Quantum statistical theory of superconductivity

Quantum Statistical Theory

of Superconductivity

SELECTED TOPICS IN SUPERCONDUCTIVITY

Series Editor: Stuart Wolf

Naval Research Laboratory

Washington, D. C.

CASE STUDIES IN SUPERCONDUCTING MAGNETS

Design and Operational Issues

Yukikazu Iwasa

INTRODUCTION TO HIGH-TEMPERATURE SUPERCONDUCTIVITY

Thomas P. Sheahen

THE NEW SUPERCONDUCTORS

Frank J. Owens and Charles P. Poole, Jr.

QUANTUM STATISTICAL THEORY OF SUPERCONDUCTIVITY

Shigeji Fujita and Salvador Godoy

STABILITY OF SUPERCONDUCTORS

Lawrence Dresner

A Continuation Order Plan is available for this series. A continuation order will bring

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Quantum Statistical Theory

of Superconductivity

Shigeji Fujita

SUNY, Buffalo

Buffalo, New York

and

Salvador Godoy

Universidad Nacional Autonoma de México

México, D. F., México

Kluwer Academic Publishers

NEW YORK, BOSTON , DORDRECHT, LONDON, MOSCOW

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0-306-45363-0

©2002 Kluwer Academic Publishers

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Preface to the Series

Since its discovery in 1911, superconductivity has been one of the most interesting topics

in physics. Superconductivity baffled some of the best minds of the 20th century and was

finally understood in a microscopic way in 1957 with the landmark Nobel Prize-winning

contribution from John Bardeen, Leon Cooper, and Robert Schrieffer. Since the early 1960s

there have been many applications of superconductivity including large magnets for medical

imaging and high-energy physics, radio-frequency cavities and components for a variety

of applications, and quantum interference devices for sensitive magnetometers and digital

circuits. These last devices are based on the Nobel Prize-winning (Brian) Josephson effect.

In 1987, a dream of many scientists was realized with the discovery of superconducting

compounds containing copper–oxygen layers that are superconducting above the boiling

point of liquid nitrogen. The revolutionary discovery of superconductivity in this class of

compounds (the cuprates) won Georg Bednorz and Alex Mueller the Nobel Prize.

This series on Selected Topics in Superconductivity will draw on the rich history of

both the science and technology of this field. In the next few years we will try to chronicle

the development of both the more traditional metallic superconductors as well as the

scientific and technological emergence of the cuprate superconductors. The series will

contain broad overviews of fundamental topics as well as some very highly focused treatises

designed for a specialized audience.

v

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Preface

Superconductivity is a striking physical phenomenon that has attracted the attention of

physicists, chemists, engineers, and also the nontechnical public. The theory of super￾conductivity is considered difficult. Lectures on the subject are normally given at the

end of Quantum Theory of Solids, a second-year graduate course.

In 1957 Bardeen, Cooper, and Schrieffer (BCS) published an epoch-making mi￾croscopic theory of superconductivity. Starting with a Hamiltonian containing electron

and hole kinetic energies and a phonon-exchange-pairing interaction Hamiltonian, they

demonstrated that (1) the ground-state energy of the BCS system is lower than that of

the Bloch system without the interaction, (2) the unpaired electron (quasi-electron) has

an energy gap ∆ 0 at 0 K, and (3) the critical temperature Tc can be related to ∆ 0 by

2∆ 0 = 3.53 k B Tc , and others. A great number of theoretical and experimental investiga￾tions followed, and results generally confirm and support the BCS theory. Yet a number

of puzzling questions remained, including why a ring supercurrent does not decay by

scattering due to impurities which must exist in any superconductor; why monovalent

metals like sodium are not superconductors; and why compound superconductors, in￾cluding intermetallic, organic, and high-Tc superconductors exhibit magnetic behaviors

different from those of elemental superconductors.

tures of both electrons and phonons in a model Hamiltonian. By doing so we were able

Recently the present authors extended the BCS theory by incorporating band struc￾to answer the preceding questions and others. We showed that under certain specific

conditions, elemental metals at low temperatures allow formation of Cooper pairs by the

phonon exchange attraction. These Cooper pairs, called the pairons, for short, move as

free bosons with a linear energy–momentum relation. They neither overlap in space nor

interact with each other. Systems of pairons undergo Bose–Einstein condensations in two

and three dimensions. The supercondensate in the ground state of the generalized BCS

system is made up of large and equal numbers of ± pairons having charges ±2e, and

it is electrically neutral. The ring supercurrent is generated by the ± pairons condensed

at a single momentum qn = 2πn L– 1 , where L is the ring length and n an integer. The

macroscopic supercurrent arises from the fact that ± pairons move with different speeds.

Josephson effects are manifestations of the fact that pairons do not interact with each

other and move like massless bosons just as photons do. Thus there is a close analogy

between a supercurrent and a laser. All superconductors, including high-Tc cuprates, can

be treated in a unified manner, based on the generalized BCS Hamiltonian.

vii

viii PREFACE

Because the supercondensate can be described in terms of independently moving

pairons, all properties of a superconductor, including ground-state energy, critical tem￾perature, quasi-particle energy spectra containing gaps, supercondensate density, specific

heat, and supercurrent density can be computed without mathematical complexities. This

simplicity is in great contrast to the far more complicated treatment required for the phase

transition in a ferromagnet or for the familiar vapor–liquid transition.

The authors believe that everything essential about superconductivity can be pre￾sented to beginning second-year graduate students. Some lecturers claim that much

physics can be learned without mathematical formulas, that excessive use of formulas

hinders learning and motivation and should therefore be avoided. Others argue that

learning physics requires a great deal of thinking and patience, and if mathematical

expressions can be of any help, they should be used with no apology. The average

physics student can learn more in this way. After all, learning the mathematics needed

for superconductor physics and following the calculational steps are easier than grasping

basic physical concepts. (The same cannot be said about learning the theory of phase

transitions in ferromagnets.) The authors subscribe to the latter philosophy and prefer

to develop theories from the ground up and to proceed step by step. This slower but

more fundamental approach, which has been well-received in the classroom, is followed

in the present text. Students are assumed to be familiar with basic differential, integral,

and vector calculuses, and partial differentiation at the sophomore–junior level. Knowl￾edge of mechanics, electromagnetism, quantum mechanics, and statistical physics at the

junior–senior level are prerequisite.

A substantial part of the difficulty students face in learning the theory of supercon￾ductivity lies in the fact that they need not only a good background in many branches

of physics but must also be familiar with a number of advanced physical concepts such

as bosons, fermions, Fermi surface, electrons and holes, phonons, Debye frequency, and

density of states. To make all of the necessary concepts clear, we include five preparatory

chapters in the present text. The first three chapters review the free-electron model of a

metal, theory of lattice vibrations, and theory of the Bose–Einstein condensation. There

follow two additional preparatory chapters on Bloch electrons and second quantization

formalism. Chapters 7–11 treat the microscopic theory of superconductivity. All basic

thermodynamic properties of type I superconductors are described and discussed, and

all important formulas are derived without omitting steps. The ground state is treated

by closely following the original BCS theory. To treat quasi-particles including Bloch

electrons, quasi-electrons, and pairons, we use Heisenberg’s equation-of-motion method,

which reduces a quantum many-body problem to a one-body problem when the system￾Hamiltonian is a sum of single-particle Hamiltonians. No Green’s function techniques

are used, which makes the text elementary and readable. Type II compounds and high-Tc

superconductors are discussed in Chapters 12 and 13, respectively. A brief summary and

overview are given in the first and last chapters.

In a typical one-semester course for beginning second-year graduate students, the

authors began with Chapter 1, omitted Chapters 2–4, then covered Chapters 5–11 in that

order. Material from Chapters 12 and 13 was used as needed to enhance the student’s

interest. Chapters 2–4 were assigned as optional readings.

The book is written in a self-contained manner so that nonphysics majors who want

to learn the microscopic theory of superconductivity step by step in no particular hurry

PREFACE ix

may find it useful as a self-study reference. Many fresh, and some provocative, views

are presented. Researchers in the field are also invited to examine the text.

Problems at the end of a section are usually of a straightforward exercise type

directly connected to the material presented in that section. By solving these problems,

the reader should be able to grasp the meanings of newly introduced subjects more firmly.

The authors thank the following individuals for valuable criticism, discussion, and

readings: Professor M. de Llano, North Dakota State University; Professor T. George,

Washington State University; Professor A. Suzuki, Science University of Tokyo; Dr. C.

L. Ko, Rancho Palos Verdes, California; Dr. S. Watanabe, Hokkaido University, Sapporo.

They also thank Sachiko, Amelia, Michio, Isao, Yoshiko, Eriko, George Redden, and

Brent Miller for their encouragement and for reading the drafts. We thank Celia García

and Benigna Cuevas for their typing and patience. We specially thank César Zepeda and

Martin Alarcón for their invaluable help with computers, providing software, hardware,

as well as advice. One of the authors (S. F.) thanks many members of the Deparatmento

de Física de la Facultad de Ciencias, Universidad Nacional Autónoma de México for

their kind hospitality during the period when most of this book was written. Finally we

gratefully acknowledge the financial support by CONACYT, México.

Shigeji Fujita

Salvador Godoy

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Contents

Constants, Signs, and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Chapter 1. Introduction

1.1. Basic Experimental Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3. Thermodynamics of a Superconductor ..................... 12

1.4. Development of a Microscopic Theory ..................... 19

1.5. Layout of the Present Book ........................... 21

References ........................................... 22

Chapter 2. Free-Electron Model for a Metal

2.1. Conduction Electrons in a Metal; The Hamiltonian . . . . . . . . . . . . 23

2.2. Free Electrons; The Fermi Energy . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3. Density of States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4. Heat Capacity of Degenerate Electrons 1; Qualitative Discussions . . 33

2.5. Heat Capacity of Degenerate Electrons 2; Quantitative Calculations . . 34

2.6. Ohm’s Law, Electrical Conductivity, and Matthiessen’s Rule . . . . . . 38

2.7. Motion of a Charged Particle in Electromagnetic Fields . . . . . . . . . . 40

Chapter 3. Lattice Vibrations: Phonons

3.1. Crystal Lattices ................................... 45

3.2. Lattice Vibrations; Einstein’s Theory of Heat Capacity ......... 46

3.3. Oscillations of Particles on a String; Normal Modes ........... 49

3.4. Transverse Oscillations of a Stretched String ................. 54

3.5. Debye’s Theory of Heat Capacity ....................... 58

References ........................................... 65

Chapter 4. Liquid Helium: Bose–Einstein Condensation

4.1. Liquid Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2. Free Bosons; Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . 68

4.3. Bosons in Condensed Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xi

xii CONTENTS

Chapter 5. Bloch Electrons; Band Structures

5.1. The Bloch Theorem ................................. 75

5.2. The Kronig–Penney Model ........................... 79

5.3. Independent-Electron Approximation; Fermi Liquid Model ...... 81

5.4. The Fermi Surface ................................ 83

5.5. Electronic Heat Capacity; The Density of States ............ 88

5.6. de-Haas–van-Alphen Oscillations; Onsager’s Formula ........ 90

5.7. The Hall Effect; Electrons and Holes .................... 93

5.8. Newtonian Equations of Motion for a Bloch Electron .......... 95

5.9. Bloch Electron Dynamics ............................ 100

5.10. Cyclotron Resonance ................................ 103

References .......................................... 106

Chapter 6. Second Quantization; Equation-of-Motion Method

6.1. Creation and Annihilation Operators for Bosons ............. 109

6.2. Physical Observables for a System of Bosons ............... 113

6.3. Creation and Annihilation Operators for Fermions ............. 114

6.4. Second Quantization in the Momentum (Position) Space ....... 115

6.5. Reduction to a One-Body Problem ....................... 117

6.6. One-Body Density Operator; Density Matrix ................. 120

6.7. Energy Eigenvalue Problem ........................... 122

6.8. Quantum Statistical Derivation of the Fermi Liquid Model ....... 124

Reference ............................................. 125

Chapter 7. Interparticle Interaction; Perturbation Methods

7.1. Electron–Ion Interaction; The Debye Screening ............... 127

7.2. Electron–Electron Interaction ........................... 129

7.3. More about the Heat Capacity; Lattice Dynamics ............. 130

7.4. Electron–Phonon Interaction; The Fröhlich Hamiltonian ......... 135

7.5. Perturbation Theory 1; The Dirac Picture ................... 138

7.6. Scattering Problem; Fermi’s Golden Rule ................... 141

7.7. Perturbation Theory 2; Second Intermediate Picture ........... 144

7.8. Electron–Impurity System; The Boltzmann Equation ........... 145

7.9. Derivation of the Boltzmann Equation ..................... 147

7.10. Phonon-Exchange Attraction ........................... 150

References ........................................... 154

Chapter 8. Superconductors at 0 K

8.1. Introduction ....................................... 155

8.2. The Generalized BCS Hamiltonian ....................... 156

8.3. The Cooper Problem 1; Ground Cooper Pairs ............... 161

8.4. The Cooper Problem 2; Excited Cooper Pairs ............... 164

8.5. The Ground State ................................... 167

8.6. Discussion ....................................... 172

8.7. Concluding Remarks ................................. 178

CONTENTS xiii

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Chapter 9. Bose–Einstein Condensation of Pairons

9.1. Pairons Move as Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.2. Free Bosons Moving in Two Dimensions with ∈ = cp . . . . . . . . . . 184

9.3. Free Bosons Moving in Three Dimensions with ∈ = cp . . . . . . . . . . 187

9.4. B–E Condensation of Pairons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Chapter 10. Superconductors below Tc

10.1. Introduction ....................................... 201

10.2. Energy Gaps for Quasi-Electrons at 0 K ................... 202

10.3. Energy Gap Equations at 0 K ........................... 204

10.4. Energy Gap Equations; Temperature-Dependent Gaps ......... 206

10.5. Energy Gaps for Pairons ............................. 208

10.6. Quantum Tunneling Experiments 1; S–I–S Systems ........... 211

10.7. Quantum Tunneling Experiments 2; S1–I–S2 and S–I–N ......... 219

10.8. Density of the Supercondensate . . . . . . . . . . . . . . . . . . . . . . . . . 222

10.9. Heat Capacity ..................................... 225

10.10. Discussion ....................................... 227

References ........................................... 232

Chapter 11. Supercurrents, Flux Quantization, and Josephson Effects

11.1. Ring Supercurrent; Flux Quantization 1 ................... 233

11.2. Josephson Tunneling; Supercurrent Interference ............... 236

11.3. Phase of the Quasi-Wave Function ....................... 239

11.4. London’s Equation and Penetration Depth; Flux Quantization 2 ... 241

11.5. Ginzburg–Landau Wave Function; More about the Supercurrent ... 245

11.6. Quasi-Wave Function: Its Evolution in Time ............... 247

11.7. Basic Equations Governing a Josephson Junction Current ....... 250

11.8. AC Josephson Effect; Shapiro Steps ..................... 253

11.9. Discussion ....................................... 255

References ........................................... 260

Chapter 12. Compound Superconductors

12.1. Introduction ....................................... 263

12.2. Type II Superconductors; The Mixed State ................. 263

12.3. Optical Phonons ................................... 268

12.4. Discussion ....................................... 270

References ........................................... 270

xiv CONTENTS

Chapter 13. High-Tc Superconductors

13.1. Introduction ....................................... 271

13.2. The Crystal Structure of YBCO; Two-Dimensional Conduction ... 271

13.3. Optical-Phonon-Exchange Attraction; The Hamiltonian ......... 274

13.4. The Ground State ................................... 276

13.5. High Critical Temperature; Heat Capacity ................... 278

13.6. Two Energy Gaps; Quantum Tunneling ................... 280

13.7. Summary ........................................ 282

References ........................................... 283

Chapter 14. Summary and Remarks

14.1. Summary ........................................ 285

14.2. Remarks ......................................... 288

Reference ............................................. 290

Appendix A. Quantum Mechanics

A. 1. Fundamental Postulates of Quantum Mechanics. . . . . . . . . . . . . . . . 291

A.2. Position and Momentum Representations; Schrödinger’s Wave

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

A.3. Schrödinger and Heisenberg Pictures . . . . . . . . . . . . . . . . . . . . . . 298

Appendix B. Permutations

B.1. Permutation Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

B.2. Odd and Even Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Appendix C. Bosons and Fermions

C.1. Indistinguishable Particles ............................. 309

C.2. Bosons and Fermions ............................... 311

C.3. More about Bosons and Fermions ....................... 313

Appendix D. Laplace Transformation; Operator Algebras

D.1. Laplace Transformation ............................... 317

D.2. Linear Operator Algebras ............................. 319

D.3. Liouville Operator Algebras; Proof of Eq. (7.9.19) ............. 320

D.4. The ν–m Representation; Proof of Eq. (7. 10. 15) ............. 322

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Index ................................................... 331