Physics of condensed matter - GBV7.4.3 Lindhard Theory of Screening 209 7.5 Friedel SumRule and...
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Physics of Condensed Matter Prasanta K. Misra Department of Physics University of Houston AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier
Transcript of Physics of condensed matter - GBV7.4.3 Lindhard Theory of Screening 209 7.5 Friedel SumRule and...
Physics ofCondensed Matter
Prasanta K. Misra
Department of Physics
University of Houston
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
Contents
Preface xxi
CHAPTER 1 Basic Properties of Crystals 1
1.1 Crystal Lattices 2
1.1.1 Primitive Cell 3
1.1.2 Unit Cell 3
1.1.3 Wigner-Seitz Cell 3
1.1.4 Lattice Point Group 3
1.2 Bravais Lattices in Two and Three Dimensions 4
1.2.1 Simple Cubic (sc) Lattice 4
1.2.2 Lattice Constants 5
1.2.3 Coordination Numbers 5
1.2.4 Body-Centered Cubic (bcc) Lattice 5
1.2.5 Face-Centered Cubic (fee) Lattice 7
1.2.6 Other Bravais Lattices 9
1.3 Lattice Planes and Miller Indices 11
1.4 Bravais Lattices and Crystal Structures 13
1.4.1 Crystal Structure 13
1.4.2 Lattice with a Basis 13
1.4.3 Packing Fraction '4
1.5 Crystal Defects and Surface Effects 14
1.5.1 Crystal Defects 14
1.5.2 Surface Effects 14
1.6 Some Simple Crystal Structures 15
1.6.1 Sodium Chloride Structure 15
1.6.2 Cesium Chloride Structure 15
1.6.3 Diamond Structure 16
1.6.4 Zincblende Structure 17
1.6.5 Hexagonal Close-Packed (hep) Stracture 17
1.7 Bragg Diffraction 19
1.8 Laue Method 20
1.9 Reciprocal Lattice 21
1.9.1 Definition 21
1.9.2 Properties of the Reciprocal Lattice 22
1.9.3 Alternative Formulation of the Laue Condition 25
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viii Contents
1.10 Brillouin Zones 27
1.10.1 Definition 27
1.10.2 One-Dimensional Latlice 28
1.10.3 Two-Dimensional Square Lattice 28
1.10.4 bec Lattice 29
1.10.5 fee Latlice 30
1.11 Diffraction by a Crystal Lattice with a Basis 31
1.11.1 Theory 31
1.11.2 Geometrical Structure Factor 32
1.11.3 Application to bec Lattice 32
1.11.4 Application lo fee Lattice 33
1.11.5 The Atomic Scattering Factor or Form Factor 33
Problems 34
References 35
CHAPTER 2 Phonons and Lattice Vibrations 37
2.1 Lattice Dynamics 37
2.1.1 Theory 37
2.1.2 Normal Modes of a One-Dimensional Monoatomic Lattice 41
2.1.3 Normal Modes of a One-Dimensional Chain with a Basis 44
2.2 Lattice Specific Heat 48
2.2.1 Theory 48
2.2.2 The Debye Model of Specific Heat 49
2.2.3 The Einstein Model of Specific Heat 52
2.3 Second Quantization 53
2.3.1 Occupation Number Representation 53
2.3.2 Creation and Annihilation Operators 54
2.3.3 Field Operators and the Hamiltonian 58
2.4 Quantization of Lattice Waves 61
2.4.1 Formulation 61
2.4.2 Quantization of Lattice Waves 65
Problems 66
References 68
CHAPTER 3 Free Electron Model 71
3.1 The Classical (Dmde) Model of a Metal 71
3.2 Sommerfeld Model 73
3.2.1 Introduction 73
3.2.2 Fermi Distribution Function 74
3.2.3 Density Operator 75
3.2.4 Free Electron Fermi Gas 77
3.2.5 Ground-State Energy of the Electron Gas 79
3.2.6 Density of Electron States 81
Contents ix
3.3 Fermi Energy and the Chemical Poiential 82
3.4 Specific Heat of the Electron Gas 84
3.5 DC Electrical Conductivity 86
3.6 The Hall Effect 87
3.7 Failures of the Free Electron Model 89
Problems 90
References 93
CHAPTER 4 Nearly Free Electron Model 95
4.1 Electrons in a Weak Periodic Potential 96
4.1.1 Introduction 96
4.1.2 Plane Wave Solutions 97
4.2 Bloch Functions and Bloch Theorem 99
4.3 Reduced, Repeated (Periodic), and Extended Zone Schemes 99
4.3.1 Reduced Zone Scheme 100
4.3.2 Repeated Zone Scheme 100
4.3.3 Extended Zone Scheme 101
4.4 Band Index 101
4.5 Effective Hamiltonian 102
4.6 Proof of Bloch's Theorem from Translational Symmetry 103
4.7 Approximate Solution Near a Zone Boundary 105
4.8 Different Zone Schemes 109
4.8.1 Reduced Zone Scheme 109
4.8.2 Extended Zone Scheme 110
4.8.3 Periodic Zone Scheme 111
4.9 Elementary Band Theory of Solids Ill
4.9.1 Introduction HI
4.9.2 Energy Bands in One Dimension 112
4.9.3 Number of States in a Band 112
4.10 Metals, Insulators, and Semiconductors 112
4.11 Brillouin Zones 117
4.12 Fermi Surface H9
4.12.1 Fermi Surface (in Two Dimensions) 119
4.12.2 Fermi Surface (in Three Dimensions) 121
4.12.3 Harrison's Method of Construction of the Fermi Surface 121
Problems 124
References•
1^0
CHAPTER 5 Band-Structure Calculations 131
5.1 Introduction 1^1
5.2 Tight-Binding Approximation 131
5.3 LCAO Method 135
x Contents
5.4 Wannier Functions
5.5 Cellular Method 142
5.6 Orthogonalized Plane-Wave (OPW) Method 145
5.7 Pseudopotentials I47
5.8 Muffin-Tin Potential 149
5.9 Augmented Plane-Wave (APW) Method 150
5.10 Green's Function (KKR) Method 152
5.11 Model Pseudopotentials 156
5.12 Empirical Pseudopotentials I57
5.13 First-Principles Pseudopotentials 158
Problems 160
References 163
CHAPTER 6 Static and Transport Properties of Solids 165
6.1 Band Picture 166
6.2 Bond Picture I67
6.3 Diamond Structure I68
6.4 Si and Ge I68
6.5 Zinc-Blende Semiconductors 170
6.6 Ionic Solids I72
6.7 Molecular Crystals I74
6.7.1 Molecular Solids 174
6.7.2 Hydrogen-Bonded Structures 174
6.8 Cohesion of Solids 174
6.8.1 Molecular Crystals: Noble Gases 174
6.8.2 Ionic Crystals 176
6.8.3 Covalent Crystals 177
6.8.4 Cohesion in Metals 178
6.9 The Semiclassical Model 179
6.10 Liouville's Theorem 182
6.11 Boltzmann Equation 183
6.12 Relaxation Time Approximation 184
6.13 Electrical Conductivity 186
6.14 Thermal Conductivity 187
6.15 Weak Scattering Theory of Conductivity 188
6.15.1 Relaxation Time and Scattering Probability 188
6.15.2 The Collision Term 188
6.15.3 Impurity Scattering 189
6.16 Resistivity Due to Scattering by Phonons 192
Problems 194
References 196
Contents xi
CHAPTER 7 Electron-Electron Interaction 199
7.1 Introduction 199
7.2 Hartree Approximation 200
7.3 Hartree-Fock Approximation 203
7.3.1 General Formulation 203
7.3.2 Hartree-Fock Theory for Jellium 204
7.4 Effect of Screening 207
7.4.1 General Formulation 207
7.4.2 Thomas-Fermi Approximation 208
7.4.3 Lindhard Theory of Screening 209
7.5 Friedel Sum Rule and Oscillations 214
7.6 Frequency and Wave-Number-Dependent Dielectric Constant 217
7.7 Mott Transition 222
7.8 Density Functional Theory 223
7.8.1 General Formulation 223
7.8.2 Local Density Approximation 224
7.9 Fermi Liquid Theory 225
7.9.1 Quasiparticles 225
7.9.2 Energy Functional 227
7.9.3 Fermi Liquid Parameters 230
7.10 Green's Function Method 232
7.10.1 General Formulation 232
7.10.2 Finite-Temperature Green's Function Formalism for Interacting Bloch
Electrons 233
7.10.3 Exchange Self-Energy in the Band Model 234
Problems 235
References 241
CHAPTER 8 Dynamics of Bloch Electrons 243
8.1 Semiclassical Model 243
8.2 Velocity Operator 244
8.3 k p Perturbation Theory 245
8.4 Quasiclassical Dynamics 246
8.5 Effective Mass 247
8.6 Bloch Electrons in External Fields 248
8.6.1 Time Evolution of Bloch Electrons in an Electric Field 250
8.6.2 Alternate Derivation for Bloch Functions in an External Electric
and Magnetic Field 252
8.6.3 Motion in an Applied DC Field 253
8.7 Bloch Oscillations 254
8.8 Holes 255
8.9 Zener Breakdown (Approximate Method) 258
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8.10 Rigorous Calculation of Zener Tunneling 261
8.11 Electron-Phonon Interaction 264
Problems 271
References 274
CHAPTER 9 Semiconductors 275
9.1 Introduction 275
9.2 Electrons and Holes 278
9.3 Electron and Hole Densities in Equilibrium 279
9.4 Intrinsic Semiconductors 283
9.5 Extrinsic Semiconductors 284
9.6 Doped Semiconductors 285
9.7 Statistics of Impurity Levels in Thermal Equilibrium 288
9.7.1 Donor Levels 288
9.7.2 Acceptor Levels 288
9.7.3 Doped Semiconductors 289
9.8 Diluted Magnetic Semiconductors 290
9.8.1 Introduction 290
9.8.2 Magnetization in Zero External Magnetic Field in DMS 291
9.8.3 Electron Paramagnetic Resonance Shift 291
9.8.4 It-Tr Model 295
9.9 Zinc Oxide 296
9.10 Amorphous Semiconductors 296
9.10.1 Introduction 296
9.10.2 Linear Combination of Hybrids Model for Tetrahedral Semiconductors 297
Problems 300
References 303
CHAPTER 10 Electronics 305
10.1 Introduction 305
10.2 p-n Junction 306
10.2.1 Introduction 306
10.2.2 p-n Junction in Equilibrium 307
10.3 Rectification by a p-n Junction 311
10.3.1 Equilibrium Case 311
10.3.2 Nonequilibrium Case (V^0) 313
10.4 Transistors 318
10.4.1 Bipolar Transistors 318
10.4.2 Field-Effect Transistor 319
10.4.3 Single-Electron Transistor 321
10.5 Integrated Circuits 325
Contents xiii
10.6 Optoelectronic Devices 325
10.7 Graphene 329
10.8 Graphene-Based Electronics 332
Problems 333
References 336
CHAPTER 11 Spintronics 339
11.1 Introduction 339
11.2 Magnetoresistance 340
11.3 Giant Magnetoresistance 340
11.3.1 Metallic Multilayers 340
11.4 Mott's Theory of Spin-Dependent Scattering of Electrons 342
11.5 Camley-Barnas Model 345
11.6 CPP-GMR 348
11.6.1 Introduction 348
11.6.2 Theory of CPP-GMR of Multilaycred Nanovvircs 350
11.7 MTJ, TMR, and MRAM 352
11.8 Spin Transfer Torques and Magnetic Switching 356
11.9 Spintronics with Semiconductors 357
11.9.1 Introduction 357
11.9.2 Theory of an FM-T-N Junction 358
11.9.3 Injection Coefficient 361
Problems 364
References 367
CHAPTER 12 Diamagnetism and Paramagnetism 369
12.1 Introduction 370
12.2 Atomic (or Ionic) Magnetic Susceptibilities 371
J 2.2.1 General Formulation 371
12.2.2 Larmor Diamagnetism 372
12.2.3 Hund's Rules 373
12.2.4 Van Vleck Paramagnetism 374
12.2.5 Lande g Factor 375
12.2.6 Curie's Law 377
12.3 Magnetic Susceptibility of Free Electrons in Metals 378
12.3.1 Genera! Formulation 378
12.3.2 Landau Diamagnetism and Pauli Paramagnetism 380
12.3.3 De Haas-van Alphen Effect 383
12.4 Many-Body Theory of Magnetic Susceptibility of Bloch Electrons in Solids. . .388
12.4.1 Introduction 388
12.4.2 Equation of Motion in the Bloch Representation 388
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12.4.3 Thermodynamic Potential 390
12.4.4 General Formula for x ^90
12.4.5 Exchange Self-Energy in the Band Model 393
12.4.6 Exchange Enhancement of/.v 394
12.4.7 Exchange and Correlation Effects on/„ 395
12.4.8 Exchange and Correlation Effects on %so 396
12.5 Quantum Hall Effect 396
12.5.1 Introduction 396
12.5.2 Two-Dimensional Electron Gas 396
12.5.3 Quantum Transport of a Two-Dimensional Electron Gas in a Strong
Magnetic Field 397
12.5.4 Quantum Hall Effect from Gauge Invariance 400
12.6 Fractional Quantum Hall Effect 400
Problems 401
References 407
CHAPTER 13 Magnetic Ordering 409
13.1 Introduction 410
13.2 Magnetic Dipole Moments 411
13.3 Models for Ferromagnetism and Antiferromagnetism 412
13.3.1 Introduction 412
13.3.2 Heitler-London Approximation 412
13.3.3 Spin Hamiltonian 414
13.3.4 Heisenberg Model 416
13.3.5 Direct, Indirect, and Superexchange 416
13.3.6 Spin Waves in Ferromagnets: Magnons 417
13.3.7 Schwinger Representation 417
13.3.8 Application to the Heisenberg Hamiltonian 418
13.3.9 Spin Waves in Antiferromagnels 421
13.4 Ferromagnetism in Solids 422
13.4.1 Ferromagnetism Near the Curie Temperature 422
13.4.2 Comparison of Spin-Wave Theory with the Weiss Field Model 424
13.4.3 Ferromagnetic Domains 425
13.4.4 Hysteresis 426
13.4.5 Ising Model 427
13.5 Ferromagnetism in Transition Metals 427
13.5.1 Introduction 427
13.5.2 Stoner Model 428
13.5.3 Ferromagnetism in Fe, Co, and Ni from Stoner's Model and
Kohn-Sham Equations 430
13.5.4 Free Electron Gas Model 431
13.5.5 Hubbard Model 433
Contents xv
13.6 Magnetization of Interacting Bloch Electrons 434
13.6.1 Introduction 434
13.6.2 Theory of Magnetization 434
13.6.3 The Quasipurtiele Contribution to Magnetization 435
13.6.4 Contribution of Correlations to Magnetization 436
13.6.5 Single-Particle Spectrum and the Criteria for Ferromagnetic Ground State 437
13.7 The Rondo Effect 439
13.8 Anderson Model 439
13.9 The Magnetic Phase Transition 440
13.9.1 Introduction 440
13.9.2 The Order Parameter 441
13.9.3 Landau Theory of Second-Order Phase Transitions 441
Problems 443
References 448
CHAPTER 14 Superconductivity 451
14.1 Properties of Superconductors 452
14.1.1 Introduction 452
14.1.2 Type 1 and Type II Superconductors 453
14.1.3 Second-Order Phase Transition 454
14.1.4 Isotope Effect 454
14.1.5 Phase Diagram 454
14.2 Meissner-Ochsenfeld Effect 455
14.3 The London Equation 455
14.4 Ginzburg-Landau Theory 456
14.4.1 Order Parameter 456
14.4.2 Boundary Conditions 457
14.4.3 Coherence Length 457
14.4.4 London Penetration Depth 458
14.5 Flux Quantization 459
14.6 Josephson Effect 460
14.6.1 Two Superconductors Separated by an Oxide Layer 460
14.6.2 AC and DC Josephson Effects 462
14.7 Microscopic Theory of Superconductivity 462
14.7.1 Introduction 462
14.7.2 Quasi-Electrons 463
14.7.3 Cooper Pairs 464
14.7.4 BCS Theory 466
14.7.5 Ground State of the Superconducting Electron Gas 466
14.7.6 Excited States at T=0 469
14.7.7 Excited States at T + 0 470
xvi Contents
14.8 Strong-Coupling Theory 472
14.8.1 Introduction 472
14.8.2 Upper Limit of the Critical Temperature, Tc 472
14.9 High-Temperature Superconductors 473
14.9.1 Introduction 473
14.9.2 Properties of Novel Superconductors (Cuprates) 474
14.9.3 Brief Review of s-, p~, and c(-wave Pairing 474
14.9.4 Experimental Confirmation of rf-wave Pairing 476
14.9.5 Search for a Theoretical Mechanism of High TL. Superconductors 481
Problems 481
References 485
CHAPTER 15 Heavy Fermions 487
15.1 Introduction 488
15.2 Kondo-Lattice, Mixed-Valence, and Heavy Fermions 490
15.2.1 Periodic Anderson and Kondo-Lattice Models 490
15.2.2 Mixed-Valence Compounds 492
15.2.3 Slave Boson Method 493
15.2.4 Cluster Calculations 494
15.3 Mean-Field Theories 498
15.3.1 The Local Impurity Self-Consistent Approximation 498
15.3.2 Application of LISA to Periodic Anderson Model 499
15.3.3 RKKY Interaction 500
15.3.4 Extended Dynamical Mean-field Theory 501
15.4 Fermi-Liquid Models 502
15.4.1 Heavy Fermi Liquids 502
15.4.2 Fractionalized Fermi Liquids 505
15.5 Metamagnetism in Heavy Fermions 506
15.6 Ce- and U-Based Superconducting Compounds 508
15.6.1 Ce-Based Compounds 508
15.6.2 U-Based Superconducting Compounds 509
15.7 Other Heavy-Fermion Superconductors 513
15.7.1 PrOs4Sbl2 513
15.7.2 PuCoGa5 513
15.7.3 PuRhGa5 515
15.7.4 Comparison between Cu and Pu Containing High-T^ Superconductors 516
15.8 Theories of Heavy-Fermion Superconductivity 516
15.9 Kondo Insulators 516
15.9.1 Brief Review 516
15.9.2 Theory of Kondo Insulators 517
Problems 519
References 524
Contents xvii
CHAPTER 16 Metallic Nanoclusters 527
16.1 Introduction 528
16.1.1 Nanosciencc and Nanoclusters 528
16.1.2 Liquid Drop Model 528
16.1.3 Size and Surface/Volume Ratio 528
16.1.4 Geometric and Electronic Shell Structures 530
16.2 Electronic Shell Structure 531
16.2.1 Spherical Jellium Model (Phciwinenologkal) -531
16.2.2 Self-Consistent Spherical Jellium Model 532
16.2.3 Ellipsoidal Shell Model 535
16.2.4 Nonalkali Clusters 535
16.2.5 Large Clusters 535
16.3 Geometric Shell Structure 537
16.3.1 Close-Packing 537
16.3.2 Wulff Construction 537
16.3.3 Polyhedra 538
16.3.4 Filling between Complete Shells 540
16.4 Cluster Growth on Surfaces 540
16.4.1 Monte Carlo Simulations 540
16.4.2 Mean-Field Rate Equations 541
16.5 Structure of Isolated Clusters 542
16.5.1 Theoretical Models 542
16.5.2 Structure of Some Isolated Clusters 546
16.6 Magnetism in Clusters 547
16.6.1 Magnetism in Isolated Clusters 547
16.6.2 Experimental Techniques for Studying Cluster Magnetism 549
16.6.3 Magnetism in Embedded Clusters 553
16.6.4 Graphite Surfaces 555
16.6.5 Study of Clusters by Scanning Tunneling Microscope 555
16.6.6 Clusters Embedded in a Matrix 557
16.7 Superconducting State of Nanoclusters 558
16.7.1 Qualitative Analysis 558
16.7.2 Thermodynamic Green's Function Formalism for Nanoclusters 559
Problems 562
References 565
CHAPTER 17 Complex Structures 567
17.1 Liquids 568
17.1.1 Introduction 568
17.1.2 Phase Diagram 568
17.1.3 Van Hove Pair Correlation Function 569
17.1.4 Correlation Function for Liquids 570
xviii Contents
17.2 Supernuid 4He 570
17.2.1 Introduction 370
17.2.2 Phase Transition in 4He 570
17.2.3 Two-Fluid Model for Liquid 4He 571
17.2.4 Theory of Superfluidity in Liquid 4He 57)
17.3 Liquid 3He 573
17.3.1 Introduction 573
17.3.2 Possibility of Superfluidity in Liquid 3He 574
17.3.3 Fermi Liquid Theory 574
17.3.4 Experimental Results of Superfluidity in Liquid 3He 575
17.3.5 Theoretical Model for the A and A, Phases 575
17.3.6 Theoretical Model for the B Phase 577
17.4 Liquid Crystals 578
17.4.1 Introduction 578
17.4.2 Three Classes of Liquid Crystals 578
17.4.3 The Order Parameter 580
17.4.4 Curvature Strains 581
17.4.5 Optical Properties of Cholesleric Liquid Crystals 581
17.5 Quasicrystals 583
17.5.1 Introduction 583
17.5.2 Penrose Tiles 583
17.5.3 Discovery of Quasicrystals 584
17.5.4 Quasiperiodic Lattice 584
17.5.5 Phonon and Phason Degrees of Freedom 586
17.5.6 Dislocation in the Penrose Lattice 589
17.5.7 Icosahedral Quasicrystals 589
17.6 Amorphous Solids 590
17.6.1 Introduction 590
17.6.2 Energy Bands in One-Dimensional Aperiodic Potentials 591
17.6.3 Density of States 593
17.6.4 Amorphous Semiconductors 593
Problems 594
References 597
CHAPTER 18 Novel Materials 599
18.1 Graphene 600
18.1.1 Introduction 600
18.1.2 Graphene Lattice 601
18.1.3 Tight-Binding Approximation 602
18.1.4 Dirac Fermions 606
18.1.5 Comprehensive View of Graphene 608
Contents xix
18.2 Fullerenes 608
18.2.1 Introduction 608
18.2.2 Discovery of CM) 609
18.3 Fullerenes and Tubules 613
18.3.1 Introduction 613
18.3.2 Carbon Nanotubeles 614
18.3.3 Three Types of Carbon Nanotubes 614
18.3.4 Symmetry Properties of Carbon Nanotubes 616
18.3.5 Band Structure of a Fullerene Nanotube 617
18.4 Polymers 617
18.4.1 Introduction 617
18.4.2 Saturated and Conjugated Polymers 618
18.4.3 Transparent Metallic Polymers 621
18.4.4 Electronic Polymers 621
18.5 Solitons in Conducting Polymers 622
18.5.1 Introduction 622
18.5.2 Electronic Structure 623
18.5.3 Tight-Binding Model 623
18.5.4 Soliton Excitations 624
18.5.5 Solitons, Polarons, and Polaron Excitations 626
18.6.6 Polarons and Bipolarons 626
18.6 Photoinduced Electron Transfer 627
Problems 627
References 630
APPENDIX A Elements of Group Theory 633
A.1 Symmetry and Its Consequences 633
A.l. 1 Symmetry of Crystals 633
A. 1.2 Definition of a Group 633
A. 1.3 Symmetry Operations in Crystal Lattices 634
A.2 Space Groups 634
A.2.1 Introduction 634
A.2.2 Space Group Operations 634
A.3 Point Group Operations 636
A.3.1 Introduction 636
A.3.2 Description of Point Groups 636
A.3.3 The Cubic Group 0„ 638
APPENDIX B Mossbauer Effect 641
B. 1 Introduction 641
B.2 Recoilless Fraction 642
B.3 Average Transferred Energy 643
Reference 644
xx Contents
APPENDIX C Introduction to Renormalization Group Approach 645
C.1 Critical Behavior 645
C.2 Theory for Scaling 646
C.3 Renormalization Group Approach 648
References 649
Index 651
