experimentally established is to set up a modulated spin structure with a period (lattice constant) a < . This condition is satisfied in the known antiferromagnetic superconductors in which   102 Å. In ferromagnetic superconductors, this condition can be satisfied by setting up a new sinusoidally modulated state that screens the exchange interaction at long wavelengths. Another way involves the presence of a negative (antiparallel) exchange field that compensates the applied magnetic field, allowing superconductivity to persist in magnetic fields much higher than the upper critical field. Yet another possibility is to utilize magnetic interactions to produce an unconventional superconducting state involving triplet spin pairing of electrons. These and other situations in which superconductivity and magnetism coexist are surveyed in this Chapter.

13.2.3

Paramagnetic Impurities in Superconductors

The effect of magnetic impurities on superconductivity has received a great deal of attention ever since the discovery that they produce a precipitous drop in the superconducting transition temperature Tc [15,20]. Herring [21] and Suhl and Matthias [22] noted that the conduction electron-impurity spin exchange interaction could account for this strong depression of Tc . Assuming that exchange scattering of conduction electrons by impurity spins may be adequately described within the first Born approximation (to second order in the exchange interaction parameter J ),Abrikosov and Gor’kov (hereafter AG) [23] developed in 1960 what is now a classic theory for superconductors containing paramagnetic impurities. Their theory successfully explained the basic features of these early experiments and further predicted the striking phenomenon of gapless superconductivity. However, studies of local moments in metals in the normal state have shown that the assumptions upon which the AG theory is founded are not applicable to many matrix-impurity systems that nonetheless exhibit a strong depression of Tc . In these systems, the hybridization of the localized d or f -electron states of the solute ion with conduction electron states results in a continuous demagnetization of the solute ion below a characteristic temperature T0. The characteristic temperature T0 can be associated with fluctuations of localized moments with a frequency sf−1 or compensation of the localized moments by the conduction electron spins (Kondo effect). Large depressions of Tc are found even when the solute ions are weakly magnetic at superconducting temperatures (i.e., T0  Tc ). Exchange Interaction For transition metal impurity ions where the orbital angular momentum L is quenched, the exchange interaction Hamiltonian is given by Hex = −2J S · s ,

(13.1)

where J is the exchange interaction parameter, S is the paramagnetic impurity spin, and s is the conduction electron spin density at the impurity site. For R

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impurities, where L is not quenched, S is replaced by its projection on the total angular momentum vector J = L + S of the Hund’s rule ground state. Then the exchange interaction Hamiltonian becomes Hex = −2J (gJ − 1)J · s ,

(13.2)

where gJ is the Land´e g-factor for the Hund’s rule ground state of the R ion. Generally, J is approximated as J ∼ J0 + J1 ,

(13.3)

where J0 (> 0) is the Heisenberg exchange term, while J1 (< 0) arises from covalent mixing between the localized d or f -electron states and conduction electron states. In terms of the Schrieffer–Wolff transformation [24], J1 is given by J1 ∼

2 Vk U < 0,  (  + U )

(13.4)

when |  |  . Here, Vk is a matrix element mixing localized states with itinerant states,  = 2 Vk N(EF ) is the Hartree–Fock (HF) half-width of the resonant state,  = E − EF , E is the energy of the centroid of the localized d or f -state, EF is the Fermi energy, and U is the intra-atomic Coulomb repulsion which splits spin-up and spin-down states in the Friedel–Anderson model [25,26]. In the limit that U  |  |, this expression takes the form J ∼

2 Vk  < 0. 

(13.5)

For partially-filled f -electron shells of rare earth ions with stable valence such as Pr, Nd, Gd, Tb, Dy, Ho, and Er, the hybridization between the 4f -electron and conduction electron states is weak and J is dominated by J0 (ferromagnetic exchange). In contrast, for partially-filled d or f -electron shells of transition metal ions, rare earth ions with unstable f -valence such as Ce, occasionally Pr, Sm, Eu, Tm, and Yb, and actinide ions, the hybridization is appreciable and J is dominated by J1 (antiferromagnetic exchange). The superconducting properties of matrixparamagnetic impurity systems for the cases of ferromagnetic and antiferromagnetic exchange are dramatically different and are described below.

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The exchange interaction also produces spin glass or long-range magnetic ordering in metals containing localized magnetic moments via the Ruderman– Kittel–Kasuya–Yosida (RKKY) mechanism. This is the primary source of long range magnetic ordering in ternary rare earth compounds, discussed later in this section, although dipolar contributions may be important in certain cases. The polarized rare earth magnetic moments can then interact with the conduction electrons in two ways: by means of the Zeeman interaction of the exchange field and the conduction electron spins, and via the electromagnetic interaction of the magnetization and the persistent current. Ferromagnetic Exchange Interaction The effect of paramagnetic impurities on superconductivity for ferromagnetic exchange is best illustrated for Gd impurity ions in situations where the interactions between Gd impurities are sufficiently weak that they may be neglected over an appreciable temperature range below Tc0, the superconducting critical temperature of the superconducting matrix into which they have been dissolved. Since Gd is an S-state ion, this also avoids complications associated with the lifting of the degeneracy of the Hund’s rule ground state by the crystalline electric field (CEF). Experimentally, it has been found that the curves of Tc /Tc0 vs n/ncr (Tc is the superconducting transition temperature of the alloy with paramagnetic impurity concentration n, and ncr is the critical concentration above which superconductivity does not occur at any temperature) and C/C0 vs Tc /Tc0 (C and C0 are the specific heat jumps at Tc and Tc0, respectively) behave in characteristic manners, both of which can be represented well by universal functions that have been calculated on the basis of the theory of AG [6]. Shown in Fig. 13.1 is a plot of Tc /Tc0 vs n/ncr for the La1−x Gdx Al2 system; Tc0 = 3.3 K is the Tc of the LaAl2 host compound (x = 0) and ncr = 0.59 at. % is the critical concentration where Tc vanishes [27]. The solid line, which has been fitted to the Tc /Tc0 vs n/ncr data, is the AG theoretical curve.Also shown in Fig.13.1 are the Curie–Weiss temperatures p ,plotted as p /Tc0 vs n/ncr, for the La1−x Gdx Al2 system where

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Fig. 13.1. Reduced superconducting critical temperature Tc /Tc0 vs reduced Gd impurity concentration n/ncr for the La1−x Gdx Al2 system. The value of the superconducting critical temperature Tc0 of the LaAl2 host compound is 3.3 K, while the value of the critical concentration ncr where Tc = 0 is 0.59 at. % Gd. The line denoted AG is the theoretical curve of Abrikosov and Gor’kov [23], which has been fitted to the data. Inset: Curie–Weiss temperatures p determined from magnetic susceptibility measurements, plotted as p /Tc0 vs n/ncr. The line through the data is a guide to the eye, after [5,27]

the solid line is a guide to the eye [5,27]. Apparently, interactions between the Gd magnetic moments are sufficiently weak in this system to allow the AG theory to be tested to concentrations in the vicinity of ncr . Shown in Fig. 13.2 is the behavior of C/C0 vs Tc /Tc0 for the La1−x Gdx Al2 system [28] and the Th1−x Gdx system [29], compared to calculations by Skalski et al. [30] based on the AG theory. As can be seen in Figs. 13.1 and 13.2, the experimental Tc /Tc0 vs n/ncr and C/C0 vs Tc /Tc0 curves are well represented by the AG theory. The basic assumptions embodied in the AG theory are that (1) the superconducting order parameter does not vary with position, (2) the spins of the impurities are fixed and randomly oriented in space, and (3) scattering of conduction electrons by the paramagnetic impurities can be calculated within the first Born approximation. The first assumption requires that the impurities be randomly distributed throughout the superconducting matrix, while the second precludes any correlations between the spins of the impurities such as those that would arise from magnetic order. If one regards the impurity spin direction as fixed, the exchange interaction is not time reversal invariant. As a result, the lifetime s of the time-reversed single particle paired states of which the superconducting wave function is comprised is no longer infinite. In the AG theory, the superconducting properties in the presence of solute spins are characterized

Fig. 13.2. Reduced specific heat jump C/C0 vs reduced critical temperature Tc /Tc0 for the matrix-impurity systems La1−x Gdx Al2 and Th1−x Gdx . The La1−x Gdx Al2 data were derived from C(T) measurements by Luengo and Maple [28] and the Th1−x Gdx data were derived from Hc2 (T) measurements by Decker and Finnemore [29]. The dashed line represents the BCS law of corresponding states, whereas the solid line is the result of the AG theory as calculated by Skalski et al. [30], after [6]

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by a pair breaking parameter ˛ = s−1 . The theory predicts a second order transition to the superconducting state and a rapid decrease of the superconducting transition temperature with ˛ given by the universal relation       Tc ˛Tc0 1 1 ln − + 0.14 = , (13.6) Tc0 2 2 ˛cr Tc where Tc0 corresponds to ˛ = 0, ˛cr = kB Tc0/4 (ln is Euler’s constant) corresponds to Tc = 0 (complete destruction of superconductivity) and is the digamma function. This expression has the linear asymptotic form   Tc ˛ , (13.7) = 1 − 0.691 Tc0 ˛cr as ˛ → 0. It is apparently sufficient to calculate the pair breaking parameter ˛ within the first Born approximation (to second order in J ), which for superconductors containing paramagnetic R ions gives the result   1 n N(EF ) ˛≡ = J 2(gJ − 1)J(J + 1) , (13.8) s  2kB where n is the paramagnetic impurity concentration and N(EF ) is the density of states at the Fermi level for the matrix. It is important to note that ˛ is proportional to n and independent of temperature. Hence, ˛/˛cr can be replaced by n/ncr in Eqs. 13.6 and 13.7 where ncr is the critical concentration for the complete suppression of superconductivity. Thus, the theory predicts that the reduced critical temperature Tc /Tc0 will be a universal function of reduced concentration n/ncr       nTc0 1 1 Tc − + 0.14 = , (13.9) ln Tc0 2 2 ncr Tc or

Tc =U Tc0



n ncr

 ,

(13.10)

The specific heat jump C at Tc as a function of Tc has also been calculated on the basis of the AG theory by Skalski et al. [30]. They find that the reduced

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specific heat jump C/C0 is a universal function V of Tc /Tc0,   C Tc , (13.11) =V C0 Tc0 which deviates markedly from the linear BCS law of corresponding states C Tc = . C0 Tc0

(13.12)

One of the most striking predictions of the AG theory is the phenomenon of gapless superconductivity. This results from the fact that the finite lifetime s corresponds to an energy broadening  ∼ /s which introduces states into the gap and spreads out the BCS peak in the density of states. As a consequence, the energy gap §G (˛) no longer corresponds to the order parameter (˛) and with increasing ˛ (or n), §G (˛) goes to zero faster that Tc (˛). For all concentrations, the superconductor is gapless at temperatures sufficiently near Tc ; while for n > 0.91 ncr , the superconductor is gapless at all temperatures.For example,the theory predicts an attenuated specific heat jump at Tc (compared to the BCS law of corresponding states; see Fig. 13.2), and a linearly temperaturedependent term in linearly temperature-dependent term in the specific heat below 0.91 ncr . Other predictions of the AG theory have been verified experimentally as described in other reviews [5,6] (Note, the recent study by Kim and Overhauser [31].) The AG theory yields an initial depression of Tc with n that is linear with a rate, for superconductors containing R impurities, given by     2 dTc  N(EF ) J 2 (gJ − 1)2J(J + 1) , =− dn n=0 2kB (13.13) where the quantity (gJ − 1)2 J(J + 1) = D(R) is called the deGennes factor [33]. The variation of (−dTc /dn)n=0 with R for the La1−x Rx system scales with the function D(R) if J 2 is assumed to decrease slightly with increasing R atomic number, except for Ce where (−dTc /dn)n=0 is anomalously large [15]. Investigations of the effect of substituted R ions on superconductivity of LaAl2 , which has a Tc of 3.3 K, revealed a variation of (−dTc /dn)n=0 with R similar to that of La1−x Rx ,with a corresponding anomalously

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from low temperature nonmagnetic behavior, where (T) approaches a constant value. A physical picture of the nonmagnetic ground state that has emerged from theories of the Kondo effect involves the gradual formation, as T decreases through TK , of a many body singlet ground state in which the spin of each paramagnetic impurity ion is screened by the antiferromagnetically aligned spins of the conduction electrons. The Kondo temperature is given by   −1 TK ∼ TF exp , (13.14) N(EF )|J |

Fig. 13.3. Initial rate of depression of the superconducting critical temperature Tc with paramagnetic impurity concentration n, (−dTc /dn)n=0 , vs rare earth R impurity for the La1−x Rx and La1−x RxAl2 systems, after [5,32]

large rate of depression of Tc with n for R = Ce [32]. The (−dTc /dn)n=0 vs R data for both La1−x Rx and La1−x Rx Al2, normalized to the value for Gd, are displayed in Fig. 13.3 [5,32]. It should be noted that, according to the AG theory, the depression of Tc depends on the magnitude of J but not on its sign. However, neglected in the AG theory are contributions to ˛ of higher order than J 2 which must be taken into account when J is negative. These contributionsgive rise to spectacular temperature dependent pair breaking effects which are considered below. Antiferromagnetic Exchange Interaction The anomalous depression of Tc with n for Ce substituents in La and LaAl2 is associated with the hybridization of the localized 4f -electron states of Ce and the conduction electron states [5]. The large depression of Tc with Ce concentration is due to the large negative contribution to the exchange interaction and the occurrence of the Kondo effect with TK ∼ Tc0, where TK is the Kondo temperature and Tc0 is the Tc of the La or LaAl2 matrix (see below). The Kondo temperature is a characteristic temperature that separates high temperature magnetic behavior, where (T) behaves as a Curie–Weiss law,

where TF is the Fermi temperature. The phenomenon of reentrant superconductivity due to the Kondo effect was discovered in the La1−x Cex Al2 system. Here, a sample within a certain range of Ce impurity concentrations becomes superconducting below a critical temperature Tc1 and then loses its superconductivity below a second critical temperature Tc2 [34, 35].The destruction of superconductivity at Tc2 is associated with the competition between singlet spin pairing of electrons in the superconducting state, with characteristic energy kB Tc , and the formation of the Kondo many body singlet state involving the conduction electrons and each Ce impurity ion, with characteristic energy kB TK . Shown in Figs. 13.4(a) and (b), respectively, are ac magnetic susceptibility vs temperature data and the reentrant Tc vs n curve of the La1−x Cex Al2 system. The values of Tc in Fig. 13.4(b) and the transition curves in Fig. 13.4(a) from which they were inferred are identified with letters [34]. In addition to the remarkable behavior exhibited by the Tc /Tc0 vs n curve, the C/C0 vs Tc /Tc0 curve also displays some interesting features which are shown in Fig. 13.5. Here, it can be seen that the curve of C/C0 vs Tc /Tc0 shows a pronounced downward deviation from both the BCS law of corresponding states and, as well, the AG theory. The two specific heat jumps for a specimen at 0.64 at.% Ce are indicated in Fig. 13.5 by the solid squares. Electrical resistivity measurements on the La1−x Cex Al2 system in the normal state revealed the occurrence of a Kondo effect, as evidenced by a minimum in (T) at low temperatures [39]. The Ce impurity contribution to (T) was found to diverge as

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Fig. 13.4. (a) Normalized transition signal, based on ac magnetic susceptibility measurements, vs temperature for the La1−x Cex Al2 system. (b) Reduced superconducting critical temperature Tc /Tc0 vs Ce impurity concentration n for the La1−x Cex Al2 system. The value of the superconducting critical temperature Tc0 of the LaAl2 host compound is 3.3 K. The data identified by letters are derived from the corresponding transition curves in (a), after [34]

− ln T with decreasing T down to ∼ 1 K, indicating a low Kondo temperature TK  1 K. Shown in Fig. 13.6 are specific heat C vs T data for a La1−x Cex Al2 alloy (x = 0.64 at.% Ce) in various magnetic fields up to 38 kOe [38]. Curves a − c, which have been drawn to fit more accurate data for a La1−x Cex Al2 alloy with x = 0.906 at.% Ce, correspond to an entropy of Rln2 per mol Ce, showing that the Ce3+ ground state is a doublet. Curve d is consistent with calculations of Bloomfield and Hamann [40] for S = 1/2 and TK = 0.42 K. The two zero field superconducting critical temperatures Tc1 and Tc2 are indicated by arrows in the figure.The Kondo anomalies in the nor-

mal state properties of the La1−x Cex Al2 system have also been well characterized and used to test theoretical models of the Kondo effect. Subsequently, reentrant superconductivity was discovered in the (La1−y Thy )1−x Cex system in the low Th concentration range 0 < y ≤ 0.25 [41]. Superconductivity was studied in superconducting matrix-impurity systems in which the impurity ions were nonmagnetic at superconducting temperatures (i.e., Tc0  T0, where T0 is a characteristic temperature that separates low temperature nonmagnetic behavior from high temperature magnetic behavior, such as the Kondo temperature TK ). These

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Fig. 13.5. Reduced specific heat jump C/C0 vs reduced critical temperature Tc /Tc0 for the La1−x Cex Al2 system. The solid triangles, circles and squares represent data from Luengo et al. [36], Armbr¨uster et al. [37], and Bader et al. [38], respectively. The dashed line represents the BCS law of corresponding states, the dot-dashed line indicates the AG result, and the solid line is a smooth curve drawn through the data, after [6]

studies revealed that the Tc vs n curves have positive curvature and can be described by a relation proposed by Kaiser [42], who considered the effect of nonmagnetic resonant d and f -electron states of impurity ions on superconductivity. In this case Tc is a modified exponential relation of the form   An Tc = Tc0 exp − , (13.15) (1 − Dn) where A and D are fitting parameters that can be related to parameters describing the conduction electrons (N(EF ) and V , where V is the electron–phonon interaction parameter) and the localized d or f electrons (N (EF ), U , and L, where N (EF ) is the density of states at EF of the localized d or f -electrons, U is the intra-atomic Coulomb interaction parameter, and L is the d or f -electron orbital angular momentum). Shown in Fig. 13.7 are plots of Tc /Tc0 vs n/n0 for three systems [6] in which T0  Tc0, Th1−x Ux [43], Th1−x Cex [44], and Al1−x Mnx [45]. Displayed in Fig. 13.8 are reduced specific heat jump

C/C0 vs reduced critical temperature Tc /Tc0 data for the Th1−x Ux [46–48], Al1−x Mnx [49, 50], and Th1−x Cex [51] systems. The nonmagnetic character of the f -electron states in these systems at superconducting temperatures where T  T0 is reflected in the conformance of the C/C0 vs Tc /Tc0 data with the BCS law of corresponding states. The Th1−x Ux system [43] provides an interesting example of the Kondo effect in the electrical resistivity. Shown in Fig. 13.9(a) are  vs T data for Th1−x Ux samples with x = 0, 0.01, 0.015, and 0.02, in which the resistivity minimum phenomenon is evident. Displayed in Fig. 13.9(b) are  vs T 2 data, where (x, T) = (x, T) − (0, T) is the U contribution to the resistivity, that can be described by the relation (x, T) = (x, 0)[1 − (T/TK )2 ] with TK ∼ 100 K, consistent with the local Fermi-liquid behavior expected for the Kondo effect at low temperatures T  TK [43]. Reentrant superconductivity associated with the Kondo effect was predicted to occur in the limit TK  Tc0, where Tc0 is the critical temperature of the superconducting host metal, by M¨uller–Hartmann and Zittartz (MHZ) [52] and Ludwig and Zuckermann (LZ) [53]. MHZ and LZ obtained exponentiallike curves of Tc vs n in the limit TK  Tc0. Calculations of Zuckermann [54] and MHZ [55] for the initial depression of Tc with n, (−dTc /dn)n=0 , yielded a maximum as a function of TK /Tc0 at TK /Tc0 ≈ 10. The magnetic state of an impurity in a superconducting matrix-impurity system can be continuously tuned by applying an external pressure P or alloying the matrix with a suitable element. An example of the first case is illustrated in Fig. 13.10, which shows Tc vs pressure for the system La1−x Cex with several concentrations x of Ce [56]. With increasing Ce concentration,a minimum in Tc as a function of pressure P develops near 15 kbar which becomes more pronounced the higher the Ce concentration. The data for the La1−x Cex sample with x = 0.02 are particularly striking; the application of pressure drives Tc to zero at about 5 kbar, Tc remains zero up to ∼ 15 kbar,whereupon superconductivity reappears and Tc increases with pressure. The initial depression of Tc , Tc /n, increases markedly with pressure up to a maximum near 15 kbar and thereafter decreases to a

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Fig. 13.6. Specific heat of a La1−x Cex Al2 alloy (0.64 at.% Ce) vs temperature in various magnetic fields up to 38 kOe. Curves a–c, which have been drawn to fit more accurate data for a La1−x Cex Al2 alloy with 0.906 at.% Ce, correspond to an entropy of Rln2 per mole Ce, showing that the Ce3+ ground state is a doublet.Curve d is consistent with calculations of Bloomfield and Hamann [40] for S = 1/2 and TK = 0.42 K. The two zero field superconducting transitions are indicated, after [38]

Fig. 13.7. Reduced superconducting critical temperature Tc /Tc0 vs reduced impurity concentration n/n0 for the Th1−x Cex , Th1−x Ux , and Al1−x Mnx systems. The solid lines represent the modified exponential relation given in the figure that describes the weakening of superconducting electron pairs by nonmagnetic localized states [42]. The dot-dashed line is the AG curve, while the dashed line is a straight line to which the other curves have been scaled in the initial linear region at low concentrations. The values of the characteristic (or Kondo) temperature T0 , superconducting critical temperature of the host Tc0 , and the scaling concentration n0 are indicated in the figure, after [6]

value that, above ∼ 100 kbar, is more than an order of magnitude smaller than the maximum depression. Isobars of Tc vs n in the inset of Fig. 13.10 evolve with increasing pressure from nearly linear depressions with slight negative curvature to depressions

with strong positive curvature. These results can be interpreted in terms of an increase of TK ,and,in turn, the ratio TK /Tc0, with pressure. As TK /Tc0 increases under pressure, the initial depression of Tc with x is expected to increase, pass through a maximum, and

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Fig. 13.8. Reduced specific heat jump C/C0 vs reduced critical temperature Tc /Tc0 for the Th1−x Cex , Th1−x Ux , and Al1−x Mnx systems. The solid line represents the BCS law of corresponding states,whereas the dot-dashed line indicates the AG result. Open circles: data from Luengo et al. [46,47]. Solid circles: data from Watson et al. [48]. Open square: data from Martin [49]. Solid squares: data from Smith [50]. Solid triangles: data from Dempesy [51], after [6]

Fig. 13.10. Superconducting critical temperature Tc vs pressure for the La1−x Cex system. Isobars of Tc vs Ce concentration are shown in the inset, after [56]

Fig. 13.9. (a) Electrical resistivity  vs temperature T of Th1−x Ux alloys with x = 0.5, 1, and 2 at. % U at low temperatures. (b) Incremental resisitivity (x, T) = (x, T) − (0, T) vs T 2 of Th1−x Ux alloys with x = 0.5, 1, and 2 at. % U, after [43]

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then decrease, according to the theory of MHZ [55]. However,at the highest pressures,such large values of TK are required (∼ 106 K!) that it was concluded that the Ce 4f -state must demagnetize within the context of the Friedel–Anderson model. An example of the second case is provided by the La1−x Cex system when Th is substituted for La. With increasing Th concentration y in the (La1−y Thy )1−x Cex system, the rate of the initial depression of Tc with Ce concentration passes through a pronounced maximum and Tc vs n changes continuously from curves with negative curvature and reentrant behavior to curves with positive curvature and exponential-like shapes [41]. Concomitantly, the normal state magnetic susceptibility [57] and electrical resistivity [58] evolve continuously from magnetic to nonmagnetic behavior.A self consistent analysis of the rates of the initial depression of Tc with n and the value of the specific heat jump at Tc demonstrated that the MHZ theory provides a good description of the data from TK /Tc0 ≈ 0.1 to 100 [59,60]. Magnetically-Ordered Superconductors The series of isostructural ternary rare earth compounds that have been investigated the most extensively in connection with the interaction between superconductivity and long-range magnetic order include the rhombohedral rare earth molybdenum chalcogenides RMo6S8 and RMo6Se8 (Chevrel phases), and the tetragonal rare earth rhodium borides RRh4 B4 [7,8].The long-range magnetic order that many of these compounds exhibit can be traced in part to the ordered R sublattice. The persistence of superconductivity, even in the presence of relatively large concentrations of R ions, can be attributed to the relatively weak exchange interaction between the conduction electron spins and the R magnetic moments. This, in turn, appears to be associated with transition metal molecular units or “clusters” which, along with the R ions, are the basic building blocks of these ternary R phases. The superconductivity is believed to be primarily associated with the transition metal d-electrons that are relatively confined within the clusters and thereby interact only weakly with the R ions. Equation 13.13 has been used to esti-

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mate the magnitude of J for the RRh4 B4 compounds from the depression of Tc of LuRh4 B4 by R impurities with partially-filled 4f electron shells [61]. The initial depression of Tc of LuRh4 B4 by Gd impurities, (dTc /dn)n=0 = −19 K per atomic fraction of Gd in Lu, yields the value |J | = 2.3 × 10−2 eV-atom, assuming N(EF ) = 0.35 states/eV-atom-spin direction [61]. Other ternary R compounds have been investigated such as RRh1.1 Sn3.5 , R2 Fe3 Si5, RRuB2 , etc. [7,8]. Antiferromagnetic Superconductors The coexistence of superconductivity and long-range antiferromagnetic order was discovered in the latter part of the 1970s in the R molybdenum selenides RMo6 Se8 (R = Gd, Tb, and Er) [8,64] and R rhodium borides RRh4 B4 (R = Nd, Sm and Tm) [8, 65] at the University of California, San Diego, and in the R molybdenum sulfides RMo6 S8 (R = Gd, Tb, Dy and Er) [8,62] at the University of Geneva.The occurrence of antiferromagnetic ordering of the R magnetic moments in the superconducting state was inferred from a -type anomaly in the heat capacity and a cusp in the magnetic susceptibility for the RMo6 Se8 compounds [8,64],and from a feature in the upper critical field Hc2 vs temperature T curve in the RMo6S8 compounds [8,62].Neutron diffraction measurements on GdMo6 Se8 [66] and RMo6S8 compounds with R = Gd, Tb, and Dy [67] confirmed the antiferromagnetic ordering of the sublattice of R ions in the Chevrel phase structure. The magnetic structure consists of alternating ferromagnetic [100] planes in the nearly cubic R sublattice in which the magnetic moments are parallel or antiparallel to the rhombohedral [111] axis. Plots of the sublattice magnetization vs temperature for RMo6 S8 compounds with R = Gd, Tb, and Dy are shown in Fig. 13.11 where they are compared with data for Hc2 vs temperature displayed in the inset of the figure [63]. There is a clear correlation between the onset of the sublattice magnetization and the depression of the Hc2 curve below TN . The anomalous depression of Hc2 in the vicinity of TN and other properties of antiferromagnetic superconductors have been addressed by numerous theories [68].Several mechanisms by means of which superconductivity is modified by antiferromagnetic

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Fig. 13.11. Temperature dependence of the sublattice magnetization for RMo6 S8 compounds with R = Gd, Tb, and Dy. The inset shows the measured upper critical field Hc2 (T) [62], after [63]

order have been considered. These include: (i) Reduction in pair breaking due to the decrease in the mean magnetization and, in turn, the conduction electron spin polarization below TN . (ii) Increase in pair breaking due to magnetic moment fluctuations

in the vicinity of TN . (iii) Decrease of the attractive phonon mediated electron–electron pairing interaction by antiferromagnetic magnons. (iv) Reduction of the available phase space for virtual pair scattering by the change in lattice periodicity associated with antiferromagnetic order.(v) Pairing of electrons with finite momentum. Finite momentum pairing of electrons in states (k, ↑; k + Q, ↓), where Q corresponds to translation by a reciprocal lattice vector, was originally proposed by Baltensperger and Strassler [69] in 1963, and has been incorporated into several subsequent theories [68]. It should be emphasized that there are some antiferromagnetic superconductors where Hc2 actually increases below TN , or, in other words, superconductivity seems to be enhanced below TN . A notable example is the compound SmRh4 B4 whose resistively determined Hc2 vs T curve, shown in Fig. 13.12, exhibits a sharp break in slope at TN = 0.87 K [9]. In contrast, the Hc2 vs T curves of the other two RRh4 B4 antiferromagnetic superconductors shown in Fig.13.12 display different behaviors with decreasing temperature; for NdRh4 B4 , which undergoes two antiferromagnetic transitions at TN1 = 1.31 K and TN2 = 0.89 K, Hc2 decreases abruptly at TN1 and then increases sharply at TN2, whereas for TmRh4 B4 , Hc2 hardly changes at TN = 0.4 K [9]. Neutron diffraction experiments indicate that the magnetic phases of NdRh4 B4 in zero magnetic field are body-centered antiferromagnetic structures in which the Nd3+ mo-

Fig. 13.12. Resistively determined upper critical field Hc2 vs temperature T for polycrystalline samples of NdRh4 B4 , SmRh4 B4 , ErRh4 B4 , and TmRh4 B4 , after [9]

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Fig. 13.13. Temperature dependence of the neutron scattering intensity of a representative satellite from each of the two magnetic phases of NdRh4 B4 , after [70]

ments are alternately aligned parallel and antiparallel to the c-axis, with a sinusoidal modulation along the [100] direction with  = 46.5 Å in the high temperature phase, and along the [110] direction with  = 45.2 Å in the low temperature magnetic phase [70]. The temperature dependences of the neutron diffraction intensity of a representative satellite from each of the two magnetic phases of NdRh4 B4 are shown in Fig. 13.13. The data in Fig. 13.12 indicate that there is no universal behavior of Hc2 vs T for antiferromagnetic superconductors. Both enhancements and depressions of Hc2 are found below TN , which appear to be determined by a combination of the mechanisms enumerated above. Recently,superconductivity was discovered in rare earth transition metal borocarbide quaternary systems [71–73]. For intermetallic compounds, these materials have high values of Tc ; Tc ≈ 13.5 K for YNi2 B3 C0.2 [71], Tc ≈ 16.5 K for RNi2 B2 C (R = Y, Tm, Er, Ho, La) [72], and Tc ≈ 23 K for YPd5 B3 C0.3 [73].

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A value Tc ≈ 21.5 K was reported for a Th based palladium borocarbide in the Th-Pd-B-C system which also has a relatively high extrapolated upper critical field Hc2(0) > 17 tesla [74]. Coexistence of superconductivity and antiferromagnetism has been reported in RNi2 B2 C for R = Tm, Er, Ho, Dy [75]. Single crystals of RNi2 B2 C compounds have been prepared and investigated extensively by means of a variety of experimental techniques [76–79]. For example, Grigereit and coworkers [78] have performed neutron scattering measurements on the reentrant antiferromagnetic superconductor HoNi2 B2 C which becomes superconducting at ∼ 7.5 K, re-enters the normal conducting state at 5 K, and quickly recovers superconductivity at lower temperature. The experiments reveal that the magnetic order that first forms upon cooling is oscillatory in nature and is directly coupled to the superconducting order parameter. However, in contrast to the ferromagnetic superconductors (discussed below), the oscillatory state in HoNi2B2 C is detrimental to superconductivity, and the superconducting state only survives at low temperatures because of a first order transition to a compensated antiferromagnet. A brief review of the phenomena that arise from the interplay between superconductivity and magnetism in the RNi2 B2 C compounds can be found in [79]. Ferromagnetic Superconductors Reentrant superconductivity due to the onset of longrange ferromagnetic ordering of the R magnetic moments was discovered in 1977 in ErRh4 B4 [80] at the University of California, San Diego, and in HoMo6S8 [81] at the University of Geneva. These two materials, which become superconducting at an upper critical temperature Tc1, lose their superconductivity at a lower critical temperature Tc2 ≈ TM , where TM is the Curie temperature. Thermal hysteresis in various physical properties and a spike-shaped feature in the heat capacity near Tc2 indicate that a first-order transition from the superconducting to the ferromagnetic normal state occurs at Tc2 [9]. Typical ac magnetic susceptibility and electrical resistance vs temperature data for ErRh4 B4 are displayed in Fig. 13.14 [82]. The thermal hysteresis at

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Fig. 13.14. Typical ac magnetic susceptibility ac and electrical resistance vs temperature data for ErRh4 B4 , after [82]

Tc2 is evident in both properties. The resistively determined Hc2 vs T curve for ErRh4 B4 is shown in Fig. 13.12 [9]. Neutron diffraction measurements established that the ground states of the compounds ErRh4 B4 [83, 84] and HoMo6 S8 [85] are ferromagnetic.In addition, small angle scattering studies of ErRh4 B4 [84,86] and HoMo6S8 [87] revealed the existence of a sinusoidally modulated magnetic state with a wavelength of the order of 100 Å that coexists with superconductivity in a narrow temperature interval above Tc2. Moreover, in ErRh4 B4 the regions within which superconductivity and the sinusoidally modulated magnetic state coexist appear to be interspersed with normal ferromagnetic domains to form a spatially inhomogeneous state. Shown in Fig. 13.15 is a plot of the heat capacity of ErRh4 B4 and the isostructural nonmagnetic compound LuRh4 B4 below 18 K [88]. The data reveal a jump in the heat capacity at Tc1 = 8.7 K on a broad background with negative curvature that is a Schottky anomaly arising from the partial lifting by the CEF of the 16-fold degeneracy of the Er3+ J = 15/2 Hund’s rule multiplet. At lower temperatures (see inset), there is a spike-shaped feature at Tc2 = 0.93 K (measured upon warming) superimposed on another anomaly that is apparently associated with the long-range ferromagnetic ordering of the Er3+ magnetic moments in the vicinity of Tc2 [67]. There is

Fig. 13.15. Specific heat C vs temperature T for ErRh4 B4 and LuRh4 B4 . The inset shows a detailed plot of C vs T for ErRh4 B4 in the vicinity of the reentrant superconducting transition at Tc2 , after [88]

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Fig. 13.16. Small angle neutron scattering results on ErRh4 B4 obtained at various temperatures. The peak at 2 = 1.4◦ indicates an oscillatory magnetization with a wavelength of 100 Å, after [84]

also a shoulder in the heat capacity above Tc2 whose origin may be attributable to the formation of a sinusoidally modulated magnetic state that coexists with superconductivity that is discussed below. Measurements of the normalized neutron scattering intensity of the [101] reflection for ErRh4 B4 [84] indicate that the ferromagnetic transition is broad, extending up to ∼ 1.4 K, well above the temperature of the reentrant superconducting transition at Tc2 ∼ 0.9 K. Also, there is definite thermal hysteresis between ∼ 0.8 K and ∼ 1.4 K. The width of the ferromagnetic transition has been attributed to a distribution of effective Curie temperatures within the material, while the hysteresis may be caused by the nucleation of normal-ferromagnetic domains within the temperature intervall between Tc2 and ∼ 1.4 K. The heat capacity data also reveal thermal hysteresis, part of which appears to be associated with the formation of ferromagnetic domains (between ∼ 0.9 K and ∼ 1.3 K), and part of which is due to the reentrant transition at Tc2. Neutron intensity vs scattering angle data at various temperatures on a polycrystalline ErRh4 B4 sample, whose reentrant superconducting transition occurs at ∼ 0.7 K, revealed the development of a peak near 1◦ which grows in intensity as the temperature is decreased and then disappears abruptly when the sample becomes normal below Tc2 (see Fig. 13.16). The data were interpreted in terms of fluctuations

into a state in which the magnetization is sinusoidally modulated with a wavelength  ∼ 100 Å that was assumed to take the form of a spiral. Subsequently, neutron scattering experiments on an ErRh4 B4 single crystal were performed by Sinha et al. [86]. Their findings are consistent with the neutron scattering experiments on polycrystalline ErRh4 B4 by Moncton et al. [84], but revealed that the sinusoidallymodulated magnetic state is a transverse linearly polarized long-range magnetic state with a wavelength of ∼ 100 Å. The linearly polarized sinusoidal modulation lies along the [010] axis and the propagation directions are at 45◦ to the [001] and the [100] axes. The neutron scattering measurements on a single crystal of ErRh4 B4 are summarized in Fig. 13.17. Neutron diffraction experiments on a HoMo6S8 single crystal by Rossat–Mignod et al. [89] revealed that the wavelength of the sinusoidally modulated phase is twice as large as observed in powdered specimens and is transverse, with the moments oriented along the [111] direction while the wave vector is along the [110] direction.The large value of the wavelength ( ∼ 570 Å at 0.1 K) and the existence of a ferromagnetic phase in microdomains of the order of 1000 Å to 1500 Å were taken as evidence that the modulated phase is actually induced by the electromagnetic interaction between the persistent current and the magnetic moments.

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on the exchange interaction, more recent theories such as those of Blount and Varma [91], Ferrell et al. [92], and Matsumoto et al. [93] are based on the electromagnetic interaction. Although, as discussed earlier, the exchange interaction is operative in these materials, the electromagnetic interaction appears to be primarily responsible for the sinusoidally modulated magnetic state that coexists with superconductivity. Other possibilities for the periodic magnetic structure above Tc2 that have been considered are (i) a spontaneous vortex lattice, (ii) a laminar structure, stabilized by the R magnetization in a selfconsistent manner, and (iii) combined spiral magnetic and spontaneous vortex states [68]. Many other techniques have been applied to investigate the physical properties of HoMo6S8 and ErRh4 B4 , and the reader is referred to several reviews on this subject that can be found in [8]. Superconductivity and Competing Magnetic Interactions

Fig. 13.17. Temperature dependence of the ferromagnetic intensity from the (101) Bragg peak, the satellite intensity, the dc resistance of ErRh4 B4 and the ratio of the satellite to the ferromagnetic intensity for the (101) reciprocal lattice point, after [86]

The sinusoidally modulated magnetic state that coexists with superconductivity in HoMo6S8 and ErRh4 B4 is reminiscent of the cryptoferromagnetic state proposed by Anderson and Suhl [90] in 1959 and has been the subject of much theoretical interest during the late 1970s and early 1980s. Whereas the original theory of Anderson and Suhl is based

Experiments on pseudoternary R compounds provide an alternative method for studying the interaction between superconductivity and long-range magnetic order, as well as for exploring the effects of competing types of magnetic moment anisotropy and/or magnetic order. Two types of RRh4 B4 pseudoternaries have been formed, one in which a second R element is substituted at the R sites, and another in which a different transition element is substituted at the Rh sites. An example of the first type of pseudoternary RRh4 B4 system is (Er1−x Hox )Rh4 B4 whose low temperature phase diagram, delineating the paramagnetic, superconducting, and magnetically ordered phases, is shown in Fig. 13.18. The phase boundaries have been determined from ac magnetic susceptibility [94,95] and neutron diffraction measurements [96]. The phase diagram displays regions in which the Er3+ and Ho3+ magnetic moments independently order ferromagnetically within the basal plane and along the tetragonal c-axis, respectively, separated by a region of mixed magnetic phases. The temperature interval above Tc2 within which the sinusoidally modulated magnetic phase in ErRh4 B4

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Fig. 13.18. Low temperature phase diagram for the (Er1−x Hox )Rh4 B4 pseudoternary system, after [94–96]

Fig. 13.19. Low temperature phase diagram for the Ho(Rh1−x Irx )4 B4 pseudoternary system, after [94,98]

coexists with normal ferromagnetic domains is also indicated in Fig. 13.18. This inhomogeneous phase presumably persists within a certain region in the T − x plane (shaded area in the figure). There is a tricritical point at the concentration xc = 0.89 at which Tc1,Tc2 and TM become coincident.The Tc2 vs x phase boundary for x < xc is depressed relative to a linear extrapolation of TM vs x for x > xc (dashed curve in Fig. 13.18). Analysis of neutron diffraction data on an (Er0.4 Ho0.6)Rh4 B4 sample [97] indicates that the actual TM of 3.67 K is about 0.2 K less than it would have been in the absence of superconductivity, in accord with the dashed-line extrapolation as well as theoretical predictions. A striking example of the second type of pseudoternary RRh4 B4 system is Ho(Rh1−x Irx )4 B4 . This system was first investigated by Ku et al. [99] and provided evidence for the coexistence of superconductivity and antiferromagnetic order with TN > Tc for x > 0.6. Subsequently, several detailed investigations of a Ho(Rh0.3 Ir0.7)4 B4 compound were carried out; heat capacity measurements [88] confirmed the bulk character of the antiferromagnetic order in this material, while neutron diffraction experiments [100] revealed the magnetic structure.The low temperature phase diagram of the Ho(Rh1−x Irx )4 B4 system [94, 98], based on low temperature specific heat, ac magnetic susceptibility, and electrical resis-

tance measurements, is shown in Fig. 13.19. The first study of this system yielded a similar phase diagram, but with no evidence for magnetic ordering at temperatures T > 1.2 K for 0.2 < x < 0.6. Several other pseudoternary RRh4 B4 systems have been investigated, and the reader is referred to more comprehensive reviews for discussion and references [8,13]. 13.2.4

Magnetic Field Induced Superconductivity

One of the most dramatic manifestations of the interaction between superconductivity and magnetism is the induction of superconductivity through the application of a high magnetic field. The mechanism that is apparently responsible for this phenomenon, referred to as the exchange field compensation effect, was proposed by Jaccarino and Peter [4] more than 20 years before magnetic field induced superconductivity (MFIS) was firmly established experimentally [101]. The best example of MFIS is illustrated in Fig. 13.20 which shows Hc2 vs T for the compound Eu0.75 Sn0.25Mo6S7.2 Se0.8 ,taken from the work of Meul et al. [101]. The normalized electrical resistance (R/RN ) vs applied magnetic field H at several temperatures is displayed in the inset of Fig. 13.20. At the lowest temperature (0.37 K), the data reveal a se-

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Fig. 13.21. Schematic diagram of the S–N–S–N transition sequence in an applied magnetic field H. HT is the net magnetic field acting on the conduction electron spins, HJ is the exchange field produced by localized magnetic moments, and Hp is the paramagnetic limiting field. The material is superconducting (S) when |HJ | < Hp and normal (N) when |HT | > Hp , after [11]

Fig. 13.20. Magnetic field H − temperature T phase diagram for the compound Eu0.75 Sn0.25 Mo6 S7.2 Se0.8 . The symbols S and N denote superconducting and normal regions, respectively. The solid circles represent the Hc2 (T) measurements of Meul et al. [101], and the solid lines are the boundaries predicted by theory. Normalized electrical resistance R/RN , vs applied magnetic field data at several temperatures are shown in the inset, after [101]

quence of transitions with increasing magnetic field H from superconducting to normal, to superconducting again, and, finally, back to normal (S–N–S– N). From a series of measurements of R/RN vs H at various temperatures,the phase diagram in the H −T plane was constructed (Fig. 13.20); two separate domains of superconductivity are found, one at low fields and another at high fields, whereas only one domain of superconductivity at low fields is present for a conventional superconductor. MFIS occurs in type II superconductors in which Hc2 (T) is determined by the paramagnetic limiting field Hp and which contain magnetic moments that are antiferromagnetically coupled to the spins of the

conduction electrons. The magnetic moments in the Eux Sn1−x Mo6 S8−y Sey compounds are carried by the divalent Eu ions and their coupling to the conduction electron spins generates an “effective” magnetic field, the exchange field HJ, that acts on the conduction electron spins in the same manner as an applied magnetic field.If the sign of the coupling between the Eu magnetic moments and the conduction electron spins is negative, the direction of H will be opposite to that of HJ; i.e., the effect of HJ will be “compensated” by H (see inset of Fig. 13.21). The net magnetic field HT is then given by HT = H − |HJ|. The sequence of S–N–S–N transitions shown in the inset of Fig.13.20 can be explained in terms of the relationship between HT and Hp as illustrated schematically in Fig. 13.21 [11]. The upper and lower curves show the applied field H and the exchange field HJ, respectively, the sum of which yields the net magnetic field HT . The variation of HT with respect to the value of H determines the superconducting behavior of the material. In zero applied field, the compound is superconducting. As the applied magnetic field is increased, the Eu magnetic moments become aligned in the di-

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Unconventional Superconductivity in Novel Materials

rection of the field at a relatively low H-value and |HJ | attains its maximum value |Jmax | of 30–50 tesla. Consequently, provided |HJ| is greater than Hp , the magnitude of HT (which is now negative) can become larger than Hp and drive the material into the normal state. As the applied magnetic field is increased further, the net magnetic field starts to decrease in magnitude, until its magnitude falls below Hp when the compound once more becomes superconducting. Thereafter, HT increases linearly with H, vanishing and then changing sign at H = |HJmax | where HJ is completely compensated, until HT (now positive) exceeds Hp , when the system once again becomes normal. Because the value of the exchange field can be rather large, the compensation effect can take place at very high magnetic fields. In actuality, the second transition from superconductivity to the normal state will also occur if H ∗ surpasses the orbital critical field Hc2 of the compound. Thus, in order to observe MFIS, it is impor∗ tant that Hc2 , which can also limit the value of the upper critical field Hc2, is large enough to allow superconductivity to occur in this magnetic field range ∗ (that is, Hc2 > |HJmax | − Hp ), since the orbital effects are not compensated by the Jaccarino–Peter mechanism. The compensation of the exchange field by the applied magnetic field in several Eux M1−x Mo6S8 systems (where M is a metal such as Sn, Pb, La [102] or Yb [103]) and in EuMo6 S8 under pressure [104], had been inferred from an enhancement and anomalous temperature dependence of Hc2 in these materials. Following the initial suggestion by Jaccarino and Peter that MFIS could occur in a ferromagnet due to exchange field compensation, several theoretical investigations of MFIS in paramagnetic systems were carried out [105–107]. One model, based on the theory of type II superconductivity including the effect of the exchange field [106], was used by Meul et al. to analyze their Hc2(T) measurements on the Eux Sn1−x Mo6S8 system.The calculated boundaries of the two superconducting domains observed for the compound Eu0.75 Sn0.25Mo6S7.2 Se0.8 are indicated by the solid lines in Fig. 13.20 where the parameters of the theory have been adjusted to give the best fit to the data. The excellent description of the H − T superconducting phase boundaries shown in Fig. 13.20

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is striking confirmation of the Jaccarino–Peter compensation mechanism. Jaccarino and Peter originally suggested MFIS in a weakly ferromagnetic material, assuming that it would be superconducting in the absence of ferromagnetic ordering. MFIS in a ferromagnet remains to be discovered.

13.3

f -Electron Heavy Fermion Superconductors

13.3.1

Introduction

In heavy-fermion compounds, strong electronic interactions between the conduction electrons and the localized f -electrons of rare earth or actinide ions result in large quasiparticle effective masses, up to several hundred times the free-electron mass. These compounds display a wide variety of striking correlated-electron phenomena including valence fluctuations, the Kondo effect, magnetic order, non-Fermi liquid behavior, and, of course, superconductivity. As discussed in Sect. 13.2, magnetic moments in conventional superconductors suppress and eventually destroy superconductivity with increasing concentration of magnetic ions. In fact, for decades it was generally accepted that magnetism and superconductivity were inimical. Thus, the 1979 discovery of superconductivity in the heavy-fermion compound CeCu2 Si2 [108], in which the sublattice of Ce ions possess well-defined local moments at high temperatures, posed a major puzzle to researchers. Since then,the list of superconducting heavy-fermion compounds has expanded to include about twenty Ce, U, Pu, and Pr-based compounds. The known heavy fermion f -electron superconductors and their crystal structure and lattice parameters are listed in Table 13.1. Many of these compounds display the coexistence of antiferromagnetism (AFM) and superconductivity, and, recently, ferromagnetic superconductors have been added to the list (UGe2 [3], URhGe [109]). The study of superconductivity in heavy-fermion compounds has led to a range of unexpected phenomena and new theories. It is now widely suspected that heavy-fermion superconductivity (HFSC) is

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Table 13.1. Crystal structure and lattice parameters of heavy-fermion superconductors Compound CeCu2 Si2 CePd2 Si2 CeRh2 Si2 CeNi2 Ge2 CeCu2 Ge2 CeIn3 CeRhIn5 CeIrIn5 CeCoIn5 Ce2 RhIn8 CePt3 Si CeRhSi3 CeIrSi3 CeNiGe3 Ce2 Ni3 Ge5 PrOs4 Sb12 UBe13 UPt3 URu2 Si2 UPd2Al3 UNi2Al3 UGe2 UIr URhGe U6 Fe PuCoGa5 PuRhGa5

Structure Type ThCr2 Si2 (Tetr. I4/mmm) ThCr2 Si2 (Tetr. I4/mmm) ThCr2 Si2 (Tetr. I4/mmm) ThCr2 Si2 (Tetr. I4/mmm) ThCr2 Si2 (Tetr. I4/mmm) AuCu3 (Cub. Pm3m) HoCoGa5 (Tetr. P4/mmm) HoCoGa5 (Tetr. P4/mmm) HoCoGa5 (Tetr. P4/mmm) Ho2 CoGa8 (Tetr. P4/mmm) CePt3 B (Tetr. P4mm) BaNiSn3 (Tetr. I4/mm) BaNiSn3 (Tetr. I4/mm) SmNiGe3 (Ortho. Cmmm) U2 Co3 Si5 (Ortho. Ibam) LaFe4 P12 (Cub. Im3¯ ) NaZn13 (Cub. Fm3¯ c) SnNi3 (Hex. P63 /mmc) ThCr2 Si2 (Tetr. I4/mmm) PrNi2 Al3 (Hex. P6/mmm) PrNi2 Al3 (Hex. P6/mmm) ZrGa2 (Orth. Cmmm) PdBi (Mono. P21 ) TiNiSi (Orth. Pnma) Mn6 Fe (Tetr. I4/mcm) HoCoGa5 (Tetr. P4/mmm) HoCoGa5 (Tetr. P4/mmm)

Lattice Parameters (Å) a b 4.103 4.24 4.09 4.150 4.18 4.689 4.652 4.668 4.62 4.665 4.072 4.269 4.252 21.808 4.135 9.809 11.838 9.3017 10.268 5.764 4.121 5.365 5.207 4.036 14.928 5.620 10.590 6.855 4.327 10.302 4.232 4.301

magnetically mediated, with an unconventional superconducting order parameter (p or d-wave symmetry) that vanishes at lines or points on the Fermi surface. In UPt3 and U1−x Thx Be13 , multiple superconducting phases are observed which is evidence for a complex, multicomponent superconducting order parameter. Very recently, superconductivity was discovered [123, 135] in PrOs4 Sb12 , in which the superconductivity may be mediated by electric quadrupolar interactions rather than magnetic interactions of the Pr ions, suggesting a new mechanism

Ref. c 9.94 9.88 10.18 9.854 10.18 7.542 7.515 7.56 12.244 5.442 9.738 9.715 4.168 5.960

4.899 9.681 4.186 4.018 4.116 5.598 (ˇ = 89.93◦ ) 7.501 5.239 6.786 6.857

[110] [111] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]

for superconductivity in heavy fermion systems. Two decades later, the question of how superconductivity and magnetism can coexist still remains a formidable challenge and further research into this subject will no doubt greatly expand our understanding of superconductivity. In this section, we briefly review the normal and superconducting properties of heavy-fermion superconductors, in general.We then focus on a few exemplary compounds such as the CeMIn5 (M = Rh, Co, Ir), UGe2, and PrOs4 Sb12. The compound UPd2Al3

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serves as an example to illustrate how various spectroscopic techniques have been used to elucidate the nature of heavy-fermion superconductivity.

1 2

C/T (J/mol K )

13.3.2

Normal-State Properties

Specific Heat For most metals, the normal-state specific heat C at low temperatures can be described by the sum of an electronic contribution that varies linearly with temperature T, and a phonon term that varies as T 3, i.e., C(T) = T + ˇT . 3

UBe

UPt

2

3

0.4

13 0.35

0.8

0.3 0

0.6

100

200

300

0.4 0.2 0

0

10

(13.16)

This equation is valid for temperatures well below the Fermi and Debye temperatures. Due to significant correlations between electrons at the Fermi-surface, the physics of heavy-fermion materials is most effectively described in terms of quasiparticle excitations. The effective mass of these quasiparticles can be much larger than that of free electrons, and is proportional to the electronic specific heat coefficient

= C/T. For an ordinary metal, is close to the value for a free-electron gas 0 = ( 2 /3kB2 )N(EF ), where N(EF ) is the density of states at the Fermi Energy; e.g., for sodium, / 0 ≈ 1.3. In heavy-fermion compounds, / 0 can range from 10–1000. In these materials, strong electronic or magnetic correlations between the electrons at the Fermi surface lead to high effective masses up to several hundred times the free-electron mass. The effective mass is proportional to the density of states at the Fermi energy, and therefore to the specific heat coefficient . Values of (T = 0) for various heavy-fermion materials are shown in Table 13.2. For many heavy-fermion compounds, C(T) deviates from Eq. (13.16) at low temperatures, such that (T) = C(T)/T increases with decreasing temperature, sometimes reaching a maximum. The specific heat of the heavy-fermion compounds UBe13 and CeCu2 Si2, plotted as C/T vs T 2 are shown in Fig.13.22.The value of of both compounds diverges at temperatures below ∼ 4 K, reaching a value of 1 J/mol K2 as T approaches 0 K. This may be an indication of NFL behavior, to be discussed in Sect. 13.3.6,

0.45

CeCu Si 2

661

20

30 2

40

50

60

70

2

T (K )

Fig. 13.22. Normal-state specific heat C(T) plotted as C/T vs T 2 for CeCu2 Si2 (circles) [108] and UBe13 (triangles) [136]. The inset shows C/T vs T 2 for the compound UPt3 [137], where the solid line is a fit to Eq. (13.17)

or a low effective Fermi temperature, below which the density of states at the Fermi energy increases. For the heavy-fermion systems UPt3 and UAl2, the specific heat at low temperatures can be described by C(T) = T + ıT 3 ln T + ˇT 3

(13.17)

for T < 20 K. The additional ıT 3 ln T term was first observed in 3 He [171], and is predicted [172, 173] for long range spin fluctuations. The specific heat of UPt3 is shown in the inset to Fig. 13.22, where the solid line is a fit to Eq. (13.17). Magnetization The magnetic susceptibility in heavy-fermion systems tends to follow a Curie–Weiss law at high temperatures (T) =

2 NA eff , 3kB (T − CW )

(13.18)

where NA is Avogadro’s number, kB is Boltzmann’s constant, eff is the effective magnetic moment, and CW is the Curie–Weiss temperature. The effective moment is often reduced somewhat with respect to the free ion value of the rare earth or actinide ion, which can be attributed to valence fluctuations, the Kondo effect, or crystalline electric field effects. The

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Table 13.2. Normal-state properties of heavy-fermion superconductors at ambient pressure, unless noted otherwise. is the electronic specific-heat coefficient at low temperatures, taken at Tc . % (300 K) and %0 are the room temperature and low temperature resistivities, respectively. TM is the magnetic ordering temperature at ambient pressure. The order type refers to antiferromagnetism (AFM), ferromagnetism (FM) or spin density wave (SDW). Tcoh is the temperature at which the screening of local moments by the Kondo effect reduces magnetic scattering, and is determined from a gradual shoulder or maximum in the electrical resistivity. Compound

(mJ/mol K2 )

300K (ܤ cm)

0 (ܤ cm)

TM (K)

Order type

Tcoh (K)

CeCu2 Si2 CePd2 Si2 CeRh2 Si2 CeNi2 Ge2 CeCu2 Ge2 CeIn3 CeRhIn5 CeIrIn5 CeCoIn5 Ce2 RhIn8 CePt3 Si CeRhSi3 CeIrSi3 CeNiGe3 Ce2 Ni3 Ge5 PrOs4 Sb12 UBe13 UPt3 URu2 Si2 UPd2Al3 UNi2Al3 UGe2 UIr URhGe U6 Fe PuCoGa5 PuRhGa5

700 [108]

90 [138] 47 17 14/16 225b

4.8 [139] 2 [141] 6a 2.6/0.9 [144] 120b [146] 0.7 [149] <1 1.25 3 65 5 0.4 1.4 1.5 1.9 6 200 [136] 0.5 30/30 [164] 10 25 0.2 0.44 2 [109] 50 20 25

0.8 [140] 10 [142] 35 [143]

SDW AFM AFM

20 [138]

4.15 [147,148] 10.1 3.8

AFM AFM AFM

2.8 2.2 1.6 5.0 5.5 4.8/4.2 [156]

AFM AFM AFM AFM AFM AFM

6 [162] 17.5 14 4.5 53 47 10

AFM AFM AFM SDW FM FM FM

a

80 350 [112]

200c [151] 720 290 ∼ 400a [152] 390 [119] 120 [120] 120 [120] 40 [121] 90 [156] 350d [123,158] 900 [159] 420 [160] 65 [126,163] 150 [127] 120 [128] 30 [165,166] 40 [167] 160 [169] 280 [170] 80d [133] 50d [134]

33c 40 17 70 90 75 [153] 140 [154] 250 [155] 180 [157] 200 140 225/100 [161] 330/170 140 1000 200 110 [168]

250 260

7.3 kbar 160 kbar c 16 kbar d determined from normal and superconducting-state properties. b

50 [145] 50 [150] 25c [115] 50 [116] 30 [117] 10a [118] 90 ∼ 200 ∼ 120 ∼ 100 10 10 50 75 100

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Unconventional Superconductivity in Novel Materials

Curie–Weiss temperature is generally large and negative due to valence fluctuations or Kondo interactions if the temperature range of the Curie–Weiss fits is much larger than the CEF splitting of the lanthanide or uranium f -electron energy levels. CW is found to be a rough measure of the Kondo temperature, i.e., TK ∼ (3 − 4)CW . At low temperatures (e.g. ∼ 50 K for CeMIn5 (M = Co, Rh, Ir), 300 K for UPt3) the magnetic susceptibility (T) typically deviates from a Curie–Weiss law. The direction of this deviation varies from compound to compound. For example, UPt3 has a larger susceptibility at low temperatures than is predicted by the Curie–Weiss law, whereas UBe13 has a smaller susceptibility. There are various mechanisms which could be responsible for these deviations, including CEF effects, magnetic ordering, quenching of the local moments by the Kondo effect (which would only explain a suppression of the magnetization), and valence fluctuations. The Pauli susceptibility of heavy-fermion materials at the lowest temperatures, (0), is much larger than that of ordinary metals, typically by a factor of 10–100. It is empirically found [174] to scale with the electronic specific heat coefficient (0). In Fig. 13.23,

(0) is plotted as a function of (0) for various heavyfermion materials. This scaling of the magnetic susceptibility and the electronic specific heat coefficient is predicted by Wilson for Kondo systems [175] where the Wilson–Sommerfeld ratio R=

 2 kB2 (0) 2

eff

(0)

(13.19)

is on the order unity for most heavy-fermion compounds. The predictions give R = 1 for the noninteracting electron model, and R = 2 for the Kondo model [175]. In a more general context the Wilson ratio is also known as the Sommerfeld ratio. Many heavy-fermion superconductors order antiferromagnetically (e.g., UPd2Al3 , CeIn3 ) or exhibit spin glass freezing at low temperatures, and a very small number also order ferromagnetically (e.g., UGe2 ). The magnetic ordering temperatures of all the heavy-fermion superconductors are listed in Table 13.3. The heavy-fermion compounds that order antiferromagnetically fall into two classes. Com-

663

Fig. 13.23. Electronic specific heat coefficient plotted as a function of magnetic susceptibility at low temperatures on a log–log scale for various heavy-fermion materials. The solid line is the free-electron prediction for the Wilson ratio (R = 1), after [174]

pounds in the first class (e.g., UPd2Al3, UNi2Al3 , CeIn3 and CeRhIn5 ) have ordered magnetic moments in the range (0.2−1) B, and appear to have standard dipolar order parameters. Compounds in the second class (URu2Si2 and UPt3) have very small ordered moments (∼ 0.01 B), although the N´eel temperatures of these compounds are comparable to those of the first class. One possible explanation for these small moments is quadrupolar ordering, although only the dipolar moment has been detected by means of neutron scattering experiments. In UPt3 and U1−x Thx Be13 , an additional tiny (0.001 B ) magnetic moment is associated with the onset of a second superconducting phase. Electrical Resistivity The electrical resistivity  of heavy-fermion materials is remarkably different than that of conventional metals. For a typical metal, (e.g., Cu), the resistivity decreases with decreasing temperature due to electron–phonon scattering for T > D , where D is the Debye temperature. At low temperatures, Fermi liquid theory predicts that (T) saturates as (T) ∝ AT 2 ∝ (T/TF)2 , with a coefficient A of the

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Table 13.3. Magnetic properties of some heavy-fermion superconductors. CW and eff are Curie–Weiss fit parameters to the susceptibility at high temperatures, and ord is the ordered moment determined from neutron diffraction. (H parallel/perpendicular to c-axis) Compound

(0) (cm3 /mol)

CW (K)

eff ( B )

CeCu2 Si2 CeNi2 Ge2 CeCu2 Ge2 CeRhIn5 CeIrIn5 CeCoIn5 Ce2 RhIn8 CePt3 Si CeRhSi3 CeIrSi3 CeNiGe3 Ce2 Ni3 Ge5 PrOs4 Sb12 UBe13 UPt3 URu2 Si2 UPd2Al3 UNi2Al3 UGe2 UIr URhGe U6 Fe PuCoGa5 PuRhGa5

0.082 > 0.009 0.3 [113] 0.022/0.005 0.018/0.009 0.014/0.008 0.013/0.016 0.013/0.012 [179] 0.095 [120] 0.032/0.062 [154] 0.08 0.044 [156] 0.06 [123] 0.015 [174] 0.19/0.10 [181] 0.01/0.06 0.015 [127] 0.023/0.055 [128,184]

140 170

2.68 [176] 2.6 [145]

16/−79 12.5/−67 −83/−54 12/−40 −45/−75 –128 –186/−109 –12 −52 −16 −53 −200 −65 −47

2.38 [115] 2.28 [116] 2.59 [117] 2.38/2.32 2.67/2.75 2.65 2.57/2.64 2.58 2.5 3.0 3.1 3.0 3.5 3.2

55 [3] –31 10 [109]

2.7 2.91 [167] 1.8

−2 −132/−160

0.68 [133] 0.85 [134,187]

0.003 [186] 0.031 0.0015/0.0013

order of 10−6 ‹§ cm/K2 . This AT 2 term is so small in simple metals that it is often obscured by other electron scattering processes. In heavy-fermion materials on the other hand, the AT 2 term can be up to five orders of magnitude larger than in ordinary metals, reflecting a low effective Fermi temperature TF∗ ∼ 10–100 K, and is often observed at low temperatures. A universal relation between the large magnitude of A and the enhanced electronic specific heat coefficient in heavy-fermion materials was found by Kadowaki and Woods [188]. They observed that most HF compounds obeyed the relation

ord ( B )

0.75 [177]

0.55 [178] 0.16 [180]

0.8 [121] 0.4 [122]

0.01 [182] 0.03 [126] 0.85 [183] 0.2 [185] 1.5 [165] 0.5 [168] 0.42 [131]

A/ 2 = 1 × 10−5 ‹§ cm (mol K/mJ)2 , as illustrated in Fig. 13.24, provided the materials are not too close to a magnetic instability. Displayed in Fig. 13.25 are the (T) curves for two typical heavy-fermion compounds UBe13 and CeAl3 . The resistivity of these and many other heavy-fermion materials is of the order of 100‹§ cm near room temperature, some 10−100 times that of normal metals, and reflects the large exchange (Kondo) interaction between the local moments and the conduction electrons.A weak temperature dependence of (T) is most often observed in these materials at higher temperatures, followed by

13

Unconventional Superconductivity in Novel Materials

665

a maximum or shoulder at Tcoh ∼ 50–100 K. The resistivity drops rapidly just below Tcoh due the formation of coherent Bloch states as the lattice of the local moments of the f -ion sublattice is screened by the conduction electrons. At the lowest temperatures, the resistivity exhibits Fermi liquid properties, 0 + AT 2 , as shown in the inset of Fig. 13.25 for CeAl3 . A departure from the T 2 behavior of the electrical resistivity at low temperatures is observed in some heavy fermion materials, and is referred to as nonFermi liquid behavior. This is discussed in detail in Sect. 13.3.7. Thermoelectric Power

Fig. 13.24. Kadowaki–Woods plot: coefficient of the T 2 term in the electrical resistivity A vs electronic specific heat coefficient . The line has a slope A/ 2 = 1 × 10−5 ‹§ cm (mol K/mJ)2 , after [188]

The thermoelectric power S is a sensitive function of the shape of the density of states near the Fermi level; therefore, large values of S are often observed in heavy-fermion compounds, compared to those of simple metals, due to the presence of a narrow f or d-band at the Fermi level.A simple expression for the thermopower due to electron diffusion, that is valid for T  D , is given by    2 kB2 T d(ln  (E)) S= , (13.20) 3e dE EF where  (E) is the value of the conductivity of the material at an energy E.A maximum in S is expected at a temperature of the order of the Kondo temperature. An anomalously large and positive value of S is often observed for Ce-based heavy-fermion compounds, with a maximum centered at TK or at the CEF splitting if CEF effects are present. For Yb-based heavyfermion materials, S is usually large and negative reflecting the fact that the electronic configuration of Yb is the one-hole counterpart to the one-electron configuration of Ce. 13.3.3

Superconducting-State Properties

Specific Heat

Fig. 13.25. Electrical resistivity  vs T of the heavy-fermion compounds UBe13 and CeAl3 . Inset:  vs T 2 for CeAl3 at low temperatures, after [136,189,190]

The specific heat in the superconducting state of heavy-fermion materials differs markedly from that of conventional superconductors. Whereas C(T) in a conventional superconductor varies as C(T) ∝ exp(−/kB T), in accordance with BCS theory, the

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specific heat of heavy-fermion materials can often be described by a power-law C(T) = 0T + ıT n ,

(13.21)

where the power-law exponent n ∼ 2−3. Both the residual specific heat 0 and the power law behavior suggest an unconventional superconducting gap that vanishes at lines (n = 2) or points (n = 3) on the Fermi surface. In heavy-fermion materials, the magnitude of the specific heat jump C at Tc is a large fraction of the value of C(Tc ). Values of C/ Tc are listed in Table 13.4. The quantity C/ Tc is predicted by BCS theory to be 1.43 (weak coupling) or 1.6 (strong coupling). The fact that C/ Tc for heavy-fermion materials is of the order of unity, where is the value derived from the normal-state specific heat, implies that the superconductivity in these materials is a bulk phenomenon, and that the same electrons responsible for the heavy-fermion behavior in the normal state are participating in the superconductivity. The specific heat jumps for the compounds UBe13 and CeCoIn5 are shown in Fig. 13.26. For CeCoIn5 , the specific heat jump is unusually large such that C/ Tc = 4.5−5 [116,191]; in addition C(T) below Tc can be fit by Eq. (13.21) with n = 2.3 and 0 = 110 mJ/mol K2 .

Fig. 13.26. Superconducting jump in the specific heat plotted as C/T vs T of UBe13 (triangles) [136] and CeCoIn5 (circles) [191]. The line below Tc is a fit to Eq. (13.21) C = T + ıT n

Upper Critical Field All of the known superconducting heavy-fermion compounds are type-II superconductors with upper critical fields Hc2 of the order of a few tens of kOe and lower critical fields Hc1 of the order of tens of Oe, corresponding to values of  ≡ /0 ∼ 50 − 100, where  is the penetration depth and 0 is the superconducting coherence length. These unusually large values of Hc2 and small values of Hc1 reflect the large effective masses of the superconducting quasiparticles which can be understood by noting that Hc1 ∝ 1/2 ∝ 1/m∗ and Hc2 ∝ 1/02 ∝ m∗2 . Most heavyfermion superconductors have a large initial slope of the upper critical field near Tc , −(dHc2 /dT)Tc , and sometimes exhibit a linear T-dependence at lower temperatures. For example, the initial slope of UBe13 is enormous, −(dHc2 /dT)Tc = 420 kOe/K [136], as shown in the inset of Fig. 13.27. Below T = 0.6 K, the upper critical field of UBe13 is linear in temperature with a slope of −91 kOe/K (Fig. 13.27). The zero

Fig. 13.27. Upper critical field Hc2 vs T of UBe13 . Inset: Expanded view of Hc2 (T) of UBe13 near Tc . The line is a fit to the data yielding a slope −(dHc2 /dT)Tc = 420 kOe/K, after [136]

13

Unconventional Superconductivity in Novel Materials

temperature value of the orbital critical field can be estimated from the initial slope of the upper critical field by the following formula [203]: ∗ Hc2 (0) = 0.693Tc (−dHc2/dT)Tc ,

(13.22)

from which the superconducting coherence length 0 can be determined from the relation [204] ∗ Hc2 (0) = ¥0 /202 ,

(13.23)

where ¥0 = 2.07 × 10−7 Oe cm2 is the flux quantum. The value of 0 can be used to estimate the Fermi velocity from the relation 0 ∝ 0.18vF /kTc ; thus, the effective mass can be obtained via m∗ = kf /vF within a spherical-Fermi-surface approximation. The values of the coherence length and the effective mass estimated in this fashion for the heavy-fermion superconductors are collected in Table 13.4. Many heavy-fermion superconductors have anisotropic upper critical fields of the order of || ⊥ Hc2 /Hc2 ≈ 1.5 near Tc (with respect to the c-axis). Heavy Fermion Superconducting Properties Section The spin susceptibility s in the superconducting state determined from Knight shift measurements is sensitive to the symmetry of the superconducting order parameter. For spin-singlet (even parity) pairing (in the absence of spin-orbit coupling), s decreases below Tc , while for spin-triplet (odd parity) pairing, the spin susceptibility remains constant (for certain directions).For nearly all of the heavy fermion superconductors,a decrease in s below Tc is observed (the notable exception is UPt3), implying that the spin part of the Cooper pair wavefunction is a spin-singlet (anti-symmetric under particle exchange). Since the orbital part of the wavefunction must then be symmetric to satisfy the Pauli exclusion principle, the total angular momentum in these materials is even, i.e., L = 0, 2, . . . (s-wave, d-wave, . . . ). Spin-lattice relaxation (1/T1) measurements that probe the shape of the density of states in the superconducting state have been used to distinguish between these two scenarios. For conventional s-wave superconductors, a Hebel-Slichter peak at Tc and an exponential temperature dependence of 1/T1 is predicted [Tinkham], while no Hebel-Slichter peak and a power law T-

667

dependence is expected for an unconventional superconductor (either T 3 for line nodes or T 5 for point nodes). A T 3 dependence of 1/T1 is generally found in the heavy fermion superconductors indicating a d-wave superconducting order parameter. 13.3.4

UPd2 Al3

Introduction The heavy-fermion compound UPd2Al3 belongs to the ever-growing class of materials which display the coexistence of antiferromagnetic order and superconductivity at low temperatures. This class, which includes such compounds as UPt3 and CeIn3 , has attracted considerable attention during the last decade due to the unusual nature of the superconductivity. The detailed nature of the superconductivity in these compounds has been investigated using an assortment of optical and spectroscopic techniques including NMR,Josephson tunneling,and inelastic neutron scattering,which suggests that the superconductivity is magnetically mediated and has d-wave order parameter symmetry. In this section, we use UPd2Al3 as an example to illustrate how spectroscopic techniques have been used to elucidate the nature of heavy-fermion superconductivity.

Fig. 13.28. Specific heat divided by temperature C/T plotted vs temperature T of polycrystalline UPd2Al3 . The N´eel temperature TN and superconducting transition temperature Tc are indicated, after [127]

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Table 13.4. Superconducting properties of heavy-fermion superconductors. Tc : superconducting transition temperature at Pc ; Hc2 : upper critical field at T = 0 K; −(dHc2 /dT)Tc : slope of the upper critical field; m∗: effective mass determined from Hc2 as discussed in Sect. 13.3.3; C/ Tc : superconducting specific heat jump C/Tc divided by electronic specific heat coefficient ; 0 : superconducting coherence length determined from Eq. (13.22). Values are listed as H parallel/perpendicular to the c-axis Compound

C/ Tc

m∗

Tc

Pc

Hc2 (0)

−(dHc2 /dT)Tc

0

(K)

(kbar)

(kOe)

(kOe/K)

(Å)

(me )

20/25

230/230 [138]

65

351 [192]

CeCu2 Si2

0.49 [108]

CePd2 Si2

0.43

28 [142]

13/7

16/12.7 [193]

230/300

97

CeRh2 Si2

0.4

6

35

13

307

118 [194]

CeNi2 Ge2

0.23

23

> 15

5 [144]

642

75

CeCu2 Ge2

∼2

165 [195]

35

11 [146]

147

37

CeIn3

0.17

25 [2]

4.5 [196]

32

300

191

CeRhIn5

2.17

16 [115]

160 [151]

140 [115]

40

0.36 [151]

104

CeIrIn5

0.40

5/9

24/48

223/141

0.76 [116]

100

CeCoIn5

2.3 [117]

4.5 [117]

Ce2 RhIn8

2 [118]

50/120

7/15 [197]

50/32

23

−/54

−/92

51

121 94

CePt3 Si

0.75 [119]

28/34 [179]

75/57

95/109

CeRhSi3

1.0 [153]

20

70

120

62

140

CeIrSi3

1.6 [154]

25

111

114

51

105

CeNiGe3

0.48 [155]

65

23

71

118

120

Ce2 Ni3 Ge5

0.26 [157]

36

8

45

200

145

PrOs4 Sb12

1.85 [123]

22

19

116

∼ 3 [158]

44

UBe13

0.85

90 [198]

420 [198]

37

1.23 [136]

351

UPt3

0.55 [199]

18

63

117

0.5 − 1.0 [160]

265

URu2 Si2

1.5 [126,163]

100

92

59

0.56

184

0.25 [119]

140

UPd2Al3

1.9

30

43

76 [200]

1.2 [127]

187

UNi2Al3

1.0

10

13

191 [200]

0.4 [128]

147

UGe2

0.7 [3,201]

10

20

58 [202]

108 [202]

0.25

235

UIr

0.14 [168]

26

URhGe

0.3 [109]

0.5

281

0.26

3

1063

7

40

199

137

U6 Fe

3.7 [170]

∼ 100

34

61

∼2

109

PuCoGa5

18.5 [133]

∼ 1270

99

16

∼1

47

PuRhGa5

8.7 [134]

∼ 270

20/35

52/40

∼1

40

UPd2Al3 is a heavy-fermion compound with an electronic specific heat coefficient of ≈ 100 mJ/mol K2 , which corresponds to an effective electron mass of m∗ ≈ 50 me . The specific heat of

UPd2Al3 reveals antiferromagnetic order at TN = 14 K, and superconductivity at Tc = 2 K, as shown in Fig. 13.28.

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Unconventional Superconductivity in Novel Materials

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Magnetic and Crystal Structure Neutron diffraction measurements [205] show that UPd2Al3 crystalizes in the hexagonal PrNi2Al3 structure with lattice parameters a = 5.37 Å and c = 4.19 Å. The antiferromagnetic structure of UPd2Al3 consists of ferromagnetic sheets in the easy hexagonal plane with a relatively large ordered magnetic moment of 0.85 B, which alternate along the c-axis. The magnetism arises from localized uranium moments with practically no spin transfer from the Pd ions, as confirmed by studies of neutron diffraction, NMR, and band-structure calculations [206–208]. The first neutron diffraction measurements [209, 210] on UPd2Al3 found that the AFM order remains almost unchanged in the superconducting state. However, recent high-resolution diffraction data indicate [211] that as the temperature is lowered below the superconducting critical temperature Tc , a small 1 % reduction occurs in the intensity of the AFM Bragg peaks. Inelastic Neutron Scattering The inelastic neutron scattering profile measured at the AFM wave vector Q = (0, 0, 0.5) is shown in Fig. 13.29. In addition to the zero energy divergence of the Bragg peak, two finite energy (inelastic) features are evident in the data: a peak at E = 1.5 meV due to the spin density wave, and a sharper peak at E = 0.36 meV which results from an energy gap in the magnetic spectrum. The spin density wave is unusual in that it persists at temperatures well below TN . It is localized about the AFM vector Q = (0, 0, 0.5), since the peak intensity of the spin density wave peak drops off rapidly away from Q. The orientation of the spin density wave has been determined to be in the a-b crystallographic plane, transverse to the sublattice magnetization, from polarized inelastic neutron scattering measurements [212]. The energy gap in the magnetic spectrum observed in Fig. 13.29 occurs only in the superconducting state below Tc = 2 K and Hc = 30 kOe. The temperature dependence of the gap is surprisingly similar to the BCS theory prediction for a superconducting energy gap. At the lowest temperatures, the

Fig. 13.29. Inelastic neutron diffraction data for UPd2Al3 at temperatures above and below Tc = 2 K. The lines are guides to the eye, after [211]

energy gap extrapolates to 0.36 meV, which corresponds to 2 = 2.2kB Tc . This is comparable to the weak-coupling BCS theory result of 2 = 3.5 Tc . The appearance of this energy gap in the magnetic spectrum in the superconducting state suggests magnetically mediated superconductivity. Superconducting energy gaps at slightly larger energies (up to 5.5kB Tc ) have been observed in UPd2Al3 by NMR [206,213,214] and electron tunneling [215] measurements. The discrepancy between the results of the different techniques is probably due to anisotropy of the energy gap; the power-law behavior of the specific heat and 1/T1 relaxation rate from NMR measurements imply that the energy gap in UPd2Al3 should vanish at lines or points on the Fermi surface. The onset of superconductivity has a marked effect on the spin density wave peak in the neutron diffraction data. As shown in Fig. 13.30, both the energy of the spin density wave peak and its line width increase abruptly at Tc.At the same time,the intensity of the peak reaches a maximum at Tc , and then decreases at lower temperatures. These effects are only apparent at fields below the critical field of UPd2Al3 , and the normal-state spin density wave reappears above Hc2. Thus, there appears to be a coupling between the superconducting and magnetic order parameters.

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[217].It has been suggested on the basis of theoretical work [216] that T0 is proportional to Tc for spin-wave mediated superconductors [217]. Electron Tunneling

Fig. 13.30. UPd2Al3 inelastic neutron diffraction data [211]. Top panel: Activation energy E of the spin density wave peak vs T. Bottom panel: Maximum of the spin density wave peak vs temperature T at various wave vectors

A temperature T0, which characterizes the frequency spread (line width) of the spin wave, can be derived by fitting the energy spectrum to a Lorentzian function, yielding T0 ≈ 28 K for UPd2Al3

Further evidence for magnetically-mediated superconductivity in UPd2Al3 , as well as details of the symmetry of the order parameter, have been elucidated by recent directional tunneling measurements on thin films [218]. The UPd2Al3 thin films, which were grown by molecular beam epitaxy in the [001] orientation, are separated from a thin film of Pb by a 4 nm layer of aluminum oxide. Applying a magnetic field H = 300 Oe, which is above the Hc2 of Pb, but well below the Hc2 of UPd2Al3 , results in a superconductor-insulator-normal tunnel junction. The resulting tunneling spectrum is shown in Fig. 13.31, where the thin line is a fit to the Dynes formula. The most important feature of this tunneling spectrum is a peak at V = 1.22 meV, which is above the gap energy, and below typical phonon frequencies (D ≈ 150 K for UPd2Al3 ). The energy of this feature is very close to the energy gap E = 1.5 meV of a dispersive spin wave at the magnetic Bragg point Q = (0, 0, 0.5), which was determined from inelastic neutron scattering measurements. Thus, these measurements strongly suggest a coupling of the superconducting order parameter to a spin-fluctuation with an energy gap E = 1.5 meV. Furthermore, the

Fig. 13.31. Normalized differential conductivity of a UPd2 Al3 -AlOx -Pb tunnel junction. The thinner line is a fit to the Dynes formula with  = 235 ‹eV and

= 35 ‹eV. The inset shows a feature associated with strong-coupling spin-fluctuations, after [218]

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directional nature of the tunneling measurements provides information about the symmetry of the superconducting order parameter.Order parameters of odd parity can be ruled out due to the strong Pauli limiting that is observed in UPd2 Al3 ,and there is only one even d-wave point group that does not show a gap node along the c-axis. The authors concluded therefore that only the fully symmetric A1g d-wave representation is consistent with the measurements. Nuclear Magnetic Resonance Two types of NMR and NQR measurements have been performed on UPd2Al3: 27Al measurements by Kyogaku and coworkers in the early 1990s [219], and more recently 105 Pd NMR measurements by Matsuda et al. [206]. Both types of measurements agree as to the nature of the superconductivity in UPd2Al3 . The spin-lattice relaxation rate T1 derived from 105Pd NQR results is plotted as 1/T1 vs temperature T in Fig. 13.32. Between Tc and TN , 1/T1 varies as T in accordance with the Curie–Weiss law. Below Tc , 1/T1 follows a T 3 temperature dependence,and the Hebel– Slichter peak usually seen in BCS superconductors is absent. These data are consistent with the presence of unconventional superconductivity with an anisotropic energy gap that has line nodes on the Fermi surface. The best fit to the 1/T1 vs T data gives a superconducting energy gap of 2(0) = 5.5 kB Tc . This energy gap is larger than that measured by neutron diffraction, perhaps due to anisotropy of the energy gap. In contrast to the NMR and NQR results, the T 3 temperature dependence of the specific heat implies point nodes on the Fermi surface.Additional NQR measurements at lower T are necessary to resolve this discrepancy. It is also possible that both point and line nodes are present and only one dominates in each measurement. The magnetic properties of UPd2Al3 can only be discerned from 105Pd NMR and not from 27Al NMR measurements. This is because the Al atoms are located interstitially between Pd-U sheets, so transferred hyperfine fields from neighboring antialigned U moments are canceled almost perfectly, and T1 reflects only the susceptibility of the conduction electrons. On the other hand, the Pd atoms are

Fig. 13.32. Spin-lattice relaxation rate 1/T1 of UPd2 Al3 as a function of temperature T determined from 105 Pd NQR measurements in the vicinity of the N´eel temperature TN and superconducting transition temperature Tc , after [206]

surrounded ferromagnetically by U moments, which affect the Pd T1 relaxation via the hyperfine interaction. The results of a recent 105 Pd NMR and NQR study are described below. 105 Pd NQR measurements show [206] a resonance line at ≈ 5 MHz which is split into two lines in the AFM state.The splitting can be explained by the presence of a 2.9 kOe internal field. The temperature dependence of this internal field, which extrapolates to zero at TN , closely matches that of the sublattice magnetization measured by neutron diffraction measurements [183,209].The presence of a local field throughout the sample below Tc confirms the microscopic coexistence of antiferromagnetism and unconventional superconductivity. The hyperfine coupling constant, deduced from the Knight shift, is very small (0.3 kOe/ B ), indicating that there is no significant polarization density of Pd 4d electrons. Furthermore, the large value of T1 above TN confirms that the Pd ions are nonmagnetic and only very weakly coupled to the U ions. The NMR and NQR measurements of UPd2Al3 confirm unconventional superconductivity with an energy gap that goes to zero at lines on the Fermi surface. The 105 Pd NMR and NQR measurements show that the antiferromagnetism is due to U local moments with an internal field H = 2.9 kOe. Furthermore,the results unambiguously demonstrate the coexistence of antiferromagnetism and superconductivity.

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13.3.5

Quantum Critical Points

Introduction Quantum phase transitions (QPT) are observed in many f -electron heavy-fermion systems, in which an antiferromagnetic or ferromagnetic transition temperature, spin glass freezing temperature, or spin density wave (SDW) is suppressed to T = 0 K by pressure or chemical substitution.When the QPT is a second-order phase transition, it is referred to as a quantum critical point (QCP). A wide variety of f -electron materials display QPTs or QCPs including CeIn3 and CePd2 Si2 (AFM),UGe2 (FM),and CeCu2 Si2 (SDW). In the vicinity of the critical pressure or concentration ıC at which the QCP occurs, so-called non-Fermi-liquid (NFL) behavior is often observed; it manifests as unusual power-law temperature dependencies of the physical properties at low temperatures. NFL behavior is discussed in more detail in Sect. 13.3.6. In some compounds, superconductivity also occurs in a narrow region around the critical pressure Pc at the QCP. The presence of superconductivity in the vicinity of the QCP suggests that the superconducting electron pairing mechanism may be mediated by magnetic fluctuations. QCPs are also observed in the high-Tc cuprate superconductors as discussed in Sect. 13.5. CePd2 Si2 and CeIn3 Lonzarich and coworkers [1, 2] reported measurements of the temperature dependence of the electrical resistivity of single crystal specimens of the AFM compounds CeIn3 and CePd2 Si2 at various pressures up to ∼ 30 kbar. The N´eel temperature TN , determined from a discontinuity in the slope d/dT at TN , was found to decrease monotonically with increasing P for both compounds and to extrapolate to zero at a critical pressure Pc of ≈ 26 kbar for CeIn3 and ≈ 28 kbar for CePd2 Si2 . At temperatures below 1 K and pressures in a narrow region near Pc , (T) was found to drop abruptly, indicative of superconductivity for both compounds. The dependence of Tc on P has an inverted parabolic shape, reminiscent of the shape of the Tc vs charge carrier concentration curve found for the high Tc cuprate superconductors.

Fig. 13.33. Temperature-pressure T − P phase diagram of CeIn3 showing the N´eel temperature TN and superconducting transition temperature Tc (plotted as 10Tc ) The inset shows the superconducting transition in the resistivity  vs temperature T at P = 24 kbar, after [1].

The N´eel temperature and superconducting transition temperature are plotted vs pressure in Fig. 13.33 for CeIn3 . The inset to this figure shows the superconducting transition in (T) at 24 kbar. At P = P  , the pressure at which the maximum in Tc occurs, the superconductivity and AFM appear to coexist. However, at pressures above P  , TN can no longer be defined from the electrical resistivity measurements. The transition from the AFM to the paramagnetic state appears to become first order or to be broadened in temperature due to pressure inhomogeneity such that it can no longer be detected in (T) measurements. The observation of superconductivity near Pc , the pressure at which TN extrapolates to zero, suggests that AFM spin fluctuations may be responsible for the pairing of superconducting electrons in these two compounds. Spin-fluctuation theories predict [220–223] that the resistivity in the vicinity of the QCP should vary as T n with n = 3/2 for 3D AFM fluctuations and n = 1 for 2D AFM fluctuations (Table 13.5). In CeIn3 and CePd2 Si2, it was observed that near Pc and above Tc , (T) varies as T n with n = 1.6 for CeIn3 and n = 1.2 for CePd2 Si2. These results

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suggest that the spin fluctuations have 3D character in cubic CeIn3 and 2D character in tetragonal CePd2 Si2 . Spin-fluctuation theory also predicts that the magnetic ordering temperature near Pc should vary as (Pc − P)y where y = 2/3 for 3D and 1 for 2D fluctuations. While a linear dependence of TN on Pc was observed in CePd2 Si2, consistent with 2D fluctuations, a power-law has not yet been distinguished for CeIn3 . CeMIn5 (M = Co, Ir, Rh) The CeMIn5 (M = Co,Rh,Ir) family of heavy-fermion compounds was discovered only a few years ago, yet has attracted an enormous amount of attention and display a variety of interesting phenomena including unconventional superconductivity, possible Ferrel– Larkin Fulde–Ovchinnikov (FFLO) superconductivity,antiferromagnetism,quantum critical points,and non-Fermi liquid behavior [115–117, 152, 178, 224, 225]. CeRhIn5 displays AFM order at TN = 3.8 K, but becomes superconducting under pressure with a Tc of about 2.2 K at 21 kbar [115]. CeIrIn5 is a superconductor at ambient pressure with a bulk Tc = 0.4 K, but a resistive superconducting transition at a higher temperature of Tc = 1.2 K [116]. Finally, CeCoIn5 becomes superconducting at 2.3 K, which is the highest Tc of any Ce-based heavy-fermion superconductor yet discovered [117]. Measurements of powerlaw T-dependencies in the specific heat [116, 117], NMR relaxation rate [226, 227], and thermal conductivity [228] show that CeCoIn5 and CeIrIn5 are unconventional superconductors with nodal gaps. In particular, Izawa and coworkers measured [229] the directional thermal conductivity of CeCoIn5 single crystals in an external magnetic field rotated in the ab plane. The thermal conductivity data show a fourfold symmetry with peaks (corresponding to a nodal direction) when the magnetic field is aligned in the [110] direction. These data are consistent with dx2 −y 2 symmetry of the superconducting energy gap. Before discussing quantum critical points in the CeMIn5 (M = Co, Rh, Ir) systems, a description of the unusual nature of the superconductivity in CeCoIn5 will be presented. Recent measurements [197, 229–232] of specific heat, magnetization, ther-

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mal conductivity, and thermal expansion show that the superconducting transition of CeCoIn5 becomes first-order for applied magnetic fields above 47 kOe in the [001] direction, and 100 kOe in the [100] direction. Evidence for the first order transition is also found in quasi-adiabatic magnetic field sweeps where the temperature of a thermally isolated sample is measured as a function of applied magnetic field [230]. For fields in the [001] direction, the temperature vs field data show a kink at temperatures above 0.72 K, corresponding to a second-order transition, and a step at temperatures below 0.72 K, corresponding to the latent heat released by a first-order transition. First-order superconducting transitions were predicted in 1964 by Maki and Tsuneto [19,233] for superconductors in the clean limit where spin-orbit scattering can be neglected. However, CeCoIn5 is the first superconductor to date to unambiguously demonstrate a first-order transition in specific heat measurements. The application of a magnetic field to a spin-singlet superconductor can destroy the superconductivity via two effects: orbital pair breaking, in which the field-induced kinetic energy of the Cooper pair exceeds the superconducting condensate energy, and Pauli paramagnetic limiting, in which it is energetically favorable for the electron spins to align with the magnetic field, thus breaking the Cooper pairs (see Sect. 13.2.). The critical field determined by orbital pair breaking is given by Eq. (13.22), whereas the critical√ field from Pauli paramagnetic limiting is Hp = 0 / 2 B . The Maki √ parameter ˛ = 2Hc2(0)/Hp is a measure of the relative strengths of these two effects.When Pauli paramagnetism is dominant and ˛ ≥ 1.8, the superconducting transition is predicted to become first order at temperatures below a reduced temperature t0 = T0 /Tc [234]. Several groups have calculated the Maki parameter for CeCoIn5 [230,235,236] with results ranging from 3.2 to 13. While there are varying techniques and experimental difficulties in determining the precise value of ˛, all the measurements agree that ˛ > 1.8 indicating that CeCoIn5 is paramagnetically limited. Furthermore, thermal conductivity measurements indicate that CeCoIn5 is in the clean limit with l/0 = 14 [228].

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Fig. 13.34. (a) Electronic specific heat of CeCoIn5 plotted as C/T vs temperature T and field H for H || [110]. (b) Contour plot of the data in (a) with the same color scale. Plots are from Bianchi et al. [232]

Recent measurements on CeCoIn5 have revealed that when Tc is suppressed below 0.3 K by magnetic fields, another transition occurs below Tc that is second-order. This transition has been observed in specific heat vs magnetic field [235], specific heat vs temperature [232], thermal conductivity [237], and in penetration-depth measurements using a selfresonant tank circuit [236]. This second transition is most prominent for H in the ab plane; however, Bianchi et al. also observe a small effect for H along the crystallographic c-axis. The shape of the phase diagram is strongly suggestive of an “FFLO” state

occurring between the second and the first-order transitions. In this state, which was independently predicted by Fulde and Ferrell [238] and Larkin and Ovchinnikov [239] in 1964, the superconductor lowers its energy in a magnetic field by forming periodic regions of superconductivity separated by domains of aligned spins. The order parameter is spatially modulated and changes sign, either sinusoidally or more abruptly. The FFLO state is predicted to occur at high fields, and at temperatures TFFLO < T < Tc in samples which are in the clean limit and for which spin-orbit scattering can be neglected. CeCoIn5 is an excellent candidate for an FFLO state. It is in the clean limit with l/0 ∼ 14 [228], has a cylindrical two-dimensional Fermi-surface [240, 241] and a d-wave superconducting order parameter. Specific heat data as a function of temperature and magnetic field by Bianchi et al. [232] exhibiting a first-order superconducting transition and the possible FFLO region is shown in Fig. 13.34. Further work such as STM measurements or neutron-diffraction will be necessary to settle the question of FFLO superconductivity in this system. The FFLO state has also been suggested to occur in CeRu2 and UPd2Al3; however, in CeRu2 the candidate FFLO transition was later found to be associated with vortex flux-line melting [242–244], and in UPd2Al3 , the transition was not consistent with theory [245]. The unconventional superconducting state in CeCoIn5 appears to evolve out of an equally unusual non-Fermi-liquid normal state. The resistivity above Tc varies linearly with temperature, and the specific heat diverges logarithmically with temperature when a magnetic field of 5 T (greater than Hc2) is applied along the [001] direction. However, at higher magnetic fields and low temperatures, Fermi-liquid behavior is recovered in the specific heat and electrical resistivity [117, 232, 246]. The high field phase diagram has been investigated by Paglione et al. [246] and by Bianchi et al. [232]. The phase diagram by Paglione et al. based on electrical resistivity and specific heat measurements is shown in Fig. 13.35. Furthermore, Bianchi et al. find scaling of the Sommerfeld coefficient = C/T with (H −Hc )/T over a broad range of temperatures and fields both in the FL and the NFL regime [232]. It is extremely unusual to find

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a QCP associated with a superconducting transition, and, in fact, the scaling behavior is indicative of a second-order QCP whereas the superconductivity is first-order near Hc2 . Both groups have therefore suggested that the QCP responsible for the NFL is in fact a second-order antiferromagnetic one, where the antiferromagnetic order is prevented or“hidden”by the superconductivity. This question of whether the QCP is the direct result of the destruction of superconductivity, or only accidentally coincident with the upper critical field in CeCoIn5 has been probed by three studies that seek to tune Hc2 and watch the evolution of the quantum critical behavior. If the superconductivity is ‘hiding’ an incipient antiferromagnetic quantum critical point, then tuning Hc2 should either reveal antiferromagnetism or drive the system towards Fermi-liquid behavior. In one study, the quantum critical behavior was investigated for both H||c where Hc2 = 4.95 T and H||ab where Hc2 = 12 T [247]. In the second study [248], the suppression of Hc2 and concomitant QCP behavior was studied in the series CeCoIn5−x Snx . Both studies found that the quantum critical behavior always occurred at Hc2, regardless of how Hc2 was varied. The nonFermi liquid behavior near the quantum critical point was evidenced in the specific heat C(H, T) as Cel /T ∼ −lnT and scaling of the Sommerfeld coefficient [( (H) − (HQCP ]/(H)˛ ∼ H/T ˇ where ˛ = 0.7(1) and ˇ = 2.5(5) for x = 0, 0.03, 0.06, with ˛ = 0.9(1) and ˇ = 3.0(5) for x = 0.12. The electrical resistivity shows non-Fermi liquid behavior in the form of (T) ∼ T ˛ where ˛ = 1.5. These two experiments show that the quantum critical behavior is directly related to the suppression of superconductivity, rather than being coincidental. In one interpretation, the quantum fluctuations present at low temperatures are responsible for both the superconducting pairing mechanism as well as the quantum critical behavior when the superconductivity is destroyed with magnetic fields. The third recent set of experiments of electrical resistivity at applied pressure and in magnetic field revealed that the field for criticality HQCP was suppressed more rapidly than the upper critical field Hc2 with applied pressure [249]. This suggests that an antiferromag-

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Fig. 13.35. Magnetic field H vs temperature T phase diagram of CeCoIn5 . Solid squares are the boundary between Fermi-liquid (FL) behavior with (T) = 0 + AT 2 and nonFermi liquid (NFL) behavior. Solid circles indicate a maximum in magnetoresistivity, solid triangles are Hc2 determined from resistivity, and open triangles and open circles are Hc2 from resistivity and specific heat data in [117]. The inset shows the coefficient A in the FL regime as a function of field, which diverges near the upper critical field Hc2 of CeCoIn5 , after [246]

netic quantum critical point is responsible for the NFL behavior in CeCoIn5 . Another quantum critical point in CeCoIn5 may occur at negative pressures [225]. Evidence for this comes from the fact that CeRhIn5 is an antiferromagnet with a larger unit cell than CeCoIn5.The N´eel temperature of CeRhIn5 can be suppressed with pressure leading to superconductivity above P = 16 kbar, with a Tc = 2 K, remarkably close to that of CeCoIn5 . At this pressure, the unit cell volume of CeRhIn5 is also close to that of CeCoIn5 . In fact, there is a general correlation in the CeMIn5 (M = Co, Rh, Ir) compounds between the superconducting transition temperature Tc and the lattice parameters [250]. It appears that, with the exception of the compounds displaying antiferromagnetism, Tc varies as a linear function of the lattice anisotropy c/a where c and a are the tetragonal lattice parameters. In Fig. 13.36,

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Fig. 13.36. Superconducting transition temperature Tc plotted vs the ratio of the tetragonal lattice constants c/a for various compounds in the CeMIn5 family (M = Co, Rh, Ir), after [250]

Tc is shown as a function of c/a for members of the CeMIn5 family both under ambient pressure and hydrostatic pressures. Thus, CeRhIn5 could be considered equivalent to CeCoIn5 with an expanded lattice parameter,or“negative pressure”.When pressure is applied to CeCoIn5 , Tc goes through a maximum at P = 15 kbar and is suppressed by P ∼ 37 kbar. When the CeRhIn5 and CeCoIn5 phase diagrams are combined, considering CeCoIn5 at ambient pressure equivalent to CeRhIn5 at P = 16 kbar, the resulting phase diagram looks remarkably similar to those of the high-Tc cuprates (Sect. 13.5). In fact, several groups suggest the formation of a pseudogap near the QCP based on electrical resistivity and 115 In NQR measurements [225, 252]. The T − P phase diagrams of CeRhIn5 and CeCoIn5 are shown in Fig. 13.37. Fermi-liquid behavior in the resistivity with a T 2 temperature-dependence is recovered at higher pressures, as shown. The details of the pressure-induced quantum critical point in CeRhIn5 have been extensively studied in an effort to understand the quantum critical point and the attendant superconductivity [151,252–257]. Pressure studies in zero applied magnetic field using specific heat, resistivity, NMR, and neutron scattering indicate that the line of antiferromagnetic tran-

Fig. 13.37. Upper figure: Temperature-pressure T − P phase diagrams of CeRhIn5 from electrical resistivity and ac susceptibility measurements (after [251]). Lower figure: T − P phase diagram of CeCoIn5 based on electrical resistivity (after [225])

sitions TN (P) vanishes abruptly at the top of the superconducting dome at Pc1. Some of the studies show that superconductivity and antiferromagnetism coexist for P < Pc1 with TN > Tc . Recent de Haas-van Alphen measurements [258] reveal a divergence of the effective mass at Pc2 ∼ 25 kbar indicating an AFM QCP at that pressure. The dHvA frequencies change dramatically from a “small” Fermi surface, where the 4f electrons of Ce are only weakly hybridized with the conduction electrons, to a “large” Fermi surface in which there is strong hybridization. This has been interpreted as a localized/itinerant crossover of the f electrons at the QCP. It is interesting to note that the dHvA frequencies above Pc2 in CeRhIn5 closely resemble those of CeCoIn5 at ambient pressure [259] indicating that CeCoIn5 is the itinerant version of localized CeRhIn5 . A recent study of

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Fig. 13.38. ac calorimetry data for CeRhIn5 in applied hydrostatic pressures of 2.1 GPa for various applied magnetic fields. Superconducting critical temperature Tc and the onset of antiferromagnetic order TN are indicated (after [260])

Fig. 13.39. Temperature-pressure phase diagram of CeRhIn5 for various magnetic fields. In applied fields, the antiferromagnetism extends into the superconducting dome, allowing antiferromagnetism and superconductivity to coexist with TN < Tc . The quantum critical point Pc1 above which antiferromagnetism vanishes for H = 0 is indicated (after [260])

CeRhIn5 in magnetic fields under pressure using ac calorimetry shows that application of magnetic field allows the line of quantum phase transitions TN (P) to extend into the superconducting dome such that AFM and SC coexist with Tc > TN [260]. The accalorimetry data is shown in Fig. 13.38 and the T − P phase diagram for various magnetic fields is shown in Fig. 13.39. An interesting question is what happens when CeRhIn5 and CeCoIn5 are mixed to form CeRh1−x Cox In5, and whether tuning the lattice parameters via concentration is equivalent to tuning the lattice parameters with pressure. Substitution studies of both CeRh1−x Irx In5 and CeRh1−x Cox In5 systems have been carried out and the phase diagrams are shown in Fig. 13.40 [191, 261]. Both systems show a strikingly similar suppression of the N´eel temperature TN with increasing x, with TN extrapolating to zero at a QCP at x ∼ 0.6 − 0.8 (see Fig. 13.37). The fact that TN depends only on the Rh concentration in these two systems, and not on the lattice parameters, strongly suggests a percolation problem.Antiferromagnetism is destroyed as Rh atoms are removed from the system.This percolation is nontrivial however,since the magnetic moment re-

sides on the Ce atom and not on the Rh atom. SR measurements [262] show that a small ordered moment may reside on the Rh atom; however, a final understanding of the role of the Rh atom in creating a three-dimensional Fermi surface and long-range AFM order in these systems is still lacking. Thus, the phase diagrams of the substituted systems are profoundly different that those of CeRhIn5 and CeCoIn5 under pressure (see Fig. 13.37). The superconductivity coexists with antiferromagnetism over a very broad range of concentrations and does not show the dome-shape typical of quantum critical points. For the CeRh1−x Cox In5 system, the specific heat plotted as C/T vs T for various Co concentrations x is shown in Fig. 13.41 [191].(These data are very similar to those of the CeRh1−x Irx In5 system in [261].) Each specific heat curve contains either one or two peaks. The peaks for T < 2 K are associated with the superconducting transition, whereas the peaks for T > 3 K are attributed to AFM transitions. The C(T)/T data for samples with 0.4 ≤ x ≤ 0.6 contain two peaks, indicating that these samples exhibit coexistence of superconductivity and antiferromagnetism. The magnitudes of the superconduct-

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Fig. 13.41. Specific heat C(T) shown as a plot of C/T vs temperature T for various CeRh1−x Cox In5 compounds. The peaks for T < 2 K are attributed to superconducting transitions, whereas the peaks for T > 3 K are ascribed to antiferromagnetic phase transitions, after [191]

Fig. 13.40. Phase diagrams of CeRh1−x Irx In5 and CeRh1−x Cox In5 . The N´eel temperature TN , superconducting transition temperature Tc , and the temperatures at which the resistivity goes to zero T=0 are shown vs Ir or Co concentration x. The lines are guides to the eye (after [191,261])

ing peaks are quite large, with C/ Tc ≈ 1 for most samples, and ∼ 5 for CeCoIn5 , indicating that the superconductivity is a bulk phenomenon involving the heavy electrons.The magnitude of the superconducting peaks is suppressed with decreasing x, whereas the magnitude of the AFM peaks is enhanced with decreasing x. The entropy at 7 K, which corresponds

to the total area under the superconducting and AFM peaks, is constant for the samples with x < 1, implying that the same heavy electrons participate in both the superconductivity and the antiferromagnetism. The question arises whether the superconductivity and magnetism coexist on a microscopic scale or if there are macroscopic concentration variations of Co and Rh within the sample. This question has been investigated for the CeRh1−x Irx In5 system by Morris and coworkers [263]. ‹SR measurements in zero field of the CeRh0.5 Ir0.5 In5 sample show that the antiferromagnetism exists over at least 85% of the sample, and ac magnetization measurements indicate at least 85% shielding of the sample. Thus, magnetism and superconductivity most likely coexist microscopically in this sample.Further discussion of the CeRh1−x Irx In5 system can be found in the multiple superconducting phases section. Noncentrosymmetric Superconductivity It was thought for a long time that spin triplet superconductivity can only occur if parity is satisfied [264]. In practice, this means that for heavy-fermion superconductors the crystal structure must have inversion symmetry (be centrosymmetric) since the strong spin-orbit coupling in the HF compounds al-

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lows the electrons at the Fermi surface to see the symmetry of the crystal. Most heavy-fermion superconductors to date have inversion symmetry. However, two HF superconductors were recently discovered, CePt3 Si [119] and UIr [168], whose crystal structures lack inversion symmetry, and yet both are strong candidates for spin-triplet superconductivity.Other noncentrosymmetric pressure-induced superconductors CeIrSi3 [154], CeIrSi3 [153], CeNiGe3 [155], and Ce2 Ni3 Ge5 [157], have also been recently reported. In UIr, ambient pressure ferromagnetism with TFM = 46 K can be suppressed with 2.6 GPa of pressure [167,168], leading to ferromagnetism coexisting with superconductivity against the background of a noncentrosymmetric orthorhombic crystal structure below Tc = 0.14 K [168]. CePt3 Si orders antiferromagnetically below TN = 2.2 K and exhibits superconductivity below Tc = 0.75 K [119]. The crystalline lattice is shown in Fig.13.42. In CePt3 Si [119], the upper critical field Hc2 = 4-5 T significantly exceeds the paramagnetic limit for singlet superconductivity Hc2 ∼ 1 T. Thus, there are strong suggestions for spin triplet superconductivity in both UIr and CePt3 Si despite the lack of inversion symmetry in the lattice structure. This apparent coexistence of spin-triplet superconductivity with a noncentrosymmetric symmetry challenges our understanding. In general, Cooper pairs form between electrons with equal and opposite momentum k. Broken inversion symmetry splits the two spin-degenerate bands forcing the electron with momentum k to pair with an electron with momentum −k + Q, creating a finite momentum Cooper pair. It has been predicted that this should result in a mixing of the spin singlet and spin triplet branches [264].These arguments were used to explain the lack of superconductivity in the noncentrosymmetric compound MnSi, which is a strong candidate for triplet superconductivity due to the presence of a ferromagnetic quantum critical point [265]. If only pure triplet superconductivity can coexist microscopically with ferromagnetism, and the noncentrosymmetric symmetry prevents a pure triplet superconductivity from forming, no superconductivity should be observed at all.

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Fig. 13.42. Tetragonal P4mm crystal structure of the heavyfermion superconductor CePt3 Si, which does not have inversion symmetry. Figure reproduced from [119].

A recent theoretical paper seeks to resolve the issue of spin triplet superconductivity in noncentrosymmetric crystals. Frigeri et al. show that under certain symmetry conditions that are satisfied by CePt3 Si, the paramagnetic limiting effect is suppressed for both spin-singlet and spin-triplet superconductivity, and either type of superconducting order parameter should be possible [265]. Another intriguing theory suggests that the paramagnetic limiting field is lifted in CePt3 Si for fields along the tetragonal a-axis through the formation of a helical vortex phase [266]. Two other d-electron systems have been discovered to our knowledge that exhibit superconductivity in a noncentrosymmetric crystalline lattice. In the family R2 C3−y (R = La, Y), there is no significant spin-orbit coupling, so the symmetry of the lattice does not necessarily affect the Cooper pairing wave function.In the heavy-fermion compound Cd2Re2 O7 , heavy-fermion superconductivity occurs in a noncentrosymmetric lattice; however experimental evidence points to nodeless s-wave superconductivity and no magnetic order, thus there is no a priori reason to assume spin triplet superconductivity (see Sergienko et al. [267] and references therein).

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In UIr, the experimental conditions of high pressure and millikelvin temperatures required to observe superconductivity have limited the number of studies to date on this compound. In CePt3 Si however, a number of studies have already been completed since its recent discovery.We will therefore review some key experimental findings here that probe the question of spin singlet or spin triplet superconductivity surviving in the absence of parity. The experimental evidence to date provides conflicting evidence for the nature of the pairing symmetry in CePt3 Si. The Sommerfeld coefficient = 390 mJ/mol K2 results in a value of C/ Tc = 0.25-0.55 that is much lower than the BCS weak coupling prediction of 1.43. A similar depressed value of C/ Tc has been observed in the spin-triplet superconductor SrRu2O4 . However, the low value may also indicate that the Fermi surface is divided between superconductivity and antiferromagnetism. Substitution studies of La on the Ce site show a strong suppression of superconductivity with only 2% La, which supports a spin-triplet scenario [268]. Neutron scattering results [269] have shown an ordered moment on the Ce atom of 0.16 B /Ce atom in the AFM phase, which is strongly reduced from the Hund’s rule prediction. This most likely indicates itinerant antiferromagnetism. On the other hand, well-defined CEF behavior in magnetization and NMR measurements indicates at least a partial localization of Ce spins [180,269].The Ce moments are ordered ferromagnetically in the basal plane and stack antiferromagnetically along the c-axis [269]. SR experiments show that magnetism occurs in the whole sample volume, indicating microscopic coexistence of superconductivity and antiferromagnetism [270]. NMR measurements [269] indicate that CePt3 Si is the first heavy-fermion superconductor to exhibit a Hebel-Slichter peak in 1/T1T below Tc . CePt3Si does not show the 1/T1 ∝ T 3 behavior that is observed in most other HF superconductors, and the behavior indicates point-nodes in the superconducting gap function that would be consistent with a spin-triplet pairing wave function. On the other hand, thermal transport and magnetic penetration [271, 272] experiments show power-law behavior at low temperatures indicating line nodes in the superconduct-

ing gap. Recent theoretical work proposes that line nodes are possible even in the absence of parity in CePt3 Si [267,273,274].The nature of the pairing symmetry and its interaction with the crystal lattice symmetry remain open questions in CePt3 Si and we look forward to many future studies probing this interesting compound. 13.3.6

Non-Fermi Liquid Behavior

At the time of their discovery, nearly all of the heavy-fermion superconductors were found to have Fermi liquid properties, i.e., (T) ∝ T 2 and C(T)/T ∝ (T) ∝ constant. However, unusual temperature dependences of the physical properties, such as a linear T-dependence of the electrical resistivity [275], known as non-Fermi liquid behavior, have been observed in many heavy-Fermion compound since their first observation in UBe13 . In particular, the relatively weak magnetic field dependence of the electrical resistivity and specific heat, and the absence of a quasi-elastic peak in neutron scattering measurements in UBe13 do not follow the expected behavior of a conventional Kondo effect, which is thought to be the origin of the heavyfermion characteristics. These experimental data led Cox to propose a two channel quadrupolar Kondo model, in which the interaction between the electric quadrupole moments of a U4+ nonmagnetic doublet ground state in a cubic crystalline electric field and the charges of the conduction electrons give rise to NFL behavior at low temperatures [276,277]. A few years later,NFL behavior was observed [278] in the Y1−x Ux Pd3 system over a wide range of U concentrations x in the paramagnetic phase below the critical concentration xcr = 0.2 where the magnetic transition temperature (later identified as the spin glass freezing temperature [279,280]) vanished. This discovery started a period of intense research in f electron materials in order to further investigate the occurrence and origin of non-Fermi liquid behavior [281–284]. Non-Fermi liquid behavior in f -electron systems is characterized by weak power-law or logarithmic temperature dependences of the physical properties at low temperatures T  T0 for most f electron systems and can be summarized as follows:

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  T n where |a| ≈ 1 and n ≈ 1 − 1.5 ;  ∝ 1−a T0 (13.24)     1 C T −1+ ∝− ln or T ( ≈ 0.7 − 0.8); T T0 T0 (13.25)   n  T , with n ≈ 0.5 , ∝ 1− T0   T , ∝ − ln (13.26) T0 



or ∝ T −1+ ( ≈ 0.7 − 0.8) ;  (!, T)

scales as

! . T

(13.27)

In several of these systems, the scaling temperature T0 can be identified with the Kondo temperature TK . The characteristic NFL temperature dependences were first established from studies of chemically substituted f -electron materials as Y1−x Ux Pd3 [278,285], UCu5−x Pdx [286, 287], and CeCu6−x Aux [288, 289], but more recently have been found in stoichiometric compounds at ambient pressure (e.g., YbRh2 Si2 [290]) and under applied pressure (e.g., CeIn3 [150], CePd2 Si2 [291]). These experiments suggest that different mechanisms may be responsible for the NFL behavior observed in these f -electron materials. Several NFL models with distinct origins have been proposed and can be grouped into three main categories: (i) single-ion models based on a multichannel Kondo effect, (ii) disorder-driven NFL models, and (iii) models involving quantum fluctuations of an order parameter at a second-order T = 0 K phase transition (QCP) Various theoretical models have been proposed based on single-ion physics involving an unconventional Kondo effect, of either magnetic or electric (quadrupolar) origin [276, 292–295]. In these multichannel Kondo models, the magnetic (quadrupolar) moments are over-compensated by a number of channels of conduction electrons due to additional orbital (spin) degrees of freedom, giving rise to power-law or logarithmic temperature dependences of the physical properties at low temperatures. Theoretical models which incorporate

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the effects of disorder include the Kondo disorder model [296–298], in which a distribution of Kondo temperatures TK due to chemical disorder with finite weight at TK = 0 K leads to NFL properties, and an inhomogeneous Griffiths–McCoy phase [299], which consists of magnetic clusters in a paramagnetic phase formed as a result of the competition between the Kondo effect and the Ruderman–Kittel– Kasuya–Yosida (RKKY) interaction in the presence of magnetic anisotropy and disorder. The critical behavior near a quantum critical point has been considered by a number of investigators [220–222,300–304]. Hertz [300] and later Millis [223,301] studied the fluctuations of an order parameter near a T = 0 K second order phase transition using renormalization techniques. They found that the critical behavior is determined by both spatial and temporal fluctuations in an effective dimension deff = d + z, where d is the spatial dimension and z is the dynamical exponent. The physical properties predicted by the Hertz/Millis theory including (T), (T), C(T)/T and the dependence of the ordering temperature on the control parameter ı are listed in Table 13.5. Similar results were obtained by Moriya [220,221] with a self consistent renormalization group model of spin fluctuations in weak itinerant magnetic systems and also by a spin fluctuation theory by Lonzarich and coworkers [222]. A recent paper by Si examines the effect of non-Gaussian fluctuations near a quantum critical point,and suggests a model in which the Kondo temperature is suppressed to zero at the same point as the second order phase transition, but from the opposite side of the phase diagram [305]. Table 13.5. Spin-fluctuation theory predictions for the temperature dependences of the resistivity  and specific heat divided by temperature C/T, and the dependence of the magnetic ordering temperature TM on the control parameter ı (pressure or carrier concentration) for 3D and 2D ferromagnetic (FM) and antiferromagnetic (AFM) spin fluctuations 3D FM  C/T TM

5/3

T − ln(T/T0 ) (ı − ıc )3/4

3D AFM 3/2

T

0 − ˇT 1/2 (ı − ıc )2/3

2D FM 4/3

T T −1/3

2D AFM T − ln(T/T0 ) ı − ıc

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In recent years, non-Fermi liquid behavior has also been observed in three systems that are based on UPd2Al3 (discussed in Sect.13.3.4): U1−x Mx Pd2 Al3 for M=Th, La, and Y. The magnetic phase diagrams of these three systems are shown in Fig. 13.43. It is remarkable that although these three systems have the same parent compound, the phase diagram of U1−x Thx Pd2Al3 is quite different from those of U1−x Lax Pd2 Al3 and U1−xYx Pd2Al3. Both the AFM

Fig. 13.43. Temperature–M concentration (T–x) phase diagrams of the U1−x Mx Pd2 Al3 (M = Th, La, Y) systems. Regions of antiferromagnetic order (AFM), spin glass freezing (SG), superconductivity (SC) and non-Fermi liquid behavior (NFL) determined from measurements of the specific heat C, magnetic susceptibility , and electrical resistivity  as a function of temperature T are indicated in the T − x plane. The temperature dependences of  and C/T observed in the NFL regions are indicated in the figure, after [306]

order and superconductivity are only slightly suppressed with Th substitution (Fig. 13.43a) at x = 0.2, and the system crosses over to an NFL regime at x ≈ 0.6. The NFL properties of U1−x Thx Pd2 Al3 at low temperatures scale with U concentration indicating the NFL behavior is associated with single-ion physics such as an unconventional Kondo effect. In the U1−xYx Pd2 Al3 system, the antiferromagnetism and superconductivity are rapidly suppressed with Y substitution (Fig. 13.43c). Similar behavior is found for U1−x Lax Pd2Al3 (Fig. 13.43b). The N´eel temperature decreases from TN ≈ 14 K in UPd2Al3 to TN ≈ 7 K by x = 0.2, after which spin glass behavior is observed at TSG ≈ 1 K. The spin glass transition is no longer observed at a critical concentration xc ≈ 0.7 near the concentration where the NFL behavior occurs [307]. It appears that TSG vanishes nearly linearly with x as x → xc . These results suggest that the NFL behavior in this system is associated with fluctuations of a magnetic order parameter above the spin glass freezing transition at T = 0 K. An example of the NFL behavior observed in these systems is shown in Fig. 13.44 for a U1−xYx Pd2 Al3 sample with a composition (x = 0.8) close to that of the QCP at xc ≈ 0.7. The physical properties of U0.2Y0.8 Pd2Al3 at low temperatures have the following T-dependences:  ∝ T, C/T ∝ (or T n with n = 0.2), and ∝ [1 − (T/T0)]1/2. Although the magnetic phase diagram of U1−x Lax Pd2Al3 is similar to that of U1−xYx Pd2Al3 , the non-Fermi liquid behavior is close to that of U1−x Thx Pd2Al3. This suggests that disorder from local expansion of the lattice by the larger Th and La ions could be driving the NFL behavior in these compounds. Substitution with Y leaves the lattice unchanged [306]. Non-Fermi liquid behavior has also been found in the vicinity of a ferromagnetic quantum critical point in systems such as Th1−x Ux Cu2 Si2 [311] and URu2−x Rex Si2 [310]. The substitution of Re or Tc for Ru in the heavy-fermion superconductor URu2Si2 has been found to suppress both hidden order (HO) [312] and superconductivity and induce ferromagnetism (FM) [308,309]. The superconducting transition temperature Tc of URu2−x Rex Si2 decreases rapidly from Tc = 1.5 K at x = 0 to below

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Fig. 13.44. Electrical resistivity  vs T, specific heat divided by temperature, C/T, vs log T (upper inset), and magnetic susceptibility vs T 1/2 (lower inset) for U0.2Y0.8 Pd2 Al3 , after [307]

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200 mK at x = 0.01, while the N`eel temperature TN associated with the formation of the small-moment (∼ 0.03 B/U atom) hidden order phase (often associated with antiferromagnetism or a spin density wave) at 17.5 K decreases less rapidly with x and appears to vanish by x = 0.15 as shown in the T − x phase diagram in Fig. 13.45. Ferromagnetic order occurs in the concentration range 0.3 ≤ x ≤ 1.0 where the Curie temperature C , saturation moment

sat , and electronic specific heat coefficient , exhibit maxima at x = 0.8 of 38 K, 0.44 B /U atom, and 160 mJ/mol K2 [308], respectively, as displayed in Fig. 13.46. Magnetic measurements indicate that the suppression of FM order occurs at xcFM = 0.3. For samples close to where features associated with the hidden order phase can no longer be resolved and the FM critical point (xcFM = 0.3) with Re concentrations 0.15 < x ≤ 0.6, the non-Fermi liquid behavior is characterized by C/T ∝ − ln T and (T) ∝ T n with 1 ≤ n < 2 at low temperatures [310]. Therefore, it appears that the NFL behavior persists well into the ferromagnetic state. NFL behavior is observed in a wide range of CeMIn5 (M = Co, Rh, Ir) compounds as discussed previously. In addition to the NFL behavior found in stoichiometric CeCoIn5 , studies of the chemically substituted systems Ce1−x (La,Y)x RhIn5 and Ce1−x Lax CoIn5 have revealed logarithmic or powerlaw divergences of the specific heat and magnetic

Fig. 13.45. Magnetic phase diagram of the URu2−x Rex Si2 system showing the hidden order (HO, filled circles), superconducting (SC, open circles), and ferromagnetic (FM, squares) regions. The power-law exponent of the electrical resistivity, n, is shown in the top part. Some data are taken from [308,309], after [310]

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Fig. 13.46. Physical properties of URu2−x Rex Si2 vs Re concentration x. (a) Saturation moment

sat (at 5 K); (b) effective moment eff ; (c) Curie temperature TC and Curie–Weiss temperature CW ; and (d) electronic specific heat coefficient

(at 1 K). The filled circles are from [308]. Data for x = 0 from [163]. The lines are guides to the eye, after [310]

susceptibility once the superconductivity or antiferromagnetism in the parent compounds is suppressed [313–315]. In Ce1−x (La,Y)x RhIn5 compounds, additional humps in the specific heat and peaks in the magnetic susceptibility are observed that coexist with NFL behavior and decrease with increasing x. These features can described by the Ce3+ J = 5/2 Hund’s rule multiplet being split by a tetragonal CEF [313]. The humps in the specific heat have also been attributed to short-range 2D antiferromagnetic correlations. [314,316]. An interesting new theory for NFL behavior has arisen out of studies of the CeMIn5 compounds. In this model, the specific heat and magnetization are decomposed into two components: a Kondo-gas part which obeys the single-ion Kondo model, as-

sociated with the coupling between Ce local moments and conduction electrons, and a Kondo-liquid portion, resulting from intersite coupling between the Ce moments. This model was proposed for the Ce1−x Lax CoIn5 system [317], and can also be applied to the Ce1−x Yx RhIn5 system [318]. For these two systems, the fraction fK of electrons participating in the single-ion Kondo effect increases roughly linearly as a function of x until it saturates at x = 0.9, at which point the Kondo-liquid portion goes to zero and the system entirely obeys the single-ion Kondo model. The fraction fK also varies as a function of temperature. For pure CeCoIn5 , the Kondo gas condenses into a Kondo liquid below T ∗ ∼ 45 K, resulting in a sudden downturn in the electrical resistivity or “coherence shoulder”. This model provides an intrigu-

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ing new possibility for the origin of NFL behavior, at least in the CeMIn5 compounds, and its applicability to other systems needs to be investigated. 13.3.7

Magnetically Ordered Heavy-Fermion Superconductors

In addition to the antiferromagnetic heavy-fermion superconductors such as CeIn3 and CePd2 Si2 discussed in Sect. 13.3.5, several ferromagnetic heavyfermion superconductors have been discovered recently and are described in the following. UGe2 Superconductivity under pressure was observed by Saxena et al. in single crystal specimens of the ferromagnetic (FM) compound UGe2 within the ferromagnetic phase up to the critical pressure Pc ≈ 16 kbar where the ferromagnetism vanishes [3]. Shown in Fig. 13.47 is the T − P phase diagram for single crystals of UGe2 [3]. The Curie temperature TFM decreases from 53 K at ambient pressure to ≈ 12 K at P = 15 kbar, above which the FM transition can no longer be determined from the (T) curves, suggesting a first-order-like transition. A small feature in the resistivity curves is observed at applied pressures P < 13 kbar, perhaps related to a spin density wave (SDW) or charge density wave (CDW), and is denoted in Fig. 13.47 as Tx [165, 201]. Superconductivity is confined to a narrow pressure region 10 kbar ≤ P < 16 kbar reaching a maximum value of Tc = 0.7 K at Px = 13 kbar where features associated with the possible SDW/CDW can no longer be observed [3] (the phase diagram of polycrystalline UGe2 samples, discussed below, is similar [201]). No superconductivity is observed at or above Pc , indicating that the ferromagnetism and superconductivity are intimately related. Recent magnetization measurements under pressure confirm the first-order FM phase transition at Pc and reveal that the transition at Tx is also first-order [319]. It has been suggested that UGe2 is a p-wave superconductor, in which the superconducting electrons do not suffer from the pair-breaking interaction caused by the large internal magnetic field,

Fig. 13.47. Temperature-pressure T − P phase diagram of UGe2 . TFM denotes the Curie temperature; Tc denotes the superconducting transition (the Tc values are scaled by a factor of 10 for clarity; Tx denotes the feature in the (P, T) curves possibly due to a SDW/CDW transition. The solid and dashed lines are guides to the eye, after [3,165]

since the mean free path l ∼ 1500 Å is much larger than the superconducting coherence length 0 ∼ 130 − 200 Å [3, 202]. Specific heat experiments under applied pressure reveal a finite value of C/T as T → 0 K, which is consistent with the superconducting gap vanishing at lines or points on the Fermi surface [166, 202]. In addition, neutron diffraction measurements yield no evidence of a change in the magnetic moment upon entering the superconducting state, consistent with p-wave superconductivity [165]. Spin-triplet superconductivity mediated by coupled SDW/CDW spin fluctuations has been put forth in recent theoretical work by Watanabe and Miyake [320]. This theory can explain the anomalous reentrant behavior observed in upper critical field measurements along the a-axis of UGe2 and the anisotropy of Hc2 [165]; however, neutron diffraction measurements do not reveal any static order associated with a SDW/CDW. While these results can be explained within a spin-triplet scenario of superconductivity,an investigation on polycrystalline samples of UGe2 with electron mean free paths of the order of the superconducting coherence length, discussed

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Fig. 13.48. Electrical resistivity  vs T 2 for UGe2 at various applied pressures P. The lines are fits to the data of the form  = 0 + AT 2 Inset (a): Coefficient of the T 2 term of (T), A, vs P. Inset (b): Residual resistivity 0 vs P, after [201]

below, indicates that the superconductivity in UGe2 may have s-wave character [201]. It is widely believed that p-wave superconductors are quite sensitive to nonmagnetic impurities, resulting in the suppression of superconductivity when l ≈ 0 , where l is the electron mean free path and 0 is the superconducting coherence length. Two of the leading candidates for p-wave superconductivity are Sr2 RuO4 [321,322] and UPt3 [323,324]. When the ratio l/0 ∼ 1, Tc is completely suppressed for Sr2 RuO4 [325] and decreases by 15 % for UPt3 [324]. However, in UGe2 superconductivity was observed in some polycrystalline samples with residual resistivities 0 as high as 3 ‹§cm, corresponding to a mean free path l ∼ 100 Å, without any apparent diminution of Tc . Since the superconducting coherence length 0 is ∼ 130 − 200 Å [3, 202], this represents a ratio l/0 ≈ 1, considerably reduced from the value l/0 ≈ 10 observed for the single crystal specimens of UGe2. This suggests that the superconductivity observed in UGe2 under pressure may have s-wave character. It is noteworthy that several theories have recently been proposed in which s-wave, rather than p-wave, superconductivity is predicted to occur in the ferromagnetic state [326–330].

In the low temperature FL-like phase, the residual resistivity 0 and the coefficient A of the T 2 contribution to the resistivity, (T) = 0 + AT 2 , are both strongly dependent on pressure as shown in Fig. 13.48 for polycrystalline samples of UGe2 and exhibit peaks at 15 kbar and 11 kbar, respectively (inset of Fig. 13.48). It is interesting that the maximum in 0 correlates with the broad maximum in Tc (P) that is centered near the same pressure. Specific heat measurements on polycrystalline samples of UGe2 reveal that increases with pressure up to 15 kbar [166]. The increase of is consistent with the behavior of A(P) according to the Kadowaki–Woods relation A = 1.0 × 10−5 2‹§ cm (mol K/mJ)2 [188]. Since Tc of a conventional s-wave superconductor increases with N(EF ), which is proportional to , and is not strongly affected by potential scattering, the correlation between Tc , , and 0 is qualitatively consistent with s-wave superconductivity. A precedent for the macroscopic coexistence of ferromagnetism and s-wave superconductivity is found in the compound ErRh4 B4 , discussed in Sect. 13.2.3, which exhibits localized moment ferromagnetism, in contrast to UGe2, which is regarded as an itinerant ferromagnet. The compound ErRh4 B4

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becomes superconducting at a critical temperature Tc1 ≈ 8.7 K, FM at a Curie temperature TFM ≈ 1.5 K, and normal again at a second critical temperature Tc2 ≈ 0.9 K [80]. There is evidence, based on neutron scattering and specific heat measurements [84], for the macroscopic coexistence of FM and superconducting regions in the temperature interval between Tc2 and c . Within the superconducting regions between Tc2 and c , a sinusoidally modulated magnetic state with a wavelength of ∼ 100 Å that coexists with superconductivity has been detected by means of small angle neutron scattering measurements [84]. The sinusoidally modulated magnetic state is believed to be produced by screening of the magnetic field associated with the ferromagnetically aligned Er magnetic moments by the supercurrents at long wavelengths [331]. Similar behavior has been found for the localized moment FM superconductor HoMo6S8 [332]. It seems reasonable that such an inhomogeneous FM s-wave superconducting state could also exist in UGe2 , in spite of the fact that it is an itinerant ferromagnet. Irrespective of whether the superconductivity in UGe2 is of s-wave (spin-singlet) or p-wave (spintriplet) variety, it appears to be strongly connected to the ferromagnetism. For s-wave superconductivity, it seems possible that the onset of superconductivity occurs when the Curie temperature drops to a value such that the superconductivity is no longer suppressed by the internal field in the ferromagnet, while near Pc where the Curie temperature vanishes, the superconductivity is suppressed by the ferromagnetic spin fluctuations. Alternatively, the longitudinal spin fluctuations may actually produce pairing of electrons in spin-singlet s-wave states as proposed recently by Suhl [329] and developed further by Abrikosov [330]. The competition between these two effects could then lead to the maximum in Tc at a pressure below Pc . For triplet-spin superconductivity, the superconductivity would seem to require the presence of the ferromagnetism and be optimized at a pressure below Pc . Further experiments will be required to develop an understanding of the origin and nature of superconductivity in this remarkable material.

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Other Ferromagnetic Heavy-Fermion Superconductors The coexistence of superconductivity and ferromagnetism was recently reported in URhGe [109], UIr [168] and ZrZn2 [333]. While bulk superconductivity appears to be well established in URhGe given the specific heat jump at Tc [109], the situation is less clear for UIr and ZrZn2 . In UIr, ferromagnetism in this material at TFM = 46 K is rapidly suppressed with pressure [167] and evolves into another smallmoment ferromagnetic phase that is destroyed at a critical pressure Pc2 = 26 kbar [168]. Near this critical pressure, the electrical resistivity follows a power-law T-dependence ( ∼ T n ) with an exponent n  5/3 as expected for critical spin fluctuations of a 3D ferromagnet [220] (see Table 13.5). In addition, superconductivity is observed at Tc = 0.14 K in the vicinity of Pc2 suggesting that ferromagnetic fluctuations may mediate the pairing in this material. As discussed in Sect. 13.3.5, UIr is a candidate for spin-triplet superconductivity,despite the lack of inversion symmetry. The large initial slope of the upper critical field and the large value of the T 2 coefficient of (T) indicate an enhancement of the effective mass [168]. However, the identical value of Tc and similar Hc2 of Ir [334] raise the possibility of filamentary superconductivity associated with Ir in UIr; additional measurements are needed to

Fig. 13.49. Pressure-temperature phase diagram of ZrZn2 . TFM : Curie temperature; Tc : Superconducting critical temperature. The values of Tc have been multiplied by a factor of 10 for clarity, after [333]

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clarify the nature of superconductivity in this interesting material. For ZrZn2 , a superconducting transition at Tc ≈ 0.3 K was observed to decrease with increasing pressure and vanish near the critical pressure Pc at which TFM vanishes as shown in Fig. 13.49. The absence of a superconducting anomaly in specific heat, and the occurrence of superconductivity within the ferromagnetic state only in very pure samples suggest the possibility of a spin-triplet pairing mechanism, although trace amounts of a superconducting impurity phase cannot be ruled out at this point. 13.3.8

Multiple Superconducting Phases

UPt3 and U1−x Thx Be13 The first direct evidence for unconventional superconductivity was found for the compound UPt3 , which displays at least three distinct superconducting phases. Multiple superconducting phases were also found in the heavy-fermion compound U1−x Thx Be13 when substituted with 2 − 4 % Th, and more recently in PrOs4Sb12 (discussed in Sect.13.3.9). The appearance of multiple superconducting phases, which cannot be explained in terms of a conventional s-wave order parameter, has prompted a flurry of investigations into these exciting compounds. UPt3 exhibits antiferromagnetic (AFM) order below TN = 5 K with a very small ordered moment of 0.02 B/U atom aligned in the basal plane [182].Early measurements on as-grown samples showed a single broad superconducting transition in the specific heat at 0.5 K; however, later measurements of annealed samples revealed two peaks in the specific heat near Tc at temperatures Tc+ and Tc− [335]. The specific heat vs temperature of a single crystal of UPt3 is plotted in Fig. 13.50. The double kink feature has been verified in many different crystals, and while the values of the Tc ’s vary between samples, the kinks are always separated by about 50 mK. These data, along with anomalies in ultrasonic attenuation [336], and a kink in Hc2 vs T point to multiple superconducting phases. The superconducting phase diagram of UPt3 determined from sound velocity measurements [337] is shown in Fig. 13.51, where “N” indicates the normal

Fig. 13.50. Specific heat of UPt3 in the vicinity of the superconducting transition plotted as C/T vs T. The dashed lines represent two idealized jumps, and the solid line represents a single jump with the same entropy, after [335]

(antiferromagnetic) state, and the three superconducting phases are labeled “A”, “B”, and “C”. The Tc+ and Tc− transitions observed in the specific heat at zero field correspond to the N-to-A and the A-to-B phase transition lines. Measurements of muon spin relaxation reveal that the transition to the B phase at Tc− is accompanied by the onset of a small internal field at 475 mK. These data might lead to the conclusion that the phase transition at Tc− just signifies the onset of magnetic order. However, the peak in the specific heat at the first transition is so large that it occupies most of the Fermi surface, and there is not enough entropy left for an equally large magnetic transition at Tc− . The second transition can only be explained in terms of a transition to a different superconducting phase. Both transitions satisfy the relation C/ Tc ∼ 1 indicating that heavy electrons are responsible for the superconductivity (see [288] for a review). The heavy-fermion compound U1−x Thx Be13 also shows a rich phase diagram with multiple superconducting phases. The phase diagram determined from specific heat, upper critical field, and ‹SR studies is shown in Fig. 13.52 where “A” and “B” indicate the two superconducting phases [338]. As in UPt3 , the second superconducting transition is accompanied by a transition to very small moment magnetism (0.001 B /U) [339].Recent point contact spectroscopy measurements on UPt3 [340] are consistent

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UPt 1.5

1.5

C N

1

3

0.5

H c

0

B A 0

100

C

1

200

300

T (mK)

400

500

0.5 B 0

0

100

200 300 T (mK)

A 400

500

Fig. 13.51. Superconducting phase diagram of UPt3 determined from ultrasonic velocity measurements. [337]. The data points delineate the second-order phase transition lines between three superconducting phases “A”, “B”, and “C”, and the normal state “N”, after [337]

Fig. 13.53. Isobars of the superconducting critical temperature Tc vs x for U1−x Thx Be13 between 0 and 12 kbar, after [341]

Fig. 13.52. Phase diagram of U1−x Thx Be13 , after [338]

with line nodes in the ab plane and point nodes along the c-axis,while NMR measurements of both systems show T 3 temperature dependence below Tc , indicating lines of nodes. Measurements of Tc (P) on the U1−x Thx Be13 system are displayed in Fig. 13.53 as isobars of Tc vs x for 0 ≤ x ≤ 12 kbar [341]. Two different types of behavior are present for each pressure, separated by

the minimum in Tc (x) at xmin . For x < xmin , a monotonic decrease of Tc (x) is observed as the pressure increases, and the magnitude of the slope dTc /dP also becomes larger with increasing x. For x > xmin , an abrupt increase of Tc occurs with a maximum and subsequent decrease for higher x. The position of this maximum is pressure dependent, moving to higher concentration as P increases. It is tempting to infer from the Tc (x) curves the existence of two distinct superconducting phases, one in the regions 0 ≤ x ≤ xmin (A phase) and the other at x > xmin (B phase), where xmin is a function of pressure. The way the Tc (x) curves evolve with pressure suggests that the Tc (x) phase boundary for x < xmin extends approximately linearly with the same slope to the region x > xmin . A variety of theories have been advanced to explain the unconventional superconductivity in UPt3

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and U1−x Thx Be13 . These theories have been extensively reviewed in [342–344]. For UPt3 , most of the data points to a spin singlet [345] or spin triplet [344,346] complex (two component) superconducting order parameter [342]. Another possibility that is difficult to rule out is the presence of two different superconducting phases with different symmetries that happen to be nearly degenerate in temperature [347, 348]. The small moment magnetism in the “B” superconducting states could be explained by a superconducting order parameter that breaks time reversal symmetry. On the other hand, the magnetism could simply result from the onset of a spin density wave. The coupling between superconductivity and magnetism is a key issue that still needs to be resolved. In the case of U1−x Thx Be13 ,there are three secondorder phase transitions that meet at a tricritical point, which is theoretically not allowed [349]. This implies the existence of other phase transition lines.An additional phase transition has indeed been observed by assuming that Th substitution is equivalent to pressure, and measuring specific heat as a function of applied pressure for a sample with x = 0.019. One of the more popular models for U1−x Thx Be13 [343,350] assumes multiple degenerate superconducting phases with different symmetries, as was mentioned for UPt3. The pure UBe13 compound is assumed to have a d-wave symmetry and an s-wave phase with a lower Tc . The d-wave phase is suppressed with Th doping more rapidly than the s-wave phase, causing the two transition lines to cross. Thus for intermediate Th concentrations there is a pure s-wave phase at high temperatures and a mixed s − d phase at lower temperatures. PrOs4 Sb12 Recently, superconductivity was observed in the filled skutterudite compound PrOs4 Sb12 at Tc = 1.85 K [123, 135]. The superconductivity evidently involves heavy-fermion quasiparticles with an effective mass m∗ ≈ 50 me , as inferred from the normal1

state electronic specific heat coefficient , the jump in the specific heat at Tc , and the slope of the upper critical field near Tc . This compound appears to be the first example of a Pr-based heavy-fermion superconductor.When this material was first discovered, the quadrupolar Kondo effect, which involves screening of Pr3+ electric quadrupole moments of a nonmagnetic 3 doublet ground state in the CEF by the charges of the conduction electrons [276, 277], analogous to the screening of the magnetic dipole moments of paramagnetic ions by conduction electron spins that occurs in the magnetic Kondo effect, was put forth as a possible scenario for the heavyfermion behavior [123,135]. Another scenario could involve virtual excitations from either a 1 singlet or a 3 nonmagnetic doublet ground state to a low lying (∼ 10 K) 5 triplet state [351]. Evidence for the heavy-fermion behavior in PrOs4 Sb12 was provided by the magnitude of the superconducting specific heat jump on a pressed pellet of single crystals as displayed in the inset of Fig. 13.54.(The C(T) data have been corrected for excess Sb derived from the molten Sb flux in which the crystals were grown.) An equal entropy construction, in which the entropy is conserved just above and below Tc , yields C/Tc = 632 mJ/mol K2 . The value of the electronic specific heat coefficient from the weakcoupling BCS prediction (C/ Tc = 1.43) is ≈ 440 mJ/mol K2. In the normal state, the specific heat can be described by C(T) = T + ˇT 3 + CSch (T), where

T and ˇT 3 are electronic and phonon contributions, respectively, and CSch (T) is a Schottky anomaly for a two level system consisting of a doublet ground state and a triplet excited state at an energy  above the ground state. The best fit of this expression to the data yields the values = 607 mJ/mol K2, ˇ = 3.95 mJ/mol K4 (corresponding to a Debye temperature D = 203 K), and  = 7.15 K. A number of measurements on PrOs4 Sb12 provide evidence for the formation of a nonmagnetic 1 ground state due to splitting of the J = 4 multiplet of Pr in a cubic crystalline electric field.1 In this scenario, hybridization between the 1 and ex-

In Oh symmetry, the J = 4 states are labeled: 1 (singlet), 3 (doublet), 4 , and 5 (triplets) [352]. The slightly reduced Th symmetry appropriate for the filled skutterudite structure, mixes the magnetic triplets (now labeled 4(1) and 4(2) , respectively), and the doublet is now labeled 23 [353]. We use the Oh notation throughout for simplicity.

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T = 1.77 K

e

C /T

1.1

c

0.9 0.7

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Γ

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Γ

0.5

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0.0 -0.5

1.6

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C(T) = γT + βT + CSch(T)

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3

0

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Pressed pellet

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2

3

4

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6

Single crystals 1.4

7

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8

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cited f 2 states (i.e., 4 ) gives rise to a heavy Fermiliquid ground state [351, 354]. The Schottky-like anomaly in specific heat and the peak in magnetic susceptibility at T ≈ 3 K just above the superconducting transition are consistent with a 1 ground state and a low-lying 5 first excited state separated by ı ≈ 8 K, followed by 4 (134 K) and 3 (205 K) excited states [123, 355, 356]. Low-temperature inelastic neutron scattering experiments [357,358] reveal peaks in intensity in the excitation spectrum at 0.72 meV (8.2 K), 11.4 meV (120 K), and 17.2 meV (200 K) corresponding to ground state transitions that are in excellent agreement with this CEF energy level scheme [123, 355, 356, 359]. Based on the near isotropy of the magnetization along the [100] and [110] directions and on the occurrence of a highfield ordered phase (discussed below), Tayama et al. suggested that such a 1 ground state scenario better described the properties of PrOs4 Sb12 than the 3 ground state scenario. Neutron diffraction measurements reveal a small antiferromagnetic moment ( AFM = 0.025 B) parallel to the [010] direction in magnetic fields above H ∼ 5 T at T = 0.25 K [360].A crystalline electric field analysis based on a 1 ground state suggests that antiferroquadrupolar order arises from a level crossing of the 1 ground state with an excited state in magnetic fields, consistent with the diffraction results. In addition, specific heat experiments up to ∼ 30 T are also consistent with the evo-

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Fig. 13.54. Specific heat C divided by temperature T, C/T, vs T for a PrOs4 Sb12 pressed pellet. The line represents a fit of the sum of electronic, lattice, and Schottky contributions to the data. Upper inset: Ce /T vs T near Tc for a PrOs4 Sb12 pressed pellet (Ce is the electronic contribution to C). Lower inset: C/T vs T near Tc for PrOs4 Sb12 single crystals, showing the structure in C near Tc . Data from [123,135]

lution of the 1 ground state CEF level scheme in magnetic field [361]. A number of experimental results suggest unconventional superconductivity in PrOs4 Sb12. Recent specific heat measurements [158,359] on single crystals of PrOs4 Sb12 show a double superconducting jump reminiscent of UPt3 (Fig. 13.50). The two superconducting transitions are estimated to occur at Tc1 = 1.75 K and Tc2 = 1.85 K using an entropy conserving construction. This double transition has been observed for several different combinations of single crystals as well as for individual single crystals [158,359]. In fact, the only specific heat measurement of PrOs4 Sb12 that does not show the double jump feature is the original measurement of a pressed powder sample, in which a single broad transition is observed, probably due to strain effects [123]. These two transitions have also been observed in a variety of other measurements including: thermal expansion [362], magnetization [356, 363, 364], and penetration depth studies [365]. The thermal expansion measurements [362] reveal that, from the Ehrenfest relation, Tc1 and Tc2 have different pressure dependences suggesting that they are associated with two distinct superconducting phases. In addition, a third superconducting phase at Tc3 ∼ 0.6 K has been proposed on the basis of a kink in the lower critical field Hc1 and the critical current Ic , possibly with a different order parameter symmetry [366].

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Fig. 13.55. (a) and (b): Angular variation of zz of PrOs4 Sb12 in a magnetic field rotating within the ab plane at 0.52 K above and below Hc2 ( 2.0 T), from [367]

Fig. 13.56. Phase diagram of the superconducting gap symmetry of PrOs4 Sb12 determined from thermal conductivity measurements. The filled circles represent the magnetic field H ∗ at which the transition from fourfold to twofold symmetry takes place. The open circles represent Hc2 . The area of the gap function with fourfold symmetry is shown in pink (A-phase) and the area of the gap function with twofold symmetry is shown in blue (B-phase), after [367]

The nature of the superconductivity of PrOs4 Sb12 was investigated by means of thermal transport measurements in aligned magnetic fields by Izawa and coworkers [367]. The thermal transport shows an anisotropy as the magnetic field is rotated in the a−b crystallographic plane as shown in Fig. 13.55, which is coupled to anisotropy in the superconducting order parameter. As the temperature is lowered below Tc1, the thermal transport changes symmetry from fourfold to twofold in the ab plane. Group theoretical arguments given the cubic crystal structure of PrOs4 Sb12 suggest that the high-temperature superconducting phase between Tc1 and Tc2 has six point nodes, whereas the low temperature phase below Tc1 has two point nodes. The compound PrOs4 Sb12 is the first heavy-fermion superconductor for which there is evidence of point nodes rather than line nodes in the superconducting order parameter. Izawa et al. also established the H − T phase diagram of the two superconducting phases from thermal transport measurements as shown in Fig. 13.56, in which Tc1 is suppressed by 25 kOe, and Tc2 is suppressed by 7 kOe. On the other hand, recent transverse field ‹SR [368] and Sb-NQR measurements [369] on PrOs4 Sb12 are consistent with an isotropic energy gap. Along with the

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693

Fig. 13.57. Magnetic field-temperature (H − T) phase diagram of PrOs4 Sb12 showing the regions exhibiting superconductivity (SC) and the high field ordered phase (HFOP) The dashed line is a measure of the splitting between the Pr3+ 3 ground state and 5 excited state (see text for further details), data from [158,359,362–364,375]

specific heat, these measurements indicate strong coupling superconductivity. These findings suggest an s-wave, or, perhaps, a Balian–Werthamer p-wave order parameter. The Balian–Werthamer p-wave order parameter, originally suggested for the “B” phase of 3 He, involves Cooper pairs with aligned spins in the L = 1 angular moment state, and results in an isotropic energy gap [370]. It is also interesting to note that zero-field ‹SR measurements reveal the presence of internal magnetic fields below Tc2 = 1.85 K indicating time-reversal symmetry breaking in the superconducting state [371] of PrOs4 Sb12 . There is also evidence for a high field ordered phase (HFOP) in PrOs4 Sb12 . The H − T phase diagram, depicting the superconducting region and the HFOP, is shown in Fig. 13.57 [364, 372]. The HFOP was first detected by means of features in (H, T) data [359,363,364]. The line that intersects the high field ordered state represents the inflection point of the “roll-off ” in (T) at low temperatures and is a measure of splitting between the Pr3+ ground state and the first excited state, which decreases with field. The HFOP has also been observed by means of large peaks in the specific heat [158,355], thermal expansion [362,373], and kinks in the magnetostriction [373] and M(H) curves [359,364,374,375]. The heavy-fermion superconductor PrOs4 Sb12 is an intriguing material because of the possibility that the heavy-fermion behavior and the supercon-

ductivity [376,377] arise from quadrupolar fluctuations. It is also only the second superconductor (after UPt3 ) that exhibits two distinct superconducting phases with different order parameter symmetries. The heavy fermion state and unconventional superconductivity observed in this material represent a challenge for theoretical description [376–380]. PuMGa5 Superconductivity has been observed in the Pubased material PuCoGa5, which crystallizes in the same tetragonal structure as the CeMIn5 materials, with a critical temperature Tc = 18.5 K [133], an order of magnitude larger than that of other heavyfermion superconductors. Shortly after the discovery of superconductivity in PuCoGa5, the isostructural compound PuRhGa5 was also found to be superconducting at Tc = 8.7 K [134] with nearly identical normal and superconducting state properties as PuCoGa5 described below. The specific heat of PuCoGa5, shown in Fig. 13.58, reveals a large specific heat jump at Tc , C/Tc = 110 mJ/mol K2, confirming bulk superconductivity in this material. If one assumes the BCS value C/ Tc = 1.43, the electronic specific heat coefficient is = 77 mJ/mol K2 , indicating a moderate enhancement of the effective mass. Further evidence for bulk superconductivity is provided by electrical resistivity and magnetic sus-

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Fig. 13.59. Linear variation of the superconducting transition temperature Tc with ratio of the tetragonal lattice parameters c/a of CeMIn5 (M=Co, Rh, Ir) and PuMGa5 (M=Co, Rh) heavy-fermion superconductors, after [385] Fig. 13.58. Specific heat over temperature C/T, electrical resistivity  and magnetic susceptibility as a function of temperature T for PuCoGa5 showing a superconducting transition at 18 K, after [133]

ceptibility measurements (Fig. 13.58). In addition to the unusually large value of Tc for an intermetallic compound, the upper critical field is correspondingly large. An initial slope dHc2 /dT = −99 kOe/K ∗ yields an orbital critical field Hc2 (0) = 1270 kOe ∗ (Eq. (13.22)). From this value of Hc2(0), the superconducting coherence length is estimated to be 0 = 16 Å, from which an effective mass m∗ = 47 me is determined.This estimate of m∗ yields = 161 mJ/mol K2 , providing further evidence for heavy-fermion behavior in PuCoGa5. The critical current density is also quite large, JC > 104 A/cm2 for T > 0.9Tc , due to pinning defects caused by the radioactive decay of Pu. In the normal state, PuCoGa5 exhibits behavior characteristic of both localized and itinerant 5f electrons, a situation well-known in the various allotropes of Pu and in the early actinides (Th-Am) [381, 382]. The magnetic susceptibility of PuCoGa5 follows a modified Curie-Weiss law all the way down to the superconducting transition temperature with an effective moment eff = 0.68 B , close to the value expected for localized Pu3+ ( eff = 0.84 B ). On the

other hand, the Sommerfeld coefficient ∼ 100 mJ/molK2 suggests some degree of delocalization. This dual nature of the 5f electrons in PuCoGa5 is consistent with photoemission experiments [383] that reveal peaks at 1.5 eV below EF and at EF characteristic of localized and itinerant behavior, respectively. The electrical resistivity of PuCoGa5 has sshaped curvature typical of materials with strong spin fluctuations. A power-law T-dependence is observed below 75 K ((T) ∼ T 4/3) [134] similar to the NFL behavior found in the related compound CeCoIn5 (Sect. 13.3.5) [384]. Recent measurements confirm d-wave superconductivity in the PuMGa5 (M=Co, Rh) materials. The striking similarity of the physical properties of PuMGa5 and CeMIn5 (M=Co, Rh, Ir), where heavyfermion d-wave superconductivity has been well established, indicates magnetically mediated superconductivity in PuMGa5. One example of the close resemblance between the two families of superconductors is shown in Fig. 13.59, where a linear relation between Tc and the ratio of tetragonal lattice parameters is found [385]. The power-law behavior of the specific heat of PuCoGa5 [385] and CeCoIn5 below Tc is also similar and is consistent with line nodes in the superconducting gap function. Perhaps the best evidence for unconventional superconductivity is provided by NQR/NMR measurements [386].

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Fig. 13.60. Spin-lattice relaxation rate 1/T1 of the Ga nuclei in PuCoGa5 . The lines are calculations for BCS s-wave (solid), pure d-wave (dotted) and “dirty” (dashed) d-wave gap functions, after [386]

pling regime (/kB Tc ∼ 4) [386]. Very similar behavior has recently been reported in PuRhGa5 [387]. Scaling of the 1/T1 data for a number of different dwave superconductors including CeCoIn5 , PuCoGa5 , and YBa2 Cu3 O7−ı , with a characteristic spin fluctuation energy scale Tsf (found to be proportional to the superconducting transition temperature) indicates that these Pu-based superconductors bridge the gap between the heavy-fermion superconductors with Tc s generally less than 1 K and the high-Tc cuprates with Tc s of order 100 K as shown in Fig. 13.61. This linear relation between Tc and the spin fluctuation energy scale suggest that other classes of materials within this magnetically mediated mechanism of superconductivity may yet be found with even higher transition temperatures.

13.3.9

Fig. 13.61. Characteristic (spin fluctuation) energy scale Tsf vs superconducting transition temperature Tc for the heavy-fermion (circles), plutonium-based (triangles), and high-Tc (squares) superconductors, after [386]

As displayed in Fig. 13.60, the spin-lattice relaxation rate 1/T1 of the Ga nuclei in PuCoGa5 exhibits a T 3 temperature dependence just below Tc followed by a T-linear dependence at the lowest temperatures due to the impurities created by the radioactive decay of Pu in the sample. The power-law behavior of 1/T1 is clearly in contrast to the expected behavior of a conventional s-wave superconductor; the best fit to the data is a “dirty d-wave” scenario in the strong cou-

695

Concluding Remarks

The past two decades of research into heavy-fermion superconductors have consistently yielded a rich variety of new compounds, new phenomena, and new insights into the nature of superconductivity. When heavy-fermion superconductivity was first discovered in CeCu2 Si2 in 1979 by Steglich and coworkers, the coexistence of magnetism and superconductivity had only been established in ternary rare earth compounds such as the rare earth molybdenum chalcogenides RMo6 X8 (X = S, Se) and the rare earth rhodium borides RRh4 B4 . Since then about 20 different Ce, Pr, U, and Pu-based heavy-fermion superconductors have been discovered and the list will continue to grow. The past few years have seen renewed interest in heavy-fermion superconductors with the discovery of the coexistence of ferromagnetism and superconductivity in UGe2 , URhSi, and, possibly, ZrZn2 . The exciting discovery of superconductivity in PrOs4Sb12 ,which contains two superconducting phases which appear to have different order parameter symmetries, and may have a quadrupolar origin, suggests a new route to unconventional superconductivity. Heavy-fermion superconductors continue to be a fertile area of research and enhance our understanding of superconductivity.

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Fig. 13.62. The molecular structures of TMTSF (tetramethyltetraselenafulvalene) and BEDTTTF (bisethylenedithio-tetrathiafulvalene, or ET for short), which are the main building blocks of crystalline molecular conductors. In contrast to TMTSF, the ET molecule is not completely planar

13.4

Organic Superconductors

13.4.1

Introduction

The first organic superconductor, (TMTSF)2PF6, was discovered more than 25 years ago by J´erome et al. [388]. It belongs to a class of compounds known as Bechgaard salts, named after the first person to synthesize them,Klaus Bechgaard.(TMTSF)2 PF6 undergoes a transition from a quasi-one-dimensional (1D) metal to a superconductor as the temperature is lowered below 1 K under an applied pressure of ∼12 kbar. At ambient pressure, however, it experiences a Peierls metal-insulator transition to a spindensity-wave (SDW) state at low temperatures. With one exception, SDW phase transitions are observed in all Bechgaard salts based on TMTSF. The molecule TMTSF (tetramethyltetraselenafulvalene) is shown in Fig. 13.62. Within a few years of this stimulating discovery, a large variety of other superconductors based on different molecular building blocks were found. Of the more than 100 different organic superconductors known to date [389], the largest number are based on the molecule BEDT-TTF (bisethylenedithio-tetrathiafulvalene or ET for short), which is also shown in Fig. 13.62. The usual stoichiometry of these quasi-two-dimensional (2D) organic metals is (ET)2 X, where X represents a monovalent anion such as I−3 or Cu(NCS)−2 [389, 390]. Charge transfer from the neutral ET molecule to the anion leads to partially filled molecular bands, resulting in metallic conductivity. To date, the highest values of Tc are 11.5 K at ambient pressure for -(ET)2 Cu[N(CN)2 ]Br [391] and about 13 K for the isostructural material

-(ET)2 Cu[N(CN)2 ]Cl under a moderate pressure of about 0.3 kbar [392]. Beyond the main building blocks, TMTSF and ET, of crystalline organic superconductors, a number of additional organic molecules are known to form superconducting species [389]. A very promising new class of 2D organic metals is based on the donor BETS (bisethylenedithio-tetraselenafulvalene), a selenium-substituted ET analogue. Molecular crystals of this donor with Fe3+ containing anions are prone to magnetic interactions between the BETS -electron system and the iron d electrons [393, 394]. This makes these materials a fascinating playground for studying the competing magnetic and superconducting interactions. For example, reentrant superconductivity upon cooling (Sect. 13.2.3) and magnetic-field-induced superconductivity (Sect. 13.2.4) can be found in these systems. Besides superconductivity, the organic metals show a wealth of different ground states such as antiferromagnetism, spin-Peierls, SDW, and chargedensity-wave (CDW) states. These ground states are accessible by tuning the structure, counter anion, magnetic field, temperature, and pressure. The study of these fertile phase diagrams has led to new theoretical concepts; however, a solid understanding of some of these states still remains a challenge. Even the normal metallic state of these electronically low-dimensional metals reveals unusual properties sometimes not in line with conventional Fermiliquid theory. Many superconducting properties of the organic superconductors, especially of the 2D materials, resemble the behavior found in the cuprate superconductors [395–398]. The H–T phase diagrams and the

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vortex dynamics are very similar in these layered strongly type-II superconductors. In contrast to the high-Tc cuprates, however, much less is known about the nature of the superconducting state in organic metals. Clear-cut experimental proof for unconventional non-BCS-like superconductivity is missing. Until very recently the 1D Bechgaard salts were believed to be triplet superconductors. This, however, is not the case as recent sophisticated experiments proved singlet pairing [711]. For the 2D organic superconductors,on the other hand,where singlet pairing has been established, an ongoing debate persists as to the exact symmetry of the order parameter (see [399–402] for recent reviews). Some experimental results hint at line nodes in the superconducting energy gap, i.e., d-wave symmetry, whereas other experiments, in particular, specific-heat data, suggest the existence of a nodeless energy gap, fully consistent with BCS-like s-wave symmetry. A further unsolved issue is the mechanism responsible for the Cooper-pair coupling. While there is some belief in the existence of a non-phononmediated coupling mechanism, as appears to be in unconventional superconductors such as the cuprates, Sr2RuO4 , and heavy-fermion compounds, this issue remains unclear and heavily debated for the organic superconductors. The proximity of antiferromagnetic or SDW states to the superconducting state spurred the notion that antiferromagnetic fluctuations are responsible for the coupling. On the other hand, a number of experiments indicate that some interaction between phonons and superconductivity exists. In any case, it is clear that the exci-

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tonic mechanism originally proposed by Little [403] is not responsible for the pairing of superconducting electrons. Nevertheless, Little’s prediction of very high transition temperatures in organic materials was an important stimulus for the search for organic superconductors. 13.4.2

Normal-State Properties

Ordinarily, the term“organic materials”implies plastics, i.e., polymeric carbon-based chains. However, the 1D and 2D organic metals discussed here are high-quality single crystals usually grown by electrocrystallization from appropriate solutions,the details of which are reviewed in depth in [390]. During this process the oxidation of the donor molecules occurs together with the crystal growth at the electrodes. The resulting charge-transfer salts weigh up to a few milligrams and are usually shaped like needles or plates. Quasi-One-Dimensional Organic Metals All 1D organic superconductors based on the donor molecule TMTSF crystallize in the same triclinic crystal structure which is depicted in Fig. 13.63 for (TMTSF)2PF6 [404]. Some crystal parameters are given in Table 13.6. TMTSF is a planar bricklike molecule that tends to stack in columns. These columns are formed along the a-axis, which is the axis with the highest conductivity since in this direction the Se–Se distances are short, resulting in a strong -orbital overlap. Along the b-axis the Se–Se

Fig. 13.63. Crystal structure of (TMTSF)2 PF6 viewed somewhat tilted relative to the b direction. a and c are projections of a, the most-conducting direction, and c, the leastconducting direction

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Fig. 13.64. (a) Calculated dispersion relation and (b) Fermi surface of (TMTSF)2 PF6 . These results are representative of all (TMTSF)2 X salts [406]

distances are similar; however, the overlap integrals are much smaller due to the side-by-side orientation of the Se atoms. Along the c-axis, the PF−6 anions (see Fig. 13.63) separate the columns. This results in only a weak electronic coupling along c with correspondingly small overlap integrals. The approximate ratio of the overlap integrals along the crystallographic directions is ta : tb : tc = 300 : 30 : 1. These numbers reveal that the Bechgaard salts are by no means ideal one-dimensional metals. It is, however, exactly this tendency towards two dimensionality which shifts the insulating SDW state to low temperatures and allows superconductivity under moderate pressure (see below). The calculation of the electronic band structure of the organic charge-transfer salts from first principles is a highly sophisticated task. Although steady progress has been made in dealing with the large unit cells in high-level first-principle studies [405], the usual method for obtaining the band structure is based on the extended H¨uckel tight-binding approximation. These calculations result in surprisingly reliable band structures for most 1D and 2D organic metals, at least for the dispersion relations in the neighborhood of the Fermi energy. The calculated band structure and the 2D Fermi surface (FS) for (TMTSF)2PF6 is shown in Fig. 13.64. For different anions, the dispersion relations and the FS are almost indistinguishable [406]. For the Bechgaard salts, there is a strong mutual interaction between the 1D electronic system and the three-dimensional lattice. In general, metals with a pronounced 1D character have a tendency to de-

velop an insulating ground state in a combined lattice and electronic phase transition. They are unstable against a Peierls transition, which is a perturbation with a wave (or nesting) vector of 2kF , where kF is the Fermi wave vector. This perturbation can either be a charge redistribution, i.e., a charge density wave (CDW), or a spin redistribution (SDW). This leads to a reconstruction of the crystallographic or magnetic unit cell resulting in a modified band structure and Fermi surface. Whether a Peierls transition occurs and at which temperature crucially depends on the dimensionality of the electronic system. For an ideal one-dimensional metal, a Peierls transition is inevitable. Most of the 1D Bechgaard salts show a Peierls transition at ambient pressure. However, this metalinsulator transition occurs at rather low temperatures, TMI ∼ 10 K (Fig. 13.65) since, as stated above, an appreciable overlap along the b direction leads to an increased dimensionality and reduces the electronic energy gained at the Peierls phase transition. The transition is clearly seen in the electrical-resistivity and magnetic-susceptibility data (Fig. 13.65) [407, 408]. These data prove the SDW nature of the distortion [409]; in contrast to an isotropic decrease of the magnetic susceptibility expected for a CDW, the susceptibility vanishes exponentially only along the b direction (inset of Fig. 13.65(b)) 13.65(b)) [408]. Such behavior is characteristic of an antiferromagnetically ordered state with alternating spin orientation along the b direction. This type of spin ordering has been confirmed by a number of additional experimental techniques,

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Fig. 13.65. (a) Temperature dependence of the resistivity for various (TMTSF)2 X salts [407].Below the Peierls transition, the resistivity shows insulating behavior. (b) Temperature dependence of the static spin susceptibility of (TMTSF)2 AsF6 . The inset shows the exponential decrease of the susceptibility in the b direction (perpendicular to a and c∗ ) below the metalinsulator transition [408]

and by the fact that the measured SDW nesting vector is compatible with the calculated band structure shown in Fig. 13.64 [389,410]. As mentioned previously, moderate pressure of a few kbar suppresses the SDW state and superconductivity evolves. The generic pressure-temperature (P − T) phase diagram for the (TM)2X salts is shown in Fig. 13.66. TM stands for TMTSF or TMTTF (= tetramethyltetrathiofulvalene), the sulfur analogue of TMTSF. For the organic metals and superconductors, the crucial parameter is pressure, not chargecarrier concentration, as is the case for the cuprate superconductors (see Sect. 13.5). An increasing pressure is equivalent to a better overlap of the molecular wave functions which results in a reduced anisotropy and,consequently,in an increased dimensionality. The only Bechgaard salt which becomes superconducting at ambient pressure (Tc = 1.2 K) is (TMTSF)2ClO4 [411]. In order for this to occur, the material has to be cooled slowly to allow the randomly oriented low-symmetry ClO4 anions to order at about 24 K, where a superlattice is formed with a doubled lattice parameter along the b direction [389]. If, on the other hand, the sample is quenched rapidly with a cooling rate greater than 50 K/min, the hightemperature anion disorder persists down to low temperature and superconductivity does not appear.

Fig. 13.66. Schematic pressure-temperature phase diagram of the (TM)2 X salts based on TMTSF and TMTTF. The arrows indicate the ambient pressure of various molecular crystals. LOC = charge-localized insulator, SP = spinPeierls, SDW = spin density wave, AF = antiferromagnetic insulator, and SC = superconductor

TMTTF-based salts are electronically more one dimensional than the TMTSF analogues. For (TMTTF)2PF6 , which is located at the far left side of the phase diagram in Fig. 13.66 at ambient pressure, an insulating state (LOC) occurs at high tempera-

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tures of about 230 K. The localization of the interacting electrons is believed to be due to a Mott–Hubbard transition. At lower temperatures, a spin-Peierls (SP) transition takes place resulting in a dimerization of the antiferromagnetically ordered spin chains. In recent sophisticated high-pressure studies [412–414], two groups were able to tune (TMTTF)2PF6 through all of the phases shown in Fig. 13.66. Indeed, at pressures of about 50 kbar superconductivity was induced. These data validate the P − T phase diagram that was originally proposed by J´erome on the basis of experimental data obtained for different (TM)2X salts [415]. More information on the directional dependence of the strongly anisotropic molecular interactions can be gained by uniaxial-pressure studies. However, these are hampered by the high fragility of the organic superconductors. One of the few uniaxial-pressure investigations was reported for (TMTSF)2PF6 [416]. Surprisingly, the strongest suppression of the SDW transition temperature was found for pressure applied along the most-conducting a direction. The larger overlap along a results in an increased bandwidth and, concomitantly, in a decreased density of states and, therefore, a reduced energy gain in the electronic system at the SDW transition. Apparently, this effect is more important than the simultaneous increase in Fermisurface nesting [417]. A more detailed overview on uniaxial-pressure studies is given in [418]. Magnetic fields can have a strong influence on the electronic properties of the organic superconductors. One fascinating phenomenon is the occurrence of field-induced spin density waves (FISDW) which become evident in Shubnikov–de Haas (SdH)-like oscillations in the longitudinal resistivity and in quantized steps of the Hall resistivity. Indeed, this was the first observation of a quantum Hall effect in a bulk crystal [419,420]. Stimulated by these intriguing results, a theory now commonly known as the “standard model” for FISDW was developed [421]. The principal effect of the applied magnetic field is an effective reduction of the electronic dimensionality. Therefore, the magnetic field can counterbalance the pressure-increased dimensionality leading to a transition into a SDW state at high enough field strengths.

For a more thorough discussion of this phenomenon, see [389]. In the 1D organic metals, a number of electricaltransport properties exist that cannot be explained by the conventional Fermi-liquid theory. In fact, for 1D metals a non-Fermi-liquid state,i.e.,a spin-charge separated Luttinger liquid,is predicted.The observed field and temperature dependences of the resistivity [422,423] as well as recent thermal-conductivity data [424] have been interpreted along these lines.At zero field,(TMTSF)2PF6 is believed to be a marginal threedimensional Fermi liquid which can be destabilized by the application of small fields in certain directions [425, 426]. Above a threshold field (∼ 0.2 T along the b direction), electronic transport shows incoherent non-Fermi-liquid properties. Consequently, the “normal state” of the Bechgaard salts behaves nonmetallically in the presence of small magnetic fields (see [427, 428] for recent reviews). This picture got support from thermal-transport experiments that for special field orientations showed Nernst signals several orders of magnitude larger than estimated from usual Fermi-liquid models [429]. Quasi-Two-Dimensional Organic Metals In contrast to the isostructural 1D Bechgaard salts, a number of different crystal structures exist for the ET-based charge-transfer salts. They mainly differ in the packing motifs of the nonplanar ET molecules and are labeled by different Greek letters (see Table 13.6). [389,390]. Figure 13.67 shows the schematic crystal structure of ˇ-(ET)2 I3 , a typical example of most (ET)2 X salts. The common structural feature is the packing of the ET molecules into layers which are separated by poorly conducting anion planes (I−3 planes in Fig. 13.67). The principal difference between the crystal structures of the ET families (˛, ˇ, ˇ  , , etc.) is the arrangement of the ET donors within the layers. There are, however, always sufficiently close contacts between the  orbitals of the sulfur atoms of neighboring intralayer ET molecules. This results in the formation of molecular bands with only relatively small in-plane anisotropies (typically within a factor of two). The anions which are responsible for

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Table 13.6. Crystallographic data and Tc for some organic metals based on the donor molecules TMTSF and ET. Z is the number of formula units per unit cell. For Tc the usually stated values are given independent from the experimental determination. The actual thermodynamic values may differ slightly. For the ET-based compounds the out-of-plane lattice parameter is underlined Compound

sp-gr.

(TMTSF)2 PF6 (TMTSF)2 ClO4 ˇ-(ET)2 I3 ˇ-(ET)2 IBr2 -(ET)2 I3 -(ET)2 Cu(NCS)2 -(ET)2 Cu[N(CN)2 ]Br -(ET)2 Cu[N(CN)2 ]Cl ˛-(ET)2 KHg(SCN)4 ˛-(ET)2 NH4 Hg(SCN)4 ˇ  -(ET)2 SF5 CH2 CF2 SO3

P 1¯ P 1¯ P 1¯ P 1¯ P21/c P21 Pnma Pnma P 1¯ P 1¯ P 1¯

a(Å)

b(Å)

c(Å)

7.297 7.266 6.615 6.593 16.387 16.256 12.949 12.977 10.082 10.091 9.260

7.711 7.678 9.100 8.975 8.466 8.456 30.016 29.977 20.565 20.595 11.635

13.522 13.275 15.286 15.093 12.832 13.143 8.539 8.480 9.933 9.963 17.572

˛(◦ )

ˇ(◦ )

(◦ )

Z

83.39 84.58 94.38 93.79 90 90 90 90 103.70 103.67 94.69

86.27 86.73 95.59 94.97 108.56 110.28 90 90 90.91 90.47 91.70

71.01 70.43 109.78 110.54 90 90 90 90 93.06 93.30 103.10

1 1 1 1 2 2 4 4 2 2 2

Tc (K) [404] [430] [431,432] [433] [434] [435] [391] [392] [436] [437] [438]

0.9a 1.2 1.4 2.8 3.5 10.4 11.5 12.8b –c 1.1 5.2

this value was reported in [388] for a pressure of 12 kbar. A pressure of ∼ 6 kbar is however sufficient to suppress the SDW state and enable superconductivity with Tc ≈ 1.1 K. See, e.g., [439,440] b at a pressure of ∼ 0.3 kbar c recently superconductivity with Tc ≈ 100 mK was reported to occur at 2.5 kbar [441] a

Fig. 13.67. Characteristic layered structure of the quasi-two-dimensional (2D) organic metals, exemplified by the crystal structure of ˇ(ET)2 I3 . The short distances between the sulfur atoms of different ET molecules result in a good conductivity within the ET layers. The ET layers are separated by the poorly conducting anion layers, schematically shown on the right side

the charge transfer lead to partially filled molecular bands, enabling metallic conductivity. The anion layers are sandwiched between the highly conducting ET layers. This results in very strong anisotropies of the physical properties parallel and perpendicular

to the layers. The electrical resistivity perpendicular to the layers is larger by about 3−6 orders of magnitude than for electronic transport within the ET layers [389,399]. One should mention, however, that the exact value of this anisotropy is difficult to deter-

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Fig. 13.68. Calculated energy dispersions and Fermi surfaces for (a) ˇ-phase and (b) -phase ET salts (after [443,444] and [445])

mine since any dislocation disturbing the in-plane transport will immediately lead to an enhanced inplane resistance [401,442]. This 2D structure, which is also reflected in the superconducting properties, is observed in all ET phases and resembles that of many layered cuprates, such as Bi2Sr2 CaCu2O8−ı [396]. Although the layered structure occurs in all ETbased superconductors,the varying in-plane packing motifs of the organic molecules lead to different 2D electronic structures. The overlap integrals vary considerably depending on whether the ET molecules are aligned face-to-face or side-by-side [443, 444]. The in-plane transfer energies can be obtained from these overlap integrals that, together with the known crystal structures, allow the in-plane electronic band structures to be calculated using the rather simple, but well-established extended H¨uckel tight-binding approximation. Two examples of the resulting energy dispersions and of the in-plane Fermi surfaces are shown in Fig. 13.68. For the ˇ phase, one electron per unit cell is transferred to the anion leaving the highest occupied molecular orbital half filled. The nearly isotropic inplane overlap integrals lead to an almost circular free-electron-like FS (Fig. 13.68(a)) [443, 444]. The

validity of this result has been verified by measurements of magnetic quantum oscillations and angular dependent magnetoresistance oscillations [410,446]. Indeed, only one closed orbit with the predicted area (shaded area in Fig. 13.68(a)) and only slightly more cornered shape [446, 447] could be reconstructed from the data. For the  phase, the resulting FS is almost as simple as for the ˇ phase. Here, the ET molecules form dimers where adjacent pairs in the plane are rotated by approximately 90◦ with respect to each other. This doubles the unit cell, resulting in two transferred electrons on two highest occupied molecular orbitals (see the dispersion relation in Fig. 13.68(b)). The resulting FS again resembles the free-electron picture where the FS cuts the first Brillouin zone with small gaps opening at the zone boundary due to Bragg reflections [445]. In fact, for -(ET)2 Cu(NCS)2 , larger gaps at the zone boundary are found due to the lack of a center of symmetry [448]. Again, de Haas– van Alphen (dHvA) and Shubnikov–de Haas (SdH) experiments confirm the predicted band structures for the -phase materials as well as for most other phases. However, the calculated effective masses are often much smaller than the experimental values,

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sometimes close to the free-electron value. This is usually ascribed to the fact that many-body effects like electron–electron and electron–phonon interactions are typically not included in band-structure calculations. For more details on Fermi-surface studies, see the reviews in [410,446,449]. Although the extended-H¨uckel treatment has been successful in describing the in-plane electronic band structure of organic metals and superconductors, it leaves the interlayer band structure unclear. The large separation of the conducting organic layers by relatively thick anion layers has led to the question of whether a real 3D Fermi surface exists at all. When the interlayer transport is incoherent, i.e., when an electron loses its phase information between successive tunneling processes, no Bloch state can evolve and the band picture breaks down [450]. Although there is no experimental proof for this scenario, certain experimental features can be utilized to verify the coherent transport and the existence of a 3D Fermi surface [450,451]. The most direct test of the 3D nature of the FS, which even gives a number for the interlayer transfer integral t⊥ , is the detection of nodes in magnetic quantum oscillations.Figure 13.69(a) shows the dHvA signal for ˇ-(ET)2 IBr2 . The fast dHvA oscillations visible in the magnetization can be ascribed to the closed FS shown in Fig. 13.68(a). The existence of a corrugated FS, sketched in the inset of Fig. 13.69(a), results in two slightly different extremal orbits for the field H normal to the layer, leading to the beating nodes observed in the dHvA signal. At certain angles  of the field relative to the normal to the ET planes, only one extremal orbit perpendicular to H remains and, consequently, no beating node appears in the dHvA signal. The careful determination of the angular dependence of these nodes allows a reliable extraction of t⊥ [452]. For ˇ-(ET)2 IBr2 with a Fermi energy EF ≈ 0.11 eV, the transfer integral t⊥ = 0.4 meV is obtained. Although solid proof exists for the coherent motion of electrons in band or Bloch states in ˇ-phase ET salts, such evidence is missing for other organic metals. For example, the magnetization of ˇ  (ET)2 SF5CH2 CF2 SO3 shows sawtooth-like dHvA oscillations which follow the behavior expected for a 2D

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metal with fixed chemical potential almost perfectly (Fig. 13.69(b)) [453]. Furthermore, from the absence of any beating node in dHvA and SdH oscillations, which start at about 1.4 T deep in the superconducting state [454], any possible FS corrugation must be extremely small. Indeed, additional tests for the existence of a 3D FS failed for this material [455], making it a likely candidate for the 2D metal with incoherent interlayer transport envisioned by McKenzie and Moses [450]. Along these lines, deviations from the conventional Bloch–Boltzmann transport theory were observed in the interlayer magnetoresistance, i.e., a field-induced metal-insulator transition was found [456]. In spite of these highly unusual properties which contradict Fermi-liquid theory, the interlayer resistance at H = 0 is metallic over the whole investigated temperature range (see Fig. 13.70) and the measured quantum oscillations are consistent with an in-plane Fermi liquid. A possible realization of the FS of ˇ  -(ET)2 SF5 CH2CF2 SO3 is depicted in the inset of Fig. 13.69(b). For some organic superconductors, such as (ET)2 Cu[N(CN)2 ]Br, the interlayer resistivity increases upon cooling, reaching a maximum before metallic behavior sets in at lower temperature [389, 399]. There were speculations that this might be caused by antiferromagnetic fluctuations, especially since NMR measurements of the spin-lattice relaxation rate 1/T1 revealed a maximum in 1/(t1T) close to the temperature where the maximum in R was observed (see also the dashed line in Fig. 13.76 below) [457,458]. However, as was shown in a recent thorough study, a significant part of the scattering contribution giving rise to the anomalous resistivity maximum is extrinsic in nature [459]. For lowresistance variants of -(ET)2 Cu[N(CN)2 ]Br, grown by a different preparation route [459], no resistance maximum occurs. This metallic behavior all the way down from room temperature is presented for such a -(ET)2 Cu[N(CN)2 ]Br sample in Fig. 13.70, where the metallic resistance for two other organic superconductors is shown as well. At high temperature, there is no generic T dependence for the different organic metals. Towards lower T, however, the interlayer resistivity can fairly well be described by  = 0 + AT 2 (inset of Fig. 13.70).

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Fig. 13.69. (a) de Haas–van Alphen (dHvA) oscillations in the magnetization of ˇ-(ET)2 IBr2 at T = 0.4 K. The nodes in the signal are caused by the slightly corrugated FS perpendicular to the conducting ET planes.The schematic sketch in the inset greatly exaggerates this corrugation. (b) Torque as a function of magnetic field of ˇ  -(ET)2 SF5 CH2 CF2 SO3 . The dHvA signal (open circles in the upper inset) can be described almost perfectly by a 2D theory (solid line in the upper inset) The FS can be regarded as ideally 2D (lower inset)

This T 2 behavior, which is also observed for many other organic metals [389], is usually taken as a sign of a well-behaved Fermi liquid where scattering is dominated by electron-electron interactions. However, although electronic correlations certainly are present in these low-dimensional metals, the wide temperature range over which the T 2 behavior can be found is not typical. Usually, this temperature dependence of  can only be observed at very low temperatures. The coefficient A depends on the strength of the electron–electron interaction and is roughly proportional to 2 , where is the Sommerfeld coefficient of the specific heat [188]. As discussed in Sect. 13.3, for narrow-band systems like

heavy-fermion metals, where electronic correlations are undoubtedly much more dominant than for organic metals, A/ 2 ≈ 10−5 § cm(mol K/J)2 . For transition metals, A/ 2 is even a factor 100 smaller. Using a Sommerfeld coefficient ≈ 25 mJ/mol K2 derived from the specific heat of the organic metals (see below), A ≈ 6 × 10−9 § cm K−2 would be expected for narrow-band metals. This is 6 to 8 orders of magnitude smaller than is actually observed for the organic metals (A ≈ 4 m§ cm K−2 for the -phase salts and about 100 times larger for the ˇ  salt). When using the in-plane resistivity data, A is still at least two orders of magnitude larger than expected. Thus, it seems highly questionable to seek the origin of

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705

Fig. 13.70. Temperature dependence of the interlayer resistance R, normalized by R at 270 K, for three different organic superconductors. The inset shows the resistivity  as a function of T 2 . Note that  for ˇ  -(ET)2 SF5 CH2 CF2 SO3 has been divided by 100

this T 2 behavior of the resistivity solely within the electronic system [460]. Scattering of 2D electrons at phonons might be the relevant mechanism. Interestingly, a number of other novel superconductors, including MgB2 and the borocarbides, reveal a T 2 dependence as well, hinting at a possible common mechanism. A further point which should be noted is the extraordinarily high interlayer resistivity  of the organic metals. For -(ET)2 Cu[N(CN)2 ]Br,  ≈ 1 § cm just above Tc , which is about a factor of 106 larger than for lead at Tc . For ˇ  -(ET)2 SF5CH2 CF2 SO3 ,  is another factor ∼ 50 larger. The mean-free path derived from these values of  according to the usual transport theory is much smaller than the interatomic distances.This effect becomes even more pronounced towards higher temperatures.Again one has to ask, in analogy to the high-Tc cuprates, whether the interlayer transport can be understood in terms of the usual Fermi-liquid theory. 13.4.3

Superconducting-State Properties

The organic superconductors, as well as all other novel superconductors discussed here, are strongly type-II superconductors. The lower critical fields, Hc1 , of the organic superconductors are only a few mT and the upper critical fields, Hc2, are in the Tesla range. These fields depend strongly on the direction of the applied field. For H applied within the planes, perpendicular to the least-conducting direction, Hc2

often exceeds the Pauli paramagnetic limit [16,17].As will be discussed below in more detail, this is taken as an indication for triplet pairing in the 1D materials. For the 2D organic superconductors, on the other hand, many-body effects seem to explain the observed moderate Hc2 enhancement. Above a certain threshold field Hirr within the Shubnikov phase, the magnetic flux can move freely leading to dissipative phenomena and phase diagrams similar to those studied in great detail for the high-Tc materials. In spite of their moderate values of Tc , the layered structure and the high purity of the organic superconductors lead to a large reversible range in the mixed state. In contrast to the high-Tc superconductors, the upper critical fields of the organic superconductors are relatively low and more experimentally accessible. Thus, we can conveniently study the complete superconducting phase diagrams and the rich physics of the vortex phase. Indeed, as previously observed in high-Tc cuprates, the first-order vortex-lattice melting transition close to Hirr was resolved in two -phase ET superconductors by means of sophisticated magnetization experiments [461,462].Another common feature observed for the highly anisotropic cuprates [463] and the equally anisotropic ET superconductors is the occurrence of Josephson plasma resonances in magnetooptical experiments [464].

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1D Organic Superconductors All the known 1D Bechgaard superconductors, except for (TMTSF)2ClO4 , need pressure above at least 6 kbar to become superconducting at transition temperatures around 1 K. The amount of pressure needed to suppress the insulating ground state depends on the quality of nesting, i.e., on the electronic energy which can be gained at a Peierls transition. In other words, a certain degree of two-dimensionality is necessary to allow superconductivity to appear. In fact, an almost linear relationship has been observed between the lattice parameter b (see Fig. 13.63) and the critical pressure which suppresses the SDW state and induces superconductivity [465]. A further increase in pressure P results in a rapid decrease of Tc on the order of dTc /dP ≈ 0.1 K/kbar.A decreasing Tc with pressure is a common feature in many conventional superconductors, however, with a much smaller dTc /dP. This can be explained by stiffening of the lattice under pressure that results in an increase of the average phonon frequency. This lattice stiffening is expected to be especially strong in soft organic crystals. Since in conventional superconductors Tc depends sensitively on the phonon frequency, Tc decreases. For the highly anisotropic organic metals, a correspondingly strong directional dependence of the critical fields is observed. The lowest upper critical fields are found for H aligned along the c ∗ direction (Hc2 ≈ 0.16 T for (TMTSF)2ClO4 ) [389]. For this field orientation, orbital effects, i.e., diamagnetic currents running in the ab plane, limit Hc2. For magnetic fields along b and a, this orbital limiting is largely suppressed due to the short c ∗ -axis coherence length. Consequently, in early experiments upper critical fields of ∼ 2 and ∼ 2.8 T, respectively, were reported [466].This result was discussed in light of a possible Pauli-limited Hc2, suggesting that the Cooper pairs in the Bechgaard salts are in the singlet state. Neglecting spin-orbit coupling and manybody effects, the Pauli (or Clogston) limit is given by Hp = (1.85 T/K)Tc [16, 17], i.e., Hp ≈ 2.3 T for (TMTSF)2ClO4 with Tc = 1.25 K. Since Hc2 displays a strong anisotropy, an exact alignment of the field with respect to the principal axes, a, b , and c ∗ is required for its correct de-

Fig. 13.71. H − T phase diagram of (TMTSF)2 PF6 at an applied pressure of 6 kbar for fields aligned along the a, b , and c∗ directions (see inset) The data are extracted from electrical-resistance measurements [439,440]

termination. Indeed, by carefully aligning H. Lee et al. proved impressively that for (TMTSF)2PF6 under pressure, Hc2 definitely exceeds the Pauli paramagnetic limit (Fig. 13.71) [439, 440]. This occurs for H oriented along the a direction but even more pronounced for H aligned along the b direction. For these field orientations, the orbital motion of the charge carriers is greatly suppressed due to the weak interplane transfer. The H −T phase diagram was extracted from electrical-resistance data. The validity of this determination has recently been verified for (TMTSF)2ClO4 utilizing torque magnetization measurements [467]. The observed strong Hc2 enhancement triggered the proposal by Lebed of triplet or p-wave pairing occurring in the 1D organic superconductors [468]. In order to provide further evidence for triplet pairing,a joint effort by several different groups tackled the challenging task of performing NMR measurements under pressure at dilution-refrigerator

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Fig. 13.72. 77 Se NMR spectra below and above Tc for (TMTSF)2 PF6 with a magnetic field H = 2.38 T aligned along the b -axis. The center of the NMR peak is virtually independent of temperature (solid line), whereas the hashed region marks the expected range where it should occur for a singlet ground state below Tc [440,469]

temperatures [469,470]. The resulting 77 Se NMR data revealed no observable change in the Knight shift between the normal metallic and the superconducting state (Fig. 13.72). This provided strong support for triplet pairing since for singlet Cooper pairs with antiparallel spins, the Knight shift, which depends linearly on the electron-spin susceptibility, should drop rapidly below Tc . Indeed, the unchanged Knight shift observed previously for Sr2 RuO4 [321] is one of the primary pieces of evidence for a triplet state in this superconductor. Stimulated by the experiments for (TMTSF)2PF6, Lebed suggested a triplet vector order parameter with the spins oriented along the baxis [471]. This proposal fits nicely with the observed features that a magnetic field applied along the b direction cannot destroy superconductivity. It further predicts a finite energy gap over the whole FS, which is consistent with thermal-conductivity results for (TMTSF)2ClO4 [472]. In another work, spin-triplet f -wave-like pairing was suggested, which should be

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in competition with singlet d-wave pairing [473]. It should be noted that previous measurements already showed a strongly enhanced Hc2 for (TMTSF)2AsF6 [474]. This and the suppression of superconductivity by defects and by chemical substitution motivated early suggestions of a possible p-wave symmetry in the 1D organic metals [475,476].Very recently, however, the same group that performed the NMR shown in Fig. 13.72 carefully checked their results. In (TMTSF)2ClO4 they found a clear change of the Knight shift at low-enough magnetic fields in the superconducting state [711].This clearly proves the singlet nature of the Cooper pairing. It remains a challenge to understand the high-field zero-resistance state in the Bechgaard salts where the Knight shift remains as in the normal state. Although the experimental data now point to a singlet scenario,no consensus on the nature of superconductivity in the Bechgaard salts has been reached [477]. In other theoretical models it was shown that even conventional s-wave pairing with an appropriate anisotropic orbital momentum may lead to the critical-field curve shown in Fig. 13.71 [478]. Another possibility is an unconventional d-wave state where the line nodes of the d-wave gap function are assumed to be eliminated by the anion ordering in (TMTSF)2ClO4 [479]. 2D Organic Superconductors Equally intense and even more controversial is the discussion on the nature of the superconducting state in the 2D organic superconductors. However, before discussing this issue in more detail some principal superconducting properties as well as some highfield peculiarities will be presented. Superconducting Properties A typical magnetic phase diagram of the 2D organic superconductors is exemplified for (ET)2 Cu[N(CN)2 ]Br in Fig. 13.73. The magnetic field is applied perpendicular to the ET planes.Hc1 and Hc2 are obtained from dc-magnetization measurements, Mdc , whereas Hirr is derived from measurements of ac susceptibility, ac [480,481]. The magnetization is

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Fig. 13.73. Temperature dependences of Hc1 , Hc2 , and of the irreversibility field, Hirr of (ET)2 Cu[N(CN)2 ]Br. The data points (•, ) are from [480]. The solid line is an exponential fit, Hirr ∝ exp(−AT/Tc ), at low T. A T 3/2 dependence near Tc is denoted by the dashed line

broadened considerably in fields due to superconducting fluctuations, and a scaling analysis was performed in order to obtain Hc2 (Fig. 13.73) [480], in a manner similar to that done for the high-Tc cuprates. The lower critical field is often extracted from the first deviation from perfect diamagnetism. For the strongly type-II superconductors, like the organic and high-Tc superconductors, however, this may overestimate Hc1 since above Hc1, flux can only penetrate into the sample over a final energy barrier. This leads to magnetic relaxation effects which become increasingly slow close to Hc1 [399,481]. Since no time dependence of M is expected to occur in the Meissner state, Hc1 is defined as the field at which the magnetic relaxation vanishes [481]. The Hc1 data are depicted in Fig. 13.73. For -(ET)2 Cu[N(CN)2 ]Br, an upper critical field of (12±2) T and a lower critical field of (3±1) mT for T = 0 can be estimated. From these results the thermodynamic critical field Hc ≈ 100 mT, the in-plane coherence length  ≈ 5 nm, and the Ginzburg– Landau parameter  ≈ 100 can be derived. With an average in-plane lattice parameter on the order d ≈ 1 nm,  /d is approximately 5, which is considerably smaller than the value of ∼103 typical for pure metals, but comparable to what is found in high-Tc cuprates and other “exotic” superconductors. The BCS coherence length is given by 0 = 0.18vF /kB Tc ≈ 6.7 nm, using an estimate of the Fermi velocity vF ≈ 5.6 × 104 m/s from dHvA data.

With a mean-free path l of 200 to 400 Å, determined from magnetic quantum oscillations,this organic superconductor is just barely in the clean limit with l/0 ≈ 5.In this case,one expects  = 0.740 ≈ 5 nm which agrees quite well with the experimental value for  . For fields parallel to the ET layers, Hc1 is considerably smaller than the earth’s field, making it nearly impossible to measure [482]. From the estimated value of Hc2 at T = 0 for in-plane field orientations, an approximate interplane coherence length of ⊥ ≈ 0.5 nm can be determined, which is much smaller than the layer distance of about 1.6 nm [396]. For other 2D organic superconductors,similar superconducting parameters have been found [402]. The physics of vortex dynamics in type-II superconductors, extensively studied for the high-Tc cuprates, has also been investigated in detail for the organic metals.Above the threshold field Hirr of typeII superconductors, vortices are free to move and, therefore, the magnetization is reversible. Below Hirr , the vortices are pinned, resulting in magnetic hysteresis, a diamagnetic ac-susceptibility signal, and a vanishing electrical resistivity. Since the depinning of the vortices is an activated process, slightly different values for Hirr are obtained depending on the method used and the criterion applied. Nonetheless, the gross temperature and angular dependences of Hirr are largely unaffected by these unambiguities [481, 482]. The Hirr data in Fig. 13.73 show a clear crossover at T/Tc ≈ 0.75, independent of the

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Fig. 13.74. The interlayer magnetoresistance of (a) ˇ  -(ET)2 SF5 CH2 CF2 SO3 and (b) (ET)2 Cu[N(CN)2 ]Br for fields applied perpendicular to the ET planes with strong peaks before superconductivity occurs

method used to obtain the data. The same behavior has been observed for another -phase organic superconductor as well [483]. A crossover from 3D vortex fluctuations to 2D fluctuations is predicted at a magnetic field Hcr ≈ 4¥0 / 2 d 2 ≈ 0.08 T, where ¥0 = 2.07 × 10−15 Tm2 is the magnetic-flux quantum, d ≈ 1.6 nm is the interlayer distance, and ≈ 200 is the anisotropy parameter estimated from magnetictorque measurements for -phase superconductors [484,485]. This prediction is in excellent agreement with the experimental value of Hcr ≈ 0.05 T. Thermally activated 3D vortices are believed to be responsible for the power-law T dependence of Hirr ∝ T 3/2 close to Tc (Fig. 13.73) [486], whereas the low-temperature exponential behavior can be understood by the breakdown of proximity-induced superconductivity (see [481] for details). This suggests that superconductivity is induced in the normalconducting anion layers by the superconducting ET layers. Above a certain threshold field Hirr for this proximity-induced superconductivity, the superconducting layers are effectively decoupled, and the vortices are weakly pinned 2D objects (pancake vortices). Measurements of the angular dependence of Hirr have verified that in the low-T 2D pinning regime, Hirr () exhibits 2D behavior as predicted by Tinkham for Hc2() of 2D films [204,487]. Close to Tc , the Lawrence–Doniach model [488] for 3D anisotropic superconductors describes the data well [481,483]. In contrast to most conventional superconductors, the short coherence lengths and large anisotropies

in 2D organic superconductors lead to strong thermal fluctuations close to Tc , clearly visible in the electrical resistivity, the diamagnetic signal, and the specific heat. Estimates for the width of the temperature regime where thermal fluctuations are relevant can be made using the Ginzburg criterion, G = |T − Tc |/Tc ∝ (Tc Hc−2 −2 ⊥−1)2 . For the ET superconductors with Tc ∼ 10 K, a value of G ≈ 10−2 results,which is comparable to values found for highTc cuprates where the larger Tc is compensated by the correspondingly larger thermodynamic critical field Hc [395,396,399,402].For elemental superconductors, G is of the order 10−8 due to the much larger coherence lengths. The fluctuation width becomes much broader in applied magnetic fields, an effect visible in resistivity and magnetization data, and it has been studied extensively in both organic and high-Tc superconductors [389,396]. One peculiar feature of the 2D organic superconductors is the occurrence of pronounced peaks in the interlayer resistance R close to the superconducting phase boundary (Fig. 13.74), which is believed to be caused by thermal fluctuations. With decreasing field, a peak in R is observed before superconductivity sets in. This peak is observed over a range of temperatures up to Tc . Although a number of models have been suggested (see [395,399] for an overview), a feasible explanation is based on the fluctuation-induced decrease of the normal density of states at the Fermi level. Combined with the other conductivity contributions caused by fluctuations (Aslamazov–Larkin and Maki–Thompson contribu-

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tions), the observed behavior can be qualitatively reproduced taking into account the high anisotropy of the superconductors [395,489,490]. Nevertheless, the origin of this peak effect in the resistance measurements is still not well understood. Measurements of the physical properties of 2D organic superconductors in high magnetic fields reveal a number of intriguing features. For the organic superconductor -(ET)2 Cu(NCS)2 , measurements using a tuned-circuit differential susceptometer revealed a change in the vortex rigidity at in-plane magnetic fields of about 22 T [491]. This has been suggested to indicate the onset of a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. This FFLO state is predicted to evolve at high magnetic fields when an attractive interaction between the Zeeman-split spin-up and spin-down quasiparticles leads to pairs with nonzero total momentum [238, 239]. (See also the discussion above for CeCoIn5 .) 2D organic superconductors are good candidates for this scenario when the magnetic field is applied in plane, which prevents the appearance of orbital pair-breaking effects. Indeed, very recently specific-heat measurements revealed clear thermodynamic evidence for the existence of an FFLO state in -(ET)2 Cu(NCS)2 [712].

A second fascinating aspect of the high-field work on organic superconductors is the observation of field-induced dissipation-less diamagnetic currents in the organic metal ˛-(ET)2 KHg(SCN)4 [492, 493]. This material is now believed to be in a CDW state at low temperatures (below 8 K) and fields below about 23 T [494,495]; the material stays metallic with only a partial nesting of the 1D FS sheets. Early experimental data were interpreted in favor of a SDW state (see [410] for details). At higher fields above 23 T, when the Zeeman splitting deteriorates the nesting condition for the spin-up and spin-down bands, a spatially modulated incommensurate socalled CDWx state is believed to evolve (analogous to the FFLO state for superconductors) [496].It is within this high-field, low-temperature state that properties resembling a superconducting vortex state appear.More precisely,a field-induced diamagnetic signal [493] and hysteresis loops comparable to the irreversible region of the Shubnikov phase are observed [492, 494, 497]. Although Fr¨ohlich superconductivity has been suggested [401], further justification of this proposal is absent. Recently, Harrison has suggested that the coexistence of the CDW with a strongly field-dependent orbital magnetization, caused by the Landau quantization of the 2D

Fig. 13.75. Left panel: In-plane field dependence of the resistance R of (BETS)2 FeCl4 for different temperatures from 0.8 to 5.4 K. Right panel: The H − T phase diagram indicates that after the suppression of an antiferromagnetic insulating (AI) low-field state with magnetic field, field-induced superconductivity (FISC), referred to in the text as magnetic field induced superconductivity (MFIS), is observed between about 18 to 41 T [500]

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FS, may lead to nonequilibrium persistent currents [498]. This effect might explain the hysteresis loops observed in ˛-(ET)2 KHg(SCN)4 . In any case, the new and interesting physics of this field-induced phase deserves further research. Finally, magnetic field induced superconductivity (MFIS) has been found between 18 and 41 T in -(BETS)2 FeCl4 [499,500]. At first glance, the evolution of a superconducting state in a magnetic field is surprising since the magnetic field usually destroys superconductivity. In the first experiment in magnetic fields up to 20 T [499], the increase of Tc with field led to speculations about unconventional pairing interactions caused by magnetic fluctuations via the Fe moments. In another experiment in fields up to 42 T, however, a more complete superconducting phase diagram evolved as shown in Fig. 13.75 [500]. At the lowest temperature (about 0.8 K), the resistance drops to zero at an in-plane field of about 18 T and becomes finite again at a field of about 41 T. The superconducting phase could be resolved up to about 4 K. It was also shown that only the in-plane field component is relevant for the occurrence of superconductivity. One possible explanation for the MFIS observed in -(BETS)2 FeCl4 is the Jaccarino–Peter fieldcompensation effect [4], as mentioned in [499]. In this scenario the applied magnetic field aligns the S = 5/2 Fe3+ moments, which impress an exchange field J S on the spins s of the conduction electrons via the exchange interaction J S · s, as described in Sect. 13.2.4. For J < 0, the applied magnetic field becomes compensated by the exchange field and may reach values less than the Pauli paramagnetic limit, thereby allowing the field-induced superconducting state to occur.One of the most striking examples of the Jaccarino–Peter effect, is provided by the Chevrel-phase superconductor Eux Sn1−x Mo6S8 containing small quantities of Br or Se (see Fig. 13.20) [101]. In this case, superconductivity with Tc ≈ 3.9 K is first destroyed at about 1 T and then restored above about 4 T but only below 1 K. In -(BETS)2 FeCl4 , the antiferromagnetic state has to be suppressed in high magnetic fields before the superconducting state can evolve out of a paramagnetic state.

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The Nature of the Superconducting State For the 2D organic superconductors, the controversial experimental situation still prevents a definite conclusion as to whether conventional BCS or unconventional superconductivity exists. It is generally accepted that the Cooper pairs are in a singlet spin state. Evidence for a superconducting ground state with singlet spin pairing includes a vanishing spin susceptibility detected by NMR experiments [501–503] and the value of the paramagnetic-limited upper critical field [504, 712]. For the latter analysis, however, the above mentioned weak-coupling estimate is not appropriate, since many-body effects have to be taken into account. An elegant determination of the Pauli limit has been performed by Zuo et al. using known thermodynamic quantities such as the superconducting condensation energy from specific-heat data and the Pauli spin susceptibility and is taken as further evidence for a singlet ground state [505]. One motivation for suggesting an unconventional order parameter, i.e., of singlet d-wave symmetry, is the nature of the P − T phase diagram of the -phase superconductors first proposed in [506] (Fig. 13.76).

Fig. 13.76. Schematic P − T phase diagram for the -(ET)2 X salts depicting regions of antiferromagnetic insulator behavior (AFM) and superconductivity (SC). The arrows indicate starting points for the different materials at ambient pressure

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Next to the superconducting state an antiferromagnetic insulating state exists, similar to that observed for the 1D Bechgaard salts and reminiscent of the phase diagram of many high-Tc materials. However, there is only one -phase material known that displays an antiferromagnetic insulating state at ambient pressure that can be suppressed by pressure. For -(ET)2 Cu[N(CN)2 ]Cl, a pressure of 300 bar is sufficient to induce superconductivity at a record Tc of about 12.8 K (Fig. 13.76) [392]. A further increase in the hydrostatic pressure rapidly suppresses the superconducting ground state. In fact, the compound -(ET)2 Cu(NCS)2 displays the largest pressure dependence of Tc yet observed for any superconductor, Tc /p ≈ −3 K/kbar [507]. Thermalexpansion measurements revealed that Tc is most sensitive to uniaxial pressure perpendicular to the planes [508, 509] indicating that, at least for the  phase, an enlargement of the anion-layer thickness might increase Tc even further. However, as evidenced by the P−T phase diagram,the -phase materials containing large anions reside close to an insulating state, which is proposed to be a Mott–Hubbard insulating state driven by the small bandwidth and a large Coulomb interaction [397, 398, 510–512]. This metal-insulator transition — under intense study now — reveals an unusual criticality beyond the known universality classes [513]. Additional studies are necessary to gain a better understanding on the nature of this metal-insulator transition in 2D organic metals. A key experiment that provided evidence for the electron–phonon interaction as the pairing interaction in the early superconductors was the detection of an isotope effect, i.e., the increase of Tc proportional to m−1/2 as predicted by BCS theory. For the organics, m is some average molecular mass. At first sight, a somewhat confusing picture exists for the 2D organic superconductors.Substitution of all eight hydrogens in the ET ethylene end groups by deuterium causes an enhancement of Tc by about 0.3 K for three different organic superconductors (Fig. 13.77) [514]. This effect is not only inverse to what is expected but, in addition, the size of the Tc shift is much larger than would be explicable by a simple molecular-mass change. Apparently, the interaction

Fig. 13.77. Representative superconducting transition curves measured by ac susceptibility for (A) L (ET)2 Ag(CF3 )4 , (B) ˇ  -(ET)2 SF5 CH2 CF2 SO3 , and (C) (ET)2 Cu(NCS)2 . Plus symbols are for hydrogenated, filled squares for deuterated samples [514]

between the outer parts of the ET molecules and the anions plays a crucial role in the pairing strength. This interaction is expected to be rather sensitive to uniaxial pressure perpendicular to the planes. As mentioned, this was indeed confirmed by thermalexpansion measurements which extracted an initial reduction of Tc by 6 K/kbar for ˇ  -(ET)2 SF5 CH2CF2 SO3 and -(ET)2 Cu(NCS)2 [509]. With this result in mind, the inverse isotope effect can be understood by the smaller zero-point displacement of the carbon-deuterium bonds compared to the lighter carbon-hydrogen bonds. This results in a reduced internal lattice pressure and, consequently, in an increased Tc for the deuterated samples [514]. In order to avoid any changes of the anionhydrogen interaction, experiments were carried out on samples in which only central atoms of the ET molecules were exchanged. Substitution of four 12 C atoms by 13 C and all eight 32 S atoms by 34 S resulted in a reduction of Tc of about 1% [515]. This agrees well with the BCS expectation and provides strong support for a Cooper-pair coupling which is at least partially mediated by phonons. Additional support for this scenario comes from Raman data which showed a clear hardening of the Raman modes below Tc for -(ET)2 Cu[N(CN)2 ]Br [516] and from inelastic neutron-scattering experiments [517]. In the latter

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Fig. 13.78. Temperature dependences of the in-plane penetration depths extracted from (a) dc-magnetization measurements [518] and (b) by use of a high-frequency tunneldiode-oscillator technique [519]. The inset in (a) shows the low-T data in an enlarged scale together with model curves for BCS (solid line) and for two triplet states, t1 and t4 (broken curves) The data in (b) are for two samples of -(ET)2 Cu[N(CN)2 ]Br (1,2) and two samples of (ET)2 Cu(NCS)2 (3,4)

measurement a significant shift of a phonon branch was observed below Tc , an effect hardly seen this clearly in any other superconductor. Consequently,

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there are a number of experiments indicating an electron-phonon mechanism for superconductivity in organic superconductors, whereas the argument for a pairing mechanism mediated by antiferromagnetic fluctuations is largely based on the P − T phase diagram (Fig. 13.76). Apart from the coupling mechanism in the organic superconductors, the symmetry of the order parameter is a highly controversial issue. The experimental situation appears to be rather inconclusive and only some aspects shall be presented. For further discussion of this topic see [395,396,399–402]. One quantity which should reflect the symmetry of the order parameter is the magnetic penetration depth, (T). For a conventional BCS s-wave gap,  should exhibit an exponential temperature dependence, whereas for a superconducting gap with nodes a power-law T dependence is expected. The serious experimental problems in determining  reliably are illustrated in Fig. 13.78. In Fig. 13.78(a) the magnetic penetration depth is determined by dc-magnetization measurements [518]; whereas, in Fig. 13.78(b) the same quantity is extracted utilizing a tunnel-diode-oscillator technique [519]. In the former experiment by Lang et al., the magnetization data can be very well described by the conventional weak-coupling BCS theory as shown by the solid line in Fig. 13.78(a). The broken lines represent model curves for two earlier suggested triplet states. In the high-resolution data of Carrington et al. [519], a T 3/2 variation of  is found, in contrast to the BCS results of Lang et al. as well as other groups, and, in addition, different from the T 2 behavior previously reported by others. This inconclusive picture prevails in a number of other experiments utilizing different techniques which report either BCS or unconventional behavior of  (see [395, 396, 399–402] for more details). The temperature dependence of the spin-lattice relaxation rate, 1/T1, from NMR measurements is often used to obtain information about nodes in the superconducting energy gap. Usually, an increase of 1/T1 just below Tc is expected, known as the Hebel–Slichter coherence peak (see Fig. 13.80). Although the occurrence of this peak is an important proof of BCS superconductivity, its absence cannot disprove conventional superconductivity as ev-

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Fig. 13.80. Temperature dependence of the normalized spin-lattice relaxation rate.The solid line depicts the behavior for the weak-coupling BCS superconductor aluminum [523]. Data from 13 C NMR for -(ET)2 Cu[N(CN)2 ]Br in fields parallel to the conducting ET planes are shown as closed circles [501] and dashed line [502]. There is no Hebel–Slichter peak and the relaxation rate decreases as T 3 rather than exponentially

Fig. 13.79. Proton NMR spin-lattice relaxation rates T1−1 . (a) Temperature dependence of 1000/T1 for ˇH -(ET)2 I3 (Tc ≈ 8 K) [520, 521]. (b) Angular dependence of 1/T1 for -(ET)2 Cu[N(CN)2 ]Br (Tc ≈ 11.5 K) above and below Tc [522]

idenced by the missing peak for the BCS superconductors Nb and V. At low temperatures, 1/T1 is predicted to vanish exponentially for a fully-gapped superconductor,and display a power-law T dependence when the superconducting energy gap goes to zero at points or lines on the Fermi surface (see Fig. 13.80). Early proton-NMR experiments received particular attention when reporting an unexpected en-

hancement of 1/T1 by about a factor of 10 at ∼ Tc /2 as seen in Fig. 13.79(a) [520, 521]. Similar results were reported for many other organic superconductors and caused considerable confusion at the time. This feature was only understood a number of years later when an NMR measurement of the angulardependence of 1/T1 showed that the strong enhancement of 1/T1 below Tc vanishes when the magnetic field is aligned within the ET planes (Fig. 13.79(b)). Below Tc , 1/T1 exhibits a sharp dip in a narrow angular region about 4◦ wide around this field orientation ( = 90◦ ). It is therefore clear that this effect is connected with the flux-line vortex dynamics in the Shubnikov state. In the reversible region, the flux lines are free to move which results in a rapidly changing magnetic field at the hydrogen sites and, consequently, causes the strongly enhanced relaxation rate. However, when the magnetic field lies within the ET planes the flux lines prefer to stay within the poorly conducting anion layers, which results from a well-known phenomenon called intrinsic pinning. For this field orientation, the magnetic field at the hydrogen sites remains approxi-

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mately constant and 1/T1 approaches the expected intrinsic value. In the organic metals, the hydrogen nucleus has a relatively weak coupling to the conduction electrons residing in the spatially distant ET layers. Therefore, the relaxation due to the moving flux lines easily dominates. This is in contrast to the situation for the cuprate superconductors, where very large relaxation rates due to the conduction electrons occur that mask the flux-line effect. After realizing this important additional relaxation channel, NMR experiments were performed with the magnetic field aligned within the ET layers and, in addition, the 13C nuclei were chosen as a local probe since they are located in the center of the ET molecules,closer to the itinerant-electron system. Three groups almost simultaneously reported comparable results [501–503], two of which are shown schematically in Fig. 13.80. There are several marked differences between the experimental results and the behavior for a BCS superconductor (shown as a solid line) [523]. First, the coherence peak just below Tc is absent (although this is not conclusive evidence for unconventional superconductivity, as mentioned above). Second, 1/T1 follows a T 3 dependence that provides strong evidence for a node-like structure of the energy gap. The rapid decrease of 1/T1 just below Tc suggests a very fast gap opening, i.e., much faster than is realized for a conventional superconductor. One should bear in mind, however, that all of the NMR experiments were performed in an applied magnetic field, which might influence the gap structure even if the vortex dynamics can be neglected. In light of the controversial issue of the symmetry of the superconducting gap in the 2D organic superconductors, all of the above measurements as well as the specific-heat measurements discussed below provide no information about the detailed node topology.Therefore,the recent attempts to measure the possible anisotropy of the order parameter directly received particular attention. The first report of gap nodes in -(ET)2 Cu(NCS)2 [524] was argued to be a misinterpretation of the millimeterwave magneto-optical data [525,526]. In other work, scanning-tunneling spectroscopy (STM) was utilized to measure the in-plane anisotropy of the supercon-

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ducting gap [527]. The STM tip was placed on various as-grown or in-air prepared surfaces perpendicular to the interlayer direction. Strongly anisotropic current–voltage spectra were recorded, which were interpreted as being consistent with d-wave pairing. In line with this result, thermal-conductivity measurements in applied magnetic fields revealed a fourfold symmetry of the electronic contribution to the thermal conductivity [528]. Interestingly, however, these two experiments disagree on the node directions in -(ET)2 Cu(NCS)2 . One should further keep in mind that both experiments can only manifest an asymmetry of their signals but cannot verify real zero points of the energy gap. Furthermore, STM investigations are naturally extremely sensitive to surface states and should, therefore, be performed on wellcharacterized surfaces. Finally, the thermal conductivity due to quasiparticle excitation had to be measured in an applied magnetic field leading to similar difficulties as mentioned for the NMR experiments above. The proposed zero-field nodes of the superconducting gap should result in power-law temperature dependences of the thermal conductivity and the specific heat and finite values at T = 0. Indeed, in one thermal-conductivity study a small residual linear contribution was reported [529]; however, it is questionable whether the chosen extrapolation from above ∼ 0.18 K towards low temperatures, assuming a linear electronic and cubic phonon contribution, is justified or not. Recent thermal-conductivity measurements showed that the phonon-scattering length down to lowest temperatures (T = 0.25 K) is strongly influenced by quasiparticle scattering preventing the evolution of the T 3 dependence expected for sampleboundary scattering of phonons [530]. Therefore, further studies are needed to clarify this issue. Specific heat, C, is a powerful technique for determining whether the superconducting energy gap goes to zero at nodes on the Fermi surface. If the quasiparticle contribution to the specific heat, Ces , vanishes with an exponential temperature dependence in the superconducting state, nodes in the energy gap can be ruled out unequivocally.On the other hand, a power-law in T dependence of Ces indicates that line or point nodes may be present. However, a

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Fig. 13.81. (a) Specific heat of (ET)2 Cu[N(CN)2 ]Br in the superconducting (H = 0) and normal (H = 14 T) state. (b) The anomaly at Tc = 11.5 K is weakly visible in the inset. (c) Normalized plot of the electronic contribution to the specific heat Ces in the superconducting state divided by Tc as a function of Tc /T. The solid line shows the exponential vanishing of Ces towards low T (fit to the data of (ET)2 Cu(NCS)2 )

power-law dependence in the specific heat needs to be checked carefully since any extrinsic effect such as a partially normal-conducting sample or an improper subtraction of the nonelectronic contributions to C can obscure the exponential temperature dependence and lead to incorrect conclusions. Measurements of C(H,T) for -(ET)2 Cu[N(CN)2]Br [531] are shown in Fig. 13.81(a) on a double-logarithmic scale. The electronic contribution to C is rather small, only about 1% at Tc = 11.5 K. Therefore, the specific-heat anomaly at Tc can only be resolved in a high-resolution experiment (see the enlargement of the data in Fig. 13.81(b)). In a magnetic field of 14 T applied perpendicular to the planes, the sample is in the normal state. Therefore, the data at H = 14 T comprises the phonon contribution and the normal-state electronic specific heat T, where the Sommerfeld coefficient ≈ 25 mJ/mol K2. Consequently, Ces can be extracted by subtracting the phonon contribution from the C data in H = 0. As evidenced in the normalized plot of Fig. 13.81(c), Ces vanishes exponentially for -(ET)2 Cu[N(CN)2 ]Br [531] as well as for three other organic superconductors [530, 532, 533]. These results are in agreement with recent reports by other groups [534,535]. However, the C(H, T) measurements are not sensitive to a

particular direction in k space and therefore cannot resolve any possible anisotropy of the superconducting gap. In a very recent specific-heat experiment a T 2 behavior in Ces was resolved at lower temperatures [713]. Although this result would be consistent with line nodes,the increase of linear with H rather points to a conventional gap symmetry. An analysis of the specific-heat jump at Tc yields an interesting trend (Fig. 13.82). For the four superconductors shown, with values of Tc between 3.4 and 11.5 K, the specific-heat jump at Tc is always larger than the weak-coupling value C/ Tc = 1.43 and increases monotonically with increasing Tc [400,531]. This provides direct evidence for an increasing coupling strength with Tc . The dashed lines in Fig. 13.82 represent the weak-coupling C dependence [536], while the solid lines describe strong-coupling behavior; a BCS-like T dependence of the energy gap (T) scaled by an appropriate parameter has been assumed [537]. Reasonable descriptions of the experimental data are achieved for the gap ratios ˛ = (0)/kB Tc given in Fig. 13.82, where ˛ = 1.76 in the weak-coupling limit. The coupling strengths  can be estimated [531] using phenomenological models, yielding values ranging from ∼ 0.8 for (ET)2 I3 (Tc = 3.4 K) to 2.5 for -(ET)2 Cu[N(CN)2 ]Br

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Fig. 13.82. Specific heat in the superconducting state minus the specific heat in the normal state for four different organic superconductors with values of Tc between 3.4 and 11.5 K. The lines show the predicted BCS behavior for weak coupling (dashed lines, ˛ = 1.76) and for strong coupling characterized by various values of the parameter ˛ (solid lines)

(Tc = 11.5 K). The large values of  are in rough agreement with the mass renormalizations extracted from magnetic quantum oscillations [410,449]. 13.4.4

Concluding Remarks

The organic metals and superconductors show a rich variety of novel ground states and cover a broad spectrum of tunable fundamental physical phenomena that are of current interest. One of the most studied, but still very controversial, issues is the nature of the superconductivity. For the 1D Bechgaard salts, earlier indications for triplet superconductivity had to be corrected by recent results. For the 2D organic superconductors, the experimental situation is unsettled. NMR and thermal-conductivity experiments suggest that the superconducting order parameter is unconventional, whereas the measurements of the specific heat indicate a nodeless gap. In an attempt to solve this puzzling situation, it was suggested that an s wave to d-wave gap-symmetry transition could be induced by an in-plane magnetic field [538]. In this theoretical treatment, phonon-mediated superconductivity with a small wavevector was assumed, which could explain such a transition. Although this proposal would resolve the controversial NMR and

specific-heat results, other predictions of this theory such as the orientation of the d-wave gap nodes are at odds with recent experimental results [528]. Further studies, especially experiments that directly probe the phase difference of the superconducting wave function, are highly desirable.

13.5

Layered Cuprate and Ruthenate Superconductors

13.5.1

Introduction

The discovery of superconductivity at ∼ 30 K in the La–Ba–Cu–O system by Bednorz and M¨uller in 1986 ignited a worldwide explosion of research on high temperature superconductivity that still persists after more than one and a half decades. The impetus behind this extraordinary episode in the history of science has been driven by two objectives: the development of a fundamental understanding of high temperature superconductivity and the wide spread use of high temperature superconductors in technological applications. Although an enormous amount of progress has been made on these challenging problems since 1986, many obstacles remain

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to be overcome before these two objectives can be achieved. In this section, we give an overview of the current knowledge and understanding of the fundamental aspects of high Tc superconductivity in cuprates. We begin with a brief introduction to the layered perovskite-like cuprate materials that have been studied the most extensively as well as the structurally related ruthenocuprate magnetic superconductors and the compound Sr2RuO4 , the leading candidate for p-wave superconductivity. The rest of this section is devoted to a brief survey of the extraordinary superconducting and normal state physical properties of the high Tc cuprate superconductors and a discussion of some of the concepts and notions that have been advanced to account for these properties. Specific topics discussed include: materials, structure and charge carrier doping, superconducting state properties (critical field and current densities, superconducting pairing mechanism, doping dependence of Tc , symmetry of the superconducting order parameter), and normal state properties (nonFermi liquid behavior, normal ground state, pseudogap, electronic inhomogeneity) and the generic phase diagram. The enormous scope of this subject

dictated a very selective choice of the examples cited to illustrate the progress made in this field since its inception in 1986. For comprehensive accounts of specific topics in high Tc superconductivity, the reader is referred to various review articles, such as those that appear in the volumes edited by Ginsberg [539] and by Gschneidner, Eyring and Maple [540]. The materials that are emphasized in this section include the hole-doped superconductors La2−x Mx CuO4 (M = Ba, Sr, Ca, Na), RBa2 Cu3 O7−ı (R = Y or Ln), Y1−x Prx Ba2 Cu3 O7−ı , Bi2Sr2 CaCu2 O8+ı , and the electron-doped superconductors Ln2−x Mx CuO4−y (Ln = Pr, Nd, Sm, Eu; M = Ce, Th). 13.5.2

The Materials

Since 1986, dramatic increases of Tc have been achieved in the layered perovskite-like cuprate superconductors, as illustrated in the plot of the maximum value of Tc vs date in Fig. 13.83. Prior to 1986, the record for the highest value of Tc (∼ 23 K) was held by the A15 compound Nb3 Ge [541]. Currently, the maximum value of Tc at atmospheric pressure is ∼ 133 K for the compound HgBa2 Ca2 CuO8 [542,543]. When this compound is subjected to high pressures

Fig. 13.83. Maximum superconducting critical temperature Tc vs date ( [546])

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of ∼ 30 GPa, the Tc onset increases to ∼ 164 K (more than halfway to room temperature!) [544,545]. The first superconducting oxide system found with Tc values exceeding the 23 K record value held by Nb3 Ge was La2−x Bax CuO4 , which has a maximum value of Tc of ∼ 30 K for x ≈ 0.15 [547,548]. Shortly thereafter, La2−x Mx CuO4 systems with M = Sr, Ca, and Na were synthesized that exhibited superconductivity with maximum values of Tc of ∼ 40 K for M = Sr [549], ∼ 20 K for M = Ca [550], and ∼ 20 K for M = Na [551,552]. The first superconductors with values of Tc exceeding the boiling point of liquid nitrogen (77 K) belong to the RBa2 Cu3 O7−ı family. Superconductivity was reported to occur for R = Y [553] and all of the lanthanides except Ce and Tb (which do not form the phase) and Pr (which forms the phase, but is insulating and thereby not superconducting) with Tc values that range from ∼ 92 K for R = Y to ∼ 95 K for R = Nd [554]. Superconductivity at ∼ 90 K in the lanthanide analogues of YBa2 Cu3 O7−ı was discovered independently in laboratories at several institutions throughout the world. (See, for example, [554] and references cited therein.) Surprisingly, superconductivity with Tc ∼ 85 K has been reported in PrBa2 Cu3 Ox samples prepared by the traveling floating zone method [555]. However, these materials show large inhomogeneity in both lattice parameters and transport and magnetic properties. In the parts of the sample that exhibit bulk superconductivity, the c-axis lattice constant is reported to be larger than that of the nonsuperconducting ceramic polycrystalline or single crystal samples prepared by standard methods. The pressure dependence of Tc of the PrBa2 Cu3 Ox samples is quite large and comparable to values observed in underdoped cuprates. Further research on this fascinating material is clearly warranted, although to date the superconductivity reported for PrBa2 Cu3Ox has not been confirmed. These results illustrate the complexity of these materials and how sensitive their properties are to the conditions of their growth. Like most high Tc cuprates, the La2−x Mx CuO4 and RBa2 Cu3 O7−ı compounds are hole-doped superconductors. The first system of electron-doped cuprate superconductors discovered was Ln2−x Mx CuO4−y

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(Ln = Pr, Nd, Sm, Eu; M = Ce, Th; x ∼ 0.1 − 0.18; y ≈ 0.02) {(Ln = Pr, Nd, Sm; M = Ce) [556]; (Ln = Nd; M = Th) [557]; (Ln = Pr, Sm; M = Th) [558]; (Ln = Eu; M = Ce) [559].} The highest values of Tc found among the electron-doped superconductors are ∼ 25 K. There has been considerable interest in systems of the type R1−x Prx Ba2 Cu3O7−ı , especially for R = Y [282]. This was initially inspired by the puzzling observation that PrBa2 Cu3 O7−ı , when prepared by standard methods, is insulating and nonsuperconducting,whereas the other RBa2 Cu3 O7−ı compounds that can be formed are metallic and superconducting with values of Tc ∼ 92 − 95 K, as noted above. The Y1−x Prx Ba2 Cu3O7−ı system is replete with extraordinary physical phenomena including a metalinsulator transition at xcr ≈ 0.55, a monotonic depression of Tc with x that vanishes near xcr in the metallic phase, a striking crossover in the pressure dependence of Tc from positive to negative with increasing x, a large T contribution to the low temperature specific heat that is reminiscent of heavy fermion behavior, and Cu2+ and Prn+ antiferromagnetic (AFM) ordering in the insulating phase with TN (Cu2+ ) > TN (Prn+ ), where TN is the N´eel temperature. It has been suggested that these and other peculiar properties of this system are related to hybridization between the Pr localized 4f states and the valence band states associated with the conducting CuO2 planes [282]. A theoretical model that involves the hybridization between the 4f states of Pr and the 2p states of neighboring oxygen atoms in the Y1−x Prx Ba2 Cu3 O7−ı system has, in fact, been developed by Fehrenbacher and Rice [560]. Similar phenomena are found in the related Y1−x Prx Ba2 Cu4 O8 system which has also been extensively investigated [561]. The striking behavior of the Y1−x Prx Ba2 Cu3O7−ı system is discussed throughout this section. In several of the compounds containing R layers, the R ions with partially-filled 4f electron shells and magnetic moments have been found to order antiferromagnetically at low temperature [554,562].Specific heat data for RBa2 Cu3 O7−ı with R = Nd, Sm, Gd, Er and Dy that exhibit AFM ordering of the R ions at low temperatures reveal pronounced peaks near the N´eel temperature TN as shown in Fig. 13.84 [562].

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Fig. 13.84. Specific heat C vs temperature between 0.5 K and 3 K for RBa2 Cu3 O7−ı compounds with R = Nd, Sm, Gd, Dy, and Er, after [562]

Several compounds containing CuO2 and RuO2 planes have been fabricated in which superconductivity and ferromagnetism, apparently confined to the CuO2 and RuO2 planes, respectively, have been reported to coexist with one another microscopically. Examples of these materials include R1.4 Ce0.6 RuSr2 Cu2O10−ı (Tc ∼ 42 K, Curie temperature c ∼ 180 K for R = Gd; Tc ∼ 32 K, c ∼ 122 K for R = Eu) [563] and GdRuSr2Cu2 O8 (Tc ∼ 46 K,c ∼ 132 K) [564, 565]. Magnetic order in the compound GdRuSr2 Cu2O8 has been studied by means of neutron diffraction measurements by Lynn et al. [566]. The Ru magnetic moments order antiferromagnetically at TN = 136 K, coincident with the previously reported onset of ferromagnetism. Neighboring spins are antiparallel in all three directions,with a low tem-

perature moment of 1.18 B along the c-axis. These measurements place an upper limit of ∼ 0.1 B to any net zero field moment, with fields exceeding 0.4 tesla needed to induce a measurable magnetization. The Gd ions order independently at TN = 2.5 K with the same spin configuration. The value TN = 136 K for AFM ordering of the Ru ions is by far the highest known N´eel temperature for antiferromagnetic ordering that coexists with superconductivity. In addition to these rare earth and actinide based cuprates, many other high Tc cuprates have been discovered, a number of which also contain rare earth elements. The superconducting compound Sr2RuO4 has the same crystal structure as the La2−x Mx CuO4 (M = Ba, Sr, Ca, Na) high Tc cuprate superconductors [571]. While the Tc of Sr2RuO4 is only ∼ 1 K, this compound is of considerable interest because it is the only layered perovskite-like superconductor without Cu and it appears to be a leading candidate for pwave superconductivity. The anisotropy of the superconducting properties of Sr2 RuO4 is very large ( = ab /c ≈ 26), where  is the superconducting coherence length. Although this anisotropy is larger than that of La2−x Mx CuO4 ,the in-plane and c-axis resistivities of Sr2RuO4 vary as T 2 at low temperature, indicative of Fermi liquid behavior. Based on similarities to 3 He and the existence of closely related oxides such as SrRuO3 that are ferromagnetic, it was suggested [572,573] that Sr2RuO4 may exhibit p-wave (L = 1) spin-triplet superconductivity. In contrast, the cuprate superconductors are generally derived by doping the CuO2 planes of antiferromagnetic parent compounds by means of chemical substitutions, apparently resulting in superconductivity with spinsinglet pairing in a d-wave (L = 2) channel. There is a considerable amount of evidence for unconventional superconductivity in Sr2 RuO4 [569, 570,574]. 17O NMR Knight shift [321] and spin polarized neutron scattering [567] measurements provide the principal evidence for spin-triplet pairing of superconducting Sr2 RuO4. These measurements reveal that there is no reduction of the conduction electron spin susceptibility s (T) in the superconducting state relative to that n (T) in the normal state. For singlet spin pairing, s (T) should vanish at low temperatures T  Tc . Although spin orbit scatter-

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Fig. 13.86. Spin triplet superconducting states corresponding to d(k) = zˆ (kx + iky ) (left) and d(k) = xk ˆ x + yˆ ky (right) The zˆ (kx + iky ) state has an angular momentum along the z-axis (thick arrow) and spins perpendicular in the plane (thin arrows) The xk ˆ x + yk ˆ y state has vanishing total angular momentum because the orbital angular momentum is compensated by the spins of the Cooper pair. Experiments indicate that the superconducting phase in Sr2 RuO4 is of the type shown in the left panel; i.e., a state with finite angular momentum, after [570]

Fig. 13.85. Conduction electron spin susceptibility in the superconducting state s for an s-wave superconductor (V3 Si) and Sr2 RuO4 , based on spin polarized neutron scattering measurements [567]. In (a), the original work of Shull and Wedgewood [568] is confirmed and the data are described well by the Yosida function. In (b), the temperature independence of x for Sr2 RuO4 is in marked contrast to that expected if Sr2 RuO4 had singlet spin pairing, after [569]

ing can increase s (T) for spin-singlet pairing, it is not expected to be strong enough in Sr2RuO4 to increase s (T) to n(T) [570].The behavior of s (T) for the s-wave superconductor V3 Si and the unconventional superconductor Sr2 RuO4, based on spin po-

larized neutron scattering measurements of Duffy et al. [567], are shown in Fig. 13.85. Other evidence for unconventional superconductivity in Sr2 RuO4 is the extreme sensitivity of the superconductivity to nonmagnetic impurities; Tc vanishes for Al or Si impurity concentrations corresponding to a mean free path that is comparable to the superconducting coherence length. One of the most intriguing results comes from ‹SR measurements on Sr2 RuO4 that reveal an enhancement in the zero field relaxation rate in the superconducting state that is consistent with a superconducting state with broken time reversal symmetry [322]. A variety of experiments indicate that the superconducting energy gap has nodes or very deep minima,although thee form of these nodes is still uncertain. The spin-triplet superconducting state that seems to be favored by experiments on Sr2 RuO4 has the form d(k) = zˆ (kx + iky ). This state and another spin-triplet state are schematically represented in Fig. 13.86 [570]. The zˆ (kx + iky ) state has angular momentum along the z-axis (thick arrow) and spins perpendicular in the plane (thin arrows), while the xˆ kx + yˆ ky state has vanishing total angular momentum because the orbital angular momentum is compensated by the spins of the Cooper pair [570]. A comprehensive review of the normal and superconducting state properties of Sr2 RuO4 and the physics of spin-triplet pairing can be found in [569]. Quantum spin ladder materials have attracted much interest [575, 576]. These materials consist of

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Table 13.7. (a) Some important classes of cuprate superconductors and the maximum value of Tc observed in each class. (b) Examples of the abbreviated names (nicknames) used to denote cuprate materials (a) Material La2−x Mx CuO4 ; M = Ba, Sr, Ca, Na Ln2−x Mx CuO4−y ; Ln = Pr, Nd, Sm, Eu; M = Ce,Th YBa2 Cu3 O7−ı LnBa2 Cu3 O7−ı Ce, Tb do not form phase. Pr forms phase; neither metallic nor superconducting. RBa2 Cu4 O8 Bi2 Sr2 Can−1 Cun O2n+4 (n = 1,2,3,4) TlBa2 Can−1 Cun O2n+3 (n = 1,2,3,4) Tl2 Ba2 Can−1 Cun O2n+4 (n = 1,2,3,4) HgBa2 Can−1 Cun O2n+2 (n = 1,2,3,4) (b) Material YBa2 Cu3 O7−ı Bi2 Sr2 Ca2 Cu3 O10 Tl2 Ba2 Ca2 Cu3 O10 HgBa2 Ca2 Cu3 O8 La1.85 Sr0.15 CuO4 Nd1.85 Ce0.15 CuO4−y

ladders made of AFM chains of S = 1/2 spins coupled by inter-chain AFM bonds. Examples of two-leg ladder materials are SrCu2 O3 and LaCuO2.5; an example of a three-leg ladder material is Sr2 Cu2 O5 . Superconductivity has been discovered in the ladder material Sr0.4 Ca13.6Cu24 O41.84 under pressure with Tc ≈ 12 K at 3 GPa [577]. Interest in quantum spin ladder materials is partly due to the fact that they are simple model systems for theories of superconductivity based on magnetic pairing mechanisms. Approximately 100 different cuprate materials, many of which are superconducting, have been discovered since 1986. Several of the more important high Tc cuprate superconductors are listed in Table 13.7, along with the maximum values of Tc observed in each class of materials.Included in the table are examples of abbreviated designations (nicknames) for

Max. Tc (K) ∼ 40 ∼ 25 92 ∼ 95

∼ 80 (n = 3) 110 (n = 4) 122 (n = 3) 122 (n = 3) 133 Nickname YBCO; YBCO–123; Y–123 BSCCO; BSCCO–2223; Bi–2223 TBCCO; TBCCO–2223; Tl–2223 HBCCO; HBCCO–1223; Hg–1223 LSCO NCCO

specific cuprate materials which we will use throughout this article (e.g., YBa2 Cu3 O7−ı = YBCO, YBCO123, Y-123). It is interesting to note that superconductivity with values of Tc in the neighborhood of 30–40 K has been found in several noncuprate materials: the cubic perovskite Ba1−x Kx BiO3 (Tc ≈ 30 K) [578,579], the fcc “buckeyball” compound Rb3 C60 (Tc ≈ 29 K) [580, 581], and the compound MgB2 (Tc ≈ 39 K) [582]. 13.5.3

Structure and Charge Carrier Doping

The high Tc cuprate superconductors have layered perovskite-like crystal structures which consist of conducting CuO2 planes separated by layers comprised of other elements, denoted as A, and oxygen,

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723

Fig. 13.87. Temperature-dopant concentration (T–x) phase diagram delineating the regions of superconductivity and antiferromagnetic ordering of the Cu2+ ions for the hole-doped La2−x Srx CuO4 and electron-doped Nd2−x Cex CuO4−y systems. AFM = antiferromagnetic phase, SG = spin-glass phase, and SC = superconducting phase, after [585]

Am On , and, in some cases, layers of Ln ions [583,584]. The mobile charge carriers, which can be electrons but are usually holes, are generally believed to reside primarily within the CuO2 planes. The Am On layers apparently function as charge reservoirs that control the doping of the CuO2 planes with charge carriers and as spacers that govern the anisotropy of the physical properties of the materials. Many of the cuprates can be doped with charge carriers and rendered superconducting by substituting appropriate elements into an AFM insulating parent compound. For example, substitution of divalent Sr for trivalent La in the AFM insulator La2 CuO4 dopes the CuO2 planes with mobile holes and produces superconductivity in La2−x Srx CuO4 with a maximum Tc of ∼ 40 K at x ≈ 0.17 [549]. Similarly, substitution of tetravalent Ce for trivalent Nd in the AFM insulating compound Nd2 CuO4 apparently dopes the CuO2 planes with electrons, resulting in superconductivity in Nd2−x Cex CuO4−y

with a maximum Tc of ∼ 25 K at x ≈ 0.15 for y ≈ 0.02 [556,585]. The temperature T vs x phase diagrams for the La2−x Srx CuO4 and Nd2−x Cex CuO4−y systems are shown in Fig. 13.87 [585]. The Ln2−x Mx CuO4−y electron-doped materials have a tetragonal crystal structure that is similar to that of the La2−x Mx CuO4 hole-doped materials, but without the apical oxygen atoms. The crystal structures of the La2 CuO4 and Nd2 CuO4 parent compounds are displayed in Fig. 13.88 [585]. The La2−x Srx CuO4 and Nd2−x Cex CuO4−y systems have one CuO2 plane per unit cell and are referred to as single CuO2 layer compounds. Other superconducting cuprate systems have more than one CuO2 plane per unit cell: LnBa2 Cu3O7−ı has two CuO2 planes per unit cell (double CuO2 layer compound), while Bi2Sr2 Can−1 CunOx has n CuO2 layers per unit cell (n CuO2 layer compound) and can be synthesized by conventional methods for n = 1, 2, or 3.

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relatively slowly with magnetic field, making them suitable for technological applications [587]. Techniques have been devised that yield values of Jc in high fields for in-plane grain oriented thin films of YBa2 Cu3 O7−ı on flexible substrates at 64 K (pumped liquid nitrogen temperatures) that exceed those of NbTi and Nb3 Sn at liquid helium temperatures [588]. Presently, two of the leading candidates for technological applications of superconductivity are the RBa2 Cu3 O7−ı and Bi2 Sr2 Can−1 Cun O2n+4 (n = 2,3) materials. The first generation superconducting tapes and wires are based on both substituted and unsubstituted Bi2Sr2 Can−1 Cun O2n+4 (n = 2,3) superconductors prepared by the powder-in-tube technique [589]. Superconducting Pairing Mechanism Fig. 13.88. Crystal structures of La2 CuO4 (T-phase) and Ln2 CuO4 (Ln = Pr, Nd, Sm, Eu, Gd; T -phase) parent compounds, after [585]

13.5.4

Superconducting Properties

Critical Field and Critical Current Density With values of Tc in excess of the boiling temperature of liquid nitrogen (77 K) for some compounds, the high Tc cuprates were immediately recognized as promising candidates for technological applications of superconductivity, since they can be cooled into the superconducting state using liquid nitrogen, closed cycle refrigerators, and other more economical refrigeration techniques. Cuprates such as the RBa2 Cu3 O7−ı compounds (Tc in the range 92 − 95 K) have enormous critical fields ∼ 102 tesla [554, 586] that are more than adequate for most high-current and high-field technological applications. Shown in Fig. 13.89 is the resistively determined upper critical field Hc2 (T) curve for the compound YbBa2 Cu3O7−ı . Extrapolation of the Hc2 (T) curves measured between 0 and ∼ 10 T to T = 0 K, using the theory developed by Werthamer, Helfand, and Hohenberg (WHH) in 1966 without paramagnetic limiting [18], yields a value of Hc2(0) ≈ 160 T [554]. Epitaxially grown thin films of YBa2 Cu3 O7−ı on single crystal SrTiO3 substrates have critical current densities Jc ≈ 106 A/cm2 in zero field that decrease

Two features in the T − x phase diagrams for La2−x Srx CuO4 and Nd2−x Cex CuO4−y in Fig. 13.87 would appear to be relevant to cuprate superconductivity: (i) the electron-hole phase-diagram may provide a constraint on viable theories of high Tc in cuprates, and (ii) the proximity of antiferromagnetism suggests that superconducting electron pairing in the cuprates may be mediated by AFM spin fluctuations. An AFM pairing mechanism is consistent with the occurrence of d-wave pairing with dx2 −y 2 symmetry that is suggested by experiments on several hole-doped cuprates (discussed later in this article) article). Several theoretical models (for example, [592, 593]), based on AFM spin fluctuations, have predicted d-wave superconductivity with dx2 −y 2 symmetry for the cuprates. In contrast to the apparent electron-hole symmetry in the T − x phase diagrams of hole and electron-doped cuprates, there appears to be an interesting electron-hole “antisymmetry” in the pressure dependence of Tc . This is illustrated in Fig. 13.90 which contains a plot of the relative change of Tc with applied pressure,dlnTc /dP, vs Tc for the electron-doped superconductors and a comparison of these data with similar data for various hole-doped cuprate superconductors [591]. Other features are consistent with non-phononmediated pairing in the hole-doped cuprates. The curve of Tc vs carrier concentration can be approximated by an inverted parabola with the max-

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Fig. 13.89. Upper critical field Hc2 vs temperature for the compound YbBa2 Cu3 O7−ı . The solid lines are based on standard WHH [18] theory for a conventional type II superconductor in the limits of maximum (spin-orbit scattering parameter so = 0) and minimum (so = ∞) paramagnetic limitation and have been fitted to the data near Tc . Inset: Normalized electrical resistivity vs temperature in several magnetic fields between 0 and 9 tesla, after [557]

Fig. 13.90. Relative rate of change of Tc with applied pressure P, dlnTc /dP, as a function of Tc for the electron-doped superconductors Ln2−x Mx CuO4−y (Ln = Pr, Nd, Sm, Eu; M = Ce, Th; x ≈ 0.15; y ≈ 0.03). Inset: Same data plotted together with similar data for hole-doped superconductors from [590]. ( : La-based “214”; +: R-based “123”; x: Bi–Sr– and Tl–Ba–Ca–Cu– O materials), after [591]

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Fig. 13.91. (a) In-plane electrical resistivity ab vs temperature T curves forY1−x Prx Ba2 Cu3 O7−ı (0 < x < 0.51) single crystals, after [282]. (b) Temperature T vs Pr concentration x phase diagram for the Y1−x Prx Ba2 Cu3 O7−ı system, delineating metallic, superconducting, insulating, and antiferromagnetically ordered regions, after [282,596]

imum value of Tc occurring at an optimal dopant concentration x0 [594]. (Note that the terminology “under-doped” refers to values of x smaller than the “optimally-doped” value x0 where Tc is maximized, whereas“over-doped”refers to values of x larger than x0 .) The isotope effect on Tc for optimally-doped material is material is essentially zero (i.e., Tc ∝ M −˛ with ˛ ≈ 0; M = ion mass) [595]. An interesting example of the inverted parabolic dependence of Tc on carrier concentration is provided by the Y1−x−y Cay Prx Ba2 Cu3O7−ı system. In this system, the Ca2+ and Pr ions that are substituted for Y3+ in YBa2 Cu3O7−ı have the effect of “counterdoping” the CuO2 planes with holes and electrons, respectively.Assuming that each substituted Ca2+ ion

generates one hole in the CuO2 planes, analysis of the Tc (x, y) data in terms of a phenomenological model described below indicates that each substituted Pr ion localizes approximately one hole in the CuO2 planes [597] (see below). The localization of holes is also reflected in the behavior of Tc as a function of Pr concentration x in the Y1−x Prx Ba2 Cu3 O7−ı system (y = 0) Here, Tc decreases with increasing x as the Y1−x Prx Ba2 Cu3O7−ı system becomes more underdoped and vanishes near the onset of the metalinsulator transition at xmi ≈ 0.55. The T − x phase diagram for the Y1−x Prx Ba2 Cu3 O7−ı system is displayed in Fig. 13.91(b) which shows the behavior of Tc (x) as well as the N´eel temperatures TN (x) for AFM ordering of Cu and Pr magnetic moments [282,596].

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Fig. 13.92. (a) Superconducting critical temperature Tc vs Ca concentration y for four Pr concentrations x in the Y1−x−y Cay Prx Ba2 Cu3 O7−ı system [597]. Curves formed by solid and dashed lines represent Eq. (13.28), as explained in the text. (b) Tc vs Pr concentration x. The dashed line represents the function Tc (x, y = 0) + (96.5 K)x, which describes pure hole localization, and the solid line represents the function Tc (x, y = 0), which includes hole localization and pair-breaking, after [282,597]

It has been argued that the depression of Tc with x is primarily due to the decrease in the number of mobile holes with increasing Pr concentration, although magnetic pair breaking by Pr may also be involved [596, 597] (see below). Shown in Fig. 13.92 (a) are plots of Tc vs Ca concentration y for four Pr concentrations x in theY1−x−y Cay Prx Ba2 Cu3O7−ı system [597]. Curves formed by solid and dashed lines represent the equation Tc (x, y) = Tc0 − A(a − bx + y)2 − Bx = Tc (97 K) − (425 K)(0.1 − 0.95x + y)2 − (96.5 K)x. (13.28) The first term is the maximum attainable value of Tc (97 K),the second term represents the effect of hole generation by Ca (y) ions and localization of holes by Pr (x) ions (0.1 is an optimal hole concentration), while the last term describes the overall depression of Tc with x due to pair breaking interactions in the linear low concentration regime. These data illustrate the inverted parabolic dependence of Tc on charge carrier concentration observed in many superconducting cuprate systems. Displayed in Fig. 13.92(b) is a plot of Tc vs Pr concentration x. The dashed line represents the function Tc (x, y = 0)+(96.5 K)x which corresponds to pure hole localization, while the solid

line corresponds to the function Tc (x, y = 0) which includes hole localization and pair breaking. The occurrence of appreciable hybridization between the Pr localized 4f electron states and the valence band states associated with the conducting CuO2 planes was first proposed by Neumeier, Maple, and Torikachvili in 1988 [598] to account for the rapid reduction of Tc and the striking crossover in the pressure dependence of Tc from positive to negative with increasing x in the Y1−x Prx Ba2 Cu3 O7−ı system. This proposal was based on similar behavior exhibited by the La1−x Cex system described in Sect. 13.2, in which the rate of depression of Tc by the Ce impurities displayed a strong dependence on pressure that was attributed to the hybridization between the Ce localized 4f states and conduction electron states [56].Direct evidence for Pr localized 4f -CuO2 valence band hybridization was obtained from resonant photoemission spectroscopy experiments (RESPES) on the Y1−x Prx Ba2 Cu3O7−ı system by Kang et al. [599]. Measurements of the pressure dependence of Tc were repeated by Maple, Paulius, and Neumeier [600] on higher quality Y1−x Prx Ba2 Cu3 O7−ı specimens studied at zero pressure by Neumeier and Maple [596]. The results of these experiments are shown in Fig. 13.93 (a) where Tc = Tc (P) − Tc (0) is plotted vs P for several values of x between 0 and 0.5 [600]. The

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Tc (P) data are in qualitative agreement with the earlier results and reveal a dramatic crossover from positive to large negative dependences of Tc on P as the Pr concentration is increased. Earlier measurements of (T, P) on a sample with x = 0.5 [596,601] showed that this sample near the metal-insulator transition becomes less metallic under pressure. An extension of the formula for Tc (x) for the Y1−x Prx Ba2 Cu3 O7−ı system to incorporate the effect of applied pressure can also account semiquantitatively for the remarkable variations in Tc with P that have been observed in the Y1−x Prx Ba2 Cu3O7−ı system. The trends in the Tc (x, P) data can be obtained by assuming that the parameters A and a are, to first approximation, independent of P, and that Tc0, b and B can be expanded in a power series to first order in P; i.e.,  Tc0(P) ≈ Tc0(0) + Tc0 (0)P (13.29)  (13.30) b(P) ≈ b(0) + b (0)P  (13.31) B(P) ≈ B(0) + B (0)P The best overall fit of the resultant expression for [dTc (x, P)/dP]P=0 to the experimental data yields the  values Tc0 (0) = 0.048 K/kbar, b (0) = 0.0041 K/kbar  and B (0) = −0.02 K/kbar [600]. The increase of b with P is consistent with an increase of the valence (decrease in the 4f electron shell occupation number) of Pr with P, which is intuitively reasonable. A decrease (increase) of the magnitude of the exchange interaction parameter J with P can be inferred from the decrease of B (0) with P,depending upon whether the complications associated with the Kondo effect are excluded (included). As discussed in Sect. 13.2, the Kondo effect is expected (and possibly observed in the specific heat [602]) in metals containing R ions that carry magnetic moments when there is strong hybridization between the localized 4f states of the R ion and conduction electron states which generates a large, negative (antiferromagnetic) exchange interaction [6]. However, inclusion of the Kondo effect in the analysis of the Tc (x, P) data is complicated. The calculated Tc (x, P) curves for x values corresponding to the experimental Tc (x, P) data presented in Fig. 13.93(a) are shown in Fig. 13.93(b) and give a semiquantitative description of the Tc (x, P) data over the range 0 ≤ x ≤ 0.5, even though

the Tc (x, P = 0) data are only described well for 0 ≤ x ≤ 0.2 (see Fig. 13.92(b)). It is interesting that superconductivity with values of Tc exceeding ∼ 40 K has only been observed in layered cuprates which have strongly anisotropic,nearly two-dimensional electronic properties. An important issue for these materials is the nature of interlayer transport and the role it plays in the superconductivity. Reflectance R(!) measurements with polarized light reveal that R(!) for the electric field E parallel to the CuO2 planes, E  CuO2 , exhibits a metallic response, while R(!) for E ⊥ CuO2 resembles that of ionic insulators with characteristic phonon peaks in the far infrared. This is illustrated in Fig. 13.94 where R vs ! data for Tl2Ba2 CuO6+x (Tl2201) [603] are displayed. For temperatures below Tc , a sharp plasma edge at ! = 37 cm−1 evolves out of a nearly “insulating” normal state spectrum since the supercurrents flow in all crystallographic directions. In an effort to explore the relationship between changes in the incoherent c-axis conductivity below Tc and the c-axis superfluid density, Basov et al. [603] have analyzed the interlayer conductivity of the cuprate high Tc superconductors Tl2 Ba2 CuO6+x , La2−x Srx CuO4 , and YBa2 Cu3 O6.6 , all of which show incoherent interlayer response in the normal state. The analysis reveals that the magnitude of the superfluid density s significantly exceeds the weight missing from the real part of the conductivity in the frequency region comparable to the superconducting energy gap 2. This indicates that a significant fraction of s is derived from mid infrared frequencies. Basov et al. [603] suggest that the discrepancy between the magnitude of s and the spectral weight that is missing from the far infrared part of the conductivity can be interpreted in terms of an interlayer kinetic energy change associated with the superconductivity. Because of their high values of Tc , short coherence lengths, long penetration depths, and large anisotropy, the cuprate superconductors exhibit a wealth of striking vortex phases and phenomena that are currently being vigorously investigated (for a review, see, for example, [604–608]).

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Fig. 13.93. Shift in Tc relative to its value at zero pressure, Tc (P) − Tc (0), of Y1−x Prx Ba2 Cu3 O7−ı samples with various Pr concentrations 0 ≤ x ≤ 0.5 vs pressure P to 18 kbar. (a) Experimental data. (b) Calculated behavior according to the phenomenological model described in the text, after [600]

Fig. 13.94. Reflectance of Tl2 Ba2 CuO6+x measured with E  c and E  ab polarizations of incident radiation. The c-axis reflectance is nearly insulating in the normal state but at T < Tc ≈ 80 K is dominated by the Josephsonlike plasma edge, after [603]

Symmetry of the Superconducting Order Parameter A great deal of effort has been expended to determine the symmetry of the superconducting order parameter of the high Tc cuprate superconductors [12, 609, 610]. The pairing symmetry provides clues to the identity of the superconducting pairing mechanism which is essential for the development of the theory of high temperature superconductivity in the cuprates.

Shortly after the discovery of high Tc superconductivity in the cuprates, it was established from flux quantization, Andreev reflection, Josephson effect, and nuclear magnetic resonance (NMR) Knight-shift measurements that the superconductivity involves electrons that are paired in singlet spin states [611]. Possible orbital pairing states include s-wave, extended s-wave, and d-wave states. In the s-wave state, the energy gap (k) is isotropic; i.e., (k) is constant over the Fermi surface. This leads to “activated”

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Fig. 13.95. Fermi surface gap functions and densities of states of a superconductor with tetragonal symmetry for various pairing symmetries. The gap functions in the kz = 0 plane (top) are represented by the light solid lines; distance from the Fermi surface (dark solid lines) gives the amplitude, a positive value being outside the Fermi surface, a negative value inside. The corresponding density of states for one-quasi-particle excitations N(E) is shown below each gap function, with N0 the normal state value. Gap node surfaces are represented by the dashed lines. Left: The classic s-wave case, where the gap function is constant, with value . This gives rise to a square-root singularity in N(E) at energy E = . Middle: The extended s-wave case derives from pairs situated on nearest-neighbor square lattice sites in real space, with an approximate k-space form of cos(kx a) − cos(ky a). For the Fermi surface shown here, the gap function has lines of nodes running out of the page. Right: A d-wave function of x 2 − y2 symmetry. The extended s-wave and d-wave functions shown here each have a linear density of states up to order , which measures the maximum gap amplitude about the Fermi surface, after [12]

behavior of the physical properties in the superconducting state for T  ; e.g., the specific heat Ce (T), ultrasonic attenuation coefficient ˛s (T), and NMR spin lattice relaxation rate 1/T1(T) vary as e−/T . For the extended s-wave state, the energy gap (T) is anisotropic; i.e., (k) exhibits a variation over the Fermi surface that has the same symmetry as the rotational symmetry of the crystal. Similarly, for the d-wave case, the energy gap (k) is anisotropic; however, the symmetry is lower than the symmetry of the crystal. The d-wave state that is consistent with most, but not all, of the experiments discussed below has dx2 −y 2 symmetry, which can be expressed as (k) = 0 [cos(kx a) − cos(ky a)]. For both the extended s-wave and d-wave cases, (k) vanishes at lines on the Fermi surface, resulting in a density of states N(E) that is linear in energy E at low values of E. This leads to “power-law” T n (n = integer) behavior of the physical properties for T  ; e.g., Ce (T) ∼ T 2, the superconducting penetration depth (T) ∼ T, and 1/T1(T) ∼ T 3 . The definitive de-

termination of the symmetry of the superconducting order parameter requires the determination of both the magnitude and the phase of (k). Shown in Fig. 13.95 is a schematic diagram of the variation of the energy gap over the Fermi surface and the density of states N(E) vs E for the “s”, “extended s”, and “dx2 −y 2 ” states [12]. Magnitude of the Superconducting Order Parameter A number of different types of measurements have been performed on the high Tc cuprate superconductors that are sensitive to |(k)|. These include microwave penetration depth (T) [612], microwave surface conductivity [613], nuclear magnetic resonance (NMR) relaxation rate 1/T1(T) [614], magnetic field dependence of the electronic specific heat Ce (T) [615], thermal conductivity [616], angle resolved photoemission spectroscopy (ARPES) [617], quasiparticle tunneling [618, 619] and Raman scattering [620]. The results of these studies are gen-

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erally consistent with a superconducting state with dx2 −y 2 or extended s-wave symmetry for the holedoped cuprates such asYBCO–123,YBCO–124,LSCO, and BSCCO. Experiments discussed below indicate a dominant component of dx2 −y 2 symmetry in the superconducting order parameter of hole–doped cuprates. Shown in Fig. 13.96 are data for [(0)/(T)]2 vs T/Tc , where  is the London penetration depth for a YBa2 Cu3O6.95 single crystal from the work of Hardy et al. [612]. The data clearly deviate from the BCS swave behavior (solid line in the figure) and are linear in T at low temperatures. The linear behavior is consistent with the existence of line nodes in the energy gap of a clean d-wave superconductor. Early electron tunneling [621], microwave penetration depth [622, 623], and Raman spectroscopy [624] measurements on the electron-doped superconductor NCCO indicated that the superconducting order parameter of this material has s-wave symmetry. Superconductivity with d-wave symmetry for hole-doped cuprates and s-wave symmetry for electron-doped cuprates seems rather surprising, in view of the similarities in the structures of the hole-doped and electron-doped superconductors and the fact that they are both derived from chemical substitution of AFM insulating parent compounds. However, more recent microwave [625, 626], angleresolved photoemission spectroscopy (ARPES) [627, 628], and phase sensitive SQUID microscope (see below) [629] experiments appear to be consistent with a d-wave superconducting order parameter.According to Blumberg et al. [630], who recently performed low energy electronic Raman scattering measurements on NCCO single crystals, these seemingly contradictory results can be rationalized in terms of a nonmonotonic form of the dx2 −y 2 order parameter. The superconducting gap opens up rapidly with departure from the diagonal nodal directions and reaches a maximum value of 4.4kBTc at the hot spots that are located closer to the nodes than to the Brillioun zone boundaries where the gap drops to 3.3kB Tc . The enhancement of the gap value in the proximity of the hot spots emphasizes the role of antiferromagnetic spin fluctuations for superconductivity in the electron-doped cuprates. Thus, despite strong

731

Fig. 13.96. Ratio [(0)/(T)]2 extracted from measurements of (T) on a YBa2 Cu3 O6.95 single crystal, a measure of the superfluid density. The different choices of (0) (square and long and short dashed curves) used have little effect on the overall shape. The strong linear behavior at low temperatures is clearly different from the s-wave BCS result, after [612]

differences between the electron- and hole-doped cuprates, their superconductivity may have the same order parameter symmetry and originate from a similar mechanism. Phase of the Superconducting Order Parameter Several different types of measurements that are sensitive to the phase of (k) have been performed. These measurements,all of which involve the Josephson effect, include superconducting quantum interference device (SQUID) interferometry [631–633], single junction modulation [634,635], tricrystal ring magnetometry [636–638], c-axis Josephson tunneling [639–643], and grain boundary tunneling [644]. The SQUID interferometry, single junction modulation, and tricrystal ring magnetometry measurements were performed on YBCO, while tricrystal magnetometry experiments have also been carried out on TBCCO. These experiments indicate that the superconducting order parameter in the YBCO and TBCCO hole-doped materials has dx2 −y 2 symmetry.

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However, c-axis Josephson tunneling studies on junctions consisting of a conventional superconductor (Pb) and twinned or untwinned single crystals of YBCO indicate that the superconducting order parameter of YBCO has a significant s-wave component [639,640]. As noted above, tricrystal ring magnetometry measurements indicate d-wave symmetry for the electron-doped superconductor NCCO [645] which, according to recent low energy polarized electronic Raman scattering studies,has a nonmonotonic dx2 −y 2 superconducting order parameter [630]. A new type of c-axis Josephson tunneling experiment in which a conventional superconductor (Pb) was deposited across a single twin boundary of a YBCO single crystal was performed by Kouznetsov et al. [646]. The Josephson critical current Ic was measured as a function of magnitude and angle  of magnetic field applied in the plane of the sample. For H aligned perpendicular to the twin boundary, a maximum in Ic as a function of H was observed at H = 0, whereas for H parallel to the twin boundary, a minimum in Ic was observed at H = 0. In all samples investigated,a clear experimental signature of an order parameter phase shift across the twin boundary was observed. The results provide evidence for mixed d and s-wave pairing in YBCO and are consistent with predominant d-wave pairing with dx2 −y 2 symmetry and a sign reversal of the s-wave component across the twin boundary. Recently, Bi2Sr2 CaCu2 O8−ı (Bi2212) bicrystal caxis twist Josephson junction experiments were performed and taken as evidence for a dominant s-wave order parameter for T < Tc [647,648]. The twist angle 0 independence of the c-axis Josephson critical current Jc across the twist junction for T just below Tc was interpreted in terms of a dominant s-wave order parameter for all T ≤ Tc . This experiment is in apparent contradiction to the results of the tricrystal experiments on Bi2212 which indicated a dominant dx2 −y 2 wave order parameter component in Bi2212 at low temperatures for both under-doped and overdoped samples [645]. It was recently suggested that these experiments would be compatible if the superconductivity exhibited by Bi2212 were mostly s-wave in the bulk and d-wave on the surface [649]. Critical discussions of these and other phase sensitive ex-

periments on Bi2212 can be found in several recent papers (for example, [650,651]) Various mixed s and d-wave pairing models that have been developed to account for the superconducting properties of the orthorhombic YBCO–123 system are reviewed by B´eal–Monod [652]. Multiple Superconducting States An anomaly in the thermal conductivity  at low temperature of the high Tc superconductor Bi2Sr2 Ca(Cu1−x Nix )2 O8 was reported by Movshovich et al. [653]. The anomaly takes the form of a sharp reduction of  at a temperature Tc∗ ≈ 200 mK, which separates a higher temperature region where  varies as T ˛ with ˛ between 1.6 and 1.75 and a lower temperature region where  is linear in T. The low temperature anomaly was found to be suppressed by the application of a small magnetic field. Movshovich et al. proposed that the observed behavior is consistent with a phase transition into a second bulk low temperature superconducting state at Tc∗ . They note that the presence of a second superconducting state would constitute direct evidence for unconventional superconductivity in Bi2Sr2 CaCu2 O8.As noted in Sect. 13.3, multiple superconducting states have been observed in the heavy fermion superconductors UPt3 and U1−x Thx Be13 in which the superconducting electrons are also believed to be paired in states with finite angular momentum. 13.5.5

Normal State Properties

It was realized at the outset that the normal state properties of the high Tc cuprate superconductors are unusual and appear to violate the Landau Fermi liquid paradigm [654–658]. Some researchers share the view that it will be necessary to develop an understanding of the normal state before the superconducting state can be understood, since the normal state properties reflect the electronic structure that underlies high Tc superconductivity. The anomalous normal state properties first identified in the high Tc cuprate superconductors include the electrical resistivity and Hall effect. The electrical resistivity ab (T) in the ab-plane of many of

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Fig. 13.97. (a) Temperature dependence of the Hall carrier number nH = V /eRH for Y1−x Prx Ba2 Cu3 O7−ı single crystals with different x values. The fact that the data for two pairs of x values (x = 0, x = 0.08, and x = 0.42, x = 0.51) are out of sequence in the V /eRH vs T plot, but not in the cot(H ) vs T 2 plot (see (b)), could be a result of the error in measuring the thickness of these crystals. The x = 0.9 value represents the nominal composition, while the x values of the superconducting samples were estimated by comparing the Tc ’s of the crystals with those of high quality polycrystalline samples. After [282]. (b) Cotangent of the Hall angle cot(H ) vs T 2 of Y1−x Prx Ba2 Cu3 O7−ı single crystals with magnetic field H = 6 tesla and H  c, after [661]

the hole-doped cuprate superconductors near optimal doping has a linear temperature dependence between Tc and high temperatures ∼ 1000 K, with an extrapolated residual resistivity ab (0) that is very small; i.e., ab (T) ≈ ab (0) + cT, with ab (0) ≈ 0 and the value of c similar within different classes of cuprate materials [659]. An example of the linear Tdependence of ab (T) is displayed in Fig. 13.91(a) for the Y1−x Prx Ba2 Cu3O7−ı system [282]. In contrast, both ab (T) and c (T) of the optimally-doped electron-doped cuprate Sm1.83Ce0.17 CuO4−y vary as T 2 , indicative of three dimensional Fermi liquid behavior in the La2−x Mx CuO4 systems [660].

The Hall coefficient RH is inversely proportional to T and the cotangent of the Hall angle H = RH / varies as T 2 ; i.e., cot H = xx /xy = AT 2 + B [662]. This is illustrated in Fig. 13.97 which contains a plot of H vs T 2 calculated for a field of 6 tesla for Y1−xPrx Ba2 Cu3 O7−ı single crystals [661]. The linear T-dependence of (T) and the quadratic Tdependence of cot(H ) have been attributed to longitudinal and transverse scattering rates −1 and t−1 that vary as T and T 2, respectively [663]. In the RVB model, the constant and T 2 terms in t−1 and, in turn, H , are ascribed to scattering of spinons by magnetic impurities and other spinons, respectively.

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Fig. 13.98. Temperature − Sr concentration (T–x) phase diagram of the La2−x Srx CuO4 system, after superconductivity has been suppressed by an intense pulsed magnetic field. The insulator-metal crossover occurs near optimum doping and a new insulating regime is revealed, in which the electrical resistivity diverges as the logarithm of the temperature. The dashed line represents the Tc vs x curve in zero field and the two solid lines delineate the insulating ab and c , metallic ab and insulating c , and metallic ab and c regions in the normal state accessed by high magnetic fields, after [666]

On the other hand, using the nearly antiferromagnetic Fermi liquid description of planar quasiparticles, Stojkovic and Pines [664] have shown that the anomalous temperature dependence of the Hall angle could result from highly anisotropic scattering at different regions of the Fermi surface. A review of magnetoresistance and Hall effect studies of cuprate superconductors, particularly the electron-doped cuprates La2−x Mx CuO4 and the hole-doped system Y1−x Prx Ba2 Cu3 O7−ı , can be found in Ref. [665]. Normal Ground State The evolution of the normal ground state of the cuprates as a function of dopant concentration is particularly interesting. This is reflected in the temperature dependences of the ab-plane and c-axis electrical resistivities ab (T) and c (T) [667]. Both ab (T) and c (T) exhibit insulating behavior (i.e., d/dT < 0) in the under-doped region, ab (T) is metallic (i.e., d/dT > 0) and c (T) is insulating or metallic in the optimally-doped region, depending on the system, and ab (T) and c (T) are both metallic in the over-doped region. The linear Tdependence of ab (T) and the insulating behavior of c (T) suggest two dimensional non-Fermi liquid behavior near the optimally-doped region, whereas the metallic (T) ∼ T n with n > 1 reflects a tendency towards three dimensional Fermi liquid behavior over-doped region. Measurements in 61-tesla

pulsed magnetic fields to quench the superconductivity have been particularly useful in elucidating the evolution of ab (T) and c (T) with dopant concentration in the La2−x Srx CuO4 system [666]. Both ab (T) and c (T) were found to exhibit a − ln T divergence in the under-doped region, indicative of a three dimensional non-Fermi liquid [668]. Shown in Fig. 13.98 is the temperature-Sr concentration (T − x) phase diagram of the La2−x Srx CuO4 system, after the superconductivity has been suppressed by an intense pulsed magnetic field [666].The insulatormetal crossover occurs near optimum doping and a new insulating regime is revealed, in which the electrical resistivity diverges as the logarithm of the temperature. As an example of the evolution of ab (T) and c (T) with doping,we again refer to theY1−x Prx Ba2 Cu3 O7−ı system. Shown in Fig. 13.99 are ab (T) and c (T) data for Y1−x Prx Ba2 Cu3 O7−ı single crystals in the range of Pr concentrations 0 ≤ x ≤ 0.55 [669]. The following features in the ab (T) and c (T) data in Fig. 13.99 are evident: a nonmonotonic evolution of c (T) with x, the transformation of both ab (T) and c (T) from metallic to semiconducting with x, and the coexistence of metallic ab(T) and semiconducting c (T) for a certain range of doping. The nonmonotonic variation of c (T) with x in Fig. 13.99(b) can be described with a phenomenological model [669] that assumes that the c-axis conductivity takes place via incoherent elastic tunneling between CuO2 bilayers

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Fig. 13.99. (a) In-plane resistivity ab (T) and (b) out-of-plane resistivity c (T) for Y1−x Prx Ba2 Cu3 O7−ı single crystals. The solid lines in (b) are fits to the data using a model described in [669]. Inset (a): The configuration of leads used in the measurements. Inset (b): Anisotropy c /ab vs Pr concentration x at different temperatures. The solid lines are guides to the eye, after [669]

and CuO chain layers with a gap in the energy spectrum of the CuO chains (solid lines in Fig. 13.99(b)). Pseudogap Perhaps the most remarkable aspect of the normal state is the pseudogap in the charge and spin excitation spectra of under-doped cuprates [610]. The pseudogap has been inferred from features in various transport, magnetic, and thermal measurements including ab (T) [671–673]; RH (T) [674], thermoelectric power S(T) [675], NMR Knight shift K(T) [676], NMR spin-lattice relaxation rate 1/T1(T) [677–679], magnetic susceptibility (T) [674], neutron scattering [680], and specific heat C(T) [681], as well as spectroscopic measurements such as infrared absorption [682–684] and ARPES [670,685].

An example of the features in ab (T) that are associated with the pseudogap can be seen in the ab (T) data displayed as in Figs.13.91(a) and 13.99(a) for the Y1−x Prx Ba2 Cu3O7−ı system. As the system becomes more under-doped with increasing x, ab(T) deviates from linear behavior at higher temperature at a characteristic temperature T ∗ which represents a crossover into the pseudogap state at T < T ∗ . The transport, thermal, magnetic, and infrared studies of the pseudogap have been carried out on several cuprate materials, including LSCO, YBCO– 123, YBCO–124, and BSCCO–2212, while the ARPES investigations of the pseudogap have mainly focused on BSCCO, although ARPES measurements have also been made on oxygen-deficient oxygendeficient YBCO.

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Fig. 13.101. Schematic phase diagram of BSCCO–2212 as a function of doping. The filled symbols represent values of Tc determined from magnetic susceptibility measurements. The open symbols are values of T ∗ at which the pseudogap closes, derived from the data shown in Fig. 13.100, after [670] Fig. 13.100. Momentum and temperature dependence of the energy gap estimated from leading edge shifts of ARPES spectra for BSCCO–2212. (a): k-dependence of the gap in the Tc = 87 K, 83 K, and 10 K samples, measured at 14 K. The inset shows the Brillouin zone with a large Fermi surface (FS) closing the ( ,  ) point, with the occupied region shaded. (b): Temperature dependence of the maximum gap in a near-optimal Tc = 87 K sample (circles), and two underdoped samples with Tc = 83 K (squares) and Tc = 10 K (triangles), after [670]

The ARPES measurements reveal several striking aspects of the pseudogap.The magnitude of the pseudogap has the same k-dependence in the ab plane as the magnitude of the superconducting energy gap, with maxima in the directions of kx and ky and minima at 45◦ to these directions. In fact, the sym-

metry is consistent with dx2 −y 2 symmetry inferred from Josephson tunneling measurements on holedoped cuprates discussed previously. Furthermore, measurements of the temperature dependence of the pseudogap at the angles where it is a maximum show that the superconducting gap grows continuously out of the pseudogap and that the value of the sum of both gaps at low temperatures is constant, independent of the temperature T ∗ at which the pseudogap opens, or Tc . This is in marked contrast to the situation in conventional superconductors where the energy gap is proportional to Tc . These features of the pseudogap are illustrated in Figs. 13.100(a) and 13.100(b) which show the k-dependence of the energy gap and the dependence of the maximum gap on temperature from the ARPES measurements of Ding et al. [670] on

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Fig. 13.102. Tunneling spectra for Bi2212 measured at various temperatures indicated in the figure. The conductance scale corresponds to the 293 K spectrum; the other spectra have been offset vertically for clarity, after [686]

BSCCO–2212 samples with values of Tc of 87 K, 83 K, and 10 K. The pseudogap and superconducting regions for BSCCO–2212 derived from the ARPES measurements of Ding et al. [670] are summarized in the temperature Tc vs charge carrier doping plot in Fig. 13.101. Recently, direct measurements of the quasiparticle density of states (DOS) by scanning tunneling spectroscopy on Bi2 Sr2CaCu2 O8+ı (Bi2212) single crystals as a function of oxygen doping and temperature were made by Renner et al. [686]. It was found that the shape of the DOS in the superconducting state is essentially doping independent, the pseudogap above Tc scales with the superconducting gap, and the pseudogap is also present in overdoped samples. Both gaps were found to be essentially temperature independent. Tunneling spectra at various temperatures between 4.2 K and 293.2 K for Bi2212 are shown in Fig. 13.102. In subsequent work, the temperature dependence of the quasiparticle density of states of over-doped

Bi2Sr2 CuO6+ı (Bi2201) single crystals was measured between 275 mK and 82 K by means of scanning tunneling spectroscopy [687].Below Tc = 120 K,the spectra exhibit a gap with well defined coherence peaks at ± greater than about 12 meV, which disappears at Tc . Above Tc , the spectra display a clear pseudogap of the same magnitude, gradually filling up and vanishing at T ∗ ≈ 68 K. This gap value is extremely large for a Tc of 10 K and is ∼ 7 times larger than the BCS d-wave prediction BCS = 1.8 meV [688]. This implies a very high ratio 2p /kB Tc ≈ 28, a value even larger than 2p /kB Tc ≈ 10 for Bi2212 at equivalent doping, showing that superconductivity is far from BCS-like in Bi2201, even in the over-doped regime. This work also revealed that the ratio of p and T ∗ is the same as found in other high Tc cuprate superconductors. This is illustrated in the plot of T ∗ /Tc vs 2p /kB Tc shown in Fig. 13.103 where it can be seen that the data are well represented by the BCS relation 2p /kBT ∗ = 4.3, where Tc has been replaced by T ∗ . This suggests that T ∗ , rather than Tc , reflects the

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Fig. 13.103. T ∗ /Tc vs 2p /kB Tc for various cuprate superconductors compared to the mean field relation 2p /kB T ∗ = 4.3, where Tc has been replaced by T ∗ . Error bars have been extracted from the references given by Kugler et al. indicated in the figure, after [687]

mean field critical temperature of cuprate superconductors and favors models which involve precursor pairing as the origin of the pseudogap phase. Electronic Inhomogeneity Stripes, a complex form of electronic selforganization, have been observed in certain cuprates at low doping near the onset of the superconducting regime [689]. The stripes are believed to consist of mobile charge carriers in the CuO2 planes that are confined to narrow one-dimensional lines (“charge stripes”) and separated by insulating regions which exhibit antiferromagnetic order,similar to that which occurs in the parent Mott insulators such as La2 CuO4 in which electrons are localized as a result of strongly repulsive electron–electron interactions. Thus, there should be large anisotropy in electrical conductivity between directions parallel and perpendicular to the stripes. The stripes are particulary prominent at low doping where Tc is low and substantially weakened at higher doping where Tc is a maximum. This suggests that stripes and superconductivity compete with one another, and many researchers believe that these two phases arise from a common underlying mechanism. Some researchers believe that the stripes are actually a precondition for high Tc superconductivity in the cuprates. The first evidence for static spin and charge order in cuprates was obtained in a neutron scatter-

ing study of a single crystal of La1.6−x Nd0.4 Srx CuO4 by Tranquada et al. [690]. In this material, the partial substitution of Nd for La results in a change in crystal structure from the usual low temperature orthorhombic (LTO) phase to the low temperature tetragonal (LTT) phase, and a depression of Tc near x = 0.125. The LTO-LTT structural transition occurs at ∼ 70 K in La1.6−x Nd0.4 Srx CuO4 for x = 0.12, and evidence for static spin and charge stripes comes from neutron scattering as well as x-ray scattering experiments [690, 691]. In the La2−x Srx CuO4 system, inelastic neutron scattering experiments have yielded evidence for dynamical two-dimensional spin correlations characterized by an incommensurate wave vector [692–694]. The characteristic wave vectors associated with magnetic scattering in La1.85Sr0.15CuO4 and La1.48Nd0.4 Sr0.12CuO4 are essentially identical. Thus, the major difference between the two systems appears to be the static vs dynamic character of the correlations. This similarity suggests that dynamical stripe correlations of both spin and charge exist in superconducting La1.85 Sr0.15CuO4 . An intriguing question is what roles these dynamical spin and charge stripe correlations play in the anomalous normal state properties and the superconducting pairing mechanism in the cuprates. Recently, Dumm et al. [695] performed infrared (IR) spectroscopy measurements on single crystals of La1.97Sr0.03 CuO4, a weakly doped phase of one of the prototypical high Tc superconducting sys-

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tems La2−x Srx CuO4. The existence of spin stripes in the La2−x Srx CuO4 system has been well documented through extensive neutron scattering and transport studies on this system by Matsuda et al. [696]. The IR spectroscopy measurements revealed anisotropy of both the electronic conductivity and the lattice dynamics. The phonon mode data indicate that the spin stripes may be accompanied by charge ordering which appears to be quasi-static on the time scale of phonon frequencies. Significant anisotropy of the electronic response within the CuO2 planes was found, with enhancement of the conductivity along the stripe direction. The authors conclude that the results uncover a more complex electronic behavior due to stripes that is beyond an idealized picture of strictly one-dimensional charge channels embedded in an antiferromagnetic insulator. Extensive low temperature scanning tunneling microscopy/spectroscopy (STM/S) studies have been performed on optimally-doped Bi2Sr2 CaCu2 O8+ı (Bi2212) [697]. The measurements reveal electronic inhomogeneities that are reflected in spatial variations in both the local density of states spectrum and the superconducting energy gap.These variations are correlated spatially and have a very short length scale of ∼ 14 Å. Recent scanning tunneling spectroscopy studies of the high Tc superconductor Bi2Sr2 CaCu2 O8+ı (Bi2212) have revealed weak, incommensurate spatial modulations in the tunneling conductance [698]. Fourier analysis of images of these energy-dependent modulations yield the dispersion wave vectors. Comparison of the dispersions with photoemission spectroscopy data indicate that quasiparticle interference, due to elastic scattering between characteristic regions of momentum space, provides a consistent explanation for the conductance modulations, without the need to invoke another order parameter. These results implicate quasiparticle scattering processes as candidates to explain other incommensurate phenomena in cuprates. However, these results do not rule out stripes in Bi2212, but they show that stripes are not needed to explain the undulations on the surface of BSCCO seen earlier by STM.

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Fig. 13.104. Generic temperature T ∗ -dopant concentration x phase diagram for cuprates (schematic) The lines labeled TN and Tc delineate the antiferromagnetic (AFM) and superconducting regions, respectively. The line denoted T ∗ represents the crossover into the pseudogap state. The dashed line is a linear extrapolation of T ∗ (x) to a possible quantum critical point (QCP)

Generic Phase Diagram Based upon investigations on LSCO, YBCO–123, YBCO–124, BSCCO, and other systems, one can construct a generic T − x phase diagram, represented schematically in Fig. 13.104. The phase diagram is very rich and contains insulating, antiferromagnetic, superconducting, pseudogap, two-dimensional (2D) non-Fermi liquid like, and three-dimensional (3D) Fermi liquid like regions. This version of the phase diagram is similar to the one proposed by Tallon and Loram [699] who examined the temperature dependence of the NMR Knight shift and relaxation rate, entropy, electrical resistivity, infrared conductivity, Raman scattering, ARPES and electron tunneling data. In this phase diagram, the maximum value of Tc occurs at a value of the hole concentration pmax ≈ 0.16, while the critical hole concentration pcr where T ∗ vanishes (quantum critical point) is pcr ≈ 0.19. A number of models and notions have been proposed to explain the part of the phase diagram delineated by the curves of T ∗ and Tc vs x (for example, [700–707]). Many of these models involve the

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Fig. 13.105. Schematic temperature T ∗ -dopant concentration x phase diagram for the cuprates. The dashed line labeled T ∗ represents the temperature below which some type of local pairing occurs leading to a suppression of low energy excitations and the formation of the pseudogap. The solid line labeled T denotes the temperature below which phase coherence develops, resulting in superconductivity. The dark solid line labeled Tc delineates the superconducting region

local pairing of electrons (or holes) at the temperature T ∗ leading to a suppression of the low lying charge and spin excitations and the formation of the normal state pseudogap, followed by the onset of phase coherence at Tc that results in superconductivity. Since the phenomenon of superconductivity involves coherent pairing, a bell shaped curve of Tc vs x results as shown schematically in Fig. 13.105. For example,in resonating valence bond (RVB) models [654,700–703,706],which incorporate spin-charge separation into “spinons” with spin  = 1/2 and charge q = 0 and “holons” with  = 0 and q = +e, where e is the charge of the electron, the spinons become paired (spin pseudogap) at T ∗ and coherent pairing of holons (Bose–Einstein condensation) occurs at T , resulting in superconductivity. As illustrated in Fig. 13.105, the temperature T ∗ at which the pseudogap forms decreases with doping level x from a value much larger than Tc at low doping to a value comparable to Tc near optimal doping level where Tc is a maximum. Extrapolation of

T ∗ (x) to 0 K suggests that there is a QCP within the superconducting region, as depicted in Fig. 13.105. A number of theories have been developed that relate the high Tc superconductivity in the cuprates to the existence of a QCP where T ∗ (x) vanishes. It is interesting that superconductivity in certain heavy fermion f -electron materials (e.g., CeIn3 , CePd2 Si2) is found in the vicinity of a magnetic QCP and Tc has an inverted parabolic dependence on pressure, the control parameter. Further evidence for a connection between the pseudogap phase and superconductivity is the similarity in the magnitude of the superconducting energy gap and its variation with direction about the Fermi surface; e.g., as discussed above, the pseudogap and the superconducting gap is BSCCO both have dx2−y 2 symmetry,according to ARPES measurements. Recent Nernst effect measurements on the La2−x Srx CuO4 system in high magnetic fields have provided evidence for vortices (or vortex-like excitations) at temperatures significantly above Tc in the pseudogap phase [708,709]. In the Nernst effect, a thermal gradient is applied to the sample in a magnetic field, and vortices are detected by the large transverse electric field produced as the vortices diffuse down the gradient. In underdoped La2−x Srx CuO4 , the vortex-like excitations extend to a Tvortex (x) is much larger than Tc and about one-half of T ∗ (x) (i.e., Tvortex (x) tracks T ∗ (x)) [708]. In overdoped La2−x Srx CuO4 , the upper critical field curve Hc2(T) does not end at the zero field critical temperature Tc , but at a much higher temperature [709]. These results suggest that Tc corresponds to a loss in phase rigidity, rather that a vanishing of the pairing amplitude. Recently, evidence for vortex-like excitations above Tc were obtained from in-plane magnetoresistivity measurements onY1−x Prx Ba2 Cu3 O7−ı single crystals [710]. 13.5.6

Concluding Remarks

The past two decades of research have consistently yielded a rich variety of new compounds, new phenomena, and new insights into the nature of superconductivity. When heavy-fermion superconductivity was first discovered in CeCu2 Si2 in 1979 by Steglich and coworkers, the coexistence of magnetism and superconductivity had only been es-

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tablished in ternary rare earth compounds such as the rare earth molybdenum chalcogenides RMo6X8 (X = S, Se) and the rare earth rhodium borides RRh4 B4 . Since then about 20 different Ce, Pr, U, and Pu-based heavy-fermion superconductors have been discovered and the list will continue to grow. The past few years have seen renewed interest in heavyfermion superconductors with the discovery of the coexistence of ferromagnetism and superconductivity in UGe2, URhGe, and, possibly, ZrZn2 . The exciting discovery of superconductivity in PrOs4 Sb12 , which contains two superconducting phases which appear to have different order parameter symmetries, and may have a quadrupolar origin, suggests a new route to unconventional superconductivity. Heavy-fermion superconductors continue to be a fertile area of research and enhance our understanding of superconductivity. The organic metals and superconductors show a rich variety of novel ground states and cover a broad spectrum of tunable fundamental physical phenomena which are of current interest. One of the most studied but still very controversial issues is the nature of the superconductivity. For the 1D Bechgaard salts a number of recent experiments indicate a triplet superconducting state. For the 2D organic superconductors, the experimental situation is unsettled. NMR and thermal-conductivity experiments suggest that the superconducting order parameter is unconventional, whereas the measurements of the specific heat indicate a nodeless gap. In an attempt to solve this puzzling situation, it was suggested that an s-wave to d-wave gap-symmetry transition could be induced by an in-plane magnetic field [538]. In this theoretical treatment, phonon-mediated superconductivity with a small wavevector was assumed, which could explain such a transition. Although this proposal would resolve the controversial NMR and specific heat results, other predictions of this theory such as the orientation of the d-wave gap nodes are at odds with recent experimental results [528]. Further studies, especially experiments that directly probe the phase difference of the superconducting wave function, are highly desirable. Many of the high Tc cuprate superconductors are derived from antiferromagnetic insulating par-

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ent compounds through chemical substitutions that dope the CuO2 planes with electrons or holes. It is widely believed that the pairing mechanism in these materials is associated with AFM spin fluctuations. The cuprates have a very rich and interesting T−x phase diagram that contains AFM, pseudogap, and superconducting phases. The pseudogap gap phase and the superconductivity appear to be intimately related and have energy gaps with the same symmetry. The preponderance of evidence indicates that the superconducting order parameter has d-wave symmetry. However, some experiments indicate that the superconducting order parameter has s-wave symmetry or, at least, a significant s-wave component. The pseudogap temperature T ∗ (x) may vanish in the superconducting phase at a QCP that may be profoundly related to the high Tc superconductivity found in these remarkable materials. An interesting feature whose role in the high Tc superconductivity of the cuprates is not yet known is electronic inhomogeneity observed in various experiments (e.g., neutron diffraction, scanning STM). Other high Tc cuprate superconductors will undoubtedly be discovered, and it is not unreasonable that some will have values of Tc greater than the current maximum value of 133 K (at atmospheric pressure). The theory of the normal and superconducting states of the cuprate superconductors represents a significant challenge. The compound Sr2 RuO4 exhibits unconventional superconductivity and is one of the leading candidates for p-wave superconductivity.

13.6

Comparison of the Properties of Different Classes of Novel Superconductors

In this chapter, we have focused on the unconventional superconductivity that is exhibited by three classes of strongly correlated electron materials: heavy fermion f -electron intermetallic compounds, organic compounds, and layered cuprates and ruthenates. For all three classes of materials, there is evidence that the unconventional superconductivity of many of these compounds is anisotropic with an energy gap that vanishes at points or along

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Fig. 13.106. Schematic temperature T ∗ -dopant concentration x and T ∗ -pressure P phase diagram for the cuprates (a); heavy fermion f -electron materials (b)–(e), and organic compounds (f)–(g)

lines on the Fermi surface. This indicates that the superconducting electrons are paired in singlet or triplet spin states with finite angular momentum L > 0. The strongest evidence for superconducting electron pairing with L > 0 comes from experiments on the layered perovskite like cuprate high Tc superconductors that probe the magnitude as well

as the phase of the superconducting order parameter (k). These experiments generally indicate that (k) has dx2 −y 2 symmetry in the hole-doped cuprates, although recent twist angle experiments mentioned above suggest s-wave pairing. As discussed earlier in this chapter, the situation with the electron-doped cuprate high Tc superconductors is less clear, with

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some experiments indicating s-wave pairing and others suggesting d-wave pairing; further research will be required to clarify the situation in this complex group of cuprate superconductors. For the heavy fermion f -electron intermetallic compounds and the organic compounds, a variety of measurements in the superconducting state that probe the magnitude of the superconducting order parameter indicate L > 0 pairing. These experiments are based on power-law T-dependences of the physical properties in the superconducting state.Since it has not yet been possible to perform experiments that are sensitive to the phase of the superconducting order parameter in these materials, as was done in the cuprates, the case for L > 0 pairing in these materials is not as strong as in the cuprates. In all three classes of materials, magnetically ordered ground states are found in close proximity to the superconducting ground states.This is illustrated by the temperature vs composition (T − x) and temperature vs pressure (T − P) phase diagrams of the high Tc cuprate, f -electron heavy fermion, and organic superconductors shown in Figs. 13.106 (a)–(g). Although these phase diagrams differ in the extent to which the magnetic and superconducting phases overlap, the overall similarity is compelling and suggests that there is some kind of relationship between magnetism and superconductivity in these materials. In many of the compounds, the curve of Tc vs x or P has an inverted parabolic shape and occurs near a magnetic quantum critical point, where the magnetic fluctuations are expected to be especially strong and extended in temperature and energy, and are thereby effective in mediating superconducting electron pairing. This is dramatically illustrated in the phase diagrams of the heavy fermion compounds CeIn3 and CePd2 Si2 as depicted in Fig. 13.106 (b). It is interesting to note that electron tunneling measurements on the heavy fermion f -electron superconductor UPd2Al3 have revealed a feature in the normal state conductance at an energy where magnetic ex-

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citations have been identified in inelastic neutron scattering studies on this compound. In addition to superconductivity, non-Fermi liquid behavior in the physical properties is found in examples in all three classes of these materials near magnetic quantum critical points. (The situation is more complex in the f -electron intermetallic compounds where non-Fermi liquid behavior in some systems occurs throughout a large range of concentration and appears to be associated with some type of unconventional Kondo effect.) Experiments on the new heavy fermion filled skutterudite compound PrOs4 Sb12 reveal unconventional superconductivity. In this compound, the heavy fermion state and the superconductivity may arise from quadrupolar fluctuations, rather than magnetic fluctuations that are believed to be responsible for the superconducting electron pairing in the unconventional superconductors considered in this chapter.The pseudogap and superconducting phases appear to be intimately related and have energy gaps with the same symmetry (apparently, d-wave symmetry). The pseudogap temperature T ∗ (x) may vanish in the superconducting phase at a QCP that could be linked to the occurrence of high Tc superconductivity in these remarkable materials.An interesting feature whose role,if any,in the high Tc superconductivity of the cuprates has not yet established is electronic inhomogeneity observed in various experiments (e.g., neutron diffraction, scanning STM). Acknowledgements We would like to thank N.P. Butch, N.A. Frederick, T.A. Sayles, and J.D. Thompson for critical comments regarding this chapter. Invaluable technical assistance in preparing this manuscript was provided by C. Rosado and C. Rayt. Research support of the U.S. Department of Energy, the National Science Foundation, and the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

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Superconductivity

K.H. Bennemann, J.B. Ketterson: Superconductivity Volume 1: Conventional and Unconventional Superconductors Volume 2: Novel Superconductors

K.H. Bennemann J.B. Ketterson Editors

Superconductivity Novel Superconductors Volume 2

123

Professor Dr. K.H. Bennemann FU Berlin Department of Physics Arnimallee 14 14195 Berlin, Germany e-mail: [email protected]

Professor Dr. John B. Ketterson Northwestern University Department of Physics & Astronomy Sheridan Road 2145 60208-3112 Evanston, USA e-mail: [email protected]

ISBN 978-3-540-73252-5

e-ISBN 978-3-540-73253-2

DOI 10.1007/978-3-540-73253-2 Library of Congress Control Number: 2008926416 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: PTP-Berlin Protago-TEX-Production GmbH, Germany Production: le-tex publishing services oHG, Leipzig, Germany Cover design: WMXDesign, Heidelberg, Germany Printed on acid-free paper 987654321 springer.com

Preface

Superconductivity is a classical problem of physics. Since the discovery by Kamerlingh–Onnes in 1911 it has undergone tremendous advances in recent years. Complimentary theoretical and experimental studies have advanced our understanding of systems with strong electronic correlations and the associated superconducting phase transitions. New experimental techniques and field-theoretical methods have emerged. The symmetry of the superconducting order-parameter in certain material classes seems different from the usual singlet, s-wave form found in conventional superconductors. There is strong evidence of a Cooper-pairing mechanism differing from the conventional one arising from the electron–phonon interaction. In view of the simultaneous occurrence of an enhanced magnetic response in many novel superconductors (cuprates, ruthenates and heavy-fermion metals), pairing due to exchange of spin-excitations has been proposed and studied.With this mechanism,triplet Cooper-pairing can also occur, in addition to singlet Cooper-pairing. This may be the case in Sr2 RuO4 and heavy-fermion metals like UPt3. Most notable of course was the discovery of high temperature superconductivity in the cuprates. This discovery resulted in an enormous activity. Combining this with the discovery of superconductivity in other materials such as heavy-Fermion metals,MgB2 , various organics, fullerenes, magnetic metals under pressure, and presumably non s-symmetry triplet Cooper-pairing in ruthenates (Sr2 RuO4) we have an enrichment of the phenomenon superconductivity. Many of the new superconductors are being called unconventional, either with respect to the pairing mechanism or the symmetry of the underlying order

parameter. A central problem is the definite identification of the pairing mechanism. Yet in spite of the new superconductors continuing studies of conventional electron–phonon based superconductivity remain interesting. With the aid of historical hindsight, we expect the field to continue to advance in unexpected directions. Nonetheless it is our view that a state-of-the-art treatise on superconductivity is justified at this time, not only to summarize the present understanding, but also to introduce newcomers to the field, as was the case with the two justly-famous 1968 “bibles of superconductivity” by our former colleague R.D. Parks for an earlier generation. It is hoped that the present book will help to point the way for future work, research and discoveries. The treatise starts with a historical introduction and with the basic phenomenological and microscopic theory of conventional superconductors. This should be particularly useful for students and beginning researchers. This and the following chapters reflect the somewhat refined present state of the understanding of phonon-mediated superconductivity. However, many of the results, in particular the Cooper-pairing theory itself, have much wider validity. The phenomenological models and many-body theories are also useful for the study of other pairing mechanisms. As an example see the treatise on electron–phonon superconductivity, studies using Ginzburg–Landau theory and fluctuation phenomena. Note that appropriately generalized Ginzburg–Landau theories are useful for analyzing multi-component and competing order parameters in unconventional superconductors and for non-equilibrium behavior in general.

VI

Preface

Several chapters discuss recent developments of superconductors containing impurities. Of particular interest are nanostructured superconductors.The important interplay of magnetism and superconductivity is also discussed. Later chapters (see Volume II) focus on novel superconductors, in particular on cuprate superconductivity with its almost universally accepted d-wave symmetry order parameter. A wide variety of important experimental results on high superconductivity are described. In particular, experiments determining the symmetry of the superconducting order-parameter are discussed. The theory analyses the pairing mechanism, important properties and in particular vortices in high-Tc superconductors. Note that the study of vortex matter is also of interest for applications. Several chapters present a theory for cuprate superconductivity assuming Cooper-pairing due to the exchange of spinexcitations. A critical comparison with experiments is presented. Moreover, superconductivity in various novel superconductors like ruthenates, heavyFermions and organics is discussed. Many of the properties involve an interesting interplay of magnetic and superconducting excitations and interactions. The use of neutrons is very important for the analysis of the elementary excitations and of magnetic activity, in particular regarding novel superconductors. Still important open problems are discussed. This should stimulate in particular current and future research.

Berlin, February 2008

The theoretical studies illustrate our present understanding of novel superconductors and should be helpful for further theoretical development towards a general theory that plays the role of the BCS theory in the past. The connections with superfluid 3He, the best understood unconventional Fermion superfluid, and with Bose–Einstein condensation are also discussed. This treatise on superconductivity is a follow up of the previous books on the physics of superconductors published a few years ago. Several important new chapters are presented and most of the other chapters have been largely rewritten, add new results and are generally updated. Thus it is hoped that this book, like Park’s famous books on conventional superconductors, describe our present more or less definite knowledge on superconductivity one of the most interesting problem in physics. It is hoped that the book will remain useful for a long time, as was the case with the Parks treatise. A parallel goal is that the book will be helpful for both students and researchers. We wish to thank the authors for their careful work and great patience.Special thanks go to Christof Bennemann for his essential help in preparing the book. Finally we thank Dr. Ascheron from Springer for his patience and help during editing.

K. H. Bennemann and J. B. Ketterson

K. H. Bennemann is Professor of Physics at the Physics Department FUB (Berlin,Germany), Sloan Fellow, and internationally known for his research on magnetism, superconductivity, and non-equilibrium phenomena. J. B. Ketterson is Professor of Physics at Northwestern University (Evanston, Ill. USA), internationally known for his research in the area of Solid State Physics, in particular of Low Temperature Physics, on Helium. Both are Editors of the well-known books The Physics of Solid and Liquid Helium.

Contents

Volume I. Superconductivity: Conventional and Unconventional Superconductors

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV 1

History of Superconductivity: Conventional, High-Transition Temperature and Novel Superconductors K. H. Bennemann and J. B. Ketterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Novel Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Granular Superconductors, Mesoscopic Systems, Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Phenomenology and Microscopic Theory: Theoretical Foundations L. Pitaevskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Off-Diagonal Long-Range Order and Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Off-Diagonal Long-Range Order in Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The London Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Thermodynamics of Superconductors in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Intermediate State of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Ginzburg–Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Surface Energy at the Boundary Between Normal and Superconducting Phases . . . . . . . . . . . . . . . . . 2.8 Superconductors of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Quantized Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Vortex–Vortex Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Cooper Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Energy Spectrum of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Thermodynamic Properties of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Elements of the Theory of Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Green’s Function of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Temperature Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Temperature Green’s Functions for a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 9 18 23 25

2

27 27 31 33 35 37 38 42 45 46 49 52 54 57 62 65 68 70 71

VIII

Contents

3

Electron–Phonon Superconductivity F. Marsiglio and J. P. Carbotte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Electron–Phonon Interaction: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Critical Temperature and the Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Thermodynamics and Critical Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Anisotropy and MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coexistence of Singlet Superconductivity and Magnetic Order in Bulk Magnetic Superconductors and SF Heterostructures M. L. Kuli´c and A. I. Buzdin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ferromagnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Antiferromagnetic Superconductors (AFS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Magnetic Superconductors in the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Josephson Effect with Bulk Magnetic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Superconductor/Ferromagnet Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74 74 92 108 118 122 139 145 148 153

4

Theory of Superconducting Alloys L. P. Gor’kov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Averages of Green Functions Over Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Superconducting Alloys with a Small Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Paramagnetic Alloys and Gapless Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Eilenberger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164 169 180 183 188 189 196 197 199

5

Impurity Nanostructures and Quantum Interference in Superconductors D. K. Morr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Review: Single Impurity Effects in Conventional and Unconventional Superconductors . . . . . . . . . 6.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Quantum Interference and Quantum Imaging in s-wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . 6.5 Quantum Interference Phenomena in dx2 −y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Molecules on the Surface of dx2 −y 2 -wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 208 216 218 223 224

6

225 225 228 231 234 244 250 253 256

Contents

Nanostructured Superconductors G. Deutscher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Nano-structured Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Granular Structure .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Normal State Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Single Grain Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Weakly Coupled Grains: Granular Insulators and Super-Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Well Coupled Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Critical Temperature of Granular Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

7

Proximity–Coupled Systems: Quasiclassical Theory of Superconductivity V. Chandrasekhar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Transport Equations in the Diffusive Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Keldysh Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Quasiclassical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Non-equilibrium Green’s Functions for Superconducting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Quasiclassical Superconducting Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 The Dirty Limit: The Usadel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Parametrization of the Quasiclassical Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Applications of the Quasiclassical Equations to Proximity-Coupled Systems . . . . . . . . . . . . . . . . . . . . . 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 260 260 262 265 269 272 275 277

8

Principles of Josephson-Junction-Based Quantum Computation S. E. Shafranjuk and J. B. Ketterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Josephson-Junction-Based Qubit Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Qubit Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Quantum Oscillations in Two Coupled Charge Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 SISIS Two-Qubit Gate with Intrinsic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 279 280 283 288 291 294 297 302 304 311 312

9

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Fluctuation Phenomena in Superconductors A. I. Larkin and A. A. Varlamov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ginzburg–Landau Formalism: Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations Below the Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ginzburg–Landau Theory of Fluctuations in Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations Near the S–I Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microscopic Derivation of the Time-Dependent Ginzburg–Landau Equation . . . . . . . . . . . . . . . . . . . . Microscopic Theory of Fluctuation Conductivity of Layered Superconductors . . . . . . . . . . . . . . . . . . . Manifestation of Fluctuations in Various Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 316 319 329 345 350 361 362 365 369 370 373 390 396 405 410 416 429

X

Contents

10.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 11

Universal Properties of Cuprate Superconductors: Evidence and Implications T. Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Critical Behavior at Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Quantum Critical Behavior and Crossover Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Concluding Remarks and Comparison with Other Layered Superconductors . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Matter G. Blatter and V. B. Geshkenbein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Ginzburg–Landau and London Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Layered Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Anisotropic Scaling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Quenched Disorder: Pinning and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Uncorrelated Disorder: Collective Pinning and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Correlated Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Surface and Geometrical Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12 Vortex Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

459 459 465 471 487 487 489

12

Unconventional Superconductivity in Novel Materials M. B. Maple, E. D. Bauer, V. S. Zapf, and J. Wosnitza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Conventional Superconductors Containing Localized Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . 13.3 f -Electron Heavy Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Layered Cuprate and Ruthenate Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Comparison of the Properties of Different Classes of Novel Superconductors . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

495 496 502 505 512 518 532 538 566 575 605 610 615 621 625

13

639 640 641 659 696 717 741 744

Contents

XI

Volume II. Superconductivity: Novel Superconductors List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV High-Tc Superconductivity H. R. Ott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Typical Structural Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Occurrence of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Physical Properties of Copper Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Physical Properties of Non-Cuprate High-Tc Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Tunneling Spectroscopy of Conventional and Unconventional Superconductors J. Zasadzinski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Basic Tunneling Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Tunneling and Strong-Coupling Effects: Microscopic Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Tunneling Spectroscopy of Conventional Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Tunneling in High-Temperature Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Heavy Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Other Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

765 765 767 774 778 813 823 823

15

Phase-Sensitive Tests of Pairing Symmetry in Cuprate Superconductors C. C. Tsuei and J. R. Kirtley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Phase Sensitive Tests: Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Phase-Sensitive Tests: Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Angle-Resolved Determination of Gap Anisotropy in YBCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Universality of the d-Wave Pair State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Implications of d-Wave Pairing Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

833 833 835 841 844 847 861 862 863 864 865

16

17 17.1 17.2 17.3 17.4 17.5 17.6

Photoemission in the High-Tc Superconductors J. C. Campuzano, M. R. Norman, and M. Randeria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics of Angle-Resolved Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Valence Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal State Dispersion and the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superconducting Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

869 869 874 880 901 902 907 912 913 923 924 924 933 935 948 954

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Contents

17.7 Photoemission Lineshapes and the Electron Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963 17.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 18

Neutron Scattering and the Magnetic Response of Superconductors and Related Compounds S. M. Hayden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993 18.2 The Neutron Scattering Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994 18.3 The Static Spin Susceptibility of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 18.4 Magnetic Excitations in Metals and Weakly Coupled Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 18.5 Excitations and Superconductive Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 18.6 High Temperature Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 18.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 18.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 19

Heavy-Fermion Superconductivity P. S. Riseborough, G. M. Schmiedeshoff, and J. L. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 19.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 19.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 19.3 Properties of the Normal State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 19.4 Properties of the Superconducting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 19.5 Heavy Fermion Superconducting Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132 19.6 The Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141 20

Organic Superconductors M. Lang and J. M¨uller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 20.2 Characteristics of Organic Charge-Transfer Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 20.3 Normal-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162 20.4 Superconducting-State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 20.5 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214 21

Concepts in High Temperature Superconductivity E. W. Carlson, V. J. Emery, S. A. Kivelson, and D. Orgad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227 21.2 High Temperature Superconductivity is Hard to Attain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1230 21.3 Superconductivity in the Cuprates: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234 21.4 Preview: Our View of the Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244 21.5 Quasi-1D Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245 21.6 Quasi-1D Physics in a Dynamical Stripe Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257 21.7 Electron Fractionalization in D > 1 as a Mechanism of High Temperature Superconductivity . . . 1259 21.8 Superconductors with Small Superfluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262 21.9 Lessons from Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272 21.10 Lessons from Strong Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277

Contents

XIII

21.11 Lessons from Numerical Studies of Hubbard and Related Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 21.12 Doped Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300 21.13 Stripes and High Temperature Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 A Spin Fluctuation Model for d-Wave Superconductivity A. V. Chubukov, D. Pines, and J. Schmalian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349 22.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349 22.2 Spin–Fermion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358 22.3 Summary of Strong-Coupling Theory for Electron–Phonon Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363 22.4 Strong-Coupling Approach to Spin–Fermion Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366 22.5 Fingerprints of Spin Fermion Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377 22.6 Comparison with the Experiments on Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 22.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 22.8 Note Added . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 22

23

Electronic Theory for Superconductivity in High-Tc Cuprates and Sr2 RuO4 D. Manske, I. Eremin, and K. H. Bennemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416 23.2 Electronic Theory for Hole-Doped and Electron-Doped Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428 23.3 Electronic Theory for Ruthenates (Sr2RuO4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1451 23.4 Results for Hole-Doped and Electron-Doped Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1460 23.5 Results for Sr2 RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 23.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511 Superfluid 3 He and the Cuprate Superconductors A. J. Leggett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 24.1 Introduction: Bose Condensation and Cooper Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 24.2 The Normal State: Is the Fermi-Liquid Picture Valid? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522 24.3 Response Functions: The MIR Peak in the Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525 24.4 The Cooper-Paired States of Superfluid 3 He and the Cuprates: General Considerations . . . . . . . . . . 1526 24.5 Symmetry of the Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1531 24.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 24.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535 24

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545

List of Contributors

Bennemann, K.H. Institut f u¨ r Theoretische Physik, Freie Universit¨at Berlin, Berlin, Germany [email protected] Campuzano, J.C. Dept. of Physics, University of Illinois at Chicago, USA and Materials Science Division, Argonne National Lab., Argonne, USA [email protected] Carlson, E.W. Department of Physics, Purdue University, 525 Northwestern Avenue West Lafayette, IN 47907-2036, USA [email protected] Chubukov, A.V. University of Wisconsin Dept. of Physics, 1150 University Ave Madison, WI 53706, USA [email protected] Emery, V.J. Brookhaven National Laboratory, Department of Physics, New York, USA [email protected] Eremin, I. Institut f u¨ r Mathematische Physik, TU-Braunschweig , Germany [email protected]

Hayden, S.M. H.H. Wills Physics Laboratory, University of Bristol, United Kingdom [email protected] Kirtley, J.B. IBM Thomas J. Watson Research Center, P.O. Box 218 Yorktown Heights, NewYork 10598, USA [email protected]. Kivelson, S.A. Stanford University, Dept. of Physics, GLAM McCullough Bldg 476 Lomita Mall Stanford University, Stanford, CA 94305-4045, California, USA [email protected] Lang, M. Physik-Institut der Johann Wolfgang Goethe Universit¨at, Frankfurt/Main, Germany [email protected] Leggett, A.J. Dept. of Physics, University of Illinois, Illinois, USA [email protected] Manske, D. Max-Plank Institute for Solid State Research, Stuttgart, Germany [email protected] M¨uller, J. Max-Planck Institute for Chemical Physics of Solids, Dresden, Germany [email protected]

XVI

List of Contributors

Norman, M. R. Materials Science Division, Argonne National Laboratory, Argonne, USA [email protected]

Schmalian, J. Ames Laboratory, Dept. of Physics and Astronomy, 1 Osborn Drive Ames, IA 50011, USA [email protected].

Orgad, D. The Hebrew Univ. of Jerusalem, Racah Institute of Physics, Israel [email protected]

Schmiedeshoff, G.M. Occidental College, Department of Physics, California, USA [email protected]

Ott, H. R. ETH Z¨urich, Laboratorium f u¨ r Festk¨orperphysik, ETH H¨onggerberg CH-8093, Switzerland [email protected] Pines, D. Los Alamos National Laboratory, CNLS Mail Stop B258 Los Alamos, NM 87545, USA [email protected]. Randeria, M. Tata Institute of Fundamental Research, Mumbai, India [email protected] Riseborough, P. S. Department of Physics, Polytech University, New York, USA [email protected]

Smith, J.L. National Laboratory Los Alamos, Nevada, USA [email protected] Tsuei, C.C. IBM Thomas J. Watson Research Center, P.O. Box 218 Yorktown Heights, New York 10598, USA [email protected]. Zasadzinski, J.F. Illinois Institute of Technology, Physics Department, 3301 Dearborn, Chicago, IL 60616, USA john [email protected]

Volume II

Superconductivity: Novel Superconductors

14

High-Tc Superconductivity

H.R. Ott ETH Z¨urich, Laboratorium f u¨ r Festk¨orperphysik, Switzerland

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 14.2 Typical Structural Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 14.2.1 BaPb1−x Bix O3 , BaPb1−x Sbx O3 , BaBi1−x Kx O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 14.2.2 Copper Oxide Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 14.2.3 Fullerites, Fullerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 14.2.4 MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 14.3 Occurrence of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 14.3.1 BaPb1−x Bix O3 , Ba1−x Kx BiO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 14.3.2 Copper Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 14.3.3 Fullerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 14.3.4 MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 14.4 Physical Properties of Copper Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 14.4.1 Normal State Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .779 14.4.2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 14.4.3 Coexistence of Superconductivity and Magnetic Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 14.5 Physical Properties of Non-Cuprate High-Tc Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 14.5.1 Ba-Based Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 14.5.2 C60-Based Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 14.5.3 MgB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 14.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

14.1

Introduction

The unexpected and really amazing discovery of J.G. Bednorz and K.A. M¨uller [1], suggesting that certain copper-oxide compounds enter a superconducting state at temperatures exceeding 30 K, provoked an avalanche of research activities worldwide. Early confirmations of the conjecture [2–4],as well as rapid progress in further enhancing the critical temperatures Tc in related materials, i.e., other types of copper oxides,to temperatures above the boiling point of nitrogen [5,6], pushed research in superconductivity

into the limelight of science and public interest. This unprecedented development in the field of superconductivity is best illustrated by plotting the critical temperature Tc as a function of time (see Fig. 14.1), exemplifying the role of the cuprate materials. It is therefore not surprising that referring to highTc superconductivity, is very often understood as considering superconductivity of copper oxides only. This is, however, not quite fair, because other materials, such as Pb/Bi-type oxides and, subsequent discoveries, such as the doped fullerites or fullerides and, more recently, MgB2, also exhibit superconduc-

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Fig. 14.1. Enhancement of critical temperatures of superconductors as function of time.

tivity at amazingly high temperatures. It is for this reason that in this review, although for the most part dedicated to the outstanding properties of a variety of cuprate materials, short overviews of physical properties related with superconductivity of Babased Bi-oxide perovskites, doped C60 materials and the amazing binary compound MgB2 , are also included. The class of quaternary borocarbide compounds for which also maximum critical temperatures for superconductivity exceeding 20 K have been reported, will be treated in a special chapter of this treatise and is therefore not included in this overview. The notion of high temperature superconductivity was introduced by V.L. Ginzburg quite some time before the essential breakthrough in 1986. Re-

search led by him, concentrating on identifying alternative electron-phonon interactions in solids that might lead to superconductivity at elevated temperatures, started in 1964 and resulted in a compendium with the title “High Temperature Superconductivity”, published 1977 in Russian in Moscow. An English translation of this work appeared 1982 in London [7]. Since at present even the highest critical temperatures that have been achieved, by common standards still have to be regarded as low temperatures (∼–140 ◦ C), we rather use the terminology of high-Tc superconductivity in our context. In spite of substantial efforts in both experimental and theoretical research, the mysteries behind the occurrence of superconductivity in cuprate materials up to temperatures exceeding 100 K are still to be regarded as conspiring to pose one of the major unsolved problems of contemporary physics. Most of the physical properties of the Cu oxides have experimentally been established with a high degree of reliability and advances in preparing the materials are such that excuses for the lack of understanding as being related to spurious effects and uncertainties in materials compositions, homogeneities and impurity content can no longer be accepted. However, it is still not possible to control the critical temperature Tc for superconductivity of these materials by suitably tailoring the chemical composition of the compounds and alloys to the extent that Tc can be enhanced at will. This unsatisfactory situation is, at least partly, due to the fact that no real basic understanding of the essential processes that lead to superconductivity in these cuprates at temperatures far above those for common metals and alloys exists and therefore no unfailing guidance for the design and subsequent synthesis of new and promising materials with even higher critical temperatures is available. Simple intuitive approaches for improving the situation are made difficult by the sheer chemical complexity, of the known materials, at least when viewed at the level of conventional standards of condensed matter physics.The simplest,i.e.,ternary compounds have to be doped with charge carriers in order to exhibit metallic conductivity, giving way to superconductivity at low temperatures. Most of these su-

14 High-Tc Superconductivity

perconductors contain four or more constituents in their chemical composition and their crystallographic structures, although being rather similar for all of them, exhibit some intricate details whose importance is still debated. It is for this reason that, before presenting and discussing a selection of physical properties of high-Tc superconductors, a section on materials aspects including chemical compositions and crystal structures is inserted at the beginning of this chapter. An additional justification for this section is the fact that more recently, solids with similar structural properties have attracted a lot of attention in other areas of condensed matter research, involving spin charge and orbital ordering phenomena [8]. With respect to physical properties in general and to features of the superconducting state in particular, we shall concentrate on some typical aspects rather than list and present many details. For instance, the entire field of vortex physics that has emerged and has attracted a lot of attention in connection with high-Tc superconductors, is discussed in a special chapter of this treatise. It was recognized very early [9] that the cuprate materials, which exhibit the highest critical temperatures for superconductivity at present, cannot simply be regarded as common metals, because even the normal state of these materials exhibits anomalous features that are difficult to understand. Therefore, some attention is also given to properties of the normal state above Tc .

14.2

Typical Structural Characteristics

Although, as pointed out above, no consensus about the real causes for superconductivity at elevated temperatures has yet been achieved, all relevant materials to be discussed here, with the exception of the fullerenes and MgB2 , in some way share a common structural feature and this is the unit cell of the perovskite structure shown in Fig. 14.2. This structure is adopted by ABO3 compounds in which A is a fairly large cation and B, a metal element, helps to form a three-dimensional array of corner-sharing BO6 octahedra. The undistorted version is cubic, as indicated in Fig. 14.2, and only very few compounds for which the ionic radii of the A atoms are of sufficient size,are

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Fig. 14.2. Schematic representation of the crystallographic unit cell of CaTiO3 (perovskite)

known to adopt the truly cubic perovskite structure. Most of the so called perovskite materials crystallize in a distorted version, the most common being an orthorhombic distortion. Most of the fullerenes also crystallize in cubic structures (fcc) but the occupation per lattice site is given by a rather large basis, formed by the C60. During the time,when investigations of superconducting materials were focused on either chemical elements or, at most binary compounds and alloys, it was argued [10] that a high symmetry of the crystal lattice was favorable for achieving high critical temperatures. Superconductivity of the cubic A15 compounds with critical temperatures between 15 and 20 K was taken as the show case for this conjecture. The most recent and also rather surprising exception from this trend is, no doubt, MgB2 which adopts a structure with hexagonal symmetry [11]. Its critical temperature, of the order of 40 K, is by now no longer a top value in general, but for simple binary compounds, it most certainly is. 14.2.1

BaPb1−x Bix O3 , BaPb1−x Sbx O3 , BaBi1−x Kx O3

The structure of these non cuprate materials of the type BaPb1−x Bix O3 and Ba1−x Kx BiO3 varies with the parameter x and different varieties of distorted perovskite type arrangements of the atoms are observed. The Pb/Bi alloy series exhibits a number of structural phase transitions, starting with an orthorhombic lattice for metallic BaPbO3 [12], changing to tetragonal at x ∼ 0.1, back to orthorhombic for x ∼ 0.35 [13]

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diate valence (between 2+ and 3+) seems appropriate. The three-dimensional character of the original perovskite structure,which may also be viewed as being built by a stacking of AO and BO2 planes, is lost because these planes are now stacked in different sequences, leading to a more or less sizeable anisotropy between the directions parallel and perpendicular to these planes. A fairly transparent case in this respect is the parent compound of the material where superconductivity of the cuprates was discovered by Bednorz and M¨uller [1]. La2−x Ax CuO4+ı (A = Sr, Ba) Fig. 14.3. Schematic representation of the crystallographic unit cell of BaBiO3 , as reported in [14]

and finally distorting to a monoclinic lattice structure at x ∼ 0.9 [14]. In the range of 0.1 < x < 0.3, also the Pb/Sb alloys adopt a tetragonally distorted perovskite-type structure. BaBiO3 is electrically insulating. Its crystallographic unit cell is shown in Fig. 14.3, revealing the distorted perovskite-type arrangement of the atoms by the quasi-octahedral coordination of oxygen atoms around the Bi atoms on inequivalent sites. On site B(1), one of the Bi-O distances is much shorter than on site B(2) and this freezing of a breathing mode is thought to be the reason for the insulating ground state. The substitution of Ba with K atoms in BaBiO3 to form Ba1−x Kx BiO3 , eventually triggers a transition to a cubic crystal lattice and a metallic ground state [15,16]. It is obvious that for these oxides, structural and electronic properties are intimately coupled, a feature that is also observed for the cuprate materials to be discussed below. 14.2.2

Copper Oxide Superconductors

The crystal structures of all superconducting Cu oxides are more or less evidently related to the perovskite structure. These Cu compounds belong to a large class of mixed-valency Cu oxides where the Cu cations may adopt different ionic configurations (2+ or 3+). In some cases rather the concept of interme-

An early study [17] of substances of this type with A = Ca, Sr, Ba and Pb revealed the metallic conductivity for these materials in the form of a decreasing electrical resistivity with decreasing temperature, i.e., ∂/∂ T > 0. For unknown reasons, this study was limited to temperatures above 200 K, but it contained also important information concerning the crystal structure of this series of compounds. Later a more detailed investigation [18] confirmed and extended these findings. As is already legend by now, it was this type of materials where Bednorz and M¨uller [1] found the first evidence for the onset of superconductivity between 30 and 40 K. Ternary La2 CuO4 , an insulating antiferromagnet, crystallizes in a tetragonal K2 NiF4 -type structure but at lower temperatures adopts an orthorhombically distorted version of this structure, induced by a cooperative alternating tilting of the CuO6 octahedra about the [110] tetragonal axis, as shown in Fig. 14.4 [19, 20]. For x = 0 the tetragonal to orthorhombic transition occurs at approximately 530 K. The onset temperature of the orthorhombic distortion Td can be reduced by partly replacing La by an alkaline earth element A and above a critical concentration of x ∼ 0.2, the tetragonal K2 NiF4 -type structure is stable down to very low temperatures [21, 22]. The tetragonal arrangement, also denoted as T-structure and shown in Fig. 14.5, may be regarded as a stacking of different planes that are also contained in the original perovskite structure. Between two CuO2 (BO2 ) planes, two LaO (AO) planes instead of only one are inserted, hence weakening the three-dimensional

14 High-Tc Superconductivity

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Fig. 14.5. Schematic representation of the tilting of the oxygen octa-hedra in La2−x Srx CuO4 (see [36])

a transition from a tetragonal high-temperature to an orthorhombic low-temperature phase [24]. The transition temperature Td is only weakly reduced upon increasing ı. For low oxygen surplus, a macroscopic phase separation phenomenon is observed below room temperature. The material separates into an oxygen-rich metallic phase and an oxygen-poor insulating phase with an antiferromagnetically ordered ground state [24]. Ln2−x Cex CuO4−ı (Ln = Pr, Nd, Sm)

Fig. 14.4. Schematic representation of the tilting of the oxygen octahedra in La2−x Srx CuO4 (see [36])

character of the structure. The essential subunits are clearly the CuO2 planes, a major characteristics of all known cuprate superconductors. For A = Ba, the appearance of a low-temperature tetragonal phase has been reported for x values around 0.125 [23]. As is well known, enhancing x eventually leads to a metallic behavior and superconductivity. Optimal conditions for superconductivity are reached for materials exhibiting the orthorhombic crystal structure at T ∼ 4Tc . Structural effects are particularly pronounced in La2 CuO4+ı , containing excess oxygen on interstitial sites.Also oxygen rich material undergoes

Cu-oxides with a 214-type composition as mentioned above also form if La is replaced by rare-earth or lanthanide (Ln) elements.These compounds,however, adopt a somewhat different crystal structure [25], the so called T’ structure and they keep the tetragonal structure down to low temperatures. It is depicted in Fig. 14.5. Here, the oxygen environment of each Cu atom in the form of a planar square is distinctly different from that of an octahedron in the T structure. As will be discussed later, the ternary compounds are again insulating antiferromagnets. Both the Cu spins and the localized Ln moments are involved in magnetic-ordering phenomena, at different temperatures, however. Metallic behavior and superconductivity is obtained by a partial replacement of the trivalent Ln element by tetravalent Ce and, in addition, a slight reduction of the oxygen content.

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Fig. 14.6. Schematic representation of the unit cells of the crystal structure of YBa2 CuO7−ı for (a) ı = 0, and (b) ı = 1

Ln2−x−y Cex Sry CuO4−ı Yet another type of Cu-O coordination is obtained in this type of compounds where part of the Ln sites are occupied by Ce and Sr [26]. The resulting unit cell of the structure is shown in Fig. 14.5. A remarkable feature of this T*-structure is the lack of an inversion center, obviously implying polarity but, nevertheless, with the proper values of x,y, and ı, superconductivity may still be achieved.The Sr content must be large enough in order to allow for a sequential ordering (Sr,Ce) and (Nd,Ce) planes as indicated in Fig. 14.5. MBa2 Cu3 O7−ı , MBa2 Cu4 O8 Structurally,these two types of compounds, where M may be Yttrium or any element of the rare-earth series except Ce and Tb, are related but chemically, the second type of materials is much more stable than the first. Again the structures may be viewed as stackings of different layers with different chemical compositions. For YBa2 Cu3 O7−ı (YBCO-123), the stacking sequence in the crystallographic unit cell is Y-CuO2 -BaO-CuOx -BaO-CuO2-Y, reminiscent of the perovskite structure [27]. The resulting orthorhombic unit cell is shown in Fig. 14.6a for the ideal case where ı = 0, i.e., x = 1. We note two inequivalent Cu sites which can most easily be distinguished by

their different oxygen environments. In detail the M atoms separate two identical blocks which contain two CuO2 planes with the Cu(2) sites, two BaO planes and a plane formed by Cu-O chains along the b direction of the orthorhombic structure involving the Cu(1) sites. As may be seen from Fig. 14.6a, the oxygen coordination of the two Cu sites is pyramidally for the Cu(2) sites and linear for the Cu(1) sites. It is the missing oxygen between the chains which gives rise to the orthorhombic distortion of the lattice. This low temperature orthorhombic structure develops out of a tetragonal high temperature structure at about 750 ◦ C by a rearrangement of atoms within the oxygen sublattice. This rearrangement can be reversed at low temperatures by reducing the oxygen content of the material. For ı = 0.6, the structure is again tetragonal also at low temperatures. Upon depletion, oxygen vacancies are formed within the Cu-O chains and they redistribute in a fashion such that on the average, oxygen atoms occupy sites along the a and b directions with equal probability [28].For ı = 1, i.e., x = 0, the former Cu-O chains are fully depleted of oxygen. The resulting structure is shown in Fig. 14.6b. This variation of the oxygen content not only influences the structure of the crystal lattice but also many other properties of different ground states, as we shall see below. A particular aspect of

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cally less fragile than the YBCO-123 variety and crystals may be grown without twin boundaries, a major problem in the growth of YBCO-123 compounds. The insertion of a second Cu-O chain element, forming Cu-O ribbons rather than chains, enhances the anisotropy, especially within the planes perpendicular to the c axis. In this Y–Ba–Cu–O quaternary system, an additional stable compound with yet another although related structure and composition, namely Y2 Ba4 Cu7 O14+ı , in short YBCO-123.5, has been identified [32,33].The corresponding unit cell is obtained by stacking the unit cells of YBCO-123 andYBCO-124 on top of each other. Concerning the Cu-O structural elements, the unit cell of the atomic arrangement thus contains, apart from the notorious Cu-O planes, both Cu-O ribbons and Cu-O chains. Fig. 14.7. Schematic representation of the unit cell of the crystal structure of YBa2 Cu4 O8 (see [36])

the crystal structure in this YBCO-123 series is the formation of superstructures in different regions of oxygen content, leading to inhomogeneities and different orthorhombic phases at different values of ı. Since a detailed discussion of these aspects is beyond the scope of this review, we refer to relevant work in the literature [29]. The unit cell of the MBa2 Cu4 O8 (YBCO-124) compounds [30,31] is shown in Fig.14.7. As mentioned above this type of compound is chemi-

Am M2 Rn−1 Cun O3n+m+1 A fairly large number of high-Tc cuprate superconductors may be classified according to the schematic chemical composition given in the subtitle. This type of compounds form with A = Hg, Tl or Bi, M = Sr or Ba and R = Ca or a heavy rare earth. These materials are of particular importance because some of them exhibit the highest critical temperatures that have been achieved until today. For A = Hg, m may be 1, for Bi only 2 and for Tl 1 or 2. Although Bi-

Fig. 14.8. Schematic representation of the crystal structure of some Bi-based Cooper oxide superconductors: Bi-2201, Bi-2212 and Bi-2223 (see [36])

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Fig. 14.9. Schematic representation of the crystallographic unit cells of Hg-12(n-1)n copper oxides

Fig. 14.10. Schematic representation of the crystallographic unit cell of HgBa2 Ca2 Cu3 O8+ı

based cuprates have been synthesized with the 1222type structure, all of them turned out to be non superconducting above 1.5 K [34]. In most cases the parameter n varies from 1 to 4 but may also adopt higher values. The parameter n indicates the number of CuO2 planes within one unit cell, separated from each other by n-1 Ca layers, and as such it is related with some physical properties to be discussed below. The structures of these compounds may be viewed as sequential stackings of elements of the perovskite as well as the NaCl structure [35]. Very often, crystallographic shears, i.e., shifts between adjacent layers are introduced in the stacking sequences. A nice com-

pilation of possible atomic arrangements in the Tlbased and Bi-based compounds is given, e.g., in [36] and therefore we present only three examples of the physically most important Bi compounds in Fig. 14.8 (on page 771). It is also possible to partially replace some of the above mentioned constituents by other elements. One such example is Pb substituting for Bi, which in some cases provides a way to stabilize a favorable phase, such as the Bi-2223 phase [37]. The latest progress in raising Tc has been made in the series HgBa2 Can−1 Cun O3n+2 [38] and it also has been possible to synthesize thin layers of material where n ≥4 [39]. A schematic representation of the stacking with increasing n is shown in Fig. 14.9 and a somewhat more detailed drawing of the unit cell of HgBa2 Ca2 Cu3 O8+ı ,the material for which the highest values of Tc have as yet been observed, is presented in Fig. 14.10. Ca1−x Srx CuO2 The structure of this type of compound is of interest here, because it may be viewed as the result of enhancing the number n of CuO2 layers per unit cell in the compounds discussed in the previous section towards infinity. Taking into account the n − 1 Ca interlayers, for large values of n, the stoichiometry will approach that of CaCuO2 . With respect to the perovskite structure, the (Ca/Sr) layers may be regarded as AO layers from which all the oxygen atoms have been removed. CaCuO2 , a hypothetical tetrago-

14 High-Tc Superconductivity

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are mentioned in Sect. 14.5.Another important compound,with all the features of the m = 2 and n = 2 Tlbased or Bi-based compounds, except for an inserted oxygen depleted Cu layer, is Pb2 Sr2YCu3 O8 [43]. 14.2.3

Fig. 14.11. Schematic representation of the crystal structure of Ca1−x Srx CuO2 (see [40])

nal compound does not form but it has been found that small amounts of Sr on Ca sites can stabilize the structure, which is shown in Fig. 14.11, and single crystals may be grown [40]. Subsequently it has been found that a high pressure synthesis helps to extend the compositional range of stability considerably [41]. General Remarks This brief overview of some crystal structures of relevant cuprate materials demonstrates quite clearly that one ingredient, namely the CuO2 planes, are a common building element for all the structures presented in the previous subsections. Even before discussing the physical properties of these materials we note that the electronic features are, to a large extent, dominated by these CuO2 planes. They are separated from each other by additional building blocks that serve both to stabilize the structure and as charge reservoirs, which control the number of itinerant charge carriers in the planes containing Cu atoms. In this sense, the structures of the compounds mentioned in the first three subsections contain one CuO2 plane per unit cell. Two CuO2 planes per unit cell are the hallmark of the compounds mentioned in the fourth subsection. Structurally the most simple member of this type of double-layer compounds, but not mentioned in detail here,is La2−x Srx CaCu2 O6 [42]. The compounds with the possibility of accommodating more than two CuO2 planes per unit cell

Fullerites, Fullerides

Fullerites are solids composed by a regular arrangement of C60 molecules, which belong to a large class of fullerenes,i.e.,stable molecules formed by a closed carbon network (C60,C70 ,C540 ,etc.) [44].For our purposes, only C60 molecules are of interest, because up to now, bulk superconductivity has only been observed in solids formed by them. A pure C60 solid is, under normal circumstances, an electrical insulator [45]. It adopts an fcc structure which, with decreasing temperature, transforms to a simple cubic structure, at 260 K, accompanied by an orientational order of the individual C60 molecules [46]. Metallic C60 -based compounds, the fullerides, may be obtained by doping [47] and the most relevant materials in our context are A3 C60 , where A is K, Rb, Cs or some combination of these three alkali elements and Na2AC60, where A may again be one of the other alkali elements or a combination of them [45]. The structure of these materials is of fcc type with

Fig. 14.12. Schematic representation of the unit cell of fcc C60

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varying space group symmetries,however.Non cubic structures of superconducting fullerides have been reported for the cases of Cs3 C60 and NH3 K3 C60 under pressure. The compounds with chemical composition of the type A4 C60 usually adopt a bct cubic structure and are insulators. A schematic drawing of the cubic structure of the A3 C60 type materials is shown in Fig. 14.12. Moreon structural aspects of C60 material may be found in [48]. 14.2.4

MgB2

Although this compound has been known for a long time, it obviously had never been tested for superconductivity until late in 2000. The discovery of its unexpectedly high critical temperature of the order of 40 K by Akimitsu and co-workers [11] came as a real surprise. MgB2 crystallizes in the hexagonal AlB2 -type structure, shown in Fig. 14.13, which exhibits clear 2D-type features. The B atoms form layers that are equivalent to the layered structure of graphite. It has been suggested [49] that this apparent two-dimensionality might be the key to the high value of Tc of this material, reminiscent of the case of the cuprates.

Fig. 14.13. Schematic representation of the crystallographic unit cell of MgB2

14.3

Occurrence of Superconductivity

For all the materials that in this review are considered in relation with high-Tc superconductivity, a superconducting ground state is achieved only in limited regions of chemical composition. In this section, we

briefly discuss the relation between chemical composition and the observation of superconductivity, including the established critical temperatures Tc , for a few instructive cases. 14.3.1

BaPb1−x Bix O3 , Ba1−x Kx BiO3

Superconductivity in the Pb/Bi compound series [50] is restricted to values of x that are close to 0.25 where a maximum critical temperature of approximately 12 K is observed [51,52].This value of Tc is unexpectedly high and remarkable because of the absence of any element of the d-transition range in the chemical composition. Figure 14.14 demonstrates that, although partial superconducting transitions, measured magnetically and resistively, have been observed around x = 0.1 and x = 0.3, narrow and complete transitions are monitored only for x = 0.25 [52]. It is therefore quite obvious that the optimal conditions for superconductivity are reached for BaPb0.75 Bi0.25 O3 and any deviation from this composition leads to a rapid deterioration of the superconducting properties and, in addition, favors inhomogeneities in the chemical composition. It is conceivable that the occurrence of superconductivity is extremely dependent on the doping level and that compositional inhomogeneities lead to the spread in Tc displayed in Fig. 14.14. Replacing Bi with Sb leads to a less favorable situation for superconductivity [53]. Maximum critical temperatures are again reached for x ∼0.25 but they barely exceed 3 K. It is not quite clear why Tc is so low in this case. More successful in this respect were experiments with material that was prepared by partly replacing the divalent Ba by the monovalent alkaline metal K in BaBiO3 . This type of substitution and the concomitant reduction of the electron concentration induces the already mentioned structural phase transition and the symmetry of the crystal lattice changes from monoclinic to cubic. Likewise, the chemical substitution leads from an insulating to a metallic state. K-doped BaBiO3 is electronically very similar to Bidoped BaPbO3 . Optimal conditions are reached at the composition Ba0.6 K0.4 BiO3 for which Tc is approximately 30 K [54,55].

14 High-Tc Superconductivity

Fig. 14.14. Onset and widths of the superconducting transition of BaPb1−x Bix O3 (see [52])

14.3.2

Copper Oxides

Very shortly after the discovery of superconductivity in Cu oxides it was realized that in these materials,the occurrence of superconductivity is intimately related with the number of itinerant charge carriers which, in most cases, can be varied rather easily by simply altering the chemical compositions of the materials in specific ways. For some types of compounds, the result of this procedure in the form of a transition from an insulating to a metallic behavior and superconductivity, or vice versa, may be illustrated very well, for others it proves to be less obvious. In what follows, four rather instructive cases, representative for the cuprate superconductors, are presented.

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of superconductivity in exactly this class of cuprates, when the physical properties of these materials were studied in great detail. The results of these investigations are being reviewed in some of the following sections but we summarize part of them in the form of low temperature phase diagrams, displaying the variation of the physical behavior as a function of the concentration of itinerant charge carriers. The charge carrier doping may be accomplished in two ways, either by enhancing the oxygen content to above four atoms per formula unit or by replacing part of the trivalent La ions by divalent ions of the alkaline earth elements Sr and Ba. No successful attempts to replace La by Ca have been reported. Figure 14.15 captures the low temperature phases that are identified if La is replaced by Sr [57]. The diagram is somewhat more complicated if Ba is chosen as the substitute. In the region of x ∼ 0.12 where a slight indentation in Tc (x) is apparent for the Sr doping (see Fig. 14.15), Tc is suppressed completely if the dopant is Ba (see also below). As mentioned above, the ternary compound La2 CuO4 is an insulator with an antiferromagnetically ordered ground state.Upon doping with Sr2+ , the N´eel temperature TN of 325 K for x = 0 is efficiently reduced, giving way to a spinglass type ground state for 0.02 < x < 0.05. Once

La2−x Srx CuO4+ı The ternary compound La2CuO4 has long been known to be an electrical insulator. Early measurements revealed peak features in the temperature dependence of the magnetic susceptibility which indicated some type of magnetic order below approximately 200 K [56]. It was only after the discovery

Fig. 14.15. Low temperature La2−x Srx CuO4 (see [57])

phase

diagram

of

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the Sr concentration exceeds 5%, superconductivity sets in at low temperatures. The optimal doping, i.e., the highest transition temperature Tc of about 40 K is reached at x ∼ 0.15. Further increasing x reduces Tc and superconductivity is lost for x > 0.25. Since, as mentioned above, the structural transition from a tetragonal to an orthorhombic structure has been reported to disappear at x ∼ 0.21, bulk superconductivity seems to persist into the tetragonal structure regime. This observation is important in view of discussing relations between the crystal structure symmetry and the occurrence of superconductivity, as illustrated in the following case. A tiny dip in the Tc (x) curve at x ∼ 0.125, displayed in Fig. 14.15, signals an effect that is much more pronounced in La1−x Bax CuO4 . Again around a concentration of x ∼ 0.125, Tc of the Ba-doped series is suppressed to zero and static magnetic order is observed instead [58,59]. Detailed studies have traced this back to the occurrence of a low temperature tetragonal structure [23,59,60]. As already mentioned in a previous section enhancing the oxygen content leads to phase separation phenomena in La2 CuO4+ı . The oxygen rich metallic

phase is superconducting at low temperatures and again, Tc values exceeding 30 K are achieved. An instructive discussion of the ground state properties of La2 CuO4+ı is offered in [57]. Ln2−x Cex CuO4−ı (Ln = Pr, Nd, Sm) This variety of compounds is, with respect to electronic properties, distinctly different from all the other Cu-oxide compounds. Here, the doping is not with holes but with electrons [61]. This difference is also reflected in the phase diagram capturing the occurrence of superconductivity [62,63].The antiferromagnetic phase is again rapidly suppressed upon doping and gives way to a superconducting ground state rather abruptly, as may be seen in Fig. 14.16. The maximum value of Tc is found next to the phase boundary and a further enhancement of doping leads to a reduction of the critical temperature and, again, to a rather abrupt disappearance of superconductivity. The range of doping where superconductivity has been observed is much more narrow than in the hole doped cuprates. The discovery of electron doped superconductors was initiated by the preceding observation of a superconducting ground state in the chemically similar but somewhat more complicated series adopting the T ∗ structure [64] (see also Sect. 14.2.2).

YBa2 Cu3 O6+ı

Fig. 14.16. Low temperature phase diagram Nd2−x Cex CuO4−ı -type compounds (see [63])

of

In this series of compounds, tetragonal YBa2 Cu3 O6 represents the electrically insulating and antiferromagnetically ordering parent compound [65]. The doping of this material is accomplished by enhancing the oxygen content. Superconductivity sets in at oxygen concentrations close to those where the structural instability leads to an orthorhombic crystal lattice, i.e., ı ∼ 0.45. The critical temperature increases roughly in two steps, with two plateaus at approximately 60 and 90 K [66]. Optimal values for Tc , of the order 91 to 92 K are obtained for ı between 0.9 and 0.95. Further enhancing the oxygen content leads to a slight decrease of Tc , by 1 or 2 K. The schematic Tc (x) variation is displayed in Fig. 14.17.

14 High-Tc Superconductivity

Fig. 14.17. Variation of the critical temperature Tc as a function of ı in YBa2 Cu3 O6+ı (see [66])

Bi-Based and Tl-Based Copper Oxides Most of what is known about superconductivity of the Bi-based compounds [67–69] is very similar to what we have discussed in Sect. 14.3.2 concerning the hole doped La-cuprates. Also here, underdoped and overdoped regimes embrace the situation of optimal doping, more or less independent of the maximum value of Tc which depends on the chemical composition of the compound, i.e., on the number of Cu-O planes per unit cell. It is rather amazing that the maximum Tc ’s of the individual series of alloys ranges from 15 to 125 K. Again it is observed that strong overdoping eventually leads to the disappearance of superconductivity. The corresponding data base for superconducting Tl-based compounds [70–72] is much less extensive but there is no evidence for a significantly different behavior. 14.3.3

Fullerides

The secret of inducing superconductivity in these types of materials is again doping, with electrons in this case [73, 74]. This is, e.g., achieved with the introduction of various numbers, between 1 and 3, of atoms of alkaline metal elements per unit cell. The

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Fig. 14.18. Variation of the critical temperature Tc of fullerides as a function of the lattice parameter a, achieved either by doping or by application of external pressure (p) (see [45])

main result of numerous investigations is the distinct variation of the critical temperature with the lattice parameter, seemingly a universal feature and, with a few exceptions,independent of the chosen material. An increase of the separation a of the atoms leads to an enhancement of Tc [75] and, vice versa, a reduction of Tc upon applying external pressure is observed [76]. This has been verified by changing the lattice parameter both via variation of the chemical composition and by applying external pressure. A schematic representation of T(a) for a selection of these materials is shown in Fig. 14.18. This plot suggests that an increase of Tc may be achieved by simply enhancing the lattice parameter a. Slight deviations from a universal T(a) behavior under pressure have, however, been reported [77,78]. Considering an extended set of data, and plotting Tc /Tmax versus the c concentration n of valence electrons, whereby Tcmax is the maximum critical temperature in a single alloy series, it has been claimed that the maxima of all the considered Tc ’s is reached if n = 3 and that no superconductivity is observed if either n < 2.5 or n > 4.0 [79]. The maximum value of the critical temperature reported for bulk material is that of Cs3 C60 ,

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apparently a non cubic compound with a Tc of about 40 K [80]. 14.3.4

MgB2

This compound does not seem to allow for large variations of the chemical composition without loosing superconductivity [81]. So far it appears that the binary compound with the proper stoichiometry reveals the highest Tc . Most attempts of alloying have led to substantial reductions of Tc and all other diborides with different cation elements have,so far,resisted to exhibit superconductivity at similarly high critical temperatures. An exception is Zn for which a tiny enhancement of Tc at low doping concentration may have been observed [82,83].

14.4

Physical Properties of Copper Oxides

As we have indicated in the preceding section, the general properties of the relevant cuprate compounds and alloys depend crucially on the chemical composition of the investigated materials. The same is true for those materials that actually do exhibit high-Tc superconductivity. Already early results of measurements probing the physical properties of these materials indicated anomalous features also in the normal state preceding the superconducting ground state and it soon became clear that the occurrence of this superconductivity at unusually high temperatures would not be explained without a sound understanding of the character of the normal state [84]. It is for this reason that we also briefly discuss those normal state properties that are difficult to understand on the basis of the very successful Fermi liquid concept that has, in the past, been used for describing the characteristic electronic properties of common metals. To keep this review tractable, only few examples that are believed to capture the essential issues of both the normal and the superconducting state of this type of materials are presented and discussed. Most of these examples, if they are not intended to emphasize the influence of the crystal structure or the chemical composition, are chosen to represent those materials with so called optimal doping, i.e., with the composition for which

a maximum Tc has been observed. In these cases, some evidence for the existence of a boundary in the quasiparticle excitation spectrum that may be identified as a Fermi surface has been obtained from both conventional [85] and angle resolved photoemission spectroscopy (ARPES) [86,87] with high energy resolution. A schematic view of the cross sections of the 3D Fermi surfaces of Bi cuprates, deduced from ARPES measurements, is shown in Fig. 14.19. This evidence is more obvious for overdoped materials than it is for optimally doped or underdoped compounds although more recently it has been claimed that also for underdoped material, resolution limited signals, indicating well defined quasiparticle excitations, have been obtained [88]. Measurements with further improved energy resolution, however, seem to reveal that quasiparticle-type signals are only well identified if the investigated material is in the superconducting state [89]. It thus appears that it is still unclear whether well defined quasiparticles do exist in the normal state. At any rate, their manifestation certainly depends on the doping level of the material. As we shall see below, the difficulties encountered with the spectra of underdoped cuprates may arise because of the formation of a pseudogap

Fig. 14.19. Mapping of the Fermi surface of Bi-based superconducting cuprates by ARPES (see [86])

14 High-Tc Superconductivity

in the quasiparticle excitation spectrum already in the normal state of these materials. In this context it is important to note that, because of the particular crystal structures, featuring the CuO planes as the essential structural element, most of the physical properties of these cuprates exhibit distinct anisotropies. 14.4.1

Normal State Properties

The first indications of anomalous features of the normal state of high-Tc cuprates were provided by measurements of the temperature dependence of the electrical resistivity (T) [90] of some of these materials, both parallel and perpendicular to the Cu-O planes [91]. These early transport results were confirmed in many cases and with improving sample quality, and complemented by additional data, mainly concerning the thermal conductivity and the Hall effect,both in the normal and the superconducting state.

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common behavior in the so called overdoped region. This linear proportionality of the resistivity with respect to the temperature is one of the common features of the normal state of cuprate superconductors, most clearly borne out in the underdoped regime. An important case in this respect is the compound Bi2Sr2 CuO6 (Bi2201) where the linear in T variation of  may be observed from low temperatures, because Tc is only of the order of 10 K, up to 700 K [93]. The persistence of this temperature dependence of  or, equivalently, of the scattering rate  −1 of the electronic quasiparticles down to Tc , is taken as evidence for an anomalous metallic state, not compatible with expectations of the Fermi liquid model, where a T2 dependence of  −1 is expected. In particular, the absence of any onset of residual resistance, definitely expected to be visible if the absolute magnitude of the measured values of  are considered, is difficult to understand. Another unusual feature of the electrical resistivity is the generally observed large anisotropies and

Electrical Resistivity First it seems instructive to consider the variation of (T), measured for currents parallel to the Cu-O planes, upon doping. A well studied case is La2−x Srx CuO4 and some selected data are shown in Fig. 14.20 [92]. What is amazing, is the strong temperature dependence of the resistivity even at large absolute values. The increasing concentration of hole doping with increasing x is reflected in both the magnitude of  at room temperature and the slope ∂/∂T. Another typical feature is the extended region of the linear variation of  with temperature for x ≤ 0.18. For common metals, the linear in T variation of the resistivity is restricted to temperatures around 200 K or above, but usually is not observed much below 100 K. It may also be seen that for higher doping levels, this particular feature is lost and the change of (T) with increasing x is much less pronounced than for lower values of x. Considering that the maximum value of Tc for this compound series is reached at optimal doping with x ≈ 0.15, (T) is clearly anomalous in what has been termed the underdoped regime and progressively adopts a more

Fig. 14.20. Temperature dependence of the electrical resistivity of La2−x Srx CuO4 in polycrystalline form (solid lines) and for currents in the basal (ab) plane of single crystalline films (see [92])

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cal attempts to understand the physical properties of these materials. The Hall effect Already early experiments [95] indicated anomalous features of the Hall response in the normal state of cuprate superconductors, exhibiting a strong increase of the Hall coefficient RH with decreasing temperature above Tc . The Hall resistivity xy and the Hall constant RH are related via xy = RH · H ,

Fig. 14.21. Temperature variation of the ratio of c and ab for various cuprate superconductors. The general trend for T−1 and T−2 behavior is also indicated (see, e.g., [94])

their temperature dependences if the resistivities are measured either parallel or perpendicular to the CuO planes [94].As may be expected, these anisotropies are strongly influenced by details of the crystal structure and examples are shown in Fig. 14.21. These large anisotropies imply that the motion of the quasiparticles is confined to the Cu-O planes and that they tend to localize in the motion perpendicular to it. The localization of quasiparticles in only one direction cannot occur in a Fermi liquid and therefore the observed anisotropies indicate the failure of the Fermi liquid model for describing the metallic state of cuprates above Tc , at least in the underdoped and optimally doped regime. Large anisotropies, as those of the electrical conductivities, are also observed in many other, physically relevant parameters. This has led to the notion that at least electronically, these cuprates should be regarded as quasi-two dimensional systems. This view has guided many theoreti-

(14.1)

where His the externally applied magnetic field.Various later experiments confirmed this trend and experiments made with single-crystalline samples of YBa2 Cu3 O7−x (YBCO) revealed that the Hall constant, for currents parallel to the Cu-O planes and external magnetic fields oriented perpendicularly to these planes, is positive and varies as T −1 [96]. The out of plane Hall constant, however, obtained for the arrangement that both the current and the field direction are in the plane but perpendicular to each other, providing a Hall field EH along the c direction of the crystal lattice, remained constant and negative between 400 K and Tc [97]. As an example for the distinct temperature dependence of the in plane Hall constant we show, in Fig. 14.22, R−1 H versus T, measured on single crystalline samples of YBCO-123 and Bi2Sr2 CaCu2 O8 (Bi2212) [98]. Equally intriguing is the temperature dependence of cotŸH = xx /xy which, as demonstrated in Fig. 14.23, varies almost as T 2 over a wide temperature range [97, 98]. The in-plane Hall angle is defined as   xy ŸH = arctg . (14.2) xx These two phenomena, apparently quite universal for high-Tc copper oxides, were interpreted by Anderson [99] to provide evidence for the anomalous features of the electronic ensemble in these cuprates. He argued that the electronic transport properties appear to be those of a two-dimensional Luttinger liquid, rather than those of a Fermi liquid. In essence there are two different scattering rates, tr−1 and H−1

14 High-Tc Superconductivity

that have to be considered. The longitudinal transport scattering rate is tr−1 ∼ T, as implied by (T) discussed above and for the transverse scattering rate

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H−1 , in this model representing the intrinsic scattering rate for quasiparticle-like excitations of the Luttinger liquid, so called spinons, a T 2 dependence is predicted. If, in addition, impurities are considered, the model predicts that cotŸH = aT 2 + C, where a is a constant and C increases with increasing impurity content. This predicted behavior has actually also been verified experimentally, supporting again the view that the normal state of these cuprates is quite different from that of a common metal. Magnetic Susceptibility

Fig. 14.22. Temperature dependence of the in plane Hall constant of single crystalline YBa2 Cu3 O7 and Bi2 Sr2 CaCu2 O8 in the form of R−1 H vs T

Fig. 14.23. Temperature dependence of the in plane Hall angle Ÿ of single crystalline YBa2 Cu3 O7 and Bi2 Sr2 CaCu2 O8 in the form of cotŸ versus T2

The temperature dependence of the bulk magnetic susceptibility (T) is very much dependent on the doping concentration. This may well be seen in Fig. 14.24, where (T) of a series of La2−x Srx CuO4 compounds with varying x is displayed. While for low doping, ∂ /∂T is positive across the whole covered temperature range, we note the appearance of a maximum in (T) with increasing Sr content x, the maximum shifting to lower temperatures with increasing x. A similar behavior of (T) with varying doping concentration was also observed for the YBCO series, where also in the underdoped regime, a strong decrease of with decreasing temperature

Fig. 14.24. Temperature dependence of the normal state magnetic susceptibility of La2−x Srx CuO4 (see [8])

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was observed. Naturally there are many ways to compare these experimental data with model calculations but it seems clear that a simple description invoking only a Pauli-type electronic magnetic susceptibility, which is a measure of the electronic density of states at the Fermi energy D(EF ), usually independent of temperature, is not appropriate. The low temperature negative slope ∂ /∂T indicates a loss of magnetic degrees of freedom with decreasing temperature and in retrospect, these measurements of (T) gave the first hint of a phenomenon that is still of high current interest: the opening of a pseudogap in the quasiparticle excitation spectrum, possibly due to the formation of pairs in some incoherent fashion, at temperatures distinctly above Tc . As we shall discuss in more detail below, a lot of additional experimental evidence for the formation of a pseudogap or, in other words, a loss of degrees of freedom in the quasiparticle excitation spectrum, has since been accumulated. NMR: Knight Shift, Spin-Lattice Relaxation In common metals the NMR response in the normal state is, to a large extent, dictated by the presence of conduction electrons. First, they provoke a shift of the resonance signal, the well known Knight shift K =  H/H0, where H0 is the resonance field for the nuclear spins in a non metallic environment, and H is the shift in magnetic field if the resonance is

probed at constant frequency. In simple metals, K is given by [100]   8 K= · < |uk (0)|2 > · P , (14.3) 3 hence is related to the spin susceptibility of the conduction electrons p = 2 B2 D(EF ) ,

(14.4)

and therefore expected to be essentially temperature independent. Second, the conduction electrons also dominate the spin-lattice relaxation process, leading to a Korringa-type behavior of the relaxation rate T1−1 , such that (T1T)−1 is approximately constant. This so called Korringa constant is related to the Knight shift via (T1T)−1 ∼ K 2 .

(14.5)

It is not at all surprising that distinct deviations from these types of behavior have been identified in NMR studies of the cuprate materials. Complications arise, particularly in the evaluation of the Knight shift, because in structurally and electronically complicated materials, such as the cuprates, this shift is the result of various different contributions, also involving orbital degrees of freedom. Since NMR is a local probe, the response of different nuclei occupying different lattice sites may be probed and all the results have to be interpreted by taking into account the local environment of chosen species of nuclei.For the cuprates,

Fig. 14.25. Temperature dependence of various components of the magnetic Knight shift of Cu and O NMR signals in YBa2 Cu4 O8 (see [101])

14 High-Tc Superconductivity

783

Fig. 14.26. Average temperature dependence of the magnetic shift of the NMR signals probing the nuclei on planar Cu sites of YBa2 Cu3 O7 (broken line) and YBa2 Cu4 O8 (solid line) (see [102])

the bulk of data has been obtained from nuclei of Cu and O isotopes. First of all, there are clear cases where the Knight shift exhibits a strong temperature dependence, not really surprising if we recall the behavior of (T) mentioned above. Examples of different components of K(T) are shown in Fig. 14.25. Here, these K components have been evaluated for Cu and O nuclei in the compound YBa2 Cu4O8 (YBCO-124) [101]. It is obvious that these different shift components are substantially reduced with decreasing temperature below 300 K, i.e., at temperatures far above Tc . Similar observations were made in NMR experiments probing underdoped YBCO and Bi-2212. The situation is quite different, if K is measured in optimally or overdoped material. In Fig. 14.26 we show this difference schematically for the planar Cu nuclei in YBCO-123 and YBCO-124 [102]. It is now accepted that the strong temperature dependence of K is related with the low doping of the corresponding materials and most likely also reflects the quasi-gap in the excitation spectrum mentioned above. It turns out that also the different experimental data concerning the temperature dependence of the spin-lattice relaxation rate T1−1 (T) are not compatible with the simplest expectations for a metal. Examples of experimental results [103] for the relaxation rate, measured for 63 Cu nuclei on planar Cu sites in var-

Fig. 14.27. Temperature dependence of 63 Cu NMR spin lattice relaxation rates for nuclei on planar Cu sites in different copper oxide compounds. The solid lines are to guide the eye and the broken lines indicate the Korringa-type variation, depending linearly on T (see [103])

ious cuprates at temperatures above the respective critical temperatures, are shown in Fig. 14.27. For comparison, the data for superconducting and non superconducting Cu oxides are shown. First we note that the relaxation is much faster in those materials which exhibit high-Tc superconductivity. It may also be seen that the relaxation in the non superconducting compounds is indeed of Korringa-type, at least at low temperatures. This is clearly not the case for the superconducting compounds. At elevated temperatures, the relaxation rate is rather well approximated by T1−1 = D + cT, indicating at least two relaxation channels. While the second contribution may be interpreted as representing some sort of Korringa relaxation, the first and temperature independent term has to be associated with another contribution, most likely due to the dynamics of localized d-electron spins on the Cu atoms. If the Korringa-type term is subtracted from the experimental T1−1 values, the resulting d-electron contributions, shown in Fig. 14.28, are obtained [103].A temperature independent relaxation rate may be related with fluctuations of local-

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Fig. 14.28. Same data as in Fig. 14.27, after subtraction of the Korringa type contribution varying linearly with temperature, as indicated by the broken lines in Fig. 14.27. These data are interpreted as representing the relaxation due to Cu d electrons. The solid lines are to guide the eye (see [103])

ized spins but may also be accounted for by spin-density fluctuations in relation with a 2D Fermi surface. Fig. 14.28, however, reveals that also this contribution suffers from a temperature dependent reduction, starting well above the corresponding critical temperatures, at least in some cases. It seems quite reasonable to ascribe this feature again to the presence of the pseudo gap discussed above. By now it is clear that the onset temperature of this gap formation tends to grow with decreasing doping. Specific Heat The outstanding problem in connection with the specific heat of copper oxide superconductors in the normal state is the evaluation of the electronic contribution to the total specific heat Ctot (T), which in the range of Tc is completely dominated by the specific heat due to lattice excitations.The lattice specific heat C ph (T) may be estimated by measurements of Ctot (T) of one particular material in external magnetic

fields or, in an attempt to only change the electronic contribution, by slightly changing the chemical composition, such as to vary the doping level. In both cases it has to be assumed, that C ph (T) is unaffected by these measures, not easy to verify in a rigorous way. The electronic part C el (T), i.e., (Ctot − C ph )(T) is usually assumed to be proportional to T or, in other words, it is assumed that the ratio C el /T is a good measure for the electronic specific heat parameter . For the formal representation of C ph (T), various approximation procedures have been suggested and a number of relevant articles discuss this issue [104, 105]. For optimally doped YBCO-123 an average and reasonable value for the electronic parameter in the normal state is = 25 mJ/moleK2 . For Bi-based copper oxides, the corresponding molar values for are somewhat smaller but still of the order of 10 mJ/moleK2 . As we shall see below, the situation is more complicated in underdoped material where C el /T is not a constant in the normal state. In comparison with simple metals, the value quoted above seems fairly large but still, the corresponding density of electronic states D(EF ) is not exceptionally high in order to justify the high values of Tc in a straightforward manner. A common way to estimate the value is based on the weak coupling BCS relation [106] (Csel − Cnel )(Tc ) = 1.43 , Cnel (Tc )

(14.6)

where Cnel (Tc ) = · Tc , and experimental data concerning the specific heat anomaly at Tc . Since this approach is valid only for mean-field type transitions and the weak coupling limit, most of these calculations may serve as order of magnitude estimates at most. As we shall see below, none of the transitions in high-Tc cuprates is really of mean-field type and in most cases the characteristics of the transition are really quite different, mainly because of rather strong fluctuation effects. In addition and complicating the situation, strong coupling effects are indicated by experimentally determined ratios of the  and Tc and therefore the use of Eq. (14.6) for estimating the parameter is more than questionable.

14 High-Tc Superconductivity

785

Characteristics of the Normal State From this brief compilation of some anomalies that are encountered in the results of measurements of some selected physical quantities, it may safely be concluded, that very simple approaches, usually employed for the description of common metals, are inadequate for treating the features of these cuprate materials. Various theoretical attempts to cope with these problems are the subject of other chapters in this treatise and are discussed in more detail there. Some of them start with the notion that the conventional Fermi liquid model for describing the metallic normal state has to be abandoned here and that other quasiparticle-like excitations have to be introduced. Many of the experimental facts mentioned above and others, not considered here, seem to justify this view. What is particularly hard to grasp, however, is the fact that the character of these excitations seems to vary with the doping level. There is not much doubt that a common quasiparticle picture is adequate for describing the properties of overdoped materials but it is still much debated how the underdoped regime should be treated. Here, the appearance of the often quoted pseudogap, to be discussed in some more detail in the next subsection, adds another complication to the state of affairs. If finally it should turn out that the electronic properties across the entire doping range where superconductivity occurs may still be understood on the basis of the Fermi liquid model, a lot of additional input concerning details of that model is probably needed. Pseudogap, Additional Evidence In this subsection we are briefly discussing additional experimental evidence for the pseudogap in the normal state of cuprate superconductors.We have chosen results of experiments whose interpretation leads to conclusions that are somewhat controversial. This should serve the purpose to indicate that these issues have not yet been settled completely. To set the stage we display a schematic phase diagram for the cuprate superconductors, where also the doping dependence of T ∗ , the temperature below which experimental evidence for the presence of a pseudogap exists, in Fig. 14.29.

Fig. 14.29. Schematic low temperature phase diagram of cuprate superconductors. Emphasized are the antiferromagnetically ordered phase at low doping, the superconducting phase and the approximate location of T∗ (x), indicating the pseudogap formation

First we mention the results of specific heat measurements [107], covering the temperature range above and below Tc of a series of samples of Y0.8 Ca0.2Ba2 Cu3 O7−ı with different oxygen contents, i.e., varying parameter ı. This choice of material allows to extend the overdoped regime for YBCO-123 type material; optimal doping conditions are reached for ı ∼ 0.32. In Fig. 14.30 we show the temperature dependence of the electronic specific heat of different samples in the form of C el /T. The electronic contribution has been obtained by subtracting the lattice contribution from the raw experimental data in an apparently controlled way [108]. In Fig. 14.30a it may be seen that in the normal state of overdoped material, C el /T is temperature independent above the respective critical temperatures and also does not depend on ı. The upturns, appearing somewhat above Tc ,most likely reflect fluctuation contributions to C el. At this point of our discussion we do not address the temperature dependence of C el at very low temperatures, leaving this point to be addressed in another section below. The situation is completely different for underdoped material. Here, as may be seen in Fig. 14.30b, the different C el /T versus T plots start to diverge above Tc in a manner that C el /T decreases with decreasing T if ı > 0.5, thus indicating a loss of entropy already above the onset of superconductiv-

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ity, distinctly different from the behavior displayed in Fig. 14.30a. An additional clear difference in the (b) panel of the figure is the sudden drop in the peak height of C el /T for ı > 0.32, although no corresponding large change in Tc is observed. In [109], a detailed discussion of the temperature dependence of the related entropies and of similar data, obtained for a series of La2−x Srx CuO4 compounds, leads to the conclusion that the pseudogap phenomenon should not be regarded as being due to a precursor formation of Cooper pairs and therefore is not related with the gap opening in the superconducting state. In particular it is argued that the pseudogap related loss of low-energy normal state spectral weight actually weakens the superconducting condensate. To some extent this is compatible with the fact that contrary to the decrease of Tc with decreasing doping, the onset temperature T ∗ increases. A quite different view has been obtained from ARPES experiments on single crystals of underdoped

Bi-2212. With this type of experiments it has been possible to map the Fermi surface of this type of materials and, because of much improved energy resolutions, to verify the onset of the gap related with the superconducting ground state. A detailed presentation and discussion of ARPES experiments on high-Tc superconductors is given elsewhere in this compendium. Nevertheless we note here that measurements of this type of material mentioned above and as a function of temperature reveal that a gap signature at EF in different directions of k space appears at different temperatures.In this sense,the data indicate a gradual loss of Fermi surface with decreasing temperature, starting well above Tc [110]. The evolution, ending in the gap features of the superconducting state below Tc appears to be smooth and therefore, it is concluded that the gap formation in the normal state is most likely due to a precursor pairing of quasiparticles, i.e., definitely related with the appearance of superconductivity below Tc [111].A schematic view of the temperature dependent shrinking of the Fermi surface in k-space is displayed in Fig. 14.31, indicating that the pseudogap in the normal state and the finally resulting gap of the superconducting state reveal the same anisotropy in k-space. A k-dependent incoherent pairing is, of course, not compatible with conventional superconductivity, where the instability of the whole Fermi surface occurs at once at Tc . Results of the same kind of experiments on some overdoped material were interpreted as to confirm the previously established view of the absence of a pseudogap above Tc [88]. Traditionally, various kinds of tunneling experiments have been instrumental for qualitative and quantitative measurements of gaps in electronic

Fig. 14.30. Temperature dependence of the electronic specific heat of Y0.8 Ca0.2 Ba2 Cu3 O7−ı in the overdoped (a) and underdoped (b) regime (see [107])

Fig. 14.31. Schematic representation of the growth of the Fermi surface of Bi-2212 with increasing temperature, as obtained from ARPES experiments (see [110])

14 High-Tc Superconductivity

quasiparticle excitation spectra, especially in connection with superconductivity [112]. With respect to the present problem we quote the results of some recent measurements employing the technique of scanning tunneling microscopy (STM),providing the possibility of local tunneling experiments. Information about gap features is obtained from measurements of the tunneling current I as a function of the applied voltage V and by plotting the tunneling conductance ∂I/∂V versus V . The voltage dependence of the conductance is, to some extent, representing the energy dependence of D(EF ), the density of states for quasiparticle excitations at the Fermi energy. Early STM experiments [113], again on single crystals of Bi-2212 compounds, confirmed the presence of a pseudogap in the normal state, but now it was claimed that this feature was present for underdoped and overdoped material [114] and that the temperature dependence of the experimental ∂I/∂V

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versus V curves was such that the pseudogap had to be a precursor of the genuine superconducting gap. More recent experiments on the single Cu-O layer material Bi-2201 confirmed this view [115]. In Fig. 14.32 we show the reported results of the temperature evolution of conductance versus voltage curves for overdoped Bi2.1Sr1.9 CuO6+d . It may well be seen that the reduction of the conductance at low energies that is believed to indicate a gap feature persists to temperatures well above the critical temperature of the bulk. An intimate relation between the two gaps in either the normal or the superconducting state is also deduced from the fact that the gap width is more or less temperature independent [113]. In addition it has been pointed out that the gap-width variations from one material to the other is the same for the superconducting and the normal state, again suggesting a common origin of the two features. The data imply a filling of the gap with increasing temperature, rather than the commonly expected narrowing. As pointed out above,the STM results seem to confirm some of the conclusions that are drawn from the ARPES data but in addition and unlike a number of other experiments, they claim the persistence of the normal state pseudogap well into the overdoped regime. 14.4.2

Superconductivity

General Remarks

Fig. 14.32. Temperature dependence of the normalized conductance vs. voltage of overdoped Bi-2201, as measured by STM. The broken line represents the background at 82 K to which the curves are normalized. The gap feature persists to temperatures substantially exceeding Tc (see [115])

The unexpected occurrence of superconductivity in these cuprate materials at unprecedented high temperatures almost immediately raised questions with regard to possible mechanisms that might trigger these transitions. Most researchers would agree that this issue is not yet definitely settled, as also the theoretical chapters of this treatise may demonstrate. The main controversy addresses the question whether lattice degrees of freedom are in any way involved in the occurrence of superconductivity in copper oxides or whether this phenomenon is entirely based on interactions within the, however anomalous, electronic subsystem. Early flux-trapping experiments [116] made it clear, at least, that the objects responsible for the superconducting state carry a charge of twice the charge of a single electron,i.e.,2|e|,just as the Cooper

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pairs in conventional superconductors. As has long been known, of course, pairing states need not be of the simplest variety with zero orbital momentum and a spin singlet configuration. Although of little interest at first, the pairing configuration became an issue in the wake of some theoretical debates,and was finally addressed experimentally by a number of different types of measurements.Another phenomenon that is intimately related with superconductivity and is not observed for normal conductors, is the screening of static external magnetic fields H from the interior of a singly connected superconducting specimen, i.e., the decay of H over a characteristic length, the London penetration depth L . Early experiments using muon-spin rotation ( SR) techniques established beyond any doubts that also for the cuprate superconductors this field expulsion, the Meissner– Ochsenfeld effect, is present [117]. The major difficulty, both in experiment and in theory, however, is still the unequivocal identification of the objects that are to be paired. At least part of our discussion concerning the normal state properties have made it clear that these objects are not yet really well defined. Because of the structural characteristics of the cuprates and the corresponding anisotropies,a large part of the discussion of their superconductivity concentrated on the properties of the Cu-O planes and the possible influence of the two-dimensionality of these systems. In what follows we try to achieve a dense presentation of some of the most significant aspects concerning superconductivity in Cu oxide materials, now concentrating on the characterization of the superconducting state. A lot of experimental evidence has been accumulated that the superconductivity of cuprates is of unconventional nature. Above we have already emphasized the importance of the concentration nc of itinerant charge carriers, dictated by the degree of doping the electrically insulating parent compounds and its influence on the magnitude of the critical temperature. That the behavior of Tc (nc ) is, to some extent, quite universal, as may be seen in Fig. 14.33, has been pointed out by Uemura [118].The magnitude of Tc is, to a large extent, also dictated by the number n of CuO planes per unit cell. It turns out that for one class of materials, Tc may be enhanced by enhancing n, but

Fig. 14.33. The variation of the critical temperature Tc of cuprate superconductors vs the low temperature SR relaxation rate  . The closed and open triangles are for La-214 compounds, the closed and open circles are for Y-123 material, the crosses represent data of Bi-2212 material and the closed diamonds are for Bi-2223 compounds. The solid lines are to guide the eye (see [118])

Fig. 14.34. Variation of the critical temperature Tc of mercury based superconductors HgBa2 Can−1 Cun O2n+2+ı as a function of n, the number of Cu-O planes (see [39])

14 High-Tc Superconductivity

this trend is usually only obeyed up to n = 3. This latter trend has very nicely been demonstrated for Hg-based cuprates, as may be seen in Fig. 14.34 [39]. Inspecting the behavior of electron or hole doped bulk cuprate superconductors, it seems obvious that hole doping of insulating cuprates results in significantly higher values of Tc than electron doping. This naturally forces the question in what way the degree of the occupation of electronic states contributing to electronic conduction influences the magnitude of the critical temperature for the onset of superconductivity in general. In this context it is at least amusing to note that even in the case of superconductivity of common metal elements most of them have a normal-state Hall constant with a positive sign, which is typical for hole dominated conductivity. A theoretical model, emphasizing “hole superconductivity”, has been promoted by Hirsch and coworkers [119]. As most non-elemental superconductors, the cuprates are type II superconductors with extremely large ratios Hc2/Hc1 between the upper and the

Fig. 14.35. Negative magnetization of a ceramic sample of superconducting YBa2 Cu3 O7−ı as a function of applied magnetic field up to 100 kOe. The arrows indicate the direction of the field variation

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lower critical fields or, in other words, resulting in a very large value of the Ginzburg–Landau parameter  = /. This may qualitatively be recognized from magnetization curves M(H), such as the one displayed in Fig. 14.35 for YBCO-123 [120]. Taking into account that by definition   Hc1 = ¥0 4L 2 ln  (14.7) and Hc2 =

¥0 , 2 2

(14.8)

the shape of the curve readily implies that the penetration depth L for magnetic fields is much larger than the superconducting coherence length . For various reasons, the experimental evaluations of these critical fields of cuprate superconductors are not very precise or even impossible.In particular, Hc2 is often regarded as a crossover field only, without manifesting a true phase transition. Because of these uncertainties we refrain from presentations and discussions of critical field data and their temperature dependence. However, both the magnetic-field penetration depth L and the coherence length  may be evaluated with an adequate certainty from M(T, H) data [121,122].It turns out that these two characteristic lengths are very anisotropic, of no surprise when considering the structural properties of these materials. This also means that the magnitudes of the critical fields for magnetic-field orientations parallel or perpendicular to the Cu-O planes are very different. In most cases, the in-plane components of the two lengths that are of interest.Since the values for the inplane penetration depths are between 1000 and 3000 Å for most of these cuprates [123], it is clear that the in-plane coherence lengths are of the order of a few in-plane interatomic distances, i.e., extremely short. The situation is even more extreme for the direction perpendicular to the Cu-O plane. Here, L may of the order of a few 104 Å and  is barely as long as an out of plane interatomic distance or even less. These anisotropies, often expressed by the ratio  = (mc /mab )1/2 between the effective masses entering the penetration depth equation, and the short coherence lengths create a situation that is very different from that encountered in conventional superconductors and, as we shall see, provoke special features in some

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of the observations to be discussed below. Because of the short coherence lengths, the coherence volume is rather small and may host only a low number of Cooper pairs. This situation favors large fluctuation effects; their importance is discussed in a special chapter of this book. Pressure Effect, Isotope Effect So far we have only considered Tc variations induced by changes of the chemical composition of the materials. Other external influences, apart from applying external magnetic fields, that may affect the critical temperature, are the application of external pressure or replacing part of the atoms with other isotopes of the same element. Historically, the strong enhancement of Tc of the first La-based copper oxide superconductors of the 214 family upon applying external pressure [124] led eventually to the discovery of the YBCO-123 superconductor, by following up the idea that the smaller ionic radius of the Y3+ ions might enhance the internal pressure of La3+ -based compounds and thus lead to higher critical temperatures [5]. Since the reduction of the volume by exerting external pressure may affect the spectra of both the lattice and the electronic excitations, direct and chemical pressure may not always induce equivalent variations of physical properties. In most common superconductors, external pressure usually reduces Tc and this trend is interpreted as being due to a pressure induced weakening of the electron phonon coupling constant ph which in essence determines the magnitude of Tc of materials where superconductivity is due to the electron phonon interaction. It has been shown [125] that to a good approximation ph =

A , (MŸD2 )

(14.9)

where A is a constant for a given crystal structure, M is the ionic mass and ŸD is the Debye temperature. The pressure induced stiffening of the lattice and the corresponding enhancement of ŸD thus account for the observed decrease of Tc with increasing pressure. Exceptions from this trend of ∂Tc /∂p < 0 are only a few, the most prominent being that of La

metal where Tc increases considerably with increasing pressure [126], and always have been considered as indicating some anomalous pressure response of the electronic subsystem. Thus, in relation with common superconductors, the generally negative influence of external pressure on Tc is usually interpreted as confirming the importance of the electron phonon interaction for triggering the phase transition. For optimally doped Cu-O superconductors, the pressure effect on Tc is, in most cases, negligible. Prominent exceptions are YBCO-124 and Hg-1223, the compound with the highest yet achieved critical temperature under ambient pressure conditions. An unusually strong enhancement of Tc from 80 K at ambient pressure to 108 K at 10 GPa has been reported for the YBCO-124 compound [127] and was interpreted as being due to a sizeable shortening of the Cu2-O1 bond, leading to a charge transfer into the Cu-O planes. Also for Hg-1223, the application of external pressure raises Tc considerably.Although different claims for the Tc enhancement, up to 160 K at pressures between 150 and 300 kbar, have been made [128–130], it is undisputed that the pressure induced enhancement is significant and can apparently not be achieved by means of variations of the chemical composition. It has been argued that the three Cu-O planes per unit cell of this compound are not all optimally doped at ambient pressure and that the situation can only be improved by a pressure induced transfer of charge carriers between the planes, not achievable by chemical means. This automatically means that the large positive value of ∂Tc /∂p is most likely not a lattice effect and certainly cannot be used to make a case for a significant influence of an electron phonon interaction in the creation of a superconducting state. Much more clean in regard of identifying the significance of lattice effects being involved in the superconductivity of any material is the controlled exchange of isotopes of some or all the atoms involved in the chemical composition of a superconductor,because the different ionic masses will only affect the lattice properties of the material. Indeed, measurements of the isotope effect on Tc gave the first unequivocal experimental evidence for the importance of the lattice dynamics on superconductivity in sim-

14 High-Tc Superconductivity

ple metals [131, 132]. For elements the situation is particularly simple because Tc ∼ M −˛

(14.10)

and in first approximation [133] it is expected that ˛ = 1/2. This result has been verified in a number of cases [134], but deviations from this most simple behavior, the most drastic for ˛-U, implying negative values of ˛ [135], have been identified even for elemental superconductors. The interpretation of isotope effect experiments is increasingly more complicated, if the chemical composition of a material involves four or more constituents, as is the case for many high-Tc superconductors. Nevertheless, the presence of an isotope effect on Tc almost inevitably indicates the influence,even if of limited significance, of the crystal lattice on the mechanism triggering the superconductivity. Reliable results on the isotope effect on Tc of cuprates are difficult to achieve. This is because apart from exchanging the chosen isotope, any other variations in the composition of the material and its structural properties must be avoided. In copper oxide compounds, the element of choice for a controlled isotope exchange is, of course, oxygen. Careful work in exchanging 16 O with 18 O and back has been made on La2−x Srx CuO4 with 0.1 ≤ x ≤ 0.15 [136]. From these experiments it is concluded that the compound with 18 O exhibits a distinctly lower Tc than if the oxygenation is accomplished with 16 O. While the analysis of the measurements indicates that there is no isotope effect on the charge carrier concentration, as one might expect, a significant shift of the effective mass of the carriers, which are identified as being of polaronic character, is deduced. Since similar results have been obtained in a number of other analogous investigations [137, 138], it has to be accepted that at least in some of the high-Tc superconductors, the isotope effect is not negligible. More recently it has been claimed that isotope variations are also affecting the onset temperature T ∗ of the pseudogap formation discussed above. The corresponding experiments involved the measurements of linewidths of crystal field excitations probed by inelastic neutron scattering on HoBa2 Cu4 O8 [139]. While the isotope effect on the

791

critical temperature is, in accordance with other work, found to be rather small with ˛ ≈ 0.05, the variation of T ∗ with the exchange of 16 O and 18 O is considerable and of opposite sign, i.e., T ∗ of the 18 O variety of the compound is distinctly higher than T∗ of the 16 O material. The corresponding ˛ parameter in T ∗ ∼ M −˛ is therefore negative. This again seems to indicate that the lattice dynamics are, to a certain extent, involved in the formation of the pseudogap of the normal state. It may well be that understanding the significance of the sign difference in the ˛ parameters for Tc and T ∗ will also clarify the relation between the pseudogap of the normal state and superconductivity below Tc . In view of these experimental results it seems difficult to deny that superconductivity in copper oxides involves, to some extent, the lattice dynamics of these materials. In this regard it has been argued [140] that these lattice dynamics lead to structural inhomogeneities in these materials, better known under the names of phase separation and stripe formation. Microscopic evidence for stripe formation in some of these materials has definitely been obtained [141, 142], but it is not yet clear to what extent this phenomenon is essential for the occurrence of superconductivity at elevated temperatures. In interpreting experimental data obtained with different techniques it should be kept in mind that the process of stripe formation may also be highly dynamical in the sense that the structural inhomogeneities are not static. This might explain some controversies that arise in comparing experimental data from measurements that probe in different energy or time windows. It seems clear that structural inhomogeneities will also generate electronic inhomogeneities and it has not yet been worked out how these have to be taken into account in the interpretation of some of the anomalous features of the normal state properties discussed above. Phase Transition Aspects The short coherence lengths, even within the Cu-O planes, of the cuprate superconductors, play an essential role in all experimental verifications of the superconducting phase transition. This is mainly be-

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cause even in zero magnetic field, fluctuation effects are non negligible to an extent that distinct deviations from the mean field behavior, usually observed for clean common superconductors, are observed. The fundamental parameter governing the strength of these fluctuations is the Ginzburg number [143]  2 Tc 2 Hc2 (0)ab c

, (14.11) 2 where ab and c are the zero temperature coherence lengths parallel and perpendicular to the CuO planes, respectively. This number, which is obviously large if the  values are small, fixes the temperature range around the critical temperature where fluctuation effects are significant, i.e., critical, via |T − Tc | < Tc ·Gi. For common superconductors, Gi is of the order of 10−6 or less and therefore, deviations from mean field behavior are usually not important. The situation is completely different in cuprates and further complications arise, because of quasi two-dimensional fluctuation effects in particularly anisotropic cuprate systems. In that case the relevant Ginzburg number is [143] Gi =

Tc Gi2D = √ , 2 · "d

(14.12)

where " = (¥0 /4)2 and d is the thickness of one layer. Fluctuation effects in the temperature dependences of the electrical resistivity (T) and the magnetization M(T) are somewhat annoying because they mainly hamper an exact evaluation of the zero field critical temperature. In Fig. 14.36 we show an

Fig. 14.36. Temperature dependence of the electrical resistivity of Bi-2212. Note the growing slope of ∂/∂T far above Tc

example of (T) of Bi-2212 which demonstrates the gradual decrease of  above Tc . A thorough discussion of fluctuation effects on the electrical transport is beyond the scope of this more general review. Instead we briefly consider the temperature dependence of the specific heat C(T) in zero magnetic field and in the vicinity of Tc .As an example we first present the result of a relevant experiment on single crystalline YBCO-123 in Fig. 14.37, revealing the anomaly in C(T) at the transition [144]. As may be seen from the scale on the left vertical border of the figure, this anomaly rides on a large background which at temperatures above the boiling point of liquid nitrogen, is due to the specific heat of the lattice. In order to achieve the experimental resolution that is demonstrated here and which is necessary to obtain any reliable comparison with theoretical models, special methods of measurement have to be involved. One particular approach, a differential thermal analysis (DTA) adapted for measurements on small single crystals, is described in [145]. From the data close to Tc both in the normal and the superconducting state, clear deviations from ideal mean field behavior may be recognized. The solid line in Fig. 14.37 is a fit on the basis of the 3D XY model for phase transitions [146]. A detailed discussion of the relevance of this model with respect to the critical regime of the zero field phase transition of cuprate supercon-

Fig. 14.37. Specific heat anomaly at the superconducting transition of single crystalline YBa2 Cu3 O7−ı in zero magnetic field. The solid line is a fit based on the 3D XY model, as explained in the text

14 High-Tc Superconductivity

793

ductors has been given in [147]. Here we only note that the experimental data can reasonably well be described on the basis of this model. The analysis is by no means straightforward. The universal temperature dependence C(T) is given by   T − C(T) = Clattice (T) + C + A ln | − 1 |, Tc for T < Tc   T −1 |, = Clattice (T) + A+ ln | Tc (14.13) for T > Tc . The parameters A+ and A− are related via the universal ratio A+ /A− ∼ = 1.05. First of all, the lattice part of the specific heat has to be evaluated with some degree of confidence. This can be achieved by additional measurements in external magnetic fields. Second it has to be tested to what extent the values of the fit parameters depend on the temperature window chosen for the fits.In the case presented here,the fit quality suggests that the critical regime appears to be extended over a range of ±8 K around Tc .Without going into further details, we note that attempts with fits on the basis of other approaches for dealing with fluctuations in the critical region around Tc gave less convincing results. As may be expected, the specific heat anomaly is considerably reduced upon the application of an external magnetic field. This is demonstrated in Fig. 14.38, where C(T, H) data, measured on the same YBCO-123 sample in various external magnetic fields, oriented perpendicularly to the Cu-O planes, are displayed. In view of the success in describing the critical fluctuations on the basis of the 3D XY model in zero field, it seems justified to test a related scaling law that is inherent in this model for non zero magnetic fields. The relevant scaling relation is    ˛ −1 T 2 2 , −1 H [C(T, H) − C(T, 0)] H = C Tc (14.14) where C(x) is the unknown scaling function [148]. The parameters ˛ ∼0.007 and  ∼ 0.669 are the 3D XY critical exponents of the specific heat and the coherence length, respectively. In Fig. 14.39, this scaling is tested by plotting the appropriately scaled quanti-

Fig. 14.38. Influence of external magnetic fields on the specific heat anomaly at the superconducting phase transition of single crystalline YBa2 Cu3 O7−ı

Fig. 14.39. Specific heat anomalies of YBa2 Cu3 O7−ı plotted in variables that are consistent with 3D XY-scaling (see Eq. (14.14)). B = 0 H

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Fig. 14.40. Specific heat of textured Bi2 Sr2 CaCu2 O8 in the temperature range around Tc = 91 K in different magnetic fields. The plots Cp (T)/T are for 0 H = 0, 0.1, 0.5, 1, 2, 3, 4 and 7T from top to bottom. Note the crossing point of all the curves at T ≈ 87 K

Fig. 14.41. Magnetic field dependence of the specific heat anomaly of superconducting Bi-2212 if plotted in the form [Cp (0) − Cp (H)]/T

ties.It may be seen,that the scaling is not well obeyed, indicating that the use of the 3D XY model for describing the critical fluctuations in external magnetic fields exceeding 1 T, is inappropriate. Results of other measurements on other single crystals of YBCO-123 are in conflict with this conclusion, claiming a 3D XY type scaling to be valid in fields up to 14 T [149,150]. Our second example is C(T) of a Bi-based copper oxide material whose anisotropy is far more excessive

than that of YBCO-123. For the latter, the anisotropy parameter  is estimated to be less than 10; for superconducting Bi-2212,  ∼ 350. In that case it may be expected that fluctuation effects are drastically influencing the temperature dependence of the specific heat in the vicinity of the transition. That this is indeed the case may be seen in Fig. 14.40, where C(T, H) of Bi-2212 is shown for a set of different fields H. Even in zero external field no trace of a discontinuity at Tc , still discernible in the zero field data of YBCO-123, may be recognized. More instructive is the plot of [C(0) − C(H)]/T versus T, shown in Fig. 14.41. It may be seen that the anomaly adopts the shape of a triangle whose apex is, amazingly, not shifted in temperature with increasing magnetic field. These data cannot be brought into agreement with predictions of the 3D XY model [150,151]. From our brief description of specific heat data of two typical cuprate superconductors with very similar critical temperatures but differing a lot with respect to their anisotropy, it is clear that the character of the phase transition depends very strongly on the anisotropy parameter  . For small values of  , i.e., less than 10, the phase transition seems, at least in zero magnetic field, to belong to the 3D XY universality class. This is certainly not the case for strongly anisotropic, quasi-2D superconductors. Unfortunately, YBCO-123 is probably the only cuprate superconductor with a low value of  ; most of the others are far more anisotropic. It has been argued that this trend might indicate that in the quasi-2D systems, the bosonic pairs are preformed and simply condense at Tc , thus implying an altogether different universality class of the phase transition. The characteristics of the phase transition may also be discussed on the basis of magnetization M(T, H) data. A particularly intriguing phenomenon has been observed in M(T, H) of cuprate superconductors with a large anisotropy parameter  . A data set of the temperature dependence of the magnetization of a single crystal of Bi-2212, measured in different external magnetic fields [152], is shown in Fig. 14.42. It may be seen that the M(T, H) curves all cross in a single point at a temperature Tcr < Tc , implying that the magnetization is field independent at this single temperature Tcr . In practice

14 High-Tc Superconductivity

it is found that Tcr is always a few degrees below the zero field Tc .The same phenomenon is also observed for Bi-2201 [153], another extremely anisotropic superconductor, but not for YBCO-123. This crossing feature indicates that the field dependence of the magnetization ∂M/∂H changes sign at Tcr , certainly not expected for the usual magnetization of a superconductor in the mixed state. The sign change may be understood, however, by considering two contributions to the magnetization, such that M = M0 + Mfl ,

(14.15)

where M0 is the usual field dependent magnetization of the mixed state, calculated using a modified London equation with the appropriate approximations, and Mfl is due to thermal fluctuations of the vortices below Tc [154]. Using a modified London model, also Mfl (T) has been calculated with different approximations in high and low fields. Inserting the appropriate expressions for M0 and Mfl into Eq. (14.15) and setting ∂M/∂lnH ≡ 0, one obtains     ˛ kB Tcr M cr ≡ M(Tcr ) = − ln √ (14.16) ¥0 s e

Fig. 14.42. Temperature dependence of the magnetization of single crystalline Bi-2212 in the range of the superconducting transition temperature in various external magnetic fields. Note the crossing of all curves at T ∗ = 86.4 K



and Tcr =

¥02 s 2 32kB  2 ab (Tcr )

795

 ,

(14.17)

where kB is the Boltzmann constant and s is taken as the interlayer distance, which is of the order of 1 nm. The numerical values of  and ˛ are not easy to eval√ uate and often it is assumed that ln(˛/ e) = 1 [155]. For this reason it is difficult to check the quantitative agreement between the experimental and calculated values of Mcr and Tcr . The crossing of the M(T, H) curves at a single temperature, experimentally observed to occur in an unexpectedly wide range of magnetic fields [152], indicates that some scaling should be obeyed. This is indeed the case, as may be seen in Fig. 14.43. The analysis is based on scaling functions that have been deduced from a Ginzburg–Landau free energy functional for quasi-2D type II superconduc-

Fig. 14.43. Scaling of the magnetization of Bi-2212 at different temperatures and in different magnetic fields. The scaling variables of the main frame are from a 2D Ginzburg– Landau, lowest Landau level (GL–LLL) type approximation. In the inset, the same data are plotted assuming a 3D version of the GL–LLL scaling (see text)

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tors [155]. In this case, the scaling function variable is [T − Tc (H)]/(TH)1/2. The corresponding plot of the renormalized magnetization in Fig. 14.43 reveals that the scaling, based on fluctuations in two dimensions,is rather well obeyed.In addition,the same plot for 3D scaling, shown in the inset of Fig. 14.43, implies that the corresponding approach is not valid in this case. A 3D scaling of this same type has, however, been verified [156] for analogous data of YBCO-123 where, as mentioned above, the crossing phenomenon is absent. A scaling plot, based on the scaling variable valid for the 3D situation is shown in Fig. 14.44. Another possibility for checking the scaling of M(T, H) in the critical regime is offered in the following way. The unknown scaling function may be eliminated by evaluating   M(T,H) 2H ∂M(T,H) − ∂H 2H   FM = ≡ (Tc − T)/ , (14.18) Fig. 14.44. Scaling of the magnetization of Y-123. The scaling variables in the main frame are compatible with the 3D XY model while those in the inset are consistent with 3D GL-LLL scaling

Fig. 14.45. FM (T), as defined in Eq. (14.18), vs T. Open triangles: 0 H = 0.75 T. Full squares, 0 H = 15 T. The plot indicates that the critical exponent  is not the same above and below Tc (see text)

∂M(T,H) ∂T

Fig. 14.46. Temperature dependence of the specific heat of single crystalline Y-123 in various external magnetic fields oriented along the orthorhombic c-axis. Note the sharp anomalies at decreasing transition temperatures upon increasing magnetic field (see [162])

14 High-Tc Superconductivity

i.e., the temperature dependence of FM is linear in T and the slope relates to the critical exponent  .A plot of FM (T), shown in Fig. 14.45, reveals that a temperature independent scaling exponent  cannot account for the data and therefore confirms the failure of the 3D XY model in external fields of 1 T and above. In relation with the superconducting transition in non zero external magnetic fields it has been realized,also on the basis of the C(T, H) data presented above,that the reduced anomalies are due to some crossover phenomenon rather than a true phase transition. Guided by theoretical insight [157] a distinct phase transition of first order has then be identified by high resolution measurements of thermodynamical properties, such as the magnetization [158, 159] and the specific heat [160, 161], both as a function of temperature in static magnetic fields. Relevant C(T, H) data [162] is shown in Fig. 14.46. The first order nature of the transition is confirmed by the observation of discontinuities in the magnetization at the same temperatures where the C(T) anomalies occur, requesting the release of some latent heat on general grounds. The transition is believed to be associated with the ensemble of the vortices of the mixed state.

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transport, i.e., ab (T,H). It may be seen from Fig. 14.48, where results of measurements [164] of the in plane resistivity of Bi-2212 are presented, that an increasing field H does not simply shift the transition to lower temperatures. Instead, the decrease of ab occurs in a temperature range of increasing width and from these data it is not a priori clear, where the resistivity actually vanishes. The extension of the temperature range of the resistivity drop is particularly large, if the anisotropy parameter is large, as in this case. This diagram confirms the above mentioned view that a rigorous definition of the upper critical field Hc2 from resistivity data is difficult. It turns out that again these features are intimately related to the behavior of the vortices in the mixed state. Naturally also c (T) is strongly affected by external magnetic fields. In Fig. 14.49 it may be seen that, because of the magnetic field shifted transition, c of Bi-2212 continues to increase with decreasing temperature, confirming the characteristically different charge transport in the two different crystallographic directions [163]. Also here, the subsequent drop of the resistivity is rather gradual, indicating a cross over behavior rather than a distinct phase transition.

Transport Below Tc (H) (a) Resistivity As has already been pointed out above, the zero field resistive transition is significantly influenced by fluctuation effects, particularly in those cases where the anisotropy parameter  is large. It is also interesting to note that in the temperature range where the inplane resistivity ab exhibits the fluctuation induced paraconductivity, often a substantial increase of the out of plane resistivity c is observed [163], i.e., at Tc , ∂c /∂T is negative. An example of this feature, observed for Bi-2212, is shown in Fig. 14.47. Nevertheless,even in these cases, c vanishes in a narrow range of temperature, confirming that even for large values of  , superconductivity is still a 3D phenomenon. Quite unusual, at least in comparison with common superconductors, is the influence of external magnetic fields on the resistive transition. First we consider the influence of external magnetic fields, oriented along the c-axis, on the in plane charge

Fig. 14.47. Anisotropy of the temperature dependence of the electrical resistivities of single crystalline Bi-2212,measured in the ab plane (ab ) and along the c-axis (c ), respectively. Note the different scales for ab and c (see [163])

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Fig. 14.49. Magnetic field dependence of the out of plane electrical resistivity c (T) of single crystalline Bi-2212 (see [163])

Fig. 14.48. Magnetic field dependence of the in plane electrical resistivity ab (T) at the superconducting phase transition of single crystalline Bi-2212. Note the difference in the features, depending on the orientation of the external field. The lower frame emphasizes the low resistance regime (see [164])

(b) Thermal conductivity Because of the gap formation in the excitation spectrum of electronic quasiparticles of superconductors, the transport of energy or heat is naturally affected by the phase transition. In common superconductors, because of the gap formation, the electronic transport of energy, i.e., the corresponding thermal conductivity el , vanishes gradually below Tc [165]. Because of the reduced rate of scattering of phonons with electronic quasiparticles, the lattice contribution to the thermal conduction is expected [165] and observed [166] to increase with decreasing temperature below Tc , at least as long as no other scatter-

ing mechanisms for phonons dictate their mean free path. The situation seems to be somewhat different for the cuprate superconductors. In Fig. 14.50 [167] we show experimental data for the temperature dependence of the thermal conductivity (T) for the transport of energy parallel to the Cu-O planes of YBCO-123.Apart from some intrinsic anisotropy between the a and b direction,the most obvious features are the discontinuous change of the slope ∂/∂T at Tc and the distinct maxima of  at approximately Tc /2, for both crystallographic directions. The anisotropy ratio of a /b is constant above Tc and the enhanced thermal conductivity along the b-direction is attributed to the additional electronic contribution due to quasiparticle transport along the Cu-O chains that are, according to Fig. 14.6a, extended only along the b direction of the crystal lattice. The slight increase of  with decreasing temperature in the normal state is interpreted as being due to itinerant lattice excitations. Although at first sight it is tempting to ascribe the anomalous increase of  below Tc to an enhanced phonon conductivity, it turns out [168] that the observed behavior is rather due to an enhanced electronic heat conduction, obviously again a rather unusual phenomenon but, to a certain ex-

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tent, also observed in strongly coupled conventional superconductors [169,170]. For the cuprates it is ascribed [168, 171] to an anomalous decrease of the quasiparticle scattering rate in the superconducting state which at first was revealed in experiments probing the surface impedance of cuprate superconductors below Tc [172, 173]. Assuming that the temperature dependence of the phonon conductivity is not significantly altered in the superconducting state, the maxima in  are then a consequence of the compensation of the scattering rate reduction by the reduced number of quasiparticle excitations because of the gap formation in the superconducting state.

Fig. 14.50. Temperature dependences and anisotropy of the thermal conductivities of single crystalline Y-123, measured along the a-axis and the b-axis, respectively (see [167])

Fig. 14.51. Temperature dependence of the in plane Hall resistivity xy of single crystalline Y-123 in various external magnetic fields oriented along the c-axis between 60 K and 150 K

(c) The Hall Effect It turns out that magnetoresistive effects in these cuprate materials, at least in experimentally achievable magnetic fields, are not really extraordinary. This is not quite true for the results of measurements of the Hall resistivity xy . Even for common type II superconductors,the Hall response in the mixed state is somewhat complicated and different model calculations of what should be expected, lead to different results [174,175]. This is even more so for the cuprate superconductors where again, some anomalous features are particularly well developed. Since the observed anomalous features seem again to be strongly linked with the properties of the vortex state, we restrict ourselves to a brief presentation of the most important observations. For this purpose we show, in Fig. 14.51, the temperature dependence of the Hall resistivity xy (T) of a single crystal of slightly underdoped YBCO-123 in various external magnetic fields [176]. Most obvious is the sign change of xy at field dependent temperatures that are close to but always less than the zero field critical temperature Tc (0) [177]. There is a clear negative shift in temperature where the minimum of xy is observed, with increasing external magnetic field. At this point it should be noted that sign changes of xy , although somewhat less pronounced, have also been observed in conventional type II superconductors [178]. It is therefore most likely, that this phenomenon is tied to the vortex state and its dynamics as such and not to special intrinsic properties of the cuprate superconductors. Nevertheless, more Although various

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of this problem. A number of publications, considering either the pinning force, fluctuation effects, a Magnus-type force, two band effects, or charge imbalances between the vortex cores and the superconducting matrix are available in the literature [182–187]. A thorough discussion of all of them is beyond the scope of this review. Characteristics of the Superconducting State (Gap Function)

Fig. 14.52. Temperature dependence of the in plane Hall resistivity xy of single crystalline Bi-2212 in various magnetic fields oriented along the c axis between 40 and 130 K. Also shown, as the solid line, is the zero field in plane electrical resistivity xx

explanations for these phenomena have been suggested, it is most likely that the Hall conductivity detailed investigations indicate that again the degree of anisotropy and, in addition, the concentration of holes introduced by doping, play an important role. Another set of xy data,obtained from measurements on a single crystal of Bi-2212 material [179], is shown in Fig. 14.52. The difference to the results shown in the previous figure is yet another sign change of xy at temperatures much below Tc [180].Apart from this observation we note that the deviations from the T−1 behavior of RH start at temperatures well above the zero field Tc . This seems to indicate that again some fluctuation effects are significant. xy =

xy 2 + 2 ) (xx xy

(14.19)

is given by the sum of at least two different contributions [181]. One of them is related to the superconducting matrix of the mixed state and the second would represent the contribution from excited quasiparticles in the cores of the vortices. There are obviously different ways for theoretical treatments

The stability of the superconducting state is, in general terms, guaranteed by the formation of a gap in the quasiparticle excitation spectrum below Tc . At present we do not consider the very special case of gapless superconductivity [188], sometimes observed in cases where the quasiparticle mean free path is much shorter than the coherence length. This condition is not fulfilled in typical cuprate superconductors. In the most simple approximation, this gap is assumed to be isotropic in k space and its magnitude at zero temperature, (0), is a measure for the free energy difference between the normal and the superconducting state, the condensation energy Econd , via [106] Econd =

D(EF ) · 2 (0) . 2

(14.20)

Since this condensation energy is another manifestation of the Meissner–Ochsenfeld effect, its experimental observation, to be discussed in more detail below,implies the existence of this gap also in cuprate superconductors. The opening of this gap at Tc is also reflected in the almost discontinuous enhancement of the specific heat at the critical temperature of YBCO-123, measured in zero magnetic field. All these results from measurements of macroscopic properties give only indirect evidence for the presence of a gap and the real way to go are experiments that probe the excitation spectrum of the electronic quasiparticles directly. Many early tunneling data revealed the expected loss of excited states in a narrow energy range around zero energy [189]. A new way of exploring the occupation of quasiparticle states near the Fermi energy of superconductors was then offered by a significant progress in improving

14 High-Tc Superconductivity

Fig. 14.53. Current and conductance vs. voltage characteristics of a break junction of a Bi-based copper oxide superconductor, reflecting the loss of available states close to EF (see [190])

the energy resolution in measurements of photoemission spectra near EF . In Fig. 14.53 we show an early result of the voltage dependence of the differential conductivity ∂I/∂V of a break junction of Bibased copper oxide, reflecting the loss of achievable states close to EF [190]. Beforewe address the issue of a possible k dependence of the gap function, we consider the information that essentially all these experiments provide, the amplitude 0 of the gap function and its temperature dependence. The zero temperature value 0 (0) is of interest because, together with the magnitude of Tc , it reveals the strength of the interaction that provides the superconducting ground state. In the weak coupling limit, the relevant ratio, following from the original BCS theory [106] is 20 (0) = 3.56 . kB Tc

(14.21)

Larger values, following from direct experimental evaluations of Tc and 0 (T) are usually interpreted as a manifestation of strong coupling effects.Without going into details, most of the results of experiments that are briefly discussed below, indicate ratios of 20 /kB Tc that are factors of 2 to 3 larger than the

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value quoted in Eq. (14.21), at least when probing the superconducting state in the Cu-O planes, i.e., we are dealing with superconductivity invoking a very strong coupling. Although there was,even before these microscopic experiments were done, never any serious doubt that the superconducting gap is well established in cuprate superconductors, there was, at first, only limited interest and correspondingly little information concerning the k dependence (k) of the gap in momentum space. The shape or symmetry of (k) is, to a certain extent at least, tied to the symmetry of the pairing configuration and the corresponding order parameter of the superconducting state. Therefore it is clear that experimental verifications of the shape of (k) are important and useful for the interpretation of a variety of experimental data, but especially also for testing predictions of theoretical models. Considering the overwhelming evidence that superconductivity in these copper oxides is mainly due to the special properties of the Cu-O planes and also taking into account the atomic arrangement of Cu and O in these planes, it seems suggestive that the most likely configuration of the gap function (k) has a dx2 −y 2 symmetry, also compatible with predictions assuming non electron–phonon pairing mechanisms [191]. In the following, a short presentation of various different experimental attempts to evaluate (k) and the corresponding main results that have been achieved, is given. Experiments Probing the Gap Anisotropy The k-dependence of the gap may have trivial causes. Its anisotropy may be tied to the anisotropy or shape of the Fermi surface. It is quite conceivable that an isotropic gap, which essentially relies on the choice of a k-independent pairing interaction, is not strictly realized even in conventional superconductors. This type of anisotropic gaps, based on a pairing interaction whose strength is not the same on all parts of the Fermi surface, may also be present in conventional superconductors and many indirect evidences for this have been accumulated a long time ago [192]. Of much more interest are cases where the anisotropy of the Fermi surface and of (k) do not

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match. This may best be discussed by invoking the concept of symmetry breaking at phase transitions, introduced by Landau [193] and, in more general terms, by Anderson [194]. For conventional superconductors, the formation of a phase-coherent pair condensate invokes the breaking of the one dimensional global gauge symmetry U(1). A superconductor is termed unconventional if the corresponding pair state breaks yet another symmetry, which may, e.g., be the time reversal symmetry T, the spin rotation symmetry R or the translational symmetry X of the crystal lattice. The resulting symmetry of the pair state is reflected in the symmetry of the order parameter and also of the gap function. Even if the symmetry of the gap function or the order parameter is known, it is still difficult or even impossible to conclude rigourously, which kind of interaction has produced the observed type of symmetry breaking.Aspects of unconventional superconductivity are also addressed in the chapter on superconductivity of heavy electron compounds and have extensively been discussed in [195]. Additional symmetry breaking and corresponding unconventional pairing configurations often lead to gap functions (k) with nodes, i.e., zero gap amplitudes at points or on lines on the Fermi surface. Such nodes have a significant influence on the temperature dependence of physical properties that are dominated by electronic excitations in the superconducting state well below Tc . For superconductors with conventional pairing, gap nodes may form only accidentally and usually the gap assumes non zero values across the entire Fermi surface. In these cases, all properties of the superconducting state which are related with the excitation of electronic quasiparticles, exhibit an exponential temperature dependence at T/Tc << 1. This is, of course, not the case if gap nodes are present. Contrary to conventional superconductors for which at zero temperature the quasiparticle density of states D(E) is zero between E = EF = 0 and E = ±, this is not so for many configurations of unconventional pairing. A well known exception is the superfluid B phase of liquid 3He, where an unconventional pair wavefunction provides a complete non-zero gap [196]. Possible forms of D(E) close to E = 0 for unconventional

Fig. 14.54. Schematic representation of the energy dependence of the electronic density of states D(E) in the gap region of unconventional superconductors. The Balian– Werthamer (BW) state is a nodeless p-wave state. The axial and the polar state imply point and lines of nodes of the gap function,respectively.The parameter  /0 is a measure for impurity induced pair breaking (see [195])

pairing states are shown in Fig. 14.54.As may be seen, D(E) vanishes in power laws of E as E → 0.The nodes are not always strictly on points or lines on the Fermi surface. Impurity scattering may widen these regions where the gap function vanishes. In some cases this happens even for tiny amounts of impurities or defects, in other cases, a critical value of impurity concentration has to be exceeded in order to achieve a gapless situation. As an immediate consequence, the above mentioned exponential temperature dependence of physical properties is replaced by power laws in T. As an example we may quote the electronic specific heat.

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rate just below Tc [197]. An example for this kind of data is shown in Fig. 14.55 [199]. The drop of the relaxation rate T1−1 , measured for Cu nuclear spins, is abrupt and quite in contrast to the expected rise of T1−1 with an onset at Tc , passing over a distinct maximum somewhat below the critical temperature and finally decreasing exponentially at T  Tc . This latter behavior, clearly identified for simple superconductors [200], is a typical manifestation of coherence effects. The same set of data, shown in Fig. 14.55, has also been claimed to provide evidence for unconventional superconductivity via the power-law type temperature dependence of the relaxation rate below Tc .

Fig. 14.55. Temperature dependence of the 63 Cu NQR spin lattice relaxation rate T1−1 (T) for three different cuprate superconductors

While for conventional superconductors it is varying as C(T) ∼exp(-/T), a power law C(T) ∼ T n is expected for unconventional superconductors. The value of the exponent n depends on the configuration of the gap nodes. It is therefore possible to identify unconventional superconductors by measurements of the temperature variation of selected physical properties well below Tc . Relevant examples are presented below. Another feature that is typical for conventional superconductors but is absent in those of unconventional varieties, is the observation of coherence effects that are a consequence of the special configuration of conventional Cooper pairs that are formed between states with opposite momenta and spins,i.e., (k ↑, −k ↓), a major ingredient of the original BCS theory [106]. The observations of these coherence effects, which depend on the interaction exerted by the measuring technique were, at the time, considered as convincing confirmations for the validity of the BCS theory [197, 198]. Some of the first indications for unconventional superconductivity in the cuprates were related with the absence of the coherence effect that is expected to be observed in measurements of the temperature dependence of NMR relaxation

(a) Penetration Depth With respect to cuprate superconductors, an early indication for unconventional superconductivity was obtained from measurements of the temperature dependence of the penetration depth L (T) for external magnetic fields well below Tc [201].In simplest terms this characteristic length is given by L2 =

mc 2 , 4ns e2

(14.22)

i.e., it depends on the ratio between the mass m of the quasiparticles and the density ns of quasiparticles condensed into pairs. The latter is temperature dependent and reaches its maximum value at T = 0K. As mentioned above, the penetration depth may be measured using the technique of SR. For high resolution measurements aiming at establishing the temperature dependence of the penetration depth at low temperatures, other experiments seem more appropriate, however. Based on microwave absorption experiments, which are well suited to measure the temperature induced variation of L with respect to a given value, arbitrarily set to zero, at some chosen temperature T0 ,quite reliable values of L = L (T)– L (T0 ) may be obtained. Corresponding results, obtained for high quality single crystals of YBCO-123 at temperatures above T0 = 1.3 K [201] (see Fig. 14.56), have demonstrated that L (Lab in this case) increases linearly with T, at least between 1.3 and 20 K, i.e., well below Tc of about 90 K. This temperature dependence of L , is a clear indication that the

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Fig. 14.56. Temperature dependence of the incremental increase of the in-plane penetration depth of single crystalline Y-123. The different symbols represent the data for different samples (see [201])

gap function of YBCO-123 has nodes, most likely on lines on the Fermi surface. The claim is quite convincing because a control experiment of the same type, using a sample of a PbSn alloy, a conventional type II superconductor, resulted in an exponential increase of L at temperatures well below Tc = 7.2 K of this material. Later experiments [202] revealed that shortening the quasiparticles’ mean free path by introducing impurities results in L ∼ T2. (b) Specific Heat Since the specific heat C(T) is solely dependent on the density of states of possible thermally induced excitations, the temperature dependence of the contribution to the specific heat due to excited electronic quasiparticles, Ce (T), is bound to reflect the presence of gap nodes. In general, point-type or line-type gap nodes are expected to result in Ce (T) being proportional to T 3 and T 2 , respectively, at very low temperatures at least. Although this verification appears as being rather trivial in principle, the practical execution is hampered by the fact, that the subtraction of not well known background contributions to C(T) is less simple as it may seem. Nevertheless, attempts of this sort have been made, mainly by invoking mea-

Fig. 14.57. Scaling of the low temperature specific heat of YBa2 Cu3 O7−ı at different temperatures and in varying magnetic fields (see [206])

surements of the specific heat of close to optimally doped YBCO-123, both in zero and non-zero magnetic fields H. In some work, a T 2 variation of Ce at T  Tc and H = 0, indicating line nodes of the gap [203], was claimed to have been verified [204]. Even more involved is the analysis of specific heat data obtained in non zero external magnetic fields, because the structure of the vortices and their normal cores add to the already present complications. However, the measured H and T dependencies may be compared with rigorous theoretical predictions for the consequences of unconventional gap functions. In particular, line nodes are predicted [205] to provoke the appearance of a new term representing the contribution due to the quasiparticle density of states Ce , varying as H 1/2T −1 and replacing the T 2 contribution at low temperatures. The predictions for line nodes include a crossover behavior at a critical value zc = Hc 2−1/2 Tc to a high-T, lowH regime with two separate contributions, an H independent T 2 term and a temperature independent contribution varying linearly with H. Both predictions have been verified experimentally [204, 206]. One of the verifications of the predicted scaling of Ce in the low temperature regime is shown in

14 High-Tc Superconductivity

Fig. 14.57. Another prediction [207,208] for the line node configuration, the scaling of the anisotropic component of the field induced specific heat, given by Ce (T, H||c) − Ce (T, H⊥c) has also been investigated and confirmed experimentally [209]. Analogous experiments with similar results and conclusions, namely the presence of line nodes in the gap function, have also been made on superconducting La1.85 Sr0.15CuO4 [210]. (c) Thermal Conductivity As we have seen in one of the previous subsections, the transport of energy or heat in the superconducting state of cuprate materials is mainly carried by excited electronic quasiparticles. It may therefore be expected that at very low temperatures, measurements of the temperature dependence of the thermal conductivity at temperatures much below Tc again provide results that are dictated by the gap configuration (k). In particular, again a rather general theoretical prediction is available for the case of dwave type gap function with line nodes in two dimensions [211]. It addresses the quasiparticle transport in such a system and states that this transport should become independent of the scattering rate as T approaches 0 K. For a d-wave type gap configuration it is known that impurities widen the zero gap regions around the line nodes, thereby enhancing the amount of remnant normal quasiparticles even at T = 0 K. This enhancement is compensated by the impurity imposed reduction of the mean free path, thus resulting in a universal limit of the heat transport. Relevant experimental data for testing this prediction are available for heat transport parallel to the Cu-O planes of YBCO-123 and Bi-2212 single crystals [212, 213]. In Fig. 14.58 we show the measured T dependence of the thermal conductivity along the a axis of YBCO-123 single crystals, doped with different amounts of Zn impurities. It may be seen that at the lowest temperatures, the ratio /T reaches a value that is independent of the amount of impurities and hence the scattering rate. From these results it is claimed that a universal value of /T does exist at very low concentrations of impurities [212]. In Fig. 14.59, a comparison of data obtained for the in plane thermal conductivity of YBCO-123 and Bi-2212

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single crystals confirms that a remnant normal component 0 , varying linearly with T, is observed in cuprate superconductors at T close to 0 K [213]. For

Fig. 14.58. Variation of the low temperature thermal conductivity along the a-axis of single crystalline Y-123 upon replacing Cu by Zn (see [212])

Fig. 14.59. Low temperature thermal conductivity of single crystalline Y-123 and Bi-2212. The solid lines emphasize the two contributions varying as T and T 3 , respectively (see [213])

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a d-wave gap configuration, the ratio 0 /T may approximately be expressed as [214]  2   0 kB · n vF = , (14.23) T 3h · d v2 where n/d is the stacking density of Cu-O planes, vF and v2 are the Fermi velocity, directed normal to the constant-energy contour, and the quasiparticle velocity tangential to the Fermi surface at each node, respectively. It turns out that the ratio vF /v2 can be deduced from direct measurements of the k-dependence of the gap function (see below) with ARPES [215]. The values of vF /v2 , calculated either from 0 /T or from the ARPES data differ only insignificantly, i.e., safely within experimental uncertainties [213]. While these experimental findings strongly support that the pairing state of cuprate superconductors implies a gap function with line nodes in general and a d-wave configuration in particular, the good agreement of the values for vF /v2 from transport and spectroscopic measurements tends to imply that the quasiparticles act like those of a Fermi liquid, at least at low energies. The pros and cons related with the Fermi liquid description of the normal state have been addressed in Sect. 14.4.1 above. (d) Nuclear Magnetic Resonance The first indications for unconventional superconductivity from NMR measurements probing the spin lattice relaxation of Cu and O nuclear spins have briefly been mentioned in the introductory to section of this subchapter. Here we first simply recall that also the relaxation of nuclear spins via interaction with electronic quasiparticles is expected to reveal the presence of a vanishing gap. This relaxation rate provided by this channel is expected to vanish following a power law in T with decreasing temperature and not exponentially as in the case of a nodeless gap. That this is indeed observed is shown in Fig. 14.55, where it may be seen that the reduction of T1−1 below Tc is definitely not exponential in 1/T. Additional relevant information concerning the gap and even the spin configuration of the pair state can be obtained from the temperature dependence of the Knight shift in the superconducting state. As

Fig. 14.60. Temperature dependence of the normalized Knight shifts of Cu and O NMR signals in the superconducting state of Y-123. The broken line represents a calculation based on gap function with d-wave symmetry (see [191])

we have mentioned in Sect. 14.4.1 the Knight shift K is, via its proportionality to the spin susceptibility P of the quasiparticles, proportional to the density of quasiparticle states at the Fermi energy. For more complex materials, such as the cuprate superconductors, other contributions to the magnetic susceptibility have to be taken into account. For conventional superconductors, where K may simply be assumed to be proportional to the magnetic susceptibility and, below Tc , the temperature dependence K(T) is given by the so called Yosida function [216]. It results from the calculation of the Pauli-type magnetic susceptibility in the superconducting state s (T) which is given by    ∂f dE , (14.24) s = −4 B2 DBCS (E) ∂E where DBCS = D(EF )·|E|/(E2 −2 ),and f is the Fermi– Dirac distribution function. If the gap function  is nodeless and the pairing is of spin singlet type, s vanishes exponentially with decreasing temperature for T → 0 K. For spin triplet pairing the Knight shift is unaffected by the superconducting transition, i.e.,

14 High-Tc Superconductivity

is not significantly reduced below Tc [217]. The main influence of a gap function with nodes is again visible in a non exponential decay of K(T) at low temperatures. In Fig. 14.60 a selection of results of Knight shift measurements involving nuclei on different lattice sites of YBCO-123 is presented [218,219]. The strong reduction of K in the superconducting state clearly reflects spin singlet pairing. Together with the information suggesting the existence of line nodes in the gap function, there is strong evidence for a d-wave type pairing being established in cuprate superconductors. Theoretical model calculations to account for the experimental data gave better agreement,if dwave rather than s-wave pairing was assumed [191]. One such attempt is shown as the broken line in Fig. 14.60. (e) Angle Resolved Photoemission As was mentioned above, significant progresses in measurements employing photoemission spectroscopy techniques, in particular ARPES with high

Fig. 14.61. Angle dependence of the leading-edge shift in ARPES spectra of Bi-2212, relative to the position of the reference Fermi edge of Pt. The solid line represents a calculation assuming a d-wave type gap function (see [87])

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energy and momentum resolution, were responsible for the evaluation of the excitation spectrum of quasiparticles in cuprate superconductors. While integrative photoemission has been able to demonstrate the formation of a gap below Tc [85], ARPES also provides information concerning the kdependence of the gap. Since a whole chapter of this compendium is dedicated to this type of experiments, it suffices for our purposes to present the results of ARPES measurements probing (k) (see Fig. 14.61). The fourfold symmetry of the node configuration expected for a gap function with dx2 −y 2 symmetry is very well borne out [87]. (f) Phase Sensitive Measurements The experiments cited and discussed above all confirm that the gap functions in the investigated cuprate superconductors exhibit nodes, most likely on lines, and that the pairing state of the quasiparticles is of a spin-singlet variety.All this taken together, also taking into account the essential electronic wave functions in the Cu-O planes and considering various theoretical arguments based on phenomenological and computational approaches [191] leads to the conjecture that the pair state and hence the order parameter as well as the gap function exhibit a dx2 −y 2 symmetry. This implies that the amplitude of the order parameter changes sign at the nodes or, in other words, the phase of the order parameter changes discontinuously by . None of the above experiments is sensitive to this phase change and other types of measurements have to be invoked to settle this issue. These measurements have to be devised such that they probe the relative phase of the gap function on different parts of the Fermi surface. Since phase information is required, this type of experiments is based on the macroscopic quantum coherence in superconductors, most distinctly manifested in the Josephson effects and the flux quantization. The concept for this kind of tests has first been put forward by Geshkenbein and Larkin [220], in connection with probing the unconventional nature of superconductivity of heavy electron metals. They suggested to investigate the phase coherence of specially tailored superconducting loops containing a combination of the superconductor under in-

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vestigation and conventional superconducting materials. This idea was later adapted by Sigrist and Rice [221] for a proposal to test the dx2 −y 2 character of the gap function of cuprate superconductors and the schematic layout of the required superconducting loop is shown in Fig. 14.62. The idea rests on the fact that at a Josephson contact between two superconductors with opposite signs of their order parameter amplitudes, a so called -junction [222], where the phase of the order parameter changes discontinuously by ,is generated.It is well known that the magnetic flux through the cavity of a closed loop of a single conventional superconductor is quantized [223]; the flux quantum is ¥0 = hc/2e.It turns out that the response to external magnetic fields of loops containing either an even (including zero) or odd number of -junctions is significantly different. In the first case the highest stability is achieved if an integer number of flux quanta ¥ =  · ¥0 is trapped in the loop. In the second case, however, the stability criterion requires that ¥ = ( + 1/2)·¥0. Essentially two different types of experiments have been designed to obtain the requested information. In this more general review,none of the intricacies of these experiments and their analyses will be addressed in detail. This has, to a large extent, been done in more specialized reviews on this topic [224,225] (see below).

Fig. 14.62. Schematic layout of a superconducting loop for phase sensitive tests of the order parameter symmetry of cuprate superconductors, employing a SQUID type configuration (see [221])

(f1) SQUID-Type Experiments Superconducting quantum interference devices (SQUIDs) are based on the Josephson effect, i.e., on the tunneling of pairs as a whole and usually contain two Josephson contacts in parallel. As the name indicates, SQUIDs offer the possibility of experimental control of the phase of the macroscopic wave function describing the superconducting state. For our purposes the chosen loop may be manufactured in a SQUID configuration, as is schematically outlined in Fig. 14.62. The design is such that one, two or none of the contacts may form as a -junction. The critical current Ic , i.e., the maximum supercurrent, varies periodically as a function of the total magnetic flux inside the loop. The location of the maxima or minima of Ic depends critically on the phase coherence around the loop. If the SQUID loop contains none or two -junctions, an Ic (¥ ) pattern as shown in Fig. 14.63a is expected. The Ic (¥ ) diagram for a SQUID with only one -contact, however, is shifted by  along the horizontal ¥ -axis, as shown in Fig. 14.63(b). The first realization of this type of SQUID experiments was accomplished by Wollman and co-workers [226] by studying the critical current behavior of SQUIDs that were fabricated from single crystalline YBCO-123 and Pb. The Pb part was in the form of a thin film, deposited in two different configurations. In the first, the film made two contacts around the corner of a single crystal, such that one of them is along the a direction and the other along the b direction of the crystal lattice.The second configuration had both contacts onto the same edge of the crystal, facing the same crystallographic direction. The schematic sample configuration for this pioneering work is shown in Fig. 14.64. Considering the crystal structure of YBCO-123 and the expected dx2 −y 2 symmetry of the order parameter,the first configuration is expected to contain one contact acting as a -junction. For the second case either none or two -junctions are likely to be established.Measurements of the dynamical resistances of these SQUIDs as a function of applied magnetic field were used to monitor the modulation of the Ic (¥ ) curves. The analysis of the data indicated a tendency of a -shift of Ic (¥ ) for loops around the corner of the crystals with respect to Ic (¥ ) obtained for edge-type SQUIDs.

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The result was thus regarded as a confirmation of the dx2 −y 2 character of the order parameter of this material. In another approach [227], a direct comparison of the field induced critical current variation of a hybrid SQUID fabricated from single crystalline YBCO123 and Nb,expected to contain one -junction,with Ic (¥ ) of two control SQUIDs made of Nb alone was made.This comparison of Ic (¥ ) of the two configurations, accomplished by simultaneous measurements of the dynamical resistances, again gave evidence for a -shift between the Ic (¥ ) curves of the two types of SQUIDs.Additional experiments [228] of the same type confirmed the main result of the quoted first two investigations.

Fig. 14.63. Interference patterns of the critical current Ic of a superconducting loop as shown schematically in Fig. 14.62, upon varying the magnetic flux through the loop (a) with an even or (b) an odd number of  -junctions (see text)

Fig. 14.64. Schematic view of the experimental realization of SQUID type configurations forming (a) a corner SQUID and (b) an edge SQUID for phase aensitive tests of the order parameter symmetry of Y-123 single crystals see [224]

(f2) Single Junction Experiments The magnetic field induced modulation of the critical current Ic of a single Josephson junction between a conventional s-wave superconductor and a cuprate superconductor may also give information concerning the order parameter symmetry of the latter. This possibility has first been discussed and finally realized again by Wollman and coworkers [229]. They argued that the Ic (¥ ) curves for single Josephson contacts, either straddling the corner separating the a and the b direction of a single crystal of YBCO123 or placed along the edge of a crystal, would again differ significantly. They also pointed to some advantages using this technique, avoiding some problems that are related with the use of the SQUID technique mentioned in (f1) above. For extended Josephson junctions between two conventional superconductors, the critical current is strongly influenced by external magnetic fields and it has since long been established that Ic (¥ ), where ¥ is the magnetic flux threading the barrier region of the contact, exhibits a Fraunhofer type pattern (see Eq.(14.25)),as shown in Fig.14.65.For junctions with extensions that are still smaller than a characteristic length, such that self-field effects can be neglected, the resulting pattern has indeed the shape described by        sin ¥   sin ¥     ¥0 ¥0  · . Ic (¥ ) ∼  (14.25)    ¥ ¥     ¥0 ¥0

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Fig. 14.65. Fraunhofer type interference pattern of an extended Josephson junction as a function of magnetic flux through the junction (see Eq. (14.25))

Fig. 14.66. Expected schematic interference pattern for a single Josephson junction straddling the corner of a Y-123 single crystal, assuming a d-wave symmetry of the superconducting order parameter (see [229])

For individual junctions based on contacts between a conventional superconductor and a cuprate material, essentially the same Ic behavior is anticipated for the above mentioned single edge contacts. In case of an anisotropic order parameter with dx2 −y 2 symmetry, in contacts straddling a crystal corner in a more or less symmetrical way, the compensation of the positive and the negative currents is expected to result in a zero critical current at ¥ = 0 (mod.) and hence [229]        sin2 ¥   sin2 ¥   2¥0   2¥0  Ic (¥ ) ∼     ·     . (14.26) ¥ ¥     2¥ 2¥ 0

0

This behavior is displayed in Fig. 14.66. Again, contacts along a single edge ought to show a flux dependence according to Eq.(14.25).Since in practice these corner contacts can never be made exactly symmetrical, some asymmetry in the patterns must be anticipated. The key feature, however, remains whether a dip or a peak in Ic (¥ ) is observed for single corner junctions at ¥ = 0. In Fig. 14.67 the result of one of this type of experiments between Pb and an underdoped single crystal of YBCO-123 is shown [230], confirming the earlier results of [229], obtained on optimally doped YBCO-123. The experimental attempts based on the techniques briefly addressed in (f1) and (f2) have been reviewed in [224].

Fig. 14.67. Magnetic field dependence of the critical current Ic of a corner junction between an underdoped Y-123 single crystal and Pb. The insets show the schematic experimental configuration and the ideal expected pattern. Note that Ic adopts a minimum at zero external magnetic field

(f3) Scanning SQUID Magnetometry For a simple closed superconducting loop, the maxima of Ic at ¥ =  · ¥0 imply that the loop may trap flux quantities given by an integer number of flux quanta, a situation with zero trapped flux is also possible. A loop with an odd number of -junctions, however,will trap fluxes ¥ = ( +1/2)·¥0.This means that in order to stabilize, the loop will always gener-

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ate a spontaneous current, thus inducing a magnetic moment corresponding to at least one half of a flux quantum if the external field is set to zero. In order to test this consequence of an odd number of -junctions in a loop arrangement, Tsuei, Kirtley and coworkers designed a really ingenious experiment [231]. In their first attempts, they deposited small ring-shaped films of YBCO-123 on suitably tailored SrTiO3 substrates. These substrates were composed of differently oriented single crystals, mended in such a way that would allow the preparation of different rings, epitaxially deposited on the substrates, with either an even, including zero, or an odd number of -junctions. The amount of trapped flux, induced in different ways, was monitored with a scanning micro SQUID probe. It was convincingly shown that loops with an odd number of -junctions behaved differently from those with an even number or none at all. The original layout of the samples is schematically shown in Fig. 14.68. With this relative orientation of the crystals,the relative orientations of the dx2 −y 2 lobes of the order parameter in the corresponding segments of the YBCO-123 films as shown in Fig. 14.69, are anticipated.With these orientations, the prediction is that only the loop with its center at the location where all three crystals meet should exhibit the trapping of half integer fluxes.This is exactly what the outcome of the experiment confirmed. A number of additional experiments with either other choices for the relative orientations of the substrate crystals [232] or with films of other cuprate superconductors, such as Tl-2201 [233] and Bi-2212 [234], all confirmed the above result and helped to rule out extrinsic causes for the observation of half integer fluxes where expected. The same type of experiments also settled an earlier debate concerning the order parameter symmetry of electron doped cuprate superconductors. For a while, based on penetration depth experiments, such as mentioned in a previous subsection, it was claimed that the superconducting state of this variety of copper oxides has to be characterized by an s-wave type pairing. Phase sensitive measurements using the scanning SQUID method [235] have shown that also electron doped cuprates adopt a superconducting state with a gap of dx2 −y 2 symmetry. An extensive review covering

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Fig. 14.68. Schematic view of superconducting loops of ring shaped films of Y-123, deposited epitaxially on a tricrystal substrate with specially chosen orientations. The central ring is expected to have one  -junction (see [231])

Fig. 14.69. Schematic relative orientation of the d-wave type order parameter contour in the ring shaped films (see [225])

the experiments based on the scanning SQUID technique is given in [225]. (f4) Related Experiments and Problems In connection with these tunneling experiments is should also be mentioned that not all of them give straightforward evidences for the pairing and hence gap anisotropies discussed above. Results of early experiments probing the Josephson tunneling through grain boundaries [236] were incompatible with dwave pairing in the sense that no complete suppression of the critical current in the relevant geometry

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of two order parameter lobes with different signs opposing each other on either side of the grain boundary could be achieved. It has been suggested [237] that in the long junction limit,the junction roughness may play an important role and that the appearance of local vortices at the grain boundary may weaken the expected frustration and therefore allow for non zero Josephson currents, i.e., currents at zero voltage, in this case. Somewhat more complicated is the interpretation of tunneling experiments where the tunneling current is directed along the c -direction of the crystal lattice.Various experiments investigating this type of arrangement with contacts between Pb and YBCO123 single crystals as well as epitaxial films have been made [238–240], and non zero Josephson currents through these junctions were observed. In case of a strict d-wave anisotropy of the pairing in the plane, such non zero currents should, of course, be absent. This apparent inconsistency has been ascribed to the orthorhombicity of the crystal lattice of YBCO-123 and the corresponding loss of the fourfold rotation symmetry in the plane [241]. This situation allows for a mixing of d and s symmetry in the pairing, a scenario that was early suggested in [242]. Therefore, the observation of non zero Josephson currents may be explained in this way. More detailed discussions of these issues are available in [224] and [225]. Consequences of d-Wave Pairing The now fairly well established dominant d-wave pairing of quasiparticles in cuprate superconductors does not automatically provide a clear cut conclusion concerning the nature of the pairing interaction. An unequivocal identification of the pairing mechanism in cuprates is a formidable task, also in view of the non Fermi liquid behavior. It was believed that, because of the appearance of gap nodes, at least an electron phonon mediated pairing could be ruled out. Recent Eliashberg-type calculations may indicate that this may not necessarily be true [243]. Nevertheless, the earliest suggestions of a dx2 −y 2 type pairing symmetry were based on the assumption that the pairing was achieved by exchange of antiferromagnetic spin excitations. This work has

been reviewed in [191] and [244], see also chapters by Chubukov et al. and Manske et al. in this volume. Since in this case in the gap equation (k) = −£

V (k − k )(k ) 2E(k )

(14.27)

the interaction V (k − k ) is always positive and has its maxima at the (,) points of the Brillouin zone, the resulting gap function adopts a dx2 −y 2 -type symmetry of the form (k) = 0 (cos kx − cos ky ) .

(14.28)

These simple considerations have been complemented by extensive numerical studies cited in [191] which, for this type of interaction, confirm the stability of a superconducting state with dx2 −y 2 pairing. Among other consequences, the d-wave pairing also has some direct influence on the configuration of the vortices in the mixed state [245, 246]. First of all, singular vortices are no longer expected to exhibit a circular cross section. Second, the vortex lattice, exhibiting a hexagonal symmetry in conventional superconductors, has been shown experimentally [247, 248] to adopt a rhombohedral symmetry where the two unit cell vectors are of approximately equal length, forming an angle of 77 degrees. One may argue that this again is a consequence of the d-wave pairing, considering both the distorted cross section of the single vortices and possible resulting changes in the interaction between the vortices. In view of some isotope effect experiments, it seems that interactions between itinerant quasiparticles and the lattice are of some importance for the superconductivity in cuprates as well. In general it is assumed that electron phonon interactions are not favorable for d-wave pairing. Therefore, it has been proposed that the pairing or gap symmetry is a combination mainly of d-wave and s-wave character [242]. Even to date the situation is far from being clear. 14.4.3

Coexistence of Superconductivity and Magnetic Order

The phenomenon of coexistence of magnetic order and superconductivity was long thought to be im-

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possible, especially in view of the pair breaking influence of magnetic fields and magnetic impurities established for conventional superconductors [249]. Nevertheless, the first examples of the coexistence of antiferromagnetic order and superconductivity were reported in the mid seventies in two classes of compounds, the Chevrel phases (molybdenum sulfides and selenides) [250] and the Rhodium–Boron compounds [251]. In these series, compounds with rare-earth ions occupying regular lattice sites may be prepared. Because of the incompletely filled 4f electron shells, most of these ions carry respectable ionic moments resulting from spin and orbital degrees of freedom. Since these materials are metallic, it is no surprise that these moments order spontaneously at low temperatures via the Ruderman–Kittel–Kasuya– Yoshida (RKKY) interaction mediated by the conduction electrons [252]. What is surprising here is that both the N´eel temperature, fixing the onset of antiferromagnetic order, and Tc for superconductivity are of similar magnitude. This situation has been studied in great detail for a number of different materials and reviews of the main results are available in [253–255], see als chapters by Kuli´c and Buzdin in volume one of this book. As we mentioned in Sect. 14.2.2, some cuprates with rare-earth ions on regular lattice sites may also be synthesized in the classes of YBCO-123 andYBCO124 compounds.Most of the rare-earth elements may be used to replace Y in these materials. At optimal doping conditions, i.e., close to seven and eight oxygen atoms per unit formula, respectively, the values of the critical temperatures of these rare-earth cuprates are very close to those of the corresponding Y compounds. In this sense there are no significant pair breaking effects to speak of. The observation of antiferromagnetic-type of ordering has been reported for some of them. The ordering temperatures are rather low and reach a maximum of approximately 2.2 K for fully oxygenated GdBCO-123 [256]. In most of the other of these compounds the ordering sets in below 1 K [257]. It is most likely that both the coexistence phenomenon and the low magnetic transition temperatures are due to the rare-earth site in the YBCO-123 crystal lattice which is situated between the two es-

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sential Cu-O planes (see Fig. 14.6(a)). These sites are electronically quite well separated from the charge carrier system in the planes and therefore have little influence on superconductivity. Likewise the RKKY interaction between these ions is, for the same reason, rather weak. Together with the reduced number of available excited quasiparticles this implies a priori a low ordering temperature. It turns out that the concentration of itinerant charge carriers at the rare-earth sites is so low that it is doubtful whether the RKKY interaction is of any significance at all. The N´eel temperature for antiferromagnetic order in HoBa2 Cu3 O7 is of the order of 0.1 K [258], most likely induced by the hyperfine field of the Ho nuclei. This is no surprise since the CEF splitting of the Ho3+ Hund’s rule J = 8 multiplet leaves a singlet ground state and the ordered moments of approximately 2 B /Ho ion turn out to be much reduced from the free ion value.

14.5

Physical Properties of Non-Cuprate High-Tc Superconductors

In this section, some selected physical properties of superconductors with unusually high critical temperatures are presented and discussed.None of them, reaches values that exceed the boiling point of liquid nitrogen. Nevertheless, the discovery of superconductivity of these materials and the magnitude of Tc were and in some cases still are rather surprising, and therefore of interest. 14.5.1

Ba-Based Oxides

BaPbO3 , in spite of in principle compensating valences of its chemical constituents, is metallic. The electrical resistivity is almost temperature independent and of the order of a few hundred §cm. Gradually replacing Pb by Bi leads to a structural transition if about 10% of the Pb content is replaced by Bi. For BaPb1−x Bix , the electrical resistivity  increases continuously with increasing x but still stays more or less temperature independent [51]. According to measurements of the Hall constant RH [51],the charge carrier concentration increases with x, passing through a maximum at x ≈ 0.25, which appears

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to be the optimal doping situation for the occurrence of superconductivity. At this x value, the density of itinerant charge carriers and the sign of RH are compatible with one extra itinerant electron per Bi atom. Conventionally viewed, this concentration of conduction electrons is rather low for a superconductor with a Tc that exceeds 10 K. In contrast to other oxide superconductors, in particular also the cuprate superconductors, the bulk magnetic susceptibility of the normal metallic phase of BaPb1−x Bix O3 is negative and largest in magnitude at x ≈ 0.25. This suggests that the paramagnetic contribution to is weakest at this composition. As shown in Fig.14.70,clear resistive transitions to superconductivity are observed for 0.15 < x < 0.35, but the sharpest transitions at maximum values for Tc are observed for x = 0.25 [51, 259]. A further increase of x provokes a reduction of Tc and eventually leads, as indicated by both  (see Fig. 14.70) and RH data, to a metal insulator transition which is accompanied by the already mentioned structural transition back to an orthorhombic symmetry of the crystal structure. Investigations of the superconducting state [52] revealed that (BPBO) BaPb0.75 Bi0.25O3 is an extreme type II superconductor with a Ginzburg Landau pa-

Fig. 14.70. Variation of the temperature dependence of the electrical resistivity of BaPb1−x Bix O3 (see [51])

rameter  of the order of 70÷80. It appears that the electronic specific parameter , estimated from a barely resolvable specific heat anomaly at Tc and using Eq. (14.6), is rather small, i.e., ≈ 1.5 mJ/molK2 , indicating a low density of electronic states D(EF ) at the Fermi energy, usually unfavorable for high critical temperatures for the onset of superconductivity. Tunneling data seem to indicate strong electron– phonon interactions extending in energy into the range of the superconducting gap in the excitation spectrum [52,259]. If strong coupling effects are important, the value quoted above is significantly reduced and D(EF ) turns out to be even smaller. From the theoretical point of view [260] there is no consensus, how the relatively high critical temperature is to be explained, because the strong-coupling concept is not compatible with band structure calculations [261] and their connection with the metal to insulator transition mentioned above. In addition, the ratio 20 (0)/kBTc has been estimated to be 3.5, practically identical with the weak coupling BCS value. Some problems are also encountered in the interpretation of measurements of the upper critical field Hc2. In the dirty limit, a reasonable choice considering the resistivity data displayed in Fig. 14.70, the limiting orbital critical field is given by [262]   ∂Hc2 Hc2(0) = −0.691 · Tc . (14.29) ∂T Tc It appears that the extrapolated experimental limit of Hc2(0), based on measurements of Hc2 (T) [52], shown in Fig. 14.71, by far exceeds the one that is calculated from the measured initial temperature dependence of Hc2 just below Tc , indicated by the broken line in Fig. 14.71. A clear result is, however, obtained from measurements of the isotope effect on Tc by exchanging 16 O and 18 O on the oxygen sites [263]. The heavier oxygen isotope leads to a significant reduction of the critical temperature and the value of the parameter ˛ entering Eq. (14.10) has been calculated to be ˛ = 0.22. Although this result seems to confirm that most likely the occurrence of superconductivity in the Pb/Bi oxides is due to an electron phonon interaction, it has been argued that electronic excitations involving atomic displacements due to charge redistributions are me-

14 High-Tc Superconductivity

diating the pairing interaction [263]. This conclusion is based on considering the relatively high value of Tc and indications that pairing in BPBO is a weak coupling effect. The observation of critical temperatures that are more than twice as large as those of the Pb/Bi alloys and comparable to some of the cuprate superconductors, i.e., of the order of 30 K, for Ba0.6 K0.4 BiO3 (BKBO) [54, 55], naturally led to additional experiments probing the physical properties of this compound [263, 264]. In particular, a comparison with some of the properties of the cuprate superconductors was attempted. The total normal state magnetic susceptibility again reveals a strong core diamagnetism which, overall, masks a very weak paramagnetic contribution [263,264]. Normal state Hall effect measurements [264] revealed that RH is weakly temperature dependent and negative at all temperatures below 300 K. This confirms that the Bi-based oxides, without Cu, however, are electronically and magnetically quite different species than the hole doped copper oxides.

The response of the superconducting state to external magnetic fields is rather similar to that of the Pb/Bi alloys, leading to almost the same values of (∂Hc2 /∂T)Tc , of the order of −5 kOe/K. Estimates of the Ginzburg–Landau parameter give  ≈ 45,again a large value. As for the Pb/Bi alloys this is compatible with a low density of itinerant carriers and consequently a large penetration depth . From available experimental results covering the upper and lower critical fields,Hc2(T) and Hc1 (T),estimates of the initial temperature dependence of the thermodynamical critical field (∂Hc /∂T)Tc can be made. Finally, using the law of corresponding states allows to calculate the electronic specific heat parameter or effective density of states D(EF ). Although somewhat larger than the corresponding value calculated for the Pb/Bi alloy, also D(EF ) for the Ba/K alloy is much too small to account for a Tc of the order of 30 K in any conventional way. Nevertheless, also for this material, a significant oxygen isotope effect confirms that some type of electron phonon interaction must be important for the onset of superconductivity [263]. The values for the ˛ parameter differ somewhat in different reports; ˛-values between 0.21 and 0.35 have been calculated [263,264].Again,conventional strong coupling effects have been ruled out. For (Ba/K)BiO3 , measurements aiming at probing the symmetry of the superconducting state have been made [265]. Corner-junction type experiments, exploring the field dependence of the critical current Ic (¥ )(see Eq. (14.26)), indicate that the superconducting state of this alloy is characterized by a nodeless gap function or order parameter. 14.5.2

Fig. 14.71. Temperature dependence of the upper critical field Hc2 of BaPb0.75 Bi0.25 O3 . The broken line is to guide the eye (see [52])

815

C60 -Based Materials

As we have briefly mentioned in Sect. 14.3.3 the simplest way to achieve a metallic state in bulk C60-based material is via electron doping with atoms of the alkaline metal elements. Most of the experimental results that are briefly discussed below have been obtained for compounds of the type A3 C60 . Some combinations of this type turn out to be electrical insulators. Based on calculations of the widths W of the essential electronic energy bands and the onsite Coulomb correlation energies U , indicating that

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Fig. 14.72. Temperature dependence of the electrical resistivity of thin films of K3 C60 and Rb3 C60 (see [269])

the ration U /W significantly exceeds the value of 1, it is argued that the metallic species are on the metallic side of a metal insulator transition of Mott– Hubbard type and therefore, correlation effects have to be taken into account [266,267].Another difficulty arises, because the energies of the relevant phonon branches are not very different from W and hence, invoking Migdal’s theorem [268] for justifying the electron pairing via a retarded electron–phonon interaction is questionable. The absolute values of the electrical resistivities  are rather high and correspond to electronic mean free paths of the order or even less than the separation between single molecules. Examples of (T) of superconducting A3 C60 films are shown in Fig. 14.72 [269]. Even at very low temperatures and for single crystalline samples, the absolute values of  are of the order of 1 m§ cm. In most cases the low temperature electrical resistivities vary as T2 [270] but also linear-in-T or T-independent variations down to low temperatures, depending on the sample configuration (film, crystalline, granular) have been reported. It is therefore clear that a theoretical treatment of this transport property is difficult, even more so, because it seems to matter whether (T) is measured at constant volume or at constant pressure. Nevertheless, calculations invoking the interactions between con-

duction electrons and various lattice modes may be brought into reasonable agreement between experiments, carried out under constant pressure conditions, revealing  ∼ T 2 , and theory. A renormalization of the same data to constant volume conditions reveals that  varies approximately linearly with T above 100 K [271]. This result has apparently not yet found a theoretical explanation. Although the very short mean free path at temperatures of the order of 300 K might suggest a saturation behavior in (T) at even more elevated temperatures, this phenomenon is, very much like in the case of the cuprates, not observed. In spite of the similarity of these features in (T) of the two types of materials, it seems quite unlikely that the same scattering processes are the cause for this observation. Quite generally, the magnetic susceptibility in the normal state is positive and almost temperature independent [272], as is also found for the resonance shift in NMR measurements [273]. Comparing the magnitude of experimental data with values calculated from band structure data suggests that a rather substantial enhancement of must be considered [272]. The corresponding density of states D(EF ) (see Eq. (14.4)) is larger by a factor of 3 than the bare density of states Db (EF ) that follows from the band structure, but in fair agreement with estimates of D(EF ) based on NMR spin-lattice relaxation data [273].As expected for metals, the spin-lattice relaxation rate T1−1 is approximately linear in T in the normal state, however, the hyperfine coupling constant is not known very well. Apart from the clear manifestations of the superconducting transition in (T) and (T) the transition is, as usual, also manifest in experimental data of a number of other properties. This is certainly the case for measurements probing the specific heat. Because of the fragility of the materials, especially with respect to air exposure,these measurements [272] are technically not trivial. As demonstrated in Fig. 14.73, the specific heat anomaly at Tc is masked by a large background of non electronic origin, but estimates concerning its magnitude can still be made. For K3 C60, the ratio of C(Tc )/Tc is of the order of 70 mJ/mole K2 , a rather small value but confirming the rather low effective electronic density of states at EF .

14 High-Tc Superconductivity

Fig. 14.73. Anomaly of the specific heat at the superconducting transition of K3 C60 . The solid line is to guide the eye and the broken line indicates the estimated background (see [272])

Considering the electronic specific heat parameter

that is related to C(Tc )/Tc and the enhancement of the magnetic susceptibility, estimates of the electron mass enhancement (1 + ph ) due to the electron phonon interaction can be made. Depending on details of the approximations, ph values between 0,5 and 0.7 have been calculated and it was concluded

817

that the pairing via the electron–phonon interaction could not be achieved by invoking low energy phonon excitations. Instead a coupling to high energy phonons and an upper bound of ph ≈ 1.0 have been postulated [272]. The participation of phonons in the pairing of electrons is put in evidence by isotope-effect experiments where 12C is replaced by 13 C.Although there is some scatter in the data, clear variations of Tc upon isotope exchange have been identified [274,275] and ˛-values between 0.3 and 0.4 seem to be the most reasonable. The importance of the phonons related with excitation modes of the C60 molecules is emphasized by the absence of an isotope effect if the isotope exchange involves the alkaline metal elements. This indicates that apart from the direct influence on the density of itinerant electrons via their ionic radii, the alkaline-metal atoms have only a weak influence on the magnitude of Tc [276,277]. The onset of superconductivity is clearly reflected in a distinct reduction of the NMR spin lattice relaxation rate T1−1 at low temperatures [273]. Results of early measurements gave no evidence for a Hebel– Slichter type enhancement of the relaxation rate at temperatures close to but below Tc , expected if the above mentioned coherence effects are effective. Even for well known conventional superconductors, the absence of this enhancement may be caused by strong coupling and other trivial effects. However, a later investigation clearly revealed an increase of T1−1 with decreasing temperature below Tc , as displayed in Fig. 14.74, providing convincing evidence

Fig. 14.74. Temperature dependence of NMR spin lattice relaxation rates, plotted as (T1 T)−1 vs T, in the normal and superconducting states of Rb2 CsC6 0, providing some evidence for coherence effects just below Tc ≈ 31 K (see [278])

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for coherence effects in a at most moderately coupled superconductor [278]. The same type of feature has been observed in SR experiments [279], and therefore it seems reasonable to qualify these alkalinemetal doped C60 materials as s-wave type superconductors with medium strength electron phonon coupling.Another manifestation of the superconducting state is apparent in the reduction of the spin susceptibility S , which in principle can be deduced from the temperature dependence of the spin part KS of the resonance shift [273,278]. Measurements of total resonance shifts of nuclei occupying inequivalent lattice sites made it possible to extract the temperature dependence of the Knight shift KS (T) and the results are compatible with expectations that are valid for weak coupling superconductors and s-wave pairing,in particular via the evaluation of the ratio 20 (0)/kB Tc [278]. Different values of this ratio have been extracted from different NMR data sets, ranging from 3.0 to 4.3. Estimates of this ratio based on optical experiments [280, 281] gave values between 3.4 and 4.2. Somewhat more enhanced values of 6.3 and 6.4 have been extracted from results of tunneling experiments [282,283].These values would imply a substantial enhancement of the effective density of states and ph ≈ 3.0.This value is far beyond any of the other estimates of this value, such that the results of the tunnelling experiments seem more than questionable. Values for the characteristic lengths of typical C60based superconductors have been evaluated from

SR [284, 285], NMR [273] and optical data [286], as well as from results of measurements of the upper critical field Hc2 [287]. Evaluations of Hc2 (T) from observing resistive transitions are hampered by a sizeable increase of the transition width in external magnetic fields [269]. The spread of the results concerning L and  is sizeable. For the penetration depth L (0), values between 4000 and 8000 Å have been obtained.The effective coherence length 0 ,taking into account the very short electronic mean free path, turns out to be of the order of 30 Å. The ratio of these two quantities, defining the GL parameter , indicates that these C60 superconductors are also of the extreme type II variety. The large values of the penetration depths are compatible with the previous indications for a low concentration of itiner-

ant charge carriers and a substantial enhancement of their effective mass. If all these experimental results providing information on the superconducting state of these C60based bulk materials are taken together [45, 288], it seems inevitable to conclude that one is dealing with conventional superconductors of the weak, or at best medium coupled variety invoking some sort of electron–phonon interaction. 14.5.3

MgB2

In the class of seemingly simple binary intermetallic compounds, MgB2 exhibits by far the highest onset temperature for superconductivity. For comparison, the cubic binary transition metal compounds of the type A3 B, crystallizing with the ˇ tungsten or A15 crystal structure, exhibit Tc ’s up to 18 K [289] and only if they are prepared under special conditions or in special forms, such as quench condensed films, they exhibit superconductivity at temperatures exceeding the temperature of the boiling point of liquid hydrogen, i.e., 20.4 K, by about 3 K [290] (see also Fig. 14.1). The again completely unexpected discovery of a critical temperature close to 40 K for a material that simple,naturally raises the question whether this feature may already be recognized in the normal state properties of the compound. The magnitude of Tc also provokes experiments aiming at clarifying the characteristics of the superconducting state and, in particular, to identify the mechanism leading to the pairing of electrons at this relatively high temperature. In this context also rather challenging questions need to be answered by theorists. In addition, for MgB2 , some properties that are crucial for potential applications seem to show favorable behavior and therefore this material is also of high current interest for more technically oriented developments. The first report on superconductivity of MgB2 dates from mid-January of 2001 [11] and, although a lot of subsequent work has appeared either in print or in e-print archives, not all the open questions that are indicated above have been answered up to now. This brief review, by far not complete and aiming at presenting typical features, considers work that has become available until the end of July 2001. Samples

14 High-Tc Superconductivity

819

for the many investigations were prepared in many different forms, including single crystals, mostly of rather small size, polycrystals, powder, films, wires and tapes. A first brief review of various properties of superconducting MgB2 has been presented in [81]. The Normal State The electrical resistivity below room temperature is that of a typical metal with  of the order of 10 §cm at room temperature and ∂/∂T > 0 at all temperatures between Tc and 300 K. Resistance ratios RR = (300)/(Tc ) of the order of 6 to 7 have been reported [291] for single crystals for currents along the basal plane of the hexagonal crystal lattice.An example of (T), measured for a current in the basal plane of a single crystal between 2 and 300 K is shown in Fig. 14.75 [292]. Amazingly large ratios of up to 25 have been obtained for polycrystalline material and even wires [293,294]. Since there are no reliable data yet available for the electrical conductivity along the hexagonal c-axis, it is not clear whether this difference might be due to a strongly anisotropic conductivity or, in other words, a much lower resistivity for currents along the c-axis. In all cases, however, the regime of residual resistivity, of the order of 1 § cm or less, seems to be reached at Tc . While practically absent at temperatures above 200 K, a fairly large positive magnetoresistance at T ≈ Tc , with almost a doubling of the zero field resistance in an external field of 10 T has been reported for polycrystalline material [293]. This effect is less pronounced for single crystals, at least for  measured in the hexagonal basal plane [291,292]. Hall effect data are scarce and they have not been obtained for well defined geometrical configurations. Nevertheless, the available set of results [81,295] is entirely consistent with holes being the itinerant charge carriers in MgB2 . No agreement has yet been established concerning the question whether the positive Hall constant RH varies with decreasing temperature between room temperature and Tc . It is interesting to note [296] that for other, isostructural metallic diboride compounds, for which no occurrence of superconductivity has yet been reported, RH is negative [297, 298].

Fig. 14.75. Temperature dependence of the in plane electrical resistivity of single crystalline MgB2 . The inset emphasizes the superconducting transition region

Significant anisotropies have been revealed in measurements of the lattice parameters, both as a function of external pressure at room temperature [299] and as a function of temperature at ambient pressure [300]. In both cases the response along the c axis is much more pronounced than along the a axis, confirming that the out-of-plane Mg-B bonds are significantly weaker than the in-plane Mg-Mg bonds.As a consequence the c/a ratio decreases with increasing pressure and with decreasing temperature. Band structure calculations [49, 301] indicate that the inplane B-B bonds are dominantly covalent. As usual in cases where superconductivity sets in at relatively high temperatures, it is not straightforward to obtain information on the normal state electronic properties, such as D(EF ), from measurements of the specific heat. This problem is also met in the case of MgB2 and several ways of solving it have been followed. An analysis [302] of measurements of Cp (T) in external fields between 0 and 9 T provides the following important parameters: (i) the electronic specific heat parameter = 2.6 mJ/mole K2 and (ii) a Debye temperature ŸD = 1050 K. Another set of similar measurements, including results obtained in magnetic fields up to 16 T give = 2.7 mJ/mole K2 [303]. Since in this work, the lattice contribution is ascribed to a continuum of modes, no

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Superconductivity

Fig. 14.76. Temperature dependence of the specific heat of a pressed-powder sample of MgB2 , plotted as C/T vs T. The solid line represents an estimated normal state background CN . The inset emphasizes the anomaly at and below the transition temperature by plotting (C − CN )/T vs T

value for ŸD is provided. In yet another approach, concentrating on a reliable estimate of the nonelectronic background contributions in zero field, the same parameters turn out to be = 3.8 mJ/mole K2 and ŸD = 750 K [304]. An example of Cp (T) of pressed non-commercial powder of MgB2 below 60 K is shown in Fig. 14.76 [305]. The normal state NMR response of MgB2 , monitored on 11 B nuclei is compatible with the overall metallic character of the compound [306, 307]. The Knight shift K is very small and positive at all temperatures. Different reports do not agree with respect to the magnitude of the shift. K(T) is approximately constant between 150 and 300 K. According to [307], K decreases steadily with decreasing temperature below 150 K and the decrease continues smoothly through the transition at Tc (H). A distinct change in the slope of K(T) at Tc has been reported, however, in [308]. The spin-lattice relaxation is well described by assuming a Korringa type relaxation via conduction electrons between room temperature and Tc with T1T ≈ 170 s, an average value resulting from a number of different investigations [306–308]. The Korringa constant, incorporating the Knight shift, is substantially smaller than the expected value for a relaxation via s-electrons and indicates that orbital degrees of freedom are dominant in the relaxation process [309].

Rather narrow transition widths T have been observed in resistivity measurements. For single crystals, values of T of the order of 0.2 K have been achieved [291, 292]. If RR, defined above, exceeds the value of 5, the reported critical temperatures only vary within a range of experimental uncertainty. However, considerable variations of Tc may be induced by applying external pressure or by exchange of isotopes, such as 10B for 11B or 24 Mg for 26Mg. At this point we may also mention that many different types of chemical substitutions have been tested. None of them, except one, have induced an enhancement of Tc .So far,the exceptional case is Zn for which a small enhancement of Tc at low concentrations of Zn atoms has been reported [310,311]. As in most common superconductors, the application of quasi-hydrostatic pressure causes a reduction of Tc . We note a considerable scatter in the available data set [81] which may be due to non hydrostatic pressure conditions in some of the experiments, combined with an almost certain anisotropic response of Tc versus uniaxial stress. On the average the pressure derivative of the critical temperature, ∂Tc /∂p is of the order of 0.8 K/GPa. Clear evidence for isotope effects have been obtained for isotope exchanges on both the B [312, 313] and the Mg [313] sites. As may be expected, the Boron isotope effect is particularly prominent and from corresponding experimental results, the measured shift of the magnetic transition is displayed in Fig. 14.77, the mass exponent ˛B in Eq. (14.10) related with the isotopes 10 B and 11 B, is deduced to lie between 0.26 and 0.3. This seems to confirm that lattice modes involving B atoms are involved in interactions providing the pairing of electrons. It is of no surprise that the exchange of the two Mg isotopes mentioned above leads to an only marginal shift of Tc and hence the Mg mass exponent ˛Mg ≈ 0.02.While the pressure dependence of Tc may partly be due to a change in the electronic excitation spectrum, the isotope effect, although reduced to below its classical value [313],provides convincing evidence that the lattice and hence the electron–phonon interaction are essential for this superconductivity at moderately high temperatures.

14 High-Tc Superconductivity

Fig. 14.77. Boron isotope effect on the magnetically probed superconducting transition of MgB2 (see [312])

Fig. 14.78. Magnetization M(H) curve of a powder sample of superconducting MgB2

Simple magnetization M(H) curves (see Fig. 14.78) [314] reveal at once that MgB2 is a type II superconductor, adopting a fairly large GL parameter . Many attempts to measure the temperature dependence of the upper critical field Hc2 (T) have

821

been made. The available data sets on Hc2 (0) and (∂Hc2 /∂T)Tc cover a wide range of values for both quantities [81]. It appears that the claimed Hc2 values are systematically higher if measured on thin film samples rather than on bulk crystalline specimens and reported values for Hc2(0) vary between 15 and 40 T. Because of these uncertainties, which partly may be caused by anisotropy effects, we concentrate on the few experimental results that have been obtained using single crystals [291,292]. Unfortunately these experiments, measuring the resistive transition for currents in the ab plane and external fields oriented either parallel or perpendicular to the c-axis, have not been carried to very high fields and therefore any extrapolation to low temperatures is questionable. Nevertheless, a clear anisotropy is discernible in these data. The destruction of the superconducting state is more easily achieved,if the external magnetic field is applied along the direction of the c-axis of the crystal lattice. Overall, an anisotropy ratio Hc2(H||ab)/Hc2(H||c) ≈ 2 is observed. Near Tc , Hc2(T) derived from monitoring resistive transitions, definitely exhibits a positive curvature for both field orientations. From these experimental data, the corresponding coherence lengths have been estimated using Eq. (14.8). The respective values are ab (0) = 65 Å and c (0) = 25 Å. A comparison with the electronic mean free path of the order of 500 to 700 Å, that may be estimated from the normal state transport properties, indicates that MgB2 may be treated as a clean-limit superconductor.Various estimates [81,303] of average penetration depths L (0) at zero temperature range between 1200 and 1850 Å. Estimates of the anisotropy of superconducting properties using electron spin resonance (ESR) and magnetization data from measurements on powder samples indicate an anisotropy parameter  between 6 and 9, i.e., much larger than what is quoted above from measurements of the upper critical fields on single crystals [315]. The bulk transition to the superconducting state has also been verified by measurements of the specific heat. Distinct anomalies indicating the discontinuity C = Csel − Cnel at Tc have been reported [302–304]. In magnitude these discontinuities are of the order C/Tc ≈ 3 mJ/mole K2, a value that is

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distinctly smaller that what is expected if the weak coupling BCS limit is considered. Electron phonon coupling parameters  between 0.6 and 1.8 have been derived. These values are of the same order of magnitude as results from band structure calculations [316]. Most estimates of the ratio 0 (0)/Tc imply that MgB2 should be regarded as a weakly coupled superconductor, rather surprising in view of the relatively high critical temperature. Specific heat data at temperatures well below Tc have been used to reach conclusions concerning details of the gap function. While some reports claim that the data are consistent with a common s-wave type pairing and a gap without nodes [304], a more detailed analysis suggests that at least two gap values, differing by a factor of 4, have to be considered [302, 303]. In [303], the Csel data obtained at very low temperatures are claimed to suggest line nodes of the gap and also the field dependence of Csel indicates a d-wave pairing. Similarly inconclusive are available tunneling data [317, 318]. In this context, however, a first principles calculation of the electron phonon coupling in MgB2 leads to the conclusion that two gaps, related with different coupling strengths on different parts of the Fermi surface, should develop [319]. The NMR response in the superconducting state has also been investigated by a number of groups. The Knight shift data [307, 308] exhibit a reduction of K below Tc but a rigorous analysis is very much hampered by the small size of the shift in the first place [309]. In one of the studies the observation of a coherence induced enhancement of the spin lattice relaxation rate T1−1 below Tc and a subsequent drastic reduction of T1−1 has been reported [307]. The result was interpreted as reflecting a strong coupling s-wave type superconducting state of MgB2 . In other work [308,309],no evidence for a coherence peak was identified and only minor reductions of T1−1 below Tc were observed in rather low fields. It cannot be ruled out that problems with sample purities and inhomogeneities are responsible for these discrepancies. First reports on probing the temperature dependence of the occupied electronic states close to EF above and below Tc using photoelectron spectroscopy on polycrystalline material [320] indicate

that the gap formation is close to isotropic in kspace, consistent with s-wave pairing. The temperature dependence of the gap amplitude is well reproduced with the original BCS expression of (T) and 2(0)/kB Tc = 3.0, a value indicating weak coupling. At this point is seems fair to say that the situation concerning the basic aspects of superconductivity of MgB2 has not yet been settled. Although there is some consensus that the electron pairing in the superconducting state of MgB2 is, in view of the high critical temperature, caused by an amazingly weak electron phonon interaction, many details need to be clarified. To some extent this situation is due to the present lack of experimental data obtained from single crystalline specimens. In particular, any aspects concerning the anisotropy of normal state and superconducting properties of this material must be treated by using single crystals. Polycrystalline or powder samples of high purity, however, may be used for exploring the potential of MgB2 for applications [293].In this context some very encouraging results have been obtained [81]. Measurements of critical current densities jc as a function of temperature and magnetic field in samples of different configurations have been made.Contrary to the Cu-oxide materials, the critical currents in MgB2 seem to be controlled by the pinning strength for trapping vortices and not by effects of weak links at grain boundaries [321]. In zero magnetic field, jc values of the order of 107 A/cm2 have been achieved in films [322] and about an order of magnitude less in bulk samples,wires and tapes [81].In wires and tapes, the critical current values are reduced rather rapidly in external magnetic fields, even at 4.2 K, much more than for the technically important superconductors of Nb-Ti alloy and Nb3 Sn, which exhibit much lower critical temperatures. It has been established, however, that at the same fields, MgB2 films may sustain much higher current densities and, in the range of liquid Helium temperatures, perform equally well as commercial Nb-Ti and Nb3 Sn wires [322]. This nurtures hopes that also the performance of wires and tapes may considerably be improved. Last but not least,the fabrication of superconducting MgB2-based conductors appears to be a rather low cost affair.

14 High-Tc Superconductivity

14.6

Final Remarks

The field of superconductivity has experienced a number of unexpected discoveries during the last 20 years. Only some of them are discussed in this chapter, others are discussed in different sections of this book. The appearance of superconductivity at unprecedented high temperatures has not only triggered research activities worldwide, but has also influenced the thinking with respect to the common understanding of the metallic state in solids in general. While there is consensus that the cuprate materials exhibit some unconventional superconductivity, it is yet not clear what is the pairing mechanisms. Note,

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recent ARPES experiments [321,323] observe interesting changes in the dispersions of some high-Tc cuprate superconductors. There is little doubt that electron phonon interactions are essential for superconductivity in the considered non-cuprate oxides, the fullerides and MgB2 . Nevertheless, there are still open questions with regard to the characterization of the superconducting state in these materials. Acknowledgement I thank H. Thomas and S. Sigrist for their help in preparing the manuscript.

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