Selected Papers of J Robert Schrieffer: In Celebration of His 70th Birthday (World Scientific Series in 20th Century Physics, V. 30)

World Scientific Series in 20th Century Physics - Vol. 30

World Scientific

Selected Papers of

JP ROBERT CHRIEFFER In Celebration of His 70th Birthday

World Scientific Series in 20th Century Physics Published Vol. 1 Gauge Theories — Past and Future edited by R. Akhoury, B. de Wit, and P. van Nieuwenhuizen Vol. 2

Scientific Highlights in Memory of Leon Van Hove edited by F. Nicodemi

Vol. 3

Selected Papers, with Commentary, of T. H. R. Skyrme edited by G. E. Brown

Vol. 4

Salamfestschrift edited by A. AH, J. Ellis and S. Randjbar-Daemi Selected Papers of Abdus Salam (with Commentary) edited by A. AH, C. Isham, T. Kibble and Riazuddin

Vol. 5 Vol. 6

Research on Particle Imaging Detectors edited by G. Charpak

Vol. 7

A Career in Theoretical Physics edited by P. W. Anderson

Vol. 8

Lepton Physics at CERN and Frascati edited by N. Cabibbo Quantum Mechanics, High Energy Physics and Accelerators — Selected Papers of J. S. Bell (with Commentary) edited by M. Bell, K. Gottfried and M. Veltman

Vol. 9

Vol. 10 How We Learn; How We Remember: Toward an Understanding of Brain and Neural Systems — Selected Papers of Leon N. Cooper edited by L N. Cooper Vol. 11 30 Years of the Landau Institute — Selected Papers edited by I. M. Khalatnikov and V. P. Mineev Vol. 12 Sir Nevill Mott — 65 Years in Physics edited by N. Mott and A. S. Alexandrov Vol. 13 Broken Symmetry — Selected Papers of Y. Nambu edited by T. Eguchi and K. Nishijima Vol. 14 Reflections on Experimental Science edited by M. L Perl Vol. 15 Encounters in Magnetic Resonances — Selected Papers of Nicolaas Bloembergen (with Commentary) edited by N. Bloembergen Vol. 16 Encounters in Nonlinear Optics — Selected Papers of Nicolaas Bloembergen (with Commentary) edited by N. Bloembergen Vol. 17 The Collected Works of Lars Onsager (with Commentary) edited by P. C. Hemmer, H. Holden and S. K. Ratkje Vol. 18 Selected Works of Hans A. Bethe (with Commentary) edited by Hans A. Bethe Vol. 19 Selected Scientific Papers of Sir Rudolf Peierls (with Commentary) edited by R. H. Dalitz and R. Peierls Vol. 20 The Origin of the Third Family edited by O. Barnabei, L Maiani, R. A. Ricci and F. R. Monaco

Vol. 21 Spectroscopy with Coherent Radiation — Selected Papers of Norman F. Ramsey (with Commentary) edited by N. F. Ramsey Vol. 22 A Quest for Symmetry — Selected Works of Bunji Sakita edited by K. Kikkawa, M. Virasoro and S. R. Wadia Vol. 23 Selected Papers of Kun Huang (with Commentary) edited by B.-F. Zhu Vol. 24 Subnuclear Physics — The First 50 Years: Highlights from Erice to ELN A. Zichichi; edited by O. Barnabei, P. Pupillo and F. Roversi Monaco Vol. 25 The Creation of Quantum Chromodynamics and the Effective Energy V. N. Gribov, G. 't Hooft, G. Veneziano and V. F. Weisskopf; edited by L N. Lipatov Vol. 26 A Quantum Legacy — Seminal Papers of Julian Schwinger edited by K. A. Milton Vol. 27 Selected Papers of Richard Feynman (with Commentary) edited by L M. Brown Vol. 28 The Legacy of Leon Van Hove edited by A. Giovannini Vol. 29 Selected Works of Emil Wolf (with Commentary) edited by E. Wolf

Forthcoming In Conclusion — A Collection of Summary Talks in High Energy Physics edited by J. D. Bjorken Formation and Evolution of Black Holes in the Galaxy edited by H. A. Bethe, G. E. Brown and C.-H. Lee Selected Papers of Kenneth G. Wilson edited by M. E. Peskin and B. Baaquie

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World Scientific Series in 20th Century Physics - Vol. 30

Selected Papers of

ROBERT CHRIEFFER In Celebration of His 70th Birthday

Editors

N. E. Bonesteel L. P. Gor'kov National High Magnetic Field Laboratory, Florida

V f e World Scientific lflb

New Jersey • London • Singapore • Hong Kong Si,

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

The editors and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to include the selected reprints found in this volume: American Institute of Physics (J. Appl. Phys., J. Vac. Sci. TechnoL, Sov. J. Low Temp. Phys.); American Physical Society (Phys. Rev., Phys. Rev. B, Phys. Rev. Lett.); Elsevier Science Publishers (Nucl. Phys., Nucl. Phys. B, Synthetic Metals, Physica B, Phys. Lett., Physica Q ; Plenum Publishing Co. (/. Low Temp. Phys.); National Academy of Sciences USA (Proc. Natl. Acad. Sci. USA); Royal Swedish Academy of Sciences (Physica Scripta); Societa Italiana di Fisica (II Nuovo Cimento A).

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

SELECTED PAPERS OF J. ROBERT SCHRIEFFER In Celebration of his 70th Birthday Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN ISBN

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Printed in Singapore by World Scientific Printers (S) Pte Ltd

Vll

PREFACE In October of 2001 a Symposium was held in Santa Fe, New Mexico in celebration of Bob Schrieffer's 70th birthday. All who were lucky enough to attend enjoyed a wonderful day of seminars on a wide range of topics — from high-Tc superconductivity to quark pairing in high-density QCD — all of which have felt the influence of Bob's ideas. It was particularly nice that day to see so many of Bob's former students and postdocs gathered to honor someone that they viewed as both a mentor and a friend. At the time of this Symposium, it was decided at the National High Magnetic Field Laboratory to publish a collection of some of Bob's papers. The result is the present volume. In preparing this volume, we asked a number of Bob's friends and colleagues to write introductory essays to put his work in a broader context. Not surprisingly, everyone we contacted readily agreed, and we are extremely grateful to Sasha Alexandrov, Jim Davenport, Ted Einstein, Steve Kivelson, Doug Scalapino, Frank Wilczek and John Wilkins for their enthusiastic efforts. As we did not want to interfere with this friendly exchange, we reprint their contributions here, essentially unchanged. Each of these essays reflects the personality of its author and, collected together, they bring about an impressive picture of the great impact on modern physics of Bob's scientific achievements. Throughout his scientific career, Bob has striven to reduce problems to their essence — always emphasizing physical thinking over formalism. That is why it is helpful so often to turn back to the original publications in order to better understand the physical ideas, rather than reading about them in digested form in textbooks. The papers collected here at least partially reflect Bob's wonderful approach to physics and we are confident that future generations of students will find this volume useful. We are grateful to Mary Layne, Ashley Remer and Alice Hobbs, as well as the editors at World Scientific, for their invaluable assistance throughout the preparation of this volume. Finally, and most importantly, we would like to take this opportunity to say in print what a great pleasure and privilege it has been to have Bob Schrieffer as a colleague and friend over the years.

National High Magnetic Field Laboratory, Tallahassee, Florida, September 2002 N. E. Bonesteel

L. P. Gor'kov

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Contents Preface

vii

I Superconductivity

1

The BCS Theory of Superconductivity Douglas J. Scalapino

3

Microscopic Theory of Superconductivity, (with J. Bardeen and L. N. Cooper), Phys. Rev. 106, 162 (1957). Theory of Superconductivity, (with J. Bardeen and L. N. Cooper), Phys. Rev. 108, 1175 (1957). Theory of the Meissner Effect, (with D. Pines), Phys. Rev. Lett. 1, 407 (1958). Excitons and Plasmons in Superconductors, (with A. Bardasis), Phys. Rev. 121, 1050 (1961). Calculation of the Quasiparticle Recombination Time in a Superconductor, (with D. M. Ginsberg), Phys. Rev. Lett. 8, 207 (1962). The Effect of Collective Excitations on the Single Particle Spectra of Nuclei, Nucl. Phys. 35, 363 (1962). Effective Tunneling Density of States in Superconductors, (with D. J. Scalapino and J. W. Wilkins), Phys. Rev. Lett. 10, 336 (1963). Strong-Coupling Superconductivity, (with D. J. Scalapino and J. W. Wilkins), Phys. Rev. 148, 263 (1966). Effect of Ferromagnetic Spin Correlations on Superconductivity, (with N. F. Berk), Phys. Rev. Lett. 17, 433 (1966). On the Description of Nearly Ferromagnetic Fermion Systems, (with N. F. Berk), Phys. Lett. 24A, 604 (1967). The Sign of the Interference Current Component in Superconducting Tunnel Junctions, (with A. B. Zorin, I. O. Kulik and K. K. Likharev), Sov. J. Low Temp. Phys. 5, 537 (1979). Rotating Superfluidity in Nuclei, (with G. F. Bertsch and R. A. Broglia), // Nuovo Cimento 100A, 283 (1988).

12 15 45 47 60

62 69

73 90

93

96

106

John Bardeen, Physica C 185 (1991). Spectral Properties of Quasiparticle Excitations Induced by Magnetic Moments in Superconductors, (with M. I. Salkola and A. V. Balatsky), Phys. Rev. B 55, 12648 (1997). Intrinsic Limits on the Q and Intermodulation of Low Power High Temperature Superconducting Microstrip Resonators, (with R. B. Hammond, E. R. Soares, B. A. Willemsen, T. Dahm and D. J. Scalapino), J. Appl. Phys. 84, 5662 (1998).

110 113

127

II Solitons and Fractional Quantum Numbers

133

Some Basic Aspects of Fractional Quantum Numbers Frank Wilczek

135

Dynamics and Statistical Mechanics of a One-Dimensional Model Hamiltonian for Structural Phase Transitions, (with J. A. Krumhansl), Phys. Rev. B 11, 3535 (1975). Brownian Motion of a Domain Wall and the Diffusion Constants, (with Y. Wada), Phys. Rev. B 18, 3897 (1978). Soliton Excitations in Polyacetylene, (with W. P. Su and A. J. Heeger), Phys. Rev. B 22, 2099 (1980). Soliton Dynamics in Polyacetylene, (with W. P. Su), Proc. Natl. Acad. Sci. USA 77, 5626 (1980). Fractionally Charged Excitations in Charge-Density-Wave Systems with Commensurability 3, (with W. P. Su), Phys. Rev. Lett. 46, 738 (1981). Solitons with Fermion Number 1/2 in Condensed Matter and Relativistic Field Theories, (with R. Jackiw), Nucl. Phys. B 190, 253 (1981). Fractional Charge, a Sharp Quantum Observable, (with S. Kivelson), Phys. Rev. B 25, 6447 (1982). Solitons in Superfluid 3He-A: Bound States on Domain Walls, (with T. L. Ho, J. R. Fulco and F. Wilczek), Phys. Rev. Lett. 52, 1524 (1984). Collective Coordinate Description of Soliton Dynamics in Trans-Polyacetylene-Like Systems, (with S. Jeyadev), Synthetic Metals 9, 451 (1984). Statistical Mechanics of Anyons, (with D. P. Arovas, F. Wilczek and A. Zee), Nucl. Phys. B 251, 117 (1985).

153

164

180 193 197

201

214 219

223

238

Lattice Relaxation Effects on the Midgap Absorption Edge in Trans-Polyacetylene, (with S. Jeyadev), Physica Scripta 32, 372 (1985). Shape of Solitons in Classically Forbidden States: "Lorentz Expansion", (with F. Guinea and R. E. Peierls), Physica Scripta 33, 282 (1986). Topological Excitations of One-Dimensional Correlated Electron Systems, (with M. I. Salkola), Phys. Rev. Lett. 82, 1752 (1999). Fractional Electrons in Liquid Helium? (with R. Jackiw and C. Rebbe), J. Low Temp. Phys. 122, 587 (2001).

248

253

255

259

III Quantum Hall Effect

263

Statistical Phases and the Fractional Quantum Hall Effect Steven A. Kivelson

265

Fractional Statistics and the Quantum Hall Effect, (with D. Arovas and F. Wilczek), Phys. Rev. Lett. 53, 722 (1984). Cooperative-Ring-Exchange Theory of the Fractional Quantized Hall Effect, (with S. Kivelson, C. Kallin and D. P. Arovas), Phys. Rev. Lett. 56, 873 (1986). Cooperative Ring Exchange and the Fractional Quantum Hall Effect, (with S. Kivelson, C. Kallin and D. P. Arovas), Phys. Rev. B 36, 1620 (1987).

270

272

276

IV Surfaces

303

Schrieffer's Contributions to Surface Physics Theodore L. Einstein and James W. Davenport

305

Effective Carrier Mobility in Surface-Space Charge Layers, Phys. Rev. 97, 641 (1955). Theory of Chemisorption, J. Vac. Sci. Technol. 9, 561 (1972). Indirect Interaction between Adatoms on a Tight-Binding Solid, (with T. L. Einstein), Phys. Rev. B 7, 3629 (1973). Errata for Phys. Rev. B 7, 3629 (1973). Theory of Vibrationally Inelastic Electron Scattering from Oriented Molecules, (with J. W. Davenport and W. Ho), Phys. Rev. B 17, 3115 (1978).

311 317 325

345 346

V Magnetism and Magnetic Impurities

359

The Schrieffer-Wolff Transformation John W. Wilkins

361

Localized Magnetic Moments in Dilute Metallic Alloys: Correlation Effects, (with D. C. Mattis), Phys. Rev. 140, 1412 (1965). Relation Between the Anderson and Kondo Hamiltonians, (with P. A. Wolff), Phys. Rev. 149, 491 (1966). Exchange Interaction in Alloys with Cerium Impurities, (with B. Coqblin), Phys. Rev. 185, 847 (1969). Theory of Itinerant Ferromagnets Exhibiting Localized-Moment Behavior Above the Curie Point, (with S. Q. Wang and W. E. Evenson), Phys. Rev. Lett. 23, 92 (1969). Theory of Itinerant Ferromagnets with Localized-Moment Characteristics: Two-Center Coupling in the Functional-Integral Scheme, (with W. E. Evenson and S. Q. Wang), Phys. Rev. B 2, 2604 (1970). New Approach to the Theory of Itinerant Electron Ferromagnets with Local-Moment Characteristics, (with W. E. Evenson and S. Q. Wang), J. Appl. Phys. 41, 1199 (1970). Generalized Ruderman-Kittel-Kasuya-Yosida Theory of Oscillatory Exchange Coupling in Magnetic Multilayers, (with F. Herman), Phys. Rev. B 46, 5806 (1992).

363

371 373 380

384

387

393

VI Electrons and Phonons

397

Schrieffer's Work on the Theory of Electrons and Phonons Alexandre S. Alexandrov

399

Collective Behavior in Solid-State Plasmas, (with D. Pines), Phys. Rev. 124, 1387 (1961). Coupled Electron-Phonon System, (with S. Engelsberg), Phys. Rev. 131, 993 (1963). Radiative Corrections to the Long-Wavelength Optical-Mode Spectrum of the Electron-Phonon Model: Absence of Mode-Splitting Effects and Hardening of the Mode, (with A. S. Alexandrov), Phys. Rev. B 56, 13731 (1997).

404 418 434

VII High-T c Superconductivity

437

Schrieffer's Papers on High-T c Superconductivity Shou-Cheng Zhang

439

Pairing-Bag Excitations in Small-Coherence-Length Superconductors, (with A. R. Bishop, P. S. Lomdahl and S. A. Trugman), Phys. Rev. Lett. 61, 2709 (1988). Dynamic Spin Fluctuations and the Bag Mechanism of High-Tc Superconductivity, (with X. G. Wen and S. C. Zhang), Phys. Rev. B 39, 11663 (1989). Pseudo Gaps and Spin Bags, (with A. Kampf), Physica B 163, 267 (1990). Pseudogap and the Spin-Bag Approach to High-Tc Superconductivity, (with A. Kampf), Phys. Rev. B 41, 6399 (1990). Constructing Quasiparticle Operators in Strongly Correlated Systems, (with E. Dagotto), Phys. Rev. B 43, 8705 (1991). Properties of Odd-Gap Superconductors, (with E. Abrahams, A. Balatsky and D. J. Scalapino), Phys. Rev. B 52, 1271 (1995). Ward's Identity and the Suppression of Spin Fluctuation Superconductivity, J. Low Temp. Phys. 99, 397 (1995). Momentum, Temperature, and Doping Dependence of Photoemission Lineshape and Implications for the Nature of the Pairing Potential in High-Tc Superconducting Materials, (with Z.-X. Shen), Phys. Rev. Lett. 78, 1771 (1997). Unusual States of Inhomogeneous dx2_yi +idxy Superconductors, (with M. I. Salkola), Phys. Rev. B 58, 5952 (1998). Collective Excitations in High-Temperature Superconductors, (with M. I. Salkola), Phys. Rev. B 58, 5944 (1998). de Haas-van Alphen Effect in Anisotropic Superconductors in Magnetic Fields Well Below Hc2, (with L. P. Gor'kov), Phys. Rev. Lett. 80, 3360 (1998).

442

446

462 466

476

480

488 494

498

502

506

I Superconductivity

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3 T H E BCS THEORY OF SUPERCONDUCTIVITY Douglas J. Scalapino Department of Physics, University of California Santa Barbara, CA 93106-9530, USA

1.1

Introduction

Bob Schrieffer began to work on superconductivity as a graduate student at the University of Illinois. John Bardeen was his thesis advisor and Leon Cooper was a postdoctoral fellow at Illinois. As is well known, the seminal work [l] of these three culminated in the solution of one of the major scientific problems of the first half of the 20 th century. Furthermore, the BCS theory published in 1957 provided a touchstone for many of the most important physical insights of the last half of that century and continues to do so today. Here in this brief commentary, I'll begin by discussing some of the background ideas that were present prior to BCS and comment on the initial reception of their new ideas. Much of this, except for the Stanford discussion, represents second-hand knowledge for which I am grateful to many colleagues. I would like to particularly thank P.W. Anderson, T. Geballe, C. Slichter, and Y. Nambu for sharing their insights and writings [2; 3]. Following this, the commentary will shift to Bob's work on the so-called strong-coupling superconductors, in which I was fortunate to participate and to the work of Bob and his student Norm Berk on spin fluctuations. There are also a number of other contributions to superconductivity collected in this section. These range from studies of collective excitations in superconductors and nuclei, to quasiparticle recombination lifetimes, the possibility of an odd-frequency gap, and the sign of the Josephson quasiparticle-pair interference tunneling current.

1.2

The BCS Theory

At the 1956 Seattle International Congress on Theoretical Physics, Richard Feynman presented a paper on "Superfluidity and Superconductivity." He noted that having worked on superconductivity he had now come to the same conclusion as Casimir, whom he quoted as saying: "There is only one way to go about working this out. It is simply to guess the quality of the answer." The last sentence of Feynman's review concludes: "The only reason that we cannot do this problem of superconductivity is that we haven't got enough imagination" [4]. Bob Schrieffer was at this conference and Leon Cooper's Letter to the Editor on "Bound Electron Pairs in a Degenerate Fermi Gas" [5] has a receipt date of September 21, 1956, the day the conference ended. But it wasn't until January of the next year, while attending a many-body theory meeting at the Stevens Institute of Technology that this young physics

4

graduate student had enough imagination to "guess the quality of the solution." As Bob was later to describe it, "While attending that meeting it occurred to me that because of the strong overlap of pairs perhaps a statistical approach analogous to a type of mean field would be appropriate to the problem. Thinking back to a paper by Sin-itiro Tomonaga that described the pion cloud around a static nucleon [6], I tried a ground-state wave function \ipo) written as

l^o) = I I {Uk + Vk CU c-fci) l°)' k

where cL. is the creation operator for an electron with momentum k and spin up, |0) is the vacuum state, and the amplitudes Uk and Vk are to be determined" [7]. Some three weeks after this breakthrough, a letter [8] on the "Microscopic Theory of Superconductivity" was submitted and by July the BCS paper [l] entitled "Theory of Superconductivity" was finished. While one doesn't explain the intuitive guess of the BCS wave function, beyond Bob's own description, it is interesting to look at the background of ideas that were present at that time. Of course with hindsight, one can pick out the significant ideas and neglect the many unproductive paths. Doing this at the time, however, was clearly much more difficult. In F. London's 1950 book [9] he had proposed that a superconductor is a "quantum structure on a macroscopic scale" with a "kind of solidification or condensation of the average momentum distribution." This picture of a condensation in momentum space leading to a wave function with long-range phase coherence was to prove insightful. Although the phenomenological Ginzburg-Landau theory [10] had also been published in 1950, it had little influence at that time in the West. After this, Abrikosov [ll] used it in a brilliant piece of work on the vortex structure of the mixed state, but the publication of his results did not appear until '57. The discovery in 1950 of the isotope effect [12] showing that Tc ~ M;~^ with a ~ 1/2 for Hg had focused attention on the importance of the electron-phonon interaction. Frohlich [13] had proposed that the virtual exchange of phonons could give rise to an attractive interaction between electrons near the Fermi surface. Bardeen and Pines [14] had extended these ideas to include the effect of the electron-electron Coulomb interaction on the electron-ion system. Finally, Cooper had found that two electrons interacting above a static Fermi sea would form a bound state no matter how weak their attraction [5]. At some point, the Illinois group decided to focus on a reduced Hamiltonian which treated the attractive interaction between only the most unstable zero center of mass pairs within the Debye energy of the Fermi surface. The wave function Bob proposed turned out to be the exact ground state for this reduced Hamiltonian as the system becomes large. Moreover, the wave function provided a remarkably good description of the superconducting properties of real materials. Looking back at this after its discovery, one knows that it was in fact the troubling strong overlapping of the pairs that make a mean-field approach so successful in treating the pairing correlations. Following the BCS publications, reactions to these new ideas varied. There were, of course, numerous results that agreed with experiment such as the magnitudes and tem-

5

perature dependences of the specific heat, the critical magnetic field, and the penetration depth. Moreover, there were the experimental consequences of the quasiparticle coherence factors which implied that the NMR spin-lattice relaxation rate should initially increase below Tc while the ultrasonic attenuation would fall continuously as the gap opened. The former experiment was being carried out by Hebel and Slichter [15] at the very time that BCS were doing their work and Morse and Bohm [16] also used the BCS prediction to analyze their ultrasonic data. The remarkable agreement of the BCS results with a variety of experiments led to a rapid acceptance of the new theory by the majority of those doing experiments. On the theoretical side, the reception was more mixed. Some leading theorists clearly recognized the significance of the BCS work while others had strong reservations. Years later, after Bob had come to UCSB, Feynman came to visit. At lunch he mentioned that he had been working on superconductivity when he received a copy of the BCS paper. He said that when he read the abstract he knew that they had solved it, and he stopped working on it. Then he added, "But I couldn't bring myself to read the rest of their paper for two years! It was such a beautiful problem." However, there were those who had their reservations. T. Geballe remembers that in the summer of '57 when the BCS paper was circulating, some well-known theorists expressed strong doubts about the theory because of the question of gauge invariance. On the other hand, at Bell Labs, Anderson focused on how to address this question. In the Soviet Union, Bogoliubov applied a canonical transformation he had previously introduced for the dilute Bose gas to rederive the BCS results [17]. Khalatnikov and Abrikosov also used this approach to calculate a number of superconducting properties, graciously publishing their work as a review after receiving the long BCS paper [18]. Gor'kov introduced his anomalous Green's function F and showed how the ideas of quantum field theory provided a natural framework for the new theory [19]. From this, he went on to obtain the Ginzburg-Landau equations and identified the order parameter as the pair field [20]. However, there were other theorists with whom the BCS theory was not so well received. I was a graduate student at Stanford and grading for a course taught by Felix Bloch. From our few interchanges about superconductivity, it was clear to me that he had serious doubts verging on actual hostility to these new ideas. He had invited Gregor Wentzel from the University of Chicago to give a series of lectures on superconductivity and told me that I would get a better picture of the BCS work by attending these. I did, but the impression that Wentzel left was that the lack of gauge invariance in the BCS analysis cast doubt not only on their derivation of the Meissner effect but moreover raised serious questions as to whether their theory could be meaningfully compared with experiment at all. In these lectures, Wentzel went on to describe his own "gauge invariant" calculations of the Meissner effect [21]. While most of this went over my head, I remember being puzzled by the fact that his result in the long wave length limit differed from London's. Several months after Wentzel's publication, Pines and Schrieffer, in a PRL included in this volume [22], pointed out the problem with Wentzel's derivation. Although I can only guess, I doubt

6 that this mollified Wentzel and as late as 1965 at his APS Presidential retirement address, Bloch [23] noted that "theorists are still looking for a satisfactory microscopic theory." Fortunately, work by Anderson [24], Bogoliubov [25], Nambu [26], and Rickayzen [27] showed that if one properly included the collective modes, gauge invariance was maintained. Furthermore, from this work flowed many new concepts that would enrich all of physics involving the idea of broken symmetry, the Anderson-Higgs phenomena [24; 28], and the Nambu-Goldstone bosons [29; 30]. Indeed, the BCS theory has had a remarkable impact on nuclear physics and high energy physics as well as on condensed matter physics. One of the most recent applications has been to quark pairing in high-density QCD [3l].

1.3

Strong Coupling Superconductors

In a 1961 review article [32] entitled Recent Developments in Superconductivity, Bardeen and Schrieffer concluded that "the agreement between experiment and predictions of the microscopic theory based on the simplified (BCS) model is in general much better than might have been expected." It was as we know, a truly remarkable theory, accounting not only for the basic thermodynamic properties, but in addition for detailed dynamic properties. For example, it could account for both the Hebel-Slichter peak in the nuclear relaxation time T-j- and the monotonic decrease in the ultrasonic attenuation below Tc. Nevertheless, as they noted, "an extension of the theory is required to account in detail for the properties of superconductors with large electron-phonon interactions and low Debye temperatures for which the weak coupling approximation is unsatisfactory. This applies particularly to Pb and Hg . . . ." In the fall of 1962, Bob joined the faculty of the University of Pennsylvania. As a postdoc I was fortunate to join Bob and a graduate student of his from Illinois, John Wilkins, in a project aimed at understanding the tunneling I(V) characteristics of Pb. Earlier that year Giaever, Hart, and Megerle [33] at GE and Rowell, Chynoweth, and Phillips [34] at Bell had found structure in the tunnel I(V) characteristic of Pb tunnel functions. Two small bumps in dl/dV at energies of order the Debye temperature had been interpreted in terms of electron-phonon processes. Bob felt that an understanding of the anomalies in the I(V) characteristics could provide detailed information on the nature of the microscopic pairing interaction. Pb has a low Debye temperature and a large electron-phonon coupling as judged from its resistivity. It was also one of the "bad actors" as Bob called those materials whose thermodynamic properties showed deviations from the law of corresponding states and hence the BCS predictions. This was both good news and bad news. The good news was that the coupling strength was sufficient that one could see structure in dl/dV. The bad news, of course, was that one would need to solve a strong-coupling theory to adequately take account of the dynamic electron-phonon interaction. Eliashberg [35] and Nambu [26] had, in fact, developed a framework for treating such a system. This framework rested upon the existence of two small parameters, Tc/Tp and

7

UDebye/EF- The first assures the validity of the BCS mean-field theory, while the second implies that the electron-phonon vertex corrections are negligible, as shown by Migdal [36]. Nevertheless, at that time there were a variety of approaches to the tunneling problem and it wasn't clear how one should best proceed. In particular, we needed to understand what determined the I(V) characteristic. Certainly a density of states would be needed, but a naive extension of the BCS model allowing for an on-shell energy dependence of the gap gave a density of states de dE

N(E) N0

E - A(E) (E2 -

^

A2(E))2

Here iVo is the normal state density of states at the Fermi surface. This form for N(E) appeared reasonable for a pairing interaction parameterized in terms of an on-shell energy transfer (ep/ — e p ). However, the actual electron-phonon interaction is short range in space due to the Fermi-Thomas screening and retarded in time. Hence it has negligible momentum dependence and a strong frequency dependence. Thus, the basic question regarding how to calculate dl/dV for a strongly-interacting superconductor remained. Several weeks passed during which we tried to understand the intricacies of tunneling. Then one day Bob came in, clearly pleased, and explained that if we just took the point of view of Cohen, Falicov, and Phillips [37], who characterize the tunneling by an effective one-body Hamiltonian and used Fermi's Golden rule to calculate the transition probability per unit time for an electron to tunnel, we would find that dl_ dV

N(U) N

N0

= ne_

u y/cu

2

-

A2{OJ)

ui=eV

Although this is well known now, it struck me at the time and I remain impressed by its elegant simplicity. The structure in dl/dV ~ N(eV) was telling us about A(UJ = eV) and this in turn reflected the underlying dynamics of the electron-phonon interaction. Using the inelastic neutron scattering data of Brockhouse et al. [38] for Pb to parameterize the electron-phonon interaction, we turned to the solution of the Eliashberg-Nambu equations for the gap. At that time, the solution of these coupled non-linear integral equations looked daunting. However, Bob flew out to Thompson-Ramo-Wooldridge (TRW) in Canoga Park, CA to use a recently developed "online" computing system and in three days returned with plots of the real and imaginary parts of A (a;). It was remarkable! It was also strange because the plots were on huge three-foot square construction paper and looked like the blue print for some exotic southern California swimming pool. However, when we read the numbers off them with a ruler and plotted N{UJ) on regular graph paper [39], we were amazed at how well it reproduced the recent tunneling data of Rowell [40]. Following this, McMillan developed a beautiful numerical method for inverting the tunneling dl/dV data to directly calculate the basic quantities a2F(uj) and fi* which determined the pairing interaction. He and John Rowell used this to obtain a quantitative description of the microscopic pairing interaction in a wide variety of materials [41]. Once a2F(u) and H* were known, the thermodynamic as well as the frequency and temperature dependence

8

of the transport properties could be determined with a high degree of accuracy [42]. In addition, the phonon density of states obtained from inelastic neutron scattering experiments and the a2F(u>) spectrum obtained from tunneling were found to be in excellent agreement for a number of materials, providing further proof of a phonon-mediated pairing mechanism.

1.4

Spin Fluctuations and Superconductivity

Bob's work on the interplay of magnetic spin fluctuations and superconductivity had an interesting history and, like so many of his ideas, a long reach. Beginning as a way to understand the suppression of Tc in some of the transition metals such as Pd, it was developed into a framework for describing nearly-ferromagnetic metals. Its reach has extended to provide a mechanism for p-wave pairing in 3He and more recently it has been suggested that antiferromagnetic spin-fluctuations may provide a mechanism for d-wave pairing in some heavy electron metals, organic superconductors, and possibly the high Tc cuprates [43]. As is well known, the latter is not without controversy and in a paper included in this volume, Bob discusses a difficulty of this approach when one gets too close to the antiferromagnetically-ordered phase [44]. In their 1961 review [32], Bardeen and Schrieffer also noted that "One of the most important questions which is still open is the criterion for superconductivity, and in particular just how the repulsive Coulomb interactions counteract the attractive electron-phonon interactions to prevent superconductivity from occurring." In the mid-sixties, Bob took up this question focusing upon the dramatic decrease in the superconducting transition temperatures that had been found to occur for metals near the end of the transition metal series. In this region, ferromagnetic exchange interactions were thought to be responsible for the so-called Stoner enhancement of the observed paramagnetic susceptibilities and based upon a phenomenological exchange interaction, Doniach [45] had suggested that these interactions could also act to suppress singlet pairing. In 1966 Norm Berk, a graduate student, and Bob published a paper [46] describing the suppression of Tc due to ferromagnetic spin fluctuations. In this paper they showed how the exchange of low-energy ferromagnetic spin fluctuations, called paramagnons, lead to an effective pairing interaction which was repulsive in the singlet channel. Within a diagrammatic RPA framework they wrote down a particle-hole t-matrix which described a pairing interaction mediated by the exchange of spin fluctuations. This was a pairing interaction mediated by the same fermions that were pairing. In this treatment, they included on the same footing phonon exchange and spin-fluctuation exchange, showing that while their contributions to the effective mass enhancement entered additively Xph + ^SF, they opposed each other \ph — XSF in the singlet pairing channel. While there were arguments regarding the magnitude of the electron mass enhancement and the Tc suppression due to paramagnons, this work marked a new way of thinking about pairing. Here, the electrons that were being paired were the same electrons involved in

9 generating the pairing interaction. This means that contrary to the case of phonon mediated pairing, where the modification of the interaction when the system becomes superconducting is negligible, there should be a significant feedback effect for the case of a spin-fluctuation mediated interaction. Following this work, in the 70's and 80's various researchers used the Berk-Schrieffer interaction to discuss the possibility that metals which had significantly smaller Stoner enhancement of their paramagnetic response might nevertheless have their Tc values limited by paramagnon exchange. For example, Herman Rietschel and co-workers [47] noted that state-of-the-art band structure and frozen phonon calculations of the electron-phonon coupling implied that Nb and V should have superconducting transition temperatures as high as 18K. They argued that while paramagnon exchange did not suppress the T c 's of Nb and V to zero, it could provide an explanation for why they were well below 18K. In 1971, Layzer and Fay [48] suggested that the Berk-Schrieffer spin-fluctuation exchange interaction could provide a p-wave pairing mechanism for 3He implementing an earlier suggestion for odd-channel pairing by Emery [49]. After the discovery of superfluid 3 He in 1973, Anderson and Brinkman [50] showed how the feedback in the superconducting state modified the paramagnetic spin fluctuation pairing interaction, favoring the formation of the A phase over the B phase. More recently, the phase diagrams of some cerium-based heavy electron materials [51] as well as the organic «- (BEDT-TTF) 2 X material [52] exhibit antiferromagnetic and non-s-wave superconducting regions in close proximity, suggesting the relevance of spin-fluctuation mediated pairing. Thus, it is clear that the Berk-Schrieffer work continues to generate new ideas in condensed matter physics.

1.5

Conclusion

Bob's scientific papers collected in this volume represent one facet of his many contributions to science. What cannot be collected and bound so easily is a proper accounting of his scientific leadership and the friendship and support he has given to so many over the years. With respect to his leadership, Ed David once noted that Bob was a "statesman of science without personal agenda, a reasonable, responsible, honest man whose ability to lead derives from a universal acceptance and belief in his ability and personal integrity" I would only add to this that Bob's determination, bordering on stubbornness and his humor when things become difficult are legendary to those who know him.

References [1] [2] [3] [4] 5

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).* P. W. Anderson, Superfluidity and Superconductivity: the Past Half-century (2002). Y. Nambu, Butsuri (Phys. Soc. Jap.) 57, 2 (2002). R. P. Feynman, Rev. Mod. Phys. 29, 205 (1957). L. N. Cooper, Phys. Rev. 104, 1189 (1956).

10 [6] [7] [8] [9] 10] 11] 12] 13] 14] 15] 16] 17] 18] 19] 20] 21] 22] 23] 24] 25] 26] 27] 28] 29] 30] 31] 32] 33] 34] 35] 36] 37] 38] [39]

S. Tomonaga, Prog. Theor. Phys. (Kyoto) 2, 6 (1947). J. R. Schrieffer, Phys. Today 45, 46 (1992). J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 162 (1957).* F. London, Superfluids, John Wiley & Sons, NY (1950). V. L. Ginzburg and L. D. Landau, J. Exptl. Theor. Phys. (USSR) 20, 1064 (1950). A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957). E. Maxwell, Phys. Rev. 78, 477 (1950); B. Serin, C. A. Reynolds, and L. B. Nesbitt, Phys. Rev. 78, 813 (1950). H. Frohlich, Phys. Rev. 79, 845 (1950). J. Bardeen and D. Pines, Phys. Rev. 99, 1104 (1955). L. C. Hebel and C. P. Slichter, Phys. Rev. 107, 901 (1957). R. W. Morse and H. V. Bohm, Phys. Rev. 108, 1094 (1957). N. N. Bogoliubov, J. Exptl. Theor. Phys. (USSR) 34, 58 (1958) [Sov. Phys. JETP 7, 41 (1958)]. I. M. Khalatnikov and A. A. Abrikosov, Phil. Mag. Sup. 8, 45 (1959). L. P. Gor'kov, J. Exptl. Theor. Phys. (USSR) 34, 735 (1958) [Sov. Phys. JETP 7, 505 (1958)]. L. P. Gor'kov, Zh. Eksp. i. Teor. Fiz. (USSR) 36, 1918 (1959) [Sov. Phys. JETP 9, 1364 (1959)]. G. Wentzel, Phys. Rev. I l l , 1488 (1958). D. Pines and J. R. Schrieffer, Phys. Rev. Lett. 1, 407 (1958).* F. Bloch, Phys. Today, 19, 27 (1966). P. W. Anderson, Phys. Rev. 112, 1900 (1958). N. N. Bogoliubov, V. V. Tolmachev, and D. V. Shirkov, A New Method in the Theory of Superconductivity, Consultants Bureau, Inc., NY (1959). Y. Nambu, Phys. Rev. 117, 648 (1960). G. Rickayzen, Phys. Rev. 115, 795 (1959). P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964). Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961). J. Goldstone, JVuovo Cimento 19, 154 (1961). F. Wilczek The Recent Excitement in High-density QCD (1999); hep-ph/9908480. J. Bardeen and J. R. Schrieffer, Prog, in Low Temp. Phys. Ill, 170 (1961). I. Giaever, H. R. Hart, Jr., and K. Megerle, Phys. Rev. 126, 941 (1962). J. M. Rowell, A. G. Chynoweth, and J. C. Phillips, Phys. Rev. Lett. 9, 59 (1962). G. M. Eliashberg, Zh. Eksp. Teor. Fiz. 38, 966 (1960) [Sov. Phys. JETP 11, 696 (I960)]. A. B. Migdal, Zh.Eksp. Teor. Fiz. 34, 1438 (1958) [Sov. Phys. JETP 7, 996 (1958)]. M. H. Cohen, L. M. Falicov, and J. C. Phillips, Phys. Rev. Lett. 8, 316 (1962). B. N. Brockhouse, T. Arase, G. Cagrioti, K. R. Rao, and A. D. B. Woods, Phys. Rev. 28, 1099 (1962). J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Lett. 10 336 (1963)*;

11

Phys. Rev. 148, 263 (1966).* [40] J. M. Rowell, P. W. Anderson, and D. E. Thomas, Phys. Rev. Lett. 10, 334 (1963). [41] W. L. McMillan and J. M. Rowell, Superconductivity I, ed. by R. Parks, Marcel Dekker, NY, Chap. 11 (1969). [42] D. J. Scalapino, Superconductivity I, ed. by R. Parks, Marcel Dekker, NY, Chap. 10 (1969). [43] D. J. Scalapino, Phys. Rep. 250, 329 (1995). [44] J. R. Schrieffer, J. Low Temp. Phys. 99, 397 (1995).* [45] S. Doniach, Proceedings of the Manchester Many-Body Conference, Sept., 1964 (unpublished). [46] N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett. 17, 433 (1966).* [47] H. Rietschel and H. Winter Phys. Rev. B 22, 4284 (1980). [48] A. Layzer and D. Fay, Int. J. Mag. 1, 135 (1971). [49] V. J. Emery, Ann. Phys. (NY) 28, 1 (1964). [50] P. W. Anderson and W. F. Brinkman, Phys. Rev. Lett. 30, 1108 (1973). [51] N. D. Mathur et al., JVature (London) 394, 39 (1998). [52] S. Lefebvre et al., Phys. Rev. Lett. 85, 5420 (2000). (The symbol * indicates a paper reprinted in this volume.)

12 162

LETTERS

TO

THE

EDITOR

expressed in the form

Ht=

L

k.k'.»..' (£ k -E k .) 2 -(fo>) 2 X t k'-K, .'Ck', ,'C k+it, ,Ck, «+-ffcoul,

(1)

2

where |M«| is the matrix element for the electronphonon interaction for the phonon wave vector tc, calculated for the zero-point amplitude of the vibrations, the c's are creation and destruction operators for the electrons in the Bloch states specified by the wave vector k and spin s, and ZTcoui represents the screened Coulomb interaction. Early attempts 3 to construct a theory were based essentially on the self-energy of the electrons, although it was recognized that a true interaction between electrons probably played an essential role. These theories gave the isotope effect, but contained various difficulties, one of which was that the calculated energy difference between what was thought to represent normal and superconducting states was far too large. It is now believed that the self-energy occurs in the normal state, and results in a slight shift of the energies of the Bloch states and a renormalization of the matrix elements. The present theory is based on the fact that the phonon interaction is negative for \Ek—E^\i)2, where N(&i) is the density of states per unit energy a t the Fermi surface. The theories of Frohlich and Bardeen mentioned above were based largely on this p a r t of the energy. The observed energy differences between superconducting and normal states at T=0°K are much smaller, of the order of -N(Sr)(kTey or about 10~ 8 ev/atom. The present theory, based on the off-diagonal elements of Hi and the screened Coulomb interaction, gives energies of the correct order of magnitude. While the self-energy terms do depend to some extent on the distribution of electrons in k space, it is now believed that this part of the energy is substantially the same in the normal and superconducting phases. T h e self-energy terms are also nearly the same for all of the various excited normal state configurations which make up the superconducting wave functions. In a preliminary communication, 17 we gave as a criterion for the occurrence of a superconducting phase that for transitions such that Ae, the attractive Hi dominate the repulsive short-range screened Coulomb interaction between electrons, so as to give a n e t attraction. We showed how an attractive interaction of this sort can give rise to a cooperative many-particle state which is lower in energy than the normal state by an amount proportional to (/zw)2, consistent with the isotope effect. We have since extended the theory to higher temperatures, have shown that it gives both a second-order transition and a Meissner effect, and have calculated specific heats and penetration depths. In the theory, the normal state is described by the Bloch individual-particle model. The ground-state wave function of a superconductor is formed b y taking a linear combination of many low-lying normal state 8 W. Heisenberg, Two Lectures (Cambridge University Press, configurations in which the Bloch states are virtually Cambridge, 1948). »H. Koppe, Ergeb. exakt. Naturw. 23, 283 (1950); Z. Physik occupied in pairs of opposite spin and momentum. If 148, 135 (1957). the state kf is occupied in any configuration, — kj, is 10 E. Maxwell, Phys. Rev. 78, 477 (1950); Reynolds, Serin, also occupied. The average excitation energy of the Wright, and Nesbitt, Phys. Rev. 78, 487 (1950). 11 virtual pairs above the Fermi sea is of the order of kTc. H. Frohlich, Phys. Rev. 79, 845 (1950). »J. Bardeen, Phys. Rev. 79, 167 (1950); 80, 567 (1950); 81, Excited states of the superconductor are formed b y 829 (1951). 13 For a review of the early work, see J. Bardeen, Revs. Modern specifying occupation of certain Bloch states and b y Phys. 23, 261 (1951). using all of the rest to form a linear combination of »H, Frohlich, Proc. Roy. Soc. (London) A21S, 291 (1952). 16 S. Nakajima, Proceedings of the International Conference on Theoretical Physics, Kyoto and Tokyo, September, 1953 (Science " J . Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955). Council of Japan, Tokyo, 1954). " Bardeen, Cooper, and Schrieffer, Phys. Rev. 106, 162 (1957).

THEORY

OF

SUPERCONDUCTIVITY

virtual pair configurations. There is thus a one-to-one correspondence between excited states of the normal and superconducting phases. T h e theory yields an energy gap for excitation of individual electrons from the superconducting ground state of about the observed order of magnitude. The most important contribution to the interaction energy is given b y short- rather than long-wavelength phonons. Our wave functions for the suerconducting phase give a coherence of short-wavelength components of the density matrix which extend over large distances in real space, so as to take maximum advantage of the attractive part of the interaction. The coherence distance, of the order of Pippard's £o, can be estimated from uncertainty principle arguments. B ' 7 If intervals of the order of A & ~ ( £ 2 V ( § F ) & F ~ 1 0 4 c m - 1 are important in k space, wave functions in real space must extend over distances of a t least Ax~'l/A*~'l{>~ 4 cm. The fraction of the total number of electrons which have energies within kTc of the Fermi surface, so that they can interact effectively, is approximately &TV<Sj«"~10~4. T h e number of these in an interaction region of volume (A*)3 is of the order of 10 2 2 X(10- 4 ) 8 X10- 4 =10 6 . Thus our wave functions must describe coherence of large numbers of electrons. 18 In the absence of a satisfactory microscopic theory, there has been considerable development of phenomenological theories for both thermal and electromagnetic properties. Of the various two-fluid models used to describe the thermal properties, the first and best known is that of Gorter and Casimir, 19 which yields a parabolic critical field curve and an electronic specific heat varying as T3. In this, as well as in subsequent theories of thermal properties, it is assumed that all of the entropy of the electrons comes from excitations of individual particles from the ground state. I n recent years, there has been considerable experimental evidence 20 for an energy gap for such excitations, decreasing from ~3kTe a t T = 0 ° K to zero at T=Tc. Two-fluid models which yield an energy gap and an exponential specific heat curve at low temperatures have been discussed b y Ginsburg 21 and by Bernardes. 22 Koppe's 16 Our picture differs from that of Schafroth, Butler, and Blatt, Helv. Phys. Acta 30, 93 (1957), who suggest that pseudomolecules of pairs of electrons of opposite spin are formed. They show if the size of the pseudomolecules is less than the average distance between them, and if other conditions are fulfilled, the system has properties similar to that of a charged Bose-Einstein gas, including a Meissner effect and a critical temperature of condensation. Our pairs are not localized in this sense, and our transition is not analogous to a Bose-Einstein condensation. » C. J. Gorter and H. B. G. Casimir, Physik. Z. 35, 963 (1934); Z. 20techn. Physik 15, S39 (1934). For discussions of evidence for an energy gap, see Blevins, Gordy, and Fairbank, Phys. Rev. 100, 1215 (1955); Corak, Goodman, Satterthwaite, and Wexler, Phys. Rev. 102,656 (1956); W. S. Corak and C. B. Satterthwaite, Phys. Rev. 102, 662 (1956); R. E. Glover and M. Tinkham, Phys. Rev. 104, 844 (1956), and to51be published. W. L. Ginsburg, Fortschr. Physik 1, 101 (1953); also see reference 7. »N. Bernardes, Phys. Rev. 107, 354 (1957).

1177

theory may also be interpreted in terms of an energygap model. 7 Our theory yields an energy gap and specific heat curve consistent with the experimental observations. The best known of the phenomenological theories for the electromagnetic properties is that of F . and H . London. 23 With an appropriate choice of gauge for the vector potential, A, the London equation for the superconducting current density, j , may be written -cAj=A.

(1.1)

The London penetration depth is given b y : X i 2 =Ac 2 /4x.

(1.2)

F. London has pointed out t h a t (1.1) would follow from quantum theory if the superconducting wave functions are so rigid that they are not modified a t all by the application of a magnetic field. For an electron density n/cm 8 , this approach gives A=m/ne?. On the basis of empirical evidence, Pippard 6 has proposed a modification of the London equation in which the current density at a point is given by an integral of the vector potential over a region surrounding the point: j(r) =

(

dr',

(1.3)

where R = r — r ' . The "coherence distance," £0, is of the order of 10 - 4 cm in a pure metal. For a very slowly varying A, the Pippard expression reduces to the London form (1.1). The present theory indicates t h a t the Meissner effect is intimately related to the existence of an energy gap, and we are led to a theory similar to, although not quite the same as, t h a t proposed by Pippard. Our theoretical values for £0 are close to those derived empirically b y Pippard. We find that while the integrand is relatively independent of temperature, the coefficient in front of the integral (in effect A) varies with T in such a way as to account for the temperature variation of penetration depth. Our theory also accounts in a qualitative way for those aspects of superconductivity associated with infinite conductivity and a persistent current flowing in a ring. When there is a net current flow, the paired states (kit,k24r) have a net momentum k i + k 2 = q , where q is the same for all virtual pairs. For each value of q, there is a metastable state with a minimum in free energy and a unique current density. Scattering of individual electrons will n o t change the value of q common to virtual pair states, and so can only produce fluctuations about the current determined by q. Nearly all fluctuations will increase the free energy; only those which involve a majority of the electrons so as to change B An excellent account may be found in F. London, Superfluids (John Wiley and Sons, Inc., New York, 1954), Vol. 1.

1178

BARDEEN,

COOPER,

the common q can decrease the free energy. These latter are presumably extremely rare, so that the metastable current carrying state can persist indefinitely. 24 It has long been recognized that there is a law of corresponding states for superconductors. The various properties can be expressed approximatly in terms of a small number of parameters. If the ratio of the electronic specific heat a t T to that of the normal state at Tc, C,(T)/Cn(Tc), is plotted on a reduced temperature scale, t=T/Tc, most superconductors fall on nearly the same curve. There are two parameters involved: (1) the density of states in energy at the Fermi surface, N(Sr), determined from Cn(T) =yT and (2) one which depends on the phonon interaction, which can be estimated from Tc. A consequence of the similarity law is that yTc2/ VmHo2 (where V„ is the molar volume and Ho the critical field at r = 0 ° K ) is approximately the same for most superconductors. A third parameter, the average velocity, v„, of electrons at the Fermi surface, »o=«r-i|3S/ak|,

(1.4)

is required for penetration phenomena. As pointed out by Faber and Pippard, 2 5 this parameter is most conveniently determined from measurements of the anomalous skin effect in normal metals in the high-frequency limit. The expression, as given by Chambers 26 for the current density when the electric field varies over a mean free path, I, may be written in the form: «»iV(S,)»o / - R [ R - A ( r ' ) > - B "

The coefficient N(§F)VO has been determined empirically for tin and aluminum. Pippard based his Eq. (1.3) on Chambers' expression. London's coefficient, A, for r = 0 ° K may be expressed in the form: A-^^NiSfW. (1.6) Faber and Pippard suggest that if £o is written: $t=ahv9/kTc,

(1.7)

the dimensionless constant a has approximately the

AND

SCHRIEFFER

same value for all superconductors and t h e y find it equal to about 0.15 for Sn and Al.27 Our theory is based on a rather idealized model in which anisotropic effects are neglected. I t contains three parameters, two corresponding to N(SF) and v0, and one dependent on the electron-phonon interaction which determines Te. The model appears to fit the law of corresponding states about as well as real metals do ( ~ 1 0 % for most properties). We find a relation corresponding to (1.7) with a = 0 . 1 8 . I t thus appears t h a t superconducting properties are not dependent on the details of the band structure but only upon the gross features. Section I I is concerned with the nature of the ground state and the energy of excited states near r = 0 ° K , Sec. I l l with excited states and thermal properties, Sec. IV with calculation of matrix elements for application to perturbation theory expansions and transition probabilities and Sec. V with electrodynamic and penetration phenomenon. Some of the computational details are given in Appendices. We give a fairly complete account of the equilibrium properties of our model, but nothing on transport or boundary effects. Starting from matrix elements of single-particle scattering operators as given in Sec. IV, it should not be difficult to determine transport properties in the superconducting state from the corresponding properties of the normal state. II. THE GROUND STATE The interaction which produces the energy difference between the normal and superconducting phases in our theory arises from the virtual exchange of phonons and the screened Coulomb repulsion between electrons. Other interactions, such as those giving rise to the single-particle self-energies, are thought to be essentially the same in both states, their effects thus cancelling in the energy difference. The problem is therefore one of calculating the ground state and excited states of a dense system of fermions interacting via two-body potentials. The Hamiltonian for the fermion system is most conveniently expressed in terms of creation and annihilation operators, based on the renormalized Bloch states specified by wave vector k and spin a, which satisfy the usual Fermi commutation relations:

"Blatt, Butler, and Schafroth, Phys. Rev. 100, 481 (1955) have introduced the concept of a "correlation length," roughly the distance over which the momenta of a pair of particles are [ c k „ c k - , . * ] + = Skk'«,.', (2.1) correlated. M. R. Schafroth, Phys. Rev. 100, 502 (1955), has argued that there is a true Meissner effect only if the correlation [ck.,Ck',-lf=0. (2.2) length is effectively infinite. In our theory, the correlation length (not to be confused with Pippard's coherence distance, fo) is most reasonably interpreted as the distance over which the The single-particle number operator » k » is defined as momentum of virtual pairs is the same. We believe that in this sense, the correlation length is effectively infinite. The value of »k» = Ck« Ck». (2.3) q is exactly zero everywhere in a simply connected body in an external field. When there is current flow, as in a torus, there is a The Hamiltonian for the electrons may be expressed in unique distribution of q values for minimum free energy. » T. E. Faber and A. B. Pippard, Proc. Roy. Soc. (London) 27 From analysis of data on transmission of microwave and far A231, 53 (1955). " See A. P. Pippard, Advances in Electronics (Academic Press, infrared radiation through superconducting films of tin and lead, Glover and Tinkham (reference 20) find o»0.27. Inc., New York, 1954), Vol. 6, p. 1.

THEORY

OF

SUPERCONDUCTIVITY

the form H= £ e k « k ,+ L |e k |(l--» k ,,)+ffcoui+* k>kr

k