Introduction to solid state physics

Introduction to solid state physics

Introduction to Solid State Physics

Charles Kittel

Professor Emeritus University of California, Berkeley

Chapter 18, Nanostructures, was written by Professor Paul McEuen of Cornell University.

John Wiley & Sons, Inc

Contents

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 1

Simple Crystal Structures 13

Sodium Chloride Structure 13 Cesium Chloride Structure 14 Hexagonal Close-Packed Structure (hep) 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 2 Problems 2

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 3

Reciprocal Lattice to sc Lattice 34 Reciprocal Lattice to bec Lattice 36 Reciprocal Lattice to fee Lattice 37 viii

Fourier Analysis of the Basis 39

Structure Factor of the bcc Lattice 40 Structure factor of the fee Lattice 40 Atomic Form Factor 41

Summary 43 Problems 43

CHAPTER 3: CRYSTAL RINDING 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 7 Elastic Energy Density 7

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 VIRRATIONS 89

Vibrations of Crystals with Monatomic Basis 91

First Brillouin Zone 93 Group Velocity 94

Contents

Long Wavelength Limit 94 Derivation of Force Constants from Experiment 94

Two Atoms per Primitive Basis 95 Quantization of Elastic Waves 9 Phonon Momentum 100 Inelastic Scattering by Phonons 100 Summary 102 Problems 102

CHAPTER 5: PHONONS 1. 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 1 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 Hall Effect

Thermal Conductivity of Metals Ratio of Thermal to Electrical Conductivity

Problems

CHAPTER 7: ENERGY RANDS

Nearly Free Electron Model Origin of the Energy Gap

Magnitude of the Energy Gap

Bloch Functions Kronig-Penney Model

Wave Equation of Electron in a

Periodic Potential Restatement of the Bloch Theorem

Crystal Momentum of an Electron Solution of the Central Equation Kronig-Penney Model in Reciprocal Space Empty Lattice Approximation Approximate Solution Near a Zone Boundary

Number of Orbitals in a Band Metals and Insulators

Summary

Problems

CHAPTER 8: SEMICONDUCTOR CRYSTALS

Band Gap

Equations of Motion Physical Derivation of hi. = F

Holes Effective Mass Physical Interpretation of the Effective Mass Effective Masses in Semiconductors Silicon and Germanium

Intrinsic Carrier Concentration Intrinsic Mobility

Impurity Conductivity

Donor States Acceptor States Thermal Ionization of Donors and Acceptors

Thermoelectric Effects

Semimetajs

Superlattices

Bloch Oscillator Zener Tunneling

Summary

Problems

CHAPTER 9: FERMI SURFACES AND METALS

Reduced Zone Scheme Periodic Zone Scheme

Construction of Fermi Surfaces Nearly Free Electrons

Electron Orbits, Hole Orbits, and Open Orbits

Calculation of Energy Bands

Tight Binding Method of Energy Bands Wigner-Seitz Method Cohesive Energy Pseudopotential Methods

Experimental Methods in Fermi Surface Studies

Quantization of Orbits in a Magnetic Field De Haas-van Alphen Effect

Extremal Orbits Fermi Surface of Copper Magnetic Breakdown

Summary

Problems

CHAPTER 10: SUPERCONDUCTIVITY

Experimental Survey

Occurrence of Superconductivity Destruction of Superconductivity of Magnetic Fields Meissner Effect Heat Capacity Energy Gap Microwave and Infrared Properties Isotope Effect

Theoretical Survey

Thermodynamics of the Superconducting Transition London Equation

Contents

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 I Superconductors 283 Vortex State 284

Estimation of Hcl and Hc2 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 1: 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

Contents

Temperature Dependence of the Saturation Magnetization 326

Saturation Magnetization at Absolute Zero 328

Magnons 330

Quantization of Spin Waves 3 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 Neel 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

CHAPTER 14:

Electron-Paramagnetic Resonance Exchange Narrowing

Zero-field Splitting

Principle of Maser Action Three-Level Maser

Lasers

Summary Problems

Dielectric Function of the Electron Gas

Definitions of the Dielectric Function Plasma Optics Dispersion Relation for Electromagnetic Waves Transverse Optical Modes in a Plasma Transparency of Metals in the Ultraviolet Longitudinal Plasma Oscillations

Plasmons

Electrostatic Screening

Screened Coulomb Potential Pseudopotential Component [7(0) Mott Metal-Insulator Transition Screening and Phonons in Metals

Polaritons LST Relation

Electron-Electron Interaction

Fermi Liquid Electron-Electron Collisions

Electron-Phonon Interaction: Polarons

Peierls Instability of Linear Metals

Summary

Problems

Optical Reflectance

Kramers-Kronig Relations Mathematical Note

Example: Conductivity of collisionless Electron Gas Electronic Interband Transitions

Excitons

Frenkel Excitons Alkali Halides Molecular Crystals Weakly Bound (Mott-Wannier) Excitons

Exciton Condensation into Electron-Hole Drops (EHD)

Raman Effects in Crystals Electron Spectroscopy with X-Rays

Energy Loss of Fast Particles in a Solid Summary Problems

CHAPTER 16: DIELECTRICS AND FERROELECTRICS

Maxwell Equations Polarization

Macroscopic Electric Field Depolarization Field, Ej

Local Electric Field at an Atom

Lorentz Field, E2 Field of Dipoles Inside Cavity, E3

Dielectric Constant and Polarizability

Electronic Polarizability Classical Theory of Electronic Polarizability

Structural Phase Transitions

Ferroelectric Crystals Classification of Ferroelectric Crystals

Displacive Transitions

Soft Optical Phonons

Landau Theory of the Phase Transition Second-Order Transition

First-Order Transition Antiferroelectricity Ferroelectric Domains Piezoelectricity

Summary

Problems

Contents

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

Heterostructure s 50 7 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 ID Systems 528 One-Dimensional Subbands 528

Spectroscopy oi Van Hove Singularities 529

ID Metals — Coluomb Interactions and Lattice Copulings 531

Electrical Transport in ID 533

Conductance Quantization and the Landauer Fonnula 533 Two Barriers in Series-resonant Tunneling 536 Incoherent Addition and Ohm's Law 538

Contents xvii

Localization 539 Voltage Probes and the Buttiker-Landauer

Formalism

Electronic Structure of OD Systems

Quantized Energy Levels Semiconductor Nanocrystals

Metallic Dots Discrete Charge States

Electrical Transport in OD

Coulomb Oscillations Spin, Mott Insulators, and the Kondo Effect Cooper Pairing in Superconducting Dots

Vibrational and Thermal Properties of

Nanostructures Quantized Vibrational Modes

Transverse Vibrations Heat Capacity and Thermal Transport

Summary Problems

CHAPTER 19: NONCRYSTALLINE SOLIDS

Diffraction Pattern

Monatomic Amorphous Materials Radial Distribution Function

Structure of Vitreous Silica, SiO2

Glasses Viscosity and the Hopping Rate

Amorphous Ferromagnets Amorphous Semiconductors

Low Energy Excitations in Amorphous Solids

Heat Capacity Calculation Thermal Conductivity

Fiber Optics Rayleigh Attenuatio n

Problems

CHAPTER 20: POINT DEFECTS Lattice Vacancies

Diffusion Metals

xviii

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 2: 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

Contents

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

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