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