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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F. Marsiglio and J. P. Carbotte

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)