Kittel′s Introduction to Solid State Physics

Chapter 1: Crystal Structure 1

Periodic Arrays 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 25

Diffraction of Waves by Crystals 27

The Bragg Law 27

Scattered Wave Amplitude 28

Fourier Analysis 29

Reciprocal Lattice Vectors 31

Diffraction Conditions 32

Laue Equations 34

Brillouin Zones 35

Reciprocal Lattice to sc Lattice 36

Reciprocal Lattice to bcc Lattice 38

Reciprocal Lattice to fcc Lattice 39

Fourier Analysis of the Basis 41

Structure Factor of the bcc Lattice 42

Structure Factor of the fcc Lattice 42

Atomic Form Factor 43

Summary 45

Problems 45

Chapter 3: Crystal Binding And Elastic Constants 49

Crystals of Inert Gases 51

Van der Waals–London Interaction 55

Repulsive Interaction 58

Equilibrium Lattice Constants 60

Cohesive Energy 61

Ionic Crystals 62

Electrostatic or Madelung Energy 62

Evaluation of the Madelung Constant 66

Covalent Crystals 69

Metals 71

Hydrogen Bonds 72

Atomic Radii 72

Ionic Crystal Radii 74

Analysis of Elastic Strains 75

Dilation 77

Stress Components 77

Elastic Compliance and Stiffness Constants 79

Elastic Energy Density 79

Elastic Stiffness Constants of Cubic Crystals 80

Bulk Modulus and Compressibility 82

Elastic Waves in Cubic Crystals 82

Waves in the [100] Direction 83

Waves in the [110] Direction 84

Summary 87

Problems 87

Chapter 4: phonons I. Crystal vibrations 91

Vibrations of Crystals with Monatomic Basis 93

First Brillouin Zone 95

Group Velocity 96

Long Wavelength Limit 96

Derivation of Force Constants from Experiment 96

Two Atoms per Primitive Basis 97

Quantization of Elastic Waves 101

Phonon Momentum 102

Inelastic Scattering by Phonons 102

Summary 104

Problems 104

Chapter 5: phonons 11. Thermal properties 107

Phonon Heat Capacity 109

Planck Distribution 109

Normal Mode Enumeration 110

Density of States in One Dimension 110

Density of States in Three Dimensions 113

Debye Model for Density of States 114

Debye T3 Law 116

Einstein Model of the Density of States 116

General Result for D( ) 119

Anharmonic Crystal Interactions 121

Thermal Expansion 122

Thermal Conductivity 123

Thermal Resistivity of Phonon Gas 125

Umklapp Processes 127

Imperfections 128

Problems 130

Chapter 6: Free Electron Fermi Gas 133

Energy Levels in One Dimension 136

Effect of Temperature on the FermiDirac Distribution 138

Free Electron Gas in Three Dimensions 139

Heat Capacity of the Electron Gas 143

Experimental Heat Capacity of Metals 147

Heavy Fermions 149

Electrical Conductivity and Ohm’s Law 149

Experimental Electrical Resistivity of Metals 150

Umklapp Scattering 153

Motion in Magnetic Fields 154

Hall Effect 155

Thermal Conductivity of Metals 158

Ratio of Thermal to Electrical Conductivity 158

Problems 159

Chapter 7: Energy Bands 163

Nearly Free Electron Model 166

Origin of the Energy Gap 167

Magnitude of the Energy Gap 169

Bloch Functions 169

Kronig-Penney Model 170

Wave Equation of Electron in a Periodic Potential 171

Restatement of the Bloch Theorem 175

Crystal Momentum of an Electron 175

Solution of the Central Equation 176

Kronig-Penney Model in Reciprocal Space 176

Empty Lattice Approximation 178

Approximate Solution Near a Zone Boundary 179

Number of Orbitals in a Band 182

Metals and Insulators 183

Summary 184

Problems 184

Chapter 8: Semiconductor Crystals 187

Band Gap 189

Equations of Motion 193

Physical Derivation of 195

Holes 196

Effective Mass 199

Physical Interpretation of the Effective Mass 200

Effective Masses in Semiconductors 202

Silicon and Germanium 204

Intrinsic Carrier Concentration 207

Intrinsic Mobility 210

Impurity Conductivity 211

Donor States 211

Acceptor States 213

Thermal Ionization of Donors and Acceptors 215

Thermoelectric Effects 216

Semimetals 217

Superlattices 218

Bloch Oscillator 219

Zener Tunneling 219

Summary 219

Problems 220

Chapter 9: Fermi Surfaces And Metals 223

Reduced Zone Scheme 225

Periodic Zone Scheme 227

Construction of Fermi Surfaces 228

Nearly Free Electrons 230

Electron Orbits, Hole Orbits, and Open Orbits 232

Calculation of Energy Bands 234

Tight Binding Method for Energy Bands 234

Wigner-Seitz Method 238

Cohesive Energy 239

Pseudopotential Methods 241

Experimental Methods in Fermi Surface Studies 244

Quantization of Orbits in a Magnetic Field 244

De Haas-van Alphen Effect 246

Extremal Orbits 250

Fermi Surface of Copper 251

Magnetic Breakdown 253

Summary 254

Problems 254

Chapter 10: Superconductivity 259

Experimental Survey 261

Occurrence of Superconductivity 262

Destruction of Superconductivity by Magnetic Fields 264

Meissner Effect 264

Heat Capacity 266

Energy Gap 268

Microwave and Infrared Properties 270

Isotope Effect 271

Theoretical Survey 272

Thermodynamics of the Superconducting Transition 272

London Equation 275

Coherence Length 278

BCS Theory of Superconductivity 279

BCS Ground State 280

Flux Quantization in a Superconducting Ring 281

Duration of Persistent Currents 284

Type II Superconductors 285

Vortex State 286

Estimation of Hc1 and Hc2 286

Single Particle Tunneling 289

Josephson Superconductor Tunneling 291

Dc Josephson Effect 291

Ac Josephson Effect 292

Macroscopic Quantum Interference 294

High-Temperature Superconductors 295

Summary 296

Problems 296

Reference 298

Chapter 11: Diamagnetism And Paramagnetism 299

Langevin Diamagnetism Equation 301

Quantum Theory of Diamagnetism of Mononuclear Systems 303

Paramagnetism 304

Quantum Theory of Paramagnetism 304

Rare Earth Ions 307

Hund Rules 308

Iron Group Ions 309

Crystal Field Splitting 309

Quenching of the Orbital Angular Momentum 310

Spectroscopic Splitting Factor 313

Van Vleck Temperature-Independent Paramagnetism 313

Cooling by Isentropic Demagnetization 314

Nuclear Demagnetization 316

Paramagnetic Susceptibility of Conduction Electrons 317

Summary 319

Problems 320

Chapter 12: Ferromagnetism And Antiferromagnetism 323

Ferromagnetic Order 325

Curie Point and the Exchange Integral 325

Temperature Dependence of the Saturation

Magnetization 328

Saturation Magnetization at Absolute Zero 330

Magnons 332

Quantization of Spin Waves 335

Thermal Excitation of Magnons 336

Neutron Magnetic Scattering 337

Ferrimagnetic Order 338

Curie Temperature and Susceptibility of Ferrimagnets 340

Iron Garnets 341

Antiferromagnetic Order 342

Susceptibility Below the Néel Temperature 345

Antiferromagnetic Magnons 346

Ferromagnetic Domains 348

Anisotropy Energy 350

Transition Region Between Domains 351

Origin of Domains 353

Coercivity and Hysteresis 354

Single-Domain Particles 356

Geomagnetism and Biomagnetism 357

Magnetic Force Microscopy 357

Summary 359

Problems 359

Chapter 13: Magnetic Resonance 363

Nuclear Magnetic Resonance 365

Equations of Motion 368

Line Width 372

Motional Narrowing 373

Hyperfine Splitting 375

Examples: Paramagnetic Point Defects 377

F Centers in Alkali Halides 378

Donor Atoms in Silicon 378

Knight Shift 379

Nuclear Quadrupole Resonance 381

Ferromagnetic Resonance 381

Shape Effects in FMR 382

Spin Wave Resonance 384

Antiferromagnetic Resonance 385

Electron Paramagnetic Resonance 388

Exchange Narrowing 388

Zero-field Splitting 388

Principle of Maser Action 388

Three-Level Maser 390

Lasers 391

Summary 392

Problems 393

Chapter 14: Dielectrics And Ferroelectrics 395

Maxwell Equations 397

Polarization 397

Macroscopic Electric Field 398

Depolarization Field, E1 400

Local Electric Field at an Atom 402

Lorentz Field, E2 404

Field of Dipoles Inside Cavity, E3 404

Dielectric Constant and Polarizability 405

Electronic Polarizability 406

Classical Theory of Electronic Polarizability 408

Structural Phase Transitions 409

Ferroelectric Crystals 409

Classification of Ferroelectric Crystals 411

Displacive Transitions 413

Soft Optical Phonons 415

Landau Theory of the Phase Transition 416

Second-Order Transition 417

First-Order Transition 419

Antiferroelectricity 421

Ferroelectric Domains 421

Piezoelectricity 423

Summary 424

Problems 425

Chapter 15: Plasmons, Polaritons, And Polarons 429

Dielectric Function of the Electron Gas 431

Definitions of the Dielectric Function 431

Plasma Optics 432

Dispersion Relation for Electromagnetic Waves 433

Transverse Optical Modes in a Plasma 434

Transparency of Metals in the Ultraviolet 434

Longitudinal Plasma Oscillations 434

Plasmons 437

Electrostatic Screening 439

Screened Coulomb Potential 442

Pseudopotential Component U(0) 443

Mott Metal-Insulator Transition 443

Screening and Phonons in Metals 445

Polaritons 446

LST Relation 450

Electron-Electron Interaction 453

Fermi Liquid 453

Electron-Electron Collisions 453

Electron-Phonon Interaction:

Polarons 456

Peierls Instability of Linear

Metals 458

Summary 460

Problems 460

Chapter 16: Optical Processes And Excitons 465

Optical Reflectance 467

Kramers-Kronig Relations 468

Mathematical Note 470

Example: Conductivity of Collisionless Electron Gas 471

Electronic Interband Transitions 472

Excitons 473

Frenkel Excitons 475

Alkali Halides 478

Molecular Crystals 478

Weakly Bound (Mott-Wannier) Excitons 479

Exciton Condensation into Electron-Hole Drops (EHD) 479

Raman Effect in Crystals 482

Electron Spectroscopy with X-Rays 485

Energy Loss of Fast Particles in a Solid 486

Summary 487

Problems 488

Chapter 17: Surface And Interface Physics 491

Reconstruction and Relaxation 493

Surface Crystallography 494

Reflection High-Energy Electron Diffraction 497

Surface Electronic Structure 498

Work Function 498

Thermionic Emission 499

Surface States 499

Tangential Surface Transport 501

Magnetoresistance in a Two-Dimensional Channel 502

Integral Quantized Hall Effect (IQHE) 503

IQHE in Real Systems 504

Fractional Quantized Hall Effect (FQHE) 507

p-n Junctions 507

Rectification 508

Solar Cells and Photovoltaic Detectors 510

Schottky Barrier 510

Heterostructures 511

n-N Heterojunction 512

Semiconductor Lasers 514

Light-Emitting Diodes 515

Problems 517

Chapter 18: Nanostructures 521

Imaging Techniques for Nanostructures 525

Electron Microscopy 526

Optical Microscopy 527

Scanning Tunneling Microscopy 529

Atomic Force Microscopy 532

Electronic Structure of 1D Systems 534

One-dimensional (1D) Subbands 534

Spectroscopy of Van Hove Singularities 535

1D Metals—Coulomb Interactions and Lattice Couplings 537

Electrical Transport in 1D 539

Conductance Quantization and the Landauer Formula 539

Two Barriers in Series-Resonant Tunneling 542

Incoherent Addition and Ohm’s Law 544

Localization 545

Voltage Probes and the Büttiker-Landauer Formalism 546

Electronic Structure of 0D Systems 551

Quantized Energy Levels 551

Semiconductor Nanocrystals 551

Metallic Dots 553

Discrete Charge States 555

Electrical Transport in 0D 557

Coulomb Oscillations 557

Spin, Mott Insulators, and the Kondo Effect 560

Cooper Pairing in Superconducting Dots 562

Vibrational and Thermal Properties 563

Quantized Vibrational Modes 563

Transverse Vibrations 565

Heat Capacity and Thermal Transport 567

Summary 568

Problems 568

Chapter 19: Noncrystalline Solids 573

Diffraction Pattern 575

Monatomic Amorphous Materials 576

Radial Distribution Function 577

Structure of Vitreous Silica, SiO2 578

Glasses 581

Viscosity and the Hopping Rate 582

Amorphous Ferromagnets 583

Amorphous Semiconductors 585

Low Energy Excitations in Amorphous Solids 586

Heat Capacity Calculation 586

Thermal Conductivity 587

Fiber Optics 589

Rayleigh Attenuation 590

Problems 590

Chapter 20: Point Defects 593

Lattice Vacancies 595

Diffusion 598

Metals 601

Color Centers 602

F Centers 602

Other Centers in Alkali Halides 603

Problems 605

Chapter 21: Dislocations 607

Shear Strength of Single Crystals 609

Slip 610

Dislocations 611

Burgers Vectors 614

Stress Fields of Dislocations 615

Low-angle Grain Boundaries 617

Dislocation Densities 620

Dislocation Multiplication and Slip 621

Strength of Alloys 623

Dislocations and Crystal Growth 625

Whiskers 626

Hardness of Materials 627

Problems 628

Chapter 22: Alloys 631

General Considerations 633

Substitutional Solid Solutions— Hume-Rothery Rules 636

Order-Disorder Transformation 639

Elementary Theory of Order 641

Phase Diagrams 644

Eutectics 644

Transition Metal Alloys 646

Electrical Conductivity 648

Kondo Effect 649

Problems 652

Appendix A: Temperature Dependence Of The Reflection Lines 653

Appendix B: Ewald Calculation Of Lattice Sums 656

Ewald-Kornfeld Method for Lattice Sums for Dipole Arrays 659

Appendix C: Quantization Of Elastic Waves: Phonons 660

Phonon Coordinates 661

Creation and Annihilation Operators 663

Appendix D: Fermi-Dirac Distribution Function 664

Appendix E: Derivation Of The Dk/Dt Equation 667

Appendix F: Boltzmann Transport Equation 668

Particle Diffusion 669

Classical Distribution 670

Fermi-Dirac Distribution 671

Electrical Conductivity 673

Appendix G: Vector Potential, Field Momentum, And Gauge Transformations 673

Lagrangian Equations of Motion 674

Derivation of the Hamiltonian 675

Field Momentum 675

Gauge Transformation 676

Gauge in the London Equation 677

Appendix H: Cooper Pairs 677

Appendix I: Ginzburg-Landau Equation 679

Appendix J: Electron-Phonon Collisions 683

Index 687