Condensed Matter Physics, 2nd Edition

Preface xix

References xxii

I ATOMIC STRUCTURE 1

1 The Idea of Crystals 3

1.1 Introduction 3

1.1.1 Why are Solids Crystalline? 4

1.2 Two-Dimensional Lattices 6

1.2.1 Bravais Lattices 6

1.2.2 Enumeration of Two-Dimensional Bravais Lattices 7

1.2.3 Lattices with Bases 9

1.2.4 Primitive Cells 9

1.2.5 Wigner-Seitz Cells 10

1.3 Symmetries 11

1.3.1 The Space Group 11

1.3.2 Translation and Point Groups 12

1.3.3 Role of Symmetry 14

Problems 14

References 16

2 Three-Dimensional Lattices 17

2.1 Introduction 17

2.2 Monatomic Lattices 20

2.2.1 The Simple Cubic Lattice 20

2.2.2 The Face-Centered Cubic Lattice 20

2.2.3 The Body-Centered Cubic Lattice 22

2.2.4 The Hexagonal Lattice 23

2.2.5 The Hexagonal Close-Packed Lattice 23

2.2.6 The Diamond Lattice 24

2.3 Compounds 24

2.3.1 Rocksalt—Sodium Chloride 25

2.3.2 Cesium Chloride 26

2.3.3 Fluorite—Calcium Fluoride 26

2.3.4 Zincblende—Zinc Sulfide 27

2.3.5 Wurtzite—Zinc Oxide 28

2.3.6 Perovskite—Calcium Titanate 28

2.4 Classification of Lattices by Symmetry 30

2.4.1 Fourteen Bravais Lattices and Seven Crystal Systems 30

2.5 Symmetries of Lattices with Bases 33

2.5.1 Thirty-Two Crystallographic Point Groups 33

2.5.2 Two Hundred Thirty Distinct Lattices 36

2.6 Some Macroscopic Implications of Microscopic Symmetries 37

2.6.1 Pyroelectricity 37

2.6.2 Piezoelectricity 37

2.6.3 Optical Activity 38

Problems 38

References 41

3 Scattering and Structures 43

3.1 Introduction 43

3.2 Theory of Scattering from Crystals 44

3.2.1 Special Conditions for Scattering 44

3.2.2 Elastic Scattering from Single Atom 46

3.2.3 Wave Scattering from Many Atoms 47

3.2.4 Lattice Sums 48

3.2.5 Reciprocal Lattice 49

3.2.6 Miller Indices 51

3.2.7 Scattering from a Lattice with a Basis 53

3.3 Experimental Methods 54

3.3.1 Laue Method 56

3.3.2 Rotating Crystal Method 57

3.3.3 Powder Method 59

3.4 Further Features of Scattering Experiments 60

3.4.1 Interaction of X-Rays with Matter 60

3.4.2 Production of X-Rays 61

3.4.3 Neutrons 63

3.4.4 Electrons 63

3.4.5 Deciphering Complex Structures 64

3.4.6 Accuracy of Structure Determinations 65

3.5 Correlation Functions 66

3.5.1 Why Bragg Peaks Survive Atomic Motions 66

3.5.2 Extended X-Ray Absorption Fine Structure (EXAFS) 67

3.5.3 Dynamic Light Scattering 68

3.5.4 Application to Dilute Solutions 70

Problems 71

References 73

4 Surfaces and Interfaces 77

4.1 Introduction 77

4.2 Geometry of Interfaces 77

4.2.1 Coherent and Commensurate Interfaces 78

4.2.2 Stacking Period and Interplanar Spacing 79

4.2.3 Other Topics in Surface Structure 81

4.3 Experimental Observation and Creation of Surfaces 82

4.3.1 Low-Energy Electron Diffraction (LEED) 82

4.3.2 Reflection High-Energy Electron Diffraction (RHEED) 84

4.3.3 Molecular Beam Epitaxy (MBE) 84

4.3.4 Field Ion Microscopy (FIM) 85

4.3.5 Scanning Tunneling Microscopy (STM) 86

4.3.6 Atomic Force Microscopy (AFM) 91

4.3.7 High Resolution Electron Microscopy (HREM) 91

Problems 91

References 94

5 Beyond Crystals 97

5.1 Introduction 97

5.2 Diffusion and Random Variables 97

5.2.1 Brownian Motion and the Diffusion Equation 97

5.2.2 Diffusion 98

5.2.3 Derivation from Master Equation 99

5.2.4 Connection Between Diffusion and Random Walks 100

5.3 Alloys 101

5.3.1 Equilibrium Structures 101

5.3.2 Phase Diagrams 102

5.3.3 Superlattices 103

5.3.4 Phase Separation 104

5.3.5 Nonequilibrium Structures in Alloys 106

5.3.6 Dynamics of Phase Separation 108

5.4 Simulations 110

5.4.1 Monte Carlo 110

5.4.2 Molecular Dynamics 112

5.5 Liquids 113

5.5.1 Order Parameters and Long-and Short-Range Order 113

5.5.2 Packing Spheres 114

5.6 Glasses 116

5.7 Liquid Crystals 120

5.7.1 Nematics, Cholesterics, and Smectics 120

5.7.2 Liquid Crystal Order Parameter 122

5.8 Polymers 123

5.8.1 Ideal Radius of Gyration 123

5.9 Colloids and Diffusing-Wave Scattering 128

5.9.1 Colloids 128

5.9.2 Diffusing-Wave Spectroscopy 128

5.10 Quasicrystals 133

5.10.1 One-Dimensional Quasicrystal 134

5.10.2 Two-Dimensional Quasicrystals—Penrose Tiles 139

5.10.3 Experimental Observations 141

5.11 Fullerenes and nanotubes 143

Problems 143

References 149

II ELECTRONIC STRUCTURE 153

6 The Free Fermi Gas and Single Electron Model 155

6.1 Introduction 155

6.2 Starting Hamiltonian 157

6.3 Densities of States 159

6.3.1 Definition of Density of States D 160

6.3.2 Results for Free Electrons 161

6.4 Statistical Mechanics of Noninteracting Electrons 163

6.5 Sommerfeld Expansion 166

6.5.1 Specific Heat of Noninteracting Electrons at Low Temper-atures 169

Problems 171

References 173

7 Non-Interacting Electrons in a Periodic Potential 175

7.1 Introduction 175

7.2 Translational Symmetry—Bloch’s Theorem 175

7.2.1 One Dimension 176

7.2.2 Bloch’s Theorem in Three Dimensions 180

7.2.3 Formal Demonstration of Bloch’s Theorem 182

7.2.4 Additional Implications of Bloch’s Theorem 183

7.2.5 Van Hove Singularities 186

7.2.6 Kronig-Penney Model 189

7.3 Rotational Symmetry—Group Representations 192

7.3.1 Classes and Characters 198

7.3.2 Consequences of point group symmetries for Schrödinger’s equation 201

Problems 203

References 206

8 Nearly Free and Tightly Bound Electrons 207

8.1 Introduction 207

8.2 Nearly Free Electrons 208

8.2.1 Degenerate Perturbation Theory 210

8.3 Brillouin Zones 211

8.3.1 Nearly Free Electron Fermi Surfaces 214

8.4 Tightly Bound Electrons 219

8.4.1 Linear Combinations of Atomic Orbitals 219

8.4.2 Wannier Functions 222

8.4.3 Geometric Phases 223

8.4.4 Tight Binding Model 226

Problems 227

References 232

9 Electron-Electron Interactions 233

9.1 Introduction 233

9.2 Hartree and Hartree-Fock Equations 234

9.2.1 Variational Principle 235

9.2.2 Hartree-Fock Equations 235

9.2.3 Numerical Implementation 239

9.2.4 Hartree-Fock Equations for Jellium 242

9.3 Density Functional Theory 244

9.3.1 Thomas-Fermi Theory 247

9.3.2 Stability of Matter 249

9.4 Quantum Monte Carlo 252

9.4.1 Integrals by Monte Carlo 252

9.4.2 Quantum Monte Carlo Methods 253

9.4.3 Physical Results 254

9.5 Kohn-Sham Equations 255

Problems 258

References 262

10 Realistic Calculations in Solids 265

10.1 Introduction 265

10.2 Numerical Methods 266

10.2.1 Pseudopotentials and Orthogonalized Planes Waves (OPW) 266

10.2.2 Linear Combination of Atomic Orbitals (LCAO) 271

10.2.3 Plane Waves 271

10.2.4 Linear Augmented Plane Waves (LAPW) 274

10.3 Definition of Metals, Insulators, and Semiconductors 277

10.4 Brief Survey of the Periodic Table 279

10.4.1 Nearly Free Electron Metals 280

10.4.2 Noble Gases 282

10.4.3 Semiconductors 283

10.4.4 Transition Metals 284

10.4.5 Rare Earths 286

Problems 286

References 291

III MECHANICAL PROPERTIES 293

11 Cohesion of Solids 295

11.1 Introduction 295

11.1.1 Radii of Atoms 297

11.2 Noble Gases 299

11.3 Tonic Crystals 301

11.3.1 EwaldSums 302

11.4 Metals 305

11.4.1 Use of Pseudopotentials 307

11.5 Band Structure Energy 308

11.5.1 Peierls Distortion 309

11.5.2 Structural Phase Transitions 311

11.6 Hydrogen-Bonded Solids 312

11.7 Cohesive Energy from Band Calculations 312

11.8 Classical Potentials 313

Problems 315

References 318

12 Elasticity 321

12.1 Introduction 321

12.2 Nonlinear Elasticity 321

12.2.1 Rubber Elasticity 322

12.2.2 Larger Extensions of Rubber 324

12.3 Linear Elasticity 325

12.3.1 Solids of Cubic Symmetry 326

12.3.2 Isotropic Solids 328

12.4 Other Constitutive Laws 332

12.4.1 Liquid Crystals 332

12.4.2 Granular Materials 335

Problems 336

References 339

13 Phonons 341

13.1 Introduction 341

13.2 Vibrations of a Classical Lattice 342

13.2.1 Classical Vibrations in One Dimension 342

13.2.2 Classical Vibrations in Three Dimensions 346

13.2.3 Normal Modes 347

13.2.4 Lattice with a Basis 348

13.3 Vibrations of a Quantum-Mechanical Lattice 351

13.3.1 Phonon Specific Heat 354

13.3.2 Einstein and Debye Models 358

13.3.3 Thermal Expansion 361

13.4 Inelastic Scattering from Phonons 363

13.4.1 Neutron Scattering 364

13.4.2 Formal Theory of Neutron Scattering 366

13.4.3 Averaging Exponentials 370

13.4.4 Evaluation of Structure Factor 372

13.4.5 Kohn Anomalies 373

13.5 The Mössbauer Effect 374

Problems 376

References 377

14 Dislocations and Cracks 379

14.1 Introduction 379

14.2 Dislocations 381

14.2.1 Experimental Observations of Dislocations 383

14.2.2 Force to Move a Dislocation 386

14.2.3 One-Dimensional Dislocations: Frehkel-Kontorova Model 386

14.3 Two-Dimensional Dislocations and Hexatic Phases 389

14.3.1 Impossibility of Crystalline Order in Two Dimensions 389

14.3.2 Orientational Order 391

14.3.3 Kosterlitz-Thouless-Berezinskii Transition 392

14.4 Cracks 399

14.4.1 Fracture of a Strip 399

14.4.2 Stresses Around an Elliptical Hole 402

14.4.3 Stress Intensity Factor 404

14.4.4 Atomic Aspects of Fracture 405

Problems 406

References 409

15 Fluid Mechanics 413

15.1 Introduction 413

15.2 Newtonian Fluids 413

15.2.1 Euler’s Equation 413

15.2.2 Navier-Stokes Equation 415

15.3 Polymeric Solutions 416

15.4 Plasticity 423

15.5 Superfluid ⁴He 427

15.5.1 Two-Fluid Hydrodynamics 430

15.5.2 Second Sound 431

15.5.3 Direct Observation of Two Fluids 433

15.5.4 Origin of Superfluidity 434

15.5.5 Lagrangian Theory of Wave Function 439

15.5.6 Superfluid ³He 442

Problems 443

References 447

IV ELECTRON TRANSPORT 451

16 Dynamics of Bloch Electrons 453

16.1 Introduction 453

16.1.1 Drude Model 453

16.2 Semiclassical Electron Dynamics 455

16.2.1 Bloch Oscillations 456

16.2.2 k-p̂ Method 457

16.2.3 Effective Mass 459

16.3 Noninteracting Electrons in an Electric Field 459

16.3.1 Zener Tunneling 462

16.4 Semiclassical Equations from Wave Packets 465

16.4.1 Formal Dynamics of Wave Packets 465

16.4.2 Dynamics from Lagrangian 467

16.5 Quantizing Semiclassical Dynamics 470

16.5.1 Wannier-Stark Ladders 472

16.5.2 de Haas-van Alphen Effect 473

16.5.3 Experimental Measurements of Fermi Surfaces 474

Problems 477

References 480

17 Transport Phenomena and Fermi Liquid Theory 4S3

17.1 Introduction 483

17.2 Boltzmann Equation 483

17.2.1 Boltzmann Equation 485

17.2.2 Including Anomalous Velocity 486

17.2.3 Relaxation Time Approximation 487

17.2.4 Relation to Rate of Production of Entropy 489

17.3 Transport Symmetries 490

17.3.1 Onsager Relations 491

17.4 Thermoelectric Phenomena 492

17.4.1 Electrical Current 492

17.4.2 Effective Mass and Holes 494

17.4.3 Mixed Thermal and Electrical Gradients 495

17.4.4 Wiedemann-Franz Law 496

17.4.5 Thermopower—Seebeck Effect 497

17.4.6 Peltier Effect 498

17.4.7 Thomson Effect 498

17.4.8 Hall Effect 500

17.4.9 Magnetoresistance 502

17.4.10 Anomalous Hall Effect 503

17.5 Fermi Liquid Theory 504

17.5.1 Basic Ideas 504

17.5.2 Statistical Mechanics of Quasi-Particles 506

17.5.3 Effective Mass 508

17.5.4 Specific Heat 510

17.5.5 Fermi Liquid Parameters 511

17.5.6 Traveling Waves 512

17.5.7 Comparison with Experiment in ³He 515

Problems 516

References 520

18 Microscopic Theories of Conduction 523

18.1 Introduction 523

18.2 Weak Scattering Theory of Conductivity 523

18.2.1 Genera] Formula for Relaxation Time 523

18.2.2 Matthiessen’s Rule 528

18.2.3 Fluctuations 529

18.3 Metal-Insulator Transitions in Disordered Solids 530

18.3.1 Impurities and Disorder 530

18.3.2 Non-Compensated Impurities and the Mott Transition . . 531

18.4 Compensated Impurity Scattering and Green’s Functions 534

18.4.1 Tight-Binding Models of Disordered Solids 534

18.4.2 Green’s Functions 536

18.4.3 Single Impurity 539

18.4.4 Coherent Potential Approximation 541

18.5 Localization 542

18.5.1 Exact Results in One Dimension 544

18.5.2 Scaling Theory of Localization 547

18.5.3 Comparison with Experiment 551

18.6 Luttinger Liquids 553

18.6.1 Density of States 557

Problems 560

References 564

19 Electronics 567

19.1 Introduction 567

19.2 Metal Interfaces 568

19.2.1 Work Functions 569

19.2.2 Schottky Barrier 570

19.2.3 Contact Potentials 572

19.3 Semiconductors 574

19.3.1 Pure Semiconductors 575

19.3.2 Semiconductor in Equilibrium 578

19.3.3 Intrinsic Semiconductor 580

19.3.4 Extrinsic Semiconductor 581

19.4 Diodes and Transistors 583

19.4.1 Surface States 586

19.4.2 Semiconductor Junctions 587

19.4.3 Boltzmann Equation for Semiconductors 590

19.4.4 Detailed Theory of Rectification 592

19.4.5 Transistor 595

19.5 Inversion Layers 598

19.5.1 Heterostructures 598

f 9,5.2 Quantum Point Contact 600

19.5.3 Quantum Dot 603

Problems 606

References 607

V OPTICAL PROPERTIES 609

20 Phenomenological Theory 611

20.1 Introduction 611

20.2 Maxwell’s Equations 613

20.2.1 Traveling Waves 615

20.2.2 Mechanical Oscillators as Dielectric Function 616

20.3 Kramers-Kronig Relations 618

20.3.1 Application to Optical Experiments 620

20.4 The Kubo-Greenwood Formula 623

20.4.1 Bom Approximation 623

20.4.2 Susceptibility 627

20.4.3 Many-Body Green Functions 628

Problems 628

References 631

21 Optical Properties of Semiconductors 633

21.1 Introduction 633

21.2 Cyclotron Resonance 633

21.2.1 Electron Energy Surfaces 636

21.3 Semiconductor Band Gaps 638

21.3.1 Direct Transitions 638

21.3.2 Indirect Transitions 639

21.4 Excitons 641

21.4.1 Mott-Wannier Excitons 641

21.4.2 Frenkel Excitons 644

21.4.3 Electron-Hole Liquid 645

21.5 Optoelectronics 645

21.5.1 SolarCells 645

21.5.2 Lasers 646

Problems 652

References 656

22 Optical Properties of Insulators 659

22.1 Introduction 659

22.2 Polarization 659

22.2.1 Ferroelectrics 659

22.2.2 Berry phase theory of polarization 661

22.2.3 Clausius-Mossotti Relation 661

22.3 Optical Modes in Ionic Crystals 664

22.3.1 Polaritons 666

22.3.2 Polarons 669

22.3.3 Experimental Observations of Polarons 674

22.4 Point Defects and Color Centers 674

22.4.1 Vacancies 675

22.4.2 F Centers 676

22.4.3 Electron Spin Resonance and Electron Nuclear Double Res-onance 677

22.4.4 Other Centers 679

22.4.5 Franck-Condon Effect 679

22.4.6 Urbach Tails 683

Problems 684

References 686

23 Optical Properties of Metals and Inelastic Scattering 689

23.1 Introduction 689

23.1.1 Plasma Frequency 689

23.2 Metals at Low Frequencies 692

23.2.1 Anomalous Skin Effect 694

23.3 Plasmons 695

23.3.1 Experimental Observation of Plasmons 696

23.4 Interband Transitions 698

23.5 Brillouin and Raman Scattering 701

23.5.1 Brillouin Scattering 702

23.5.2 Raman Scattering 703

23.5.3 Inelastic X-Ray Scattering 703

23.6 Photoemission 703

23.6.1 Measurement of Work Functions 703

23.6.2 Angle-Resolved Photoemission 706

23.6.3 Core-Level Photoemission and Charge-Transfer Insulators 710

Problems 716

References 719

VI MAGNETISM 721

24 Classical Theories of Magnetism and Ordering 723

24.1 Introduction 723

24.2 Three Views of Magnetism 723

24.2.1 From Magnetic Moments 723

24.2.2 From Conductivity 724

24.2.3 From a Free Energy 725

24.3 Magnetic Dipole Moments 727

24.3.1 Spontaneous Magnetization of Ferromagnets 730

24.3.2 Ferrimagnets 731

24.3.3 Antiferromagnets 733

24.4 Mean Field Theory and the Ising Model 734

24.4.1 Domains 736

24.4.2 Hysteresis 739

24.5 Other Order-Disorder Transitions 740

24.5.1 Alloy Superlattices 740

24.5.2 Spin Glasses 743

24.6 Critical Phenomena 743

24.6.1 Landau Free Energy 744

24.6.2 Scaling Theory 750

Problems 754

References 757

25 Magnetism of Ions and Electrons 759

25.1 Introduction 759

25.2 Atomic Magnetism 761

25.2.1 Hund’s Rules 762

25.2.2 Curie’s Law 766

25.3 Magnetism of the Free-El ectron Gas 769

25.3.1 Pauli Paramagnetism 770

25.3.2 Landau Diamagnetism 771

25.3.3 Aharonov-Bohm Effect 774

25.4 Tightly Bound Electrons in Magnetic Fields Ill

25.5 Quantum Hall Effect 780

25.5.1 Integer Quantum Hall Effect 780

25.5.2 Fractional Quantum Hall Effect 785

Problems 791

References 794

26 Quantum Mechanics of Interacting Magnetic Moments 797

26.1 Introduction 797

26.2 Origin of Ferromagnetism 797

26.2.1 Heitler-London Calculation 797

26.2.2 Spin Hamiltonian 802

26.3 Heisenberg Model 802

26.3.1 Indirect Exchange and Superexchange 804

26.3.2 Ground State 805

26.3.3 Spin Waves 805

26.3.4 Spin Waves in Antiferromagnets 808

26.3.5 Comparison with Experiment 811

26.4 Ferromagnetism in Transition Metals 811

26.4.1 Stoner Model 811

26.4.2 Calculations Within Band Theory 813

26.5 Spintronics 815

26.5.1 Giant Magnetoresistance 815

26.5.2 Spin Torque 816

26.6 Kondo Effect 819

26.6.1 Scaling Theory 824

26.7 Hubbard Model 828

26.7.1 Mean-Field Solution 829

Problems 832

References 835

27 Superconductivity 839

27.1 Introduction 839

27.2 Phenomenology of Superconductivity 840

27.2.1 Phenomenological Free Energy 841

27.2.2 Thermodynamics of Superconductors 843

27.2.3 Landau-Ginzburg Free Energy 844

27.2.4 Type I and Type II Superconductors 845

27.2.5 Flux Quantization 850

27.2.6 The Josephson Effect 852

27.2.7 Circuits with Josephson Junction Elements 854

27.2.8 SQUIDS 855

27.2.9 Origin of Josephson’s Equations 856

27.3 Microscopic Theory of Superconductivity 858

27.3.1 Electron-Ion Interaction 859

27.3.2 Instability of the Normal State: Cooper Problem 863

27.3.3 Self-Consistent Ground State 865

27.3.4 Thermodynamics of Superconductors 869

27.3.5 Superconductor in External Magnetic Field 873

27.3.6 Derivation of Meissner Effect 876

27.3.7 Comparison with Experiment 879

27.3.8 High-Temperature Superconductors 881

Problems 888

References 890

APPENDICES 895

A Lattice Sums and Fourier Transforms 897

A. l One-Dimensional Sum 897

A. 2 Area Under Peaks 897

A. 3 Three-Dimensional Sum 898

A. 4 Discrete Case 899

A.5 Convolution 900

A. 6 Using the Fast Fourier Transform 900

References 902

B Variational Techniques 903

B. l Functionals and Functional Derivatives 903

B. 2 Time-Independent Schrodinger Equation 904

B. 3 Time-Dependent Schrodinger Equation 905

B. 4 Method of Steepest Descent 906

References 906

C Second Quantization 907

C. l Rules 907

C. 1.1 States 907

C. l.2 Operators 907

C. l.3 Hamiltonians 908

C.2 Derivations 909

C.2.1 Bosons 909

C.2.2 Fermions 910

Index