Introduction to Solid State Physics, 8th Edition

Chapter 1: Crystal Structure 1

Periodic Array of Atoms 3

Lattice Translation Vectors 4

Basis and the Crystal Structure 5

Primitive Lattice Cell 6

Fundamental Types of Lattices 6

Two-Dimensional Lattice Types 8

Three-Dimensional Lattice Types 9

Index Systems for Crystal Planes 11

Simple Crystal Structures 13

Sodium Chloride Structure 13

Cesium Chloride Structure 14

Hexagonal Close-Packed Structure (hcp) 15

Diamond Structure 16

Cubic Zinc Sulfide Structure 17

Direct Imaging of Atomic Structure 18

Nonideal Crystal Structures 18

Random Stacking and Polytypism 19

Crystal Structure Data 19

Summary 22

Problems 22

Chapter 2: Wave Diffraction and the Reciprocal Lattice 23

Diffraction of Waves by Crystals 25

Bragg Law 25

Scattered Wave Amplitude 26

Fourier Analysis 27

Reciprocal Lattice Vectors 29

Diffraction Conditions 30

Laue Equations 32

Brillouin Zones 33

Reciprocal Lattice to sc Lattice 34

Reciprocal Lattice to bcc Lattice 36

Reciprocal Lattice to fcc Lattice 37

Fourier Analysis of the Basis 39

Structure Factor of the bcc Lattice 40

Structure factor of the fcc Lattice 40

Atomic Form Factor 41

Summary 43

Problems 43

Chapter 3: Crystal Binding and Elastic Constants 47

Crystals of Inert Gases 49

Van der Waals—London Interaction 53

Repulsive Interaction 56

Equilibrium Lattice Constants 58

Cohesive Energy 59

Ionic Crystals 60

Electrostatic or Madelung Energy 60

Evaluation of the Madelung Constant 64

Covalent Crystals 67

Metals 69

Hydrogen Bonds 70

Atomic Radii 70

Ionic Crystal Radii 72

Analysis of Elastic Strains 73

Dilation 75

Stress Components 75

Elastic Compliance and Stiffness Constants 77

Elastic Energy Density 77

Elastic Stiffness Constants of Cubic Crystals 78

Bulk Modulus and Compressibility 80

Elastic Waves in Cubic Crstals 80

Waves in the [100] Direction 81

Waves in the [110] Direction 82

Summary 85

Problems 85

Chapter 4: Phonons I. Crystal Vibrations 89

Vibrations of Crystals with Monatomic Basis 91

First Brillouin Zone 93

Group Velocity 94

Long Wavelength Limit 94

Derivation of Force Constants from Experiment 94

Two Atoms per Primitive Basis 95

Quantization of Elastic Waves 99

Phonon Momentum 100

Inelastic Scattering by Phonons 100

Summary 102

Problems 102

Chapter 5: Phonons 11. Thermal Properties 105

Phonon Heat Capacity 107

Planck Distribution 107

Normal Mode Enumeration 108

Density of States in One Dimension 108

Density of States in Three Dimensions 111

Debye Model for Density of States 112

Debye T₃ Law 114

Einstein Model of the Density of States 114

General Result for D(w) 117

Anharmonic Crystal Interactions 119

Thermal Expansion 120

Thermal Conductivity 121

Thermal Resistivity of Phonon Gas 123

Umklapp Processes 125

Imperfecions 126

Problems 128

Chapter 6: Free Electron Fermi Gas 131

Energy Levels in One Dimension 134

Effect of Temperature on the Fermi-Dirac Distribution 136

Free Electron Gas in Three Dimensions 137

Heat Capacity of the Electron Gas 141

Experimental Heat Capacity of Metals 145

Heavy Fermions 147

Electrical Conductivity and Ohm’s Law 147

Experimental Electrical Resistivity of Metals 148

Umklapp Scattering 151

Motion in Magnetic Fields 152

Hall Effect 153

Thermal Conductivity of Metals 156

Ratio of Thermal to Electrical Conductivity 156

Problems 157

Chapter 7: Energy Bands 161

Nearly Free Electron Model 164

Origin of the Energy Gap 165

Magnitude of the Energy Gap 167

Bloch Functions 167

Kronig-Penney Model 168

Wave Equation of Electron in a Periodic Potential 169

Restatement of the Bloch Theorem 173

Crystal Momentum of an Electron 173

Solution of the Central Equation 174

Kronig-Penney Model in Reciprocal Space 174

Empty Lattice Approximation 176

Approximate Solution Near a Zone Boundary 177

Number of Orbitals in a Band 180

Metals and Insulators 181

Summary 182

Problems 182

Chapter 8: Semiconductor Crystals 185

Band Gap 187

Equations of Motion 191

Physical Derivation of ħk̇ = F 193

Holes 194

Effective Mass 197

Physical Interpretation of the Effective Mass 198

Effective Masses in Semiconductors 200

Silicon and Germanium 202

Intrinsic Carrier Concentration 205

Intrinsic Mobility 208

Impurity Conductivity 209

Donor States 209

Acceptor States 211

Thermal Ionization of Donors and Acceptors 213

Thermoelectric Effects 214

Semimetals 215

Superlattices 216

Bloch Oscillator 217

Zener Tunneling 217

Summary 217

Problems 218

Chapter 9: Fermi Surfaces and Metals 221

Reduced Zone Scheme 223

Periodic Zone Scheme 225

Construction of Fermi Surfaces 226

Nearly Free Electrons 228

Electron Orbits, Hole Orbits, and Open Orbits 230

Calculation of Energy Bands 232

Tight Binding Method of Energy Bands 232

Wigner-Seitz Method 236

Cohesive Energy 237

Pseudopotential Methods 239

Experimental Methods in Fermi Surface Studies 242

Quantization of Orbits in a Magnetic Field 242

De Haas-van Alphen Effect 244

Extremal Orbits 248

Fermi Surface of Copper 249

Magnetic Breakdown 251

Summary 252

Problems 252

Chapter 10: Superconductivity 257

Experimental Survey 259

Occurrence of Superconductivity 260

Destruction of Superconductivity of Magnetic Fields 262

Meissner Effect 262

Heat Capacity 264

Energy Gap 266

Microwave and Infrared Properties 268

Isotope Effect 269

Theoretical Survey 270

Thermodynamics of the Superconducting Transition 270

London Equation 273

Coherence Length 276

BCS Theory of Superconductivity 277

BCS Ground State 278

Flux Quantization in a Superconducting Ring 279

Duration of Persistent Currents 282

Type II Superconductors 283

Vortex State 284

Estimation of H_c1 and H_c2 284

Single Particle Tunneling 287

Josephson Superconductor Tunneling 289

Dc Josephson Effect 289

Ac Josephson Effect 290

Macroscopic Quantum Interference 292

High-Temperature Superconductors 293

Summary 294

Problems 294

Reference 296

Chapter 11: Diamagnetism and Paramagnetism 297

Langevin Diamagnetism Equation 299

Quantum Theory of Diamagnetism of

Mononuclear Systems 301

Paramagnetism 302

Quantum Theory of Paramagnetism 302

Rare Earth Ions 305

Hund Rules 306

Iron Group Ions 307

Crystal Field Splitting 307

Quenching of the Orbital Angular Momentum 308

Spectroscopic Splitting Factor 311

Van Vleck Temperature-Independent Paramagnetism 311

Cooling by Isentropic Demagnetization 312

Nuclear Demagnetization 314

Paramagnetic Susceptibility of Conduction Electrons 315

Summary 317

Problems 318

Chapter 12: Ferromagnetism and Antiferromagnetism 321

Ferromagnetic Order 323

Curie Point and the Exchange Integral 323

Temperature Dependence of the Saturation Magnetization 326

Saturation Magnetization at Absolute Zero 328

Magnons 330

Quantization of Spin Waves 333

Thermal Excitation of Magnons 334

Neutron Magnetic Scattering 335

Ferrimagnetic Order 336

Curie Temperature and Susceptibility of Ferrimagnets 338

Iron Garnets 339

Antiferromagnetic Order 340

Susceptibility Below the Néel Temperature 343

Antiferromagnetic Magnons 344

Ferromagnetic Domains 346

Anisotropy Energy 348

Transition Region between Domains 349

Origin of Domains 351

Coercivity and Hysteresis 352

Single Domain Particles 354

Geomagnetism and Biomagnetism 355

Magnetic Force Microscopy 355

Summary 356

Problems 357

Chapter 13: Magnetic Resonance 361

Nuclear Magnetic Resonance 363

Equations of Motion 366

Line Width 370

Motional Narrowing 371

Hyperfine Splitting 373

Examples: Paramagnetic Point Defects 375

F Centers in Alkali Halides 376

Donor Atoms in Silicon 376

Knight Shift 377

Nuclear Quadrupole Resonance 379

Ferromagnetic Resonance 379

Shape Effects in FMR 380

Spin Wave Resonance 382

Antiferromagnetic Resonance 383

Electron Paramagnetic Resonance 386

Exchange Narrowing 386

Zero-field Splitting 386

Principle of Maser Action 386

Three-Level Maser 388

Lasers 389

Summary 390

Problems 391

Chapter 14: Plasmons, Polaritons, and Polarons 393

Dielectric Function of the Electron Gas 395

Definitions of the Dielectric Function 395

Plasma Optics 396

Dispersion Relation for Electromagnetic Waves 397

Transverse Optical Modes in a Plasma 398

Transparency of Metals in the Ultraviolet 398

Longitudinal Plasma Oscillations 398

Plasmons 401

Electrostatic Screening 403

Screened Coulomb Potential 406

Pseudopotential Component U(0) 407

Mott Metal-Insulator Transition 407

Screening and Phonons in Metals 409

Polaritons 410

LST Relation 414

Electron-Electron Interaction 417

Fermi Liquid 417

Electron-Electron Collisions 417

Electron-Phonon Interaction: Polarons 420

Peierls Instability of Linear Metals 422

Summary 424

Problems 424

Chapter 15: Optical Processes and Excitons 427

Optical Reflectance 429

Kramers-Kronig Relations 430

Mathematical Note 432

Example: Conductivity of collisionless Electron Gas 433

Electronic Interband Transitions 434

Excitons 435

Frenkel Excitons 437

Alkali Halides 440

Molecular Crystals 440

Weakly Bound (Mott-Wannier) Excitons 441

Exciton Condensation into Electron-Hole Drops (EHD) 441

Raman Effects in Crystals 444

Electron Spectroscopy with X-Rays 447

Energy Loss of Fast Particles in a Solid 448

Summary 449

Problems 450

Chapter 16: Dielectrics And Ferroelectrics 453

Maxwell Equations 455

Polarization 455

Macroscopic Electric Field 456

Depolarization Field, E₁ 458

Local Electric Field at an Atom 460

Lorentz Field, E₂ 462

Field of Dipoles Inside Cavity, E₃ 462

Dielectric Constant and Polarizability 463

Electronic Polarizability 464

Classical Theory of Electronic Polarizability 466

Structural Phase Transitions 467

Ferroelectric Crystals 467

Classification of Ferroelectric Crystals 469

Displacive Transitions 471

Soft Optical Phonons 473

Landau Theory of the Phase Transition 474

Second-Order Transition 475

First-Order Transition 477

Antiferroelectricity 479

Ferroelectric Domains 479

Piezoelectricity 481

Summary 482

Problems 483

Chapter 17: Surface and Interface Physics 487

Reconstruction and Relaxation 489

Surface Crystallography 490

Reflection High-Energy Electron Diffraction 493

Surface Electronic Structure 494

Work Function 494

Thermionic Emission 495

Surface States 495

Tangential Surface Transport 497

Magnetoresistance in a Two-Dimensional Channel 498

Integral Quantized Hall Effect (IQHE) 499

IQHE in Real Systems 500

Fractional Quantized Hall Effect (FQHE) 503

p-n Junctions 503

Rectification 504

Solar Cells and Photovoltaic Detectors 506

Schottky Barrier 506

Heterostructures 507

n-N Heterojunction 508

Semiconductor Lasers 510

Light-Emitting Diodes 511

Problems 513

Chapter 18: Nanostructures 515

Imaging Techniques for Nanostructures 519

Electron Microscopy 520

Optical Microscopy 521

Scanning Tunneling Microscopy 523

Atomic Force Microscopy 526

Electronic Structure of 1D Systems 528

One-Dimensional Subbands 528

Spectroscopy of Van Hove Singularities 529

1D Metals — Coluomb Interactions and Lattice Copulings 531

Electrical Transport in 1D 533

Conductance Quantization and the Landauer Formula 533

Two Barriers in Series-resonant Tunneling 536

Incoherent Addition and Ohm’s Law 538

Localization 539

Voltage Probes and the Buttiker-Landauer

Formalism 540

Electronic Structure of 0D Systems 545

Quantized Energy Levels 545

Semiconductor Nanocrystals 545

Metallic Dots 547

Discrete Charge States 549

Electrical Transport in 0D 551

Coulomb Oscillations 551

Spin, Mott Insulators, and the Kondo Effect 554

Cooper Pairing in Superconducting Dots 556

Vibrational and Thermal Properties of Nanostructures 557

Quantized Vibrational Modes 557

Transverse Vibrations 559

Heat Capacity and Thermal Transport 561

Summary 562

Problems 562

Chapter 19: Noncrystalline Solids 565

Diffraction Pattern 567

Monatomic Amorphous Materials 568

Radial Distribution Function 569

Structure of Vitreous Silica, SiO₂ 570

Glasses 573

Viscosity and the Hopping Rate 574

Amorphous Ferromagnets 575

Amorphous Semiconductors 577

Low Energy Excitations in Amorphous Solids 578

Heat Capacity Calculation 578

Thermal Conductivity 579

Fiber Optics 581

Rayleigh Attenuation 582

Problems 582

Chapter 20: Point Defects 583

Lattice Vacancies 585

Diffusion 588

Metals 591

Color Centers 592

F Centers 592

Other Centers in Alkali Halides 593

Problems 595

Chapter 21: Dislocations 597

Shear Strength of Single Crystals 599

Slip 600

Dislocations 601

Burgers Vectors 604

Stress Fields of Dislocations 605

Low-angle Grain Boundaries 607

Dislocation Densities 610

Dislocation Multiplication and Slip 611

Strength of Alloys 613

Dislocations and Crystal Growth 615

Whiskers 616

Hardness of Materials 617

Problems 618

Chapter 22: Alloys 619

General Considerations 621

Substitutional Solid Solutions—Hume-Rothery Rules 624

Order-Disorder Transformation 627

Elementary Theory of Order 629

Phase Diagrams 632

Eutectics 632

Transition Metal Alloys 634

Electrical Conductivity 636

Kondo Effect 637

Problems 640

Appendix A: Temperature Dependence of the Reflection Lines 641

Appendix B: Ewald Calculation of Lattice Sums 644

Ewald-Kornfeld Method for Lattice Sums for Dipole Arrays 647

Appendix C: Quantization of Elastic Waves: Phonons 648

Phonon Coordinates 649

Creation and Annihilation Operators 651

Appendix D: Fermi-Dirac Distribution Function 652

Appendix E: Derivation of the dk/dt Equation 655

Appendix F: Boltzmann Transport Equation 656

Particle Diffusion 657

Classical Distribution 658

Fermi-Dirac Distribution 659

Electrical Conductivity 661

Appendix G: Vector Potential, Field Momentum, and Gauge Transformations 661

Lagrangian Equations of Motion 662

Derivation of the Hamiltonian 663

Field Momentum 663

Gauge Transformation 664

Gauge in the London Equation 665

Appendix H: Cooper Pairs 665

Appendix I: Ginzburg-Landau Equation 667

Appendix J: Electron-Phonon Collisions 671

Index 675