Fundamentals of the Physics of Solids (eBook)

Preface 6
Contents 9
Contents Volume 1: Structure and Dynamics 17
Contents Volume 2: Electronic Properties 20
28 Electron–Electron Interaction and Correlations 23
28.1 Models of the Interacting Electron System 24
28.1.1 The Hamiltonian 24
28.1.2 Second-Quantized Form of the Hamiltonian 26
28.1.3 The Homogeneous Electron Gas 28
28.1.4 Interaction Between Bloch Electrons 30
28.1.5 The Hubbard Model 32
28.2 Normal Fermi Systems 36
28.3 Simple Approximate Treatments of the Interaction 39
28.3.1 Hartree Approximation 40
28.3.2 Hartree Approximation as a Mean-Field Theory 42
28.3.3 Hartree Equations Derived from the Variational Principle 45
28.3.4 Hartree–Fock Approximation 46
28.3.5 Hartree–Fock Approximation as a Mean-Field Theory 48
28.3.6 Quasiparticles in the Hartree–Fock Approximation 52
28.3.7 Total Energy in the Hartree–Fock Approximation 54
28.3.8 Hartree–Fock Theory of the Uniform Electron Gas 55
28.3.9 Hartree–Fock Theory of the Hubbard Model 60
28.4 Spatial and Temporal Correlations 61
28.4.1 The n-Particle Density Matrix 61
28.4.2 Pair Distribution Functions 66
28.4.3 Correlations in the Homogeneous Electron Gas 69
28.4.4 The Structure Factor 73
28.4.5 Dynamical Correlations Between Electrons 75
28.4.6 Dynamical Structure Factor and Scattering Cross Section 80
28.4.7 Magnetic Correlations 81
Further Reading 82
29 Electronic Response to External Perturbations 83
29.1 The Dielectric Function 83
29.1.1 Dielectric Response of the Electron System 85
29.1.2 Density–Density Response Function 87
29.1.3 Relationship to the Dynamical Structure Factor 89
29.1.4 Self-Consistent Treatment of the Interaction 90
29.2 Dielectric Function of the Uniform Electron Gas 93
29.2.1 Thomas–Fermi Approximation 93
29.2.2 The RPA 95
29.2.3 The Lindhard Dielectric Function 97
29.2.4 Alternative Derivation of the Lindhard Function 99
29.2.5 Explicit Form of the Lindhard Dielectric Function 102
29.2.6 Corrections Beyond the RPA 105
29.2.7 Effect of Finite Relaxation Time 108
29.3 Static Screening 111
29.3.1 Thomas–Fermi Screening 112
29.3.2 Friedel Oscillations 114
29.4 Dielectric Function of Metals and Semiconductors 115
29.4.1 Dielectric Function of Bloch Electrons 116
29.4.2 Dielectric Constant of Semiconductors 117
29.5 Dielectric Function in Special Cases 119
29.5.1 Dielectric Function of the Two-Dimensional Electron Gas 120
29.5.2 Dielectric Function of the One-Dimensional Electron Gas 121
29.5.3 Materials with Nested Fermi Surface 124
29.6 Response to Electromagnetic Field 126
29.6.1 Interaction with the Electromagnetic Field 126
29.6.2 Current–Current Correlations and the Kubo Formula 128
29.6.3 Transverse and Longitudinal Response 132
29.6.4 Dielectric Tensor and Conductivity 137
29.6.5 Transverse Dielectric Function of the Electron Gas 138
29.7 Optical and DC Conductivity 139
29.7.1 Optical Conductivity 139
29.7.2 Optical Conductivity of the Electron Gas 141
29.7.3 DC Conductivity 143
29.7.4 The Kubo–Greenwood Formula 145
29.8 Response to Magnetic Perturbations 147
29.8.1 Stoner Enhancement of the Susceptibility 147
29.8.2 Dynamical Susceptibility 149
29.8.3 Transverse Dynamical Susceptibility 155
29.8.4 Ruderman–Kittel Oscillations 157
Further Reading 160
30 Cohesive Energy of the Electron System 161
30.1 Total Energy of the Dense Electron Gas 161
30.1.1 Total Energy in the Hartree–Fock Approximation 161
30.1.2 The Exchange Potential 164
30.1.3 Higher Order Corrections to the Energy 166
30.1.4 Relationship Between Energy and Correlation Function 170
30.1.5 Correlation Energy in the RPA 172
30.2 The Total Energy at Lower Densities 173
30.2.1 The Low-Density Electron Gas, Wigner Crystal 174
30.2.2 Parametrization of the Correlation Energy 178
30.3 The Density-Functional Theory 180
30.3.1 Hohenberg–Kohn Theorems 181
30.3.2 Kohn--Sham Equations 185
30.3.3 Local-Density Approximation 188
30.3.4 Spin-Polarized Systems 189
30.3.5 Applications of the Density-Functional Theory 195
Further Reading 196
31 Excitations in the Interacting Electron Gas 197
31.1 One-Particle and Electron–Hole Pair Excitations 198
31.1.1 One-Particle Elementary Excitations 198
31.1.2 Effective Mass of Quasiparticles 204
31.1.3 Lifetime of Electron States 206
31.1.4 Electron–Hole Pair Excitations 208
31.2 Collective Excitations 213
31.2.1 Dispersion Relation of Plasmons 214
31.2.2 Study of Plasmons with Inelastic Scattering of Electrons 216
31.2.3 Transverse Excitations in the Electron Gas 219
31.3 Bound Electron–Hole Pairs, Excitons 221
31.3.1 Electron–Hole Pairs in Semiconductors and Insulators 221
31.3.2 Wannier Excitons 222
31.3.3 Frenkel Excitons 225
31.4 Magnetic Excitations 227
31.4.1 Paramagnons in Nearly Ferromagnetic Metals 227
31.4.2 Spin Waves in Magnetic Field 229
Further Reading 231
32 Fermion Liquids 232
32.1 Ground State and Excited States of Normal Fermi Systems 233
32.1.1 Ground State of Normal Fermi Systems 235
32.1.2 Quasiparticles in Normal Fermi Systems 235
32.2 Landau's Theory of Fermi Liquids 237
32.2.1 Energy of Quasiparticles and Their Interaction 238
32.2.2 Distribution Function of Quasiparticles 242
32.2.3 Thermodynamic Properties of Fermi Liquids 243
32.2.4 Creation of Quasiparticles by External Perturbation 245
32.2.5 Susceptibility of Fermi Liquids 247
32.2.6 Effective Mass of Quasiparticles 250
32.2.7 Stability Condition of Fermi Liquids 254
32.2.8 3He as a Normal Fermi Liquid 256
32.2.9 Charged Fermi Liquid in Metals 260
32.3 Tomonaga–Luttinger Model 263
32.3.1 Linearized Dispersion Relation 264
32.3.2 Bosonic Electron–Hole Excitations 267
32.3.3 Bosonic Form of the Noninteracting Hamiltonian 272
32.3.4 Spin–Charge Separation 275
32.3.5 Interactions in the Tomonaga–Luttinger Model 278
32.3.6 Excitations in the Interacting Model 281
32.3.7 Thermodynamic Properties and Correlation Functions 285
32.3.8 Absence of the Fermi Edge 290
32.4 The Hubbard Model in One Dimension 293
32.4.1 Bethe-Ansatz Solution 294
32.4.2 Ground State of the Hubbard Chain 297
32.4.3 Low-Energy Excitations 299
32.4.4 Correlation Functions in a Hubbard Chain 310
32.4.5 Mapping Between the Hubbard and TL Models 311
32.5 Luttinger Liquids 313
32.5.1 Low-Energy Spectrum of the XXZ Chain 314
32.5.2 Generic Properties 321
32.5.3 Scaling Theory of the One-Dimensional Electron Gas 323
32.5.4 Experimental Results 327
32.5.5 Luttinger Liquids in Higher Dimensions 330
32.6 Alternatives to Luttinger-Liquid Behavior 330
32.6.1 Mott Insulator 330
32.6.2 Luther–Emery Liquid 333
32.6.3 Phase separation 334
32.7 Quantum Hall Liquid 334
32.7.1 Fractional Quantum Hall Effect 334
32.7.2 Laughlin State 336
32.7.3 Quasiparticles in the Quantum Hall Liquid 339
32.7.4 Anisotropic Hall Liquids 340
Further Reading 341
33 Electronic Phases with Broken Symmetry 342
33.1 Ferromagnetic Instability 344
33.1.1 Ferromagnetism in the Homogeneous Electron Gas 344
33.1.2 Stoner Model 345
33.1.3 Stoner Excitations 348
33.1.4 Stoner Model at Finite Temperatures 350
33.1.5 Failure of the Stoner Model 351
33.1.6 Spin Waves in the Ferromagnetic Electron Gas 351
33.1.7 Role of Spin Waves in the Ferromagnetic Electron Gas 355
33.2 Itinerant Antiferromagnets 356
33.2.1 Slater's Theory of Antiferromagnetism 357
33.2.2 Antiferromagnetic Exchange 360
33.3 Spin-Density Waves 363
33.3.1 Susceptibility of the One-Dimensional Model 364
33.3.2 The Spin-Density-Wave Ground State 367
33.3.3 One-Particle Excitation in the SDW State 369
33.3.4 The Energy Gap 373
33.3.5 Collective Excitations 376
33.4 Charge-Density Waves 378
33.4.1 Peierls Transition 378
33.4.2 The CDW State 381
33.4.3 Determination of the Gap 383
33.4.4 Collective Excitations 385
33.4.5 Dynamics of Charge-Density Waves 386
33.4.6 Topological Excitations 389
33.4.7 Soliton Lattice 394
33.4.8 Electrodynamics of Charge-Density Waves 396
33.4.9 The Role of Fluctuations and Interchain Couplings 398
33.5 Density Waves in Quasi-One-Dimensional Materials 401
33.5.1 Quasi-One-Dimensional Materials 402
33.5.2 Nonlinear and Oscillation Phenomena 409
Further Reading 413
34 Microscopic Theory of Superconductivity 414
34.1 Instability Against Pair Formation 414
34.1.1 Cooper Pairs 415
34.1.2 Instability at Finite Temperatures 419
34.2 The Bardeen–Cooper–Schrieffer Theory 425
34.2.1 BCS Hamiltonian and BCS Ground State 425
34.2.2 Variational Calculation of the Coherence Factors 429
34.2.3 Coherence Length 432
34.2.4 Energy of the Superconducting State 433
34.2.5 Excited States of Superconductors 435
34.2.6 Quasiparticles in the Superconducting State 437
34.2.7 BCS Theory at Finite Temperatures 441
34.2.8 Critical Temperature and the Gap 446
34.3 Thermodynamics and Electrodynamics of Superconductors 447
34.3.1 Thermodynamic Properties 448
34.3.2 Infinite Conductivity 452
34.3.3 The Meissner–Ochsenfeld Effect 453
34.4 Inhomogeneous Superconductors and Retardation Effects 455
34.4.1 Bogoliubov Equations 455
34.4.2 Derivation of the Ginzburg–Landau Equations 459
34.4.3 Eliashberg Equations 462
34.5 Unconventional Superconductors 466
34.5.1 Non-s-Wave Superconductors 466
34.5.2 High-Temperature Superconductors 470
34.5.3 Heavy-Fermion Superconductors 473
34.5.4 Organic Superconductors 476
34.5.5 Coexistence of Superconductivity and Ferromagnetism 477
34.6 Tunneling Phenomena 478
34.6.1 General Description of Tunneling 478
34.6.2 Tunneling in SIN Junctions 481
34.6.3 Tunneling in SIS Junctions 484
34.6.4 Microscopic Calculation of the Current 486
34.6.5 Green-Function Theory of Tunneling 491
Further Reading 491
35 Strongly Correlated Systems 493
35.1 The Mott Metal–Insulator Transition 494
35.1.1 Physical Picture for the Mott Transition 497
35.1.2 Simple Treatment of the Hubbard Model 500
35.1.3 The Gutzwiller–Brinkman–Rice Approach 504
35.1.4 Numerical Results 507
35.1.5 Other Phases of the Hubbard Model 510
35.2 Magnetic Impurities in Metals 512
35.2.1 The Anderson Model 513
35.2.2 Formation of the Localized Moment 514
35.2.3 Better Treatment of the Anderson Model 518
35.2.4 Kondo Model 521
35.2.5 Perturbative Treatment of the Kondo Problem 522
35.2.6 Scaling Theory of the Kondo Problem 524
35.2.7 Wilson's Solution of the Kondo Problem 530
35.2.8 Low-Temperature Behavior of the Kondo Model 533
35.2.9 Nozières's Local-Fermi-Liquid Theory 534
35.3 Mixed-Valence and Heavy-Fermion Compounds 537
35.3.1 Mixed-Valence Compounds 538
35.3.2 Heavy-Fermion Materials 539
35.3.3 Periodic Anderson Model 541
35.3.4 Kondo Lattice 546
35.3.5 Open Problems 548
Further Reading 549
36 Disordered Systems 550
36.1 Disordered Alloys 551
36.1.1 Averaged T-Matrix Approximation 552
36.1.2 Coherent-Potential Approximation 555
36.2 The Anderson Metal–Insulator Transition 556
36.2.1 Anderson Localization 557
36.2.2 Continuous or Discontinuous Transition 559
36.2.3 Phase Coherence and Interference of Electrons 561
36.2.4 Oscillation Phenomena due to Phase Coherence 564
36.2.5 Quantum Corrections to Conductivity, Weak Localization 566
36.2.6 Strong Localization, Hopping Conductivity 571
36.2.7 Scaling Theory of Localization 573
36.2.8 The Role of Electron–Electron Interaction 579
36.3 Spin Glasses 580
36.3.1 Experimental Findings 581
36.3.2 Models of Spin Glasses 584
36.3.3 Quenched Disorder 585
36.3.4 Frustration 586
36.3.5 Edwards–Anderson Model of Spin Glasses 588
36.3.6 Sherrington–Kirkpatrick Model 590
36.3.7 Recent Developments 592
Further Reading 594
J Response to External Perturbations 595
J.1 Linear Response Theory 595
J.1.1 Time-Dependent Response 596
J.1.2 Generalized Susceptibilities 598
J.1.3 Kubo Formula 600
J.1.4 Alternative Form of the Kubo Formula 602
J.1.5 Analytic Properties of Susceptibilities 603
J.1.6 Kramers–Kronig Relations 606
J.1.7 Response Functions and Correlation Functions 607
J.1.8 Fluctuation–Dissipation Theorem 609
J.2 Density–Density Response Function 611
J.2.1 Sum Rules 612
J.2.2 Equation-of-Motion Method 616
J.2.3 Decoupling Procedures 619
J.2.4 Alternative Derivation 624
Further Reading 628
K Green Functions of the Many-Body Problem 629
K.1 Green Functions 629
K.1.1 One-Particle Green Function 630
K.1.2 Phonon Propagator 634
K.1.3 Spectral Representation 634
K.1.4 Green Function and Density of States 640
K.1.5 Temperature Green Function 642
K.1.6 Relation Between the Retarded, Advanced, and Temperature Green Functions 645
K.2 Calculating the Green Functions 646
K.2.1 Equation of Motion for Green Functions 646
K.2.2 Perturbation Theory at Zero Temperature 653
K.2.3 Finite-Temperature Diagram Technique 660
K.3 Green Functions in Superconductivity 663
K.3.1 Gorkov Equations 664
K.3.2 Temperature Green Functions for Superconductors 666
K.3.3 Derivation of the Ginzburg–Landau Equations 668
Further Reading 670
L Field Theory of Luttinger Liquids 671
L.1 Field Theory of the Harmonic Chain 671
L.2 Fermion–Boson Equivalence 673
L.2.1 Phase Field for Spinless Fermions 673
L.2.2 Klein Factors 681
L.2.3 Bosonized Form of the Fermion Field Operators 683
L.2.4 Boson Representation of the Spin Operators 686
L.2.5 Fermions with Spin 687
L.3 Boson Representation of the Hamiltonian 689
L.3.1 Free Spinless Fermions 689
L.3.2 Boson Form of the Full Hamiltonian 691
L.3.3 Boson Form of the Umklapp Scattering 692
L.3.4 Fermions with Spin 692
L.4 Correlation Functions 694
L.4.1 Noninteracting Spinless Fermions 695
L.4.2 Interacting Spinless Fermions 697
L.4.3 Fermions with Spin 699
Further Reading 700
M Renormalization and Scaling in Solid-State Physics 701
M.1 Poor Man's Scaling 701
M.1.1 General Considerations 702
M.1.2 Scaling Theory of the One-Dimensional Electron Gas 704
M.1.3 Scaling Theory of the Kondo Problem 710
M.2 Numerical Renormalization Group 714
Further Reading 721
Figure Credits 722
Name Index 724
Subject Index 729