Solid State Physics

Gerald Burns IBM Thomas J. Watson Research Center

Yorktown Heights, New York

ЯЙ ACADEMIC PRESS, INC.

Harcourt Brace Jovanovich, Pubiishers Boston San Diego New York

London Sydney Tokyo Toronto

Contents

Symmetry Operations 1-1 A Symmetry Operation 4 1-2 Point Symmetry Operations 5 1-3 The Point Groups of a Molecule 9 1-4 Other Symmetry Operations of Crystals 17

Notes 20 Problems 20

Symmetry Description of Crystals 2-1 Lattice 25 2-2 Primitive Unit Cell 26 2-3 The 7 Crystal Systems 26 2-4 The 14 Bravais Lattices 29 2-5 The 32 Crystallographic Point Groups 35 2-6 Space Groups 38 2-7 Definitions of Directions, Coordinates, and Planes 43

Appendix to Chapter 2 46 Notes 47 Problems 48

Simple Crystal Structures 3-1 Introduction 51 3-2 Several Cubic Symmorphic Structures 51

*1

2

3

The book may be started in Chapter 1, Chapter 6, or Chapter 9.

IX

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3-3 Diamond and Zinc Blende Structures 56 3-4 Point Group of a Space Group (S) 58 3-5 Examples of Defect Structures 60 3-6 Different Points of View of a Structure 61 3-7 Close Packing (and the Hexagonal Close-Packed

Structure) 62 3-8 Volume Effects for Simple Structures 65 3-9 Wurtzite Structure 66 3-10 Site Symmetry (S) 67

Notes 68 Problems 69

T" X-Ray Diffraction 4-1 Electron, Neutron, and X-ray Diffraction 73 4-2 Bragg's Law 75 4-3 The Laue Formulation 77 4-4 Experimental X-ray Diffraction Methods (S) 81

Notes 83 Problems 83

Э Crystal Symmetry and Physical Properties (S) 5-1 Introduction 87 5-2 Neumann's Principle 88 5-3 Tensors 88 5-4 Crystal Symmetry and Physical Properties 90 5-5 Nonlinear Optics 96

Notes 98 Problems 98

О Classification of Solids 6-1 Summary of Chapters 1-3 103 6-2 Introduction to Classification of Solids 112 6-3 Five Types of Bonds 112 6-4 Repulsive Potential Energy 115 6-5 Molecular Bond 118 6-6 Hydrogen Bond (S) 124

Notes 127 Problems 128

CONTENTS

The Ionic Bond 7-1 Transfer of Electrons 131 7-2 Ionic Radii 133 7-3 Typical Structures 134 7-4 Cohesive Energies of Ionic Crystals 138

Notes 143 Problems 144

The Covalent Bond 8-1 Introduction 149 8-2 Bonding and Antibonding 150 8-3 The Hydrogen Molecule 154 8-4 Maximum Overlap 157 8-5 The Formation of a Crystal 164 8-6 "Classical" Semiconductors 168 8-7 Continuous Range of Bonding (S) 175

Appendix 183 Notes 184 Problems 185

Metals PART A DRUDE'S MODEL 191 9-1 Drude's Free Electron Theory 191 9-2 Drude's Assumptions 195 9-3 DC Conductivity 196 9-4 Wiedemann-Franz Law 197 9-5 Frequency-Dependent Conductivity (S, A) 198 9-6 Problems of Drude's Model 201

PART В QUANTUM MECHANICS APPLIED 203 9-7 Eigenfunctions of Free Electrons in a Metal 203 9-8 Fermi Energy, Density of States, and Fermi

Surface 208 9-9 Soft X-rays, Heat Capacities 213 9-10 Fermi-Dirac Statistics 215 9-11 Low Temperature Expansion Using F-D Statistics 216 9-12 Thermal Properties of the Electron Gas 217 9-13 DC Conductivity (with F-D Statistics) 223 9-14 Electron-Electron Collisions (S) 225 9-15 Hall Effect (and Other Magnetic Field Effects) (S) 228

Xll

9-16 Landau Levels (S, A) 233 Notes 235 Problems 236

Band Theory PART A QUALITATIVE DISCUSSION 243 10-1 Nearly Free Electrons 243 10-2 Classifications of Solids 247 10-3 Effective Mass 248

PART В WAVE FUNCTIONS AND ENERGY LEVELS 10-4 Bloch Functions 252 10-5 Nearly Free Electrons 257 10-6 Brillouin Zones 260 10-7 Examples of Brillouin Zones 263 10-8 Wigner-Seitz Approximation — The Binding

Energy (S) 273 10-9 The Tight Binding Approximation (S) 276 10-10 Crystal Momentum 280

PART С SEMICONDUCTORS, REAL BANDS, AND RELATED CONCEPTS 281

10-11 Holes 281 10-12 Band Preliminaries (A) 289 10-13 / (k) for a Two-Dimensional Square Lattice 293 10-14 Body-Centered Cubic Lattice — Sodium (S, A) 302 10-15 Si,Ge, GaAs, andGaP 304 10-16 Carrier Concentration at Thermal Equilibrium 313 10-17 p-n Junctions 323 10-18 Metal-Semiconductor Junctions 334 10-19 The Gunn Effect (S) 337 10-20 Other Topcis(S) 339 10-21 Summary 345

Notes 348 Problems 349

Some Thermal Effects in Solids PART A HEAT CAPACITY 355 11-1 Specific Heat at Constant Volume and Pressure 355 11-2 Energy and Cv from Statistical Mechanics 357 11-3 Classical Results for Cv 360 11-4 Einstein's Model 362 11-5 Debye's Calculation of Cv 365

CONTENTS

PART В EFFECTS ASSOCIATED WITH DISORDER 371 11-6 Orientational Disorder in Molecular and Ionic

Crystals 371 11-7 Polarization by Orientation (S) 379 11-8 Point Imperfections in Crystals 385 11-9 Diffusion (S) 389 11-10 Color Centers in Ionic Crystals (S) 396 11-11 Localized Vibrational Modes (S) 398

Notes 399 Problems 401

Lattice Vibrations 12-1 Introduction 407 12-2 Vibrations of a One-Dimensional Monatomic

Chain 408 12-3 Vibrations of a One-Dimensional Diatomic Chain 412 12-4 Real Crystal Systems 419 12-5 Phonons (A) 425 12-6 Crystal Momentum (A) 428 12-7 Neutron Diffraction from Phonons 430 12-8 Thermal Conductivity (S) 433

Notes 441 Problems 443

Optical Properties of Crystals PART A MACROSCOPIC THEORY 450 13-1 Dielectric Polarization 450 13-2 Oscillating Fields 452 13-3 Electromagnetic Waves in Solids 454 13-4 Reflectivity at an Interface 457 13-5 Kramers-Kronig Relations (S, A) 458 13-6 Damped Harmonic Oscillator 461 13-7 Dielectric Response of a Quantum System 464

PART В LATTICE VIBRATIONS 465 13-8 Introduction 465 13-9 Long Wavelength Optical Vibrations 466 13-10 Measurements and Results 471 13-11 Polaritons (S) 476 13-12 A Microscopic Model (S) 480 13-13 Clausius-Mossotti (Lorenz-Lorentz) Equations (S) 484

XIV CONTENTS

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15

PART С FREE CARRIER ABSORPTION 486 13-14 Introduction 486 13-15 Oscillator Model 487 13-16 Experimental Results 490 13-17 Transverse and Longitudinal Free Electron

Modes (S) 495

PARTD INTERBAND TRANSITIONS 498 13-18 Introduction 498 13-19 Fundamental Absorption Near £% 500 13-20 Excitons (Mostly Weakly Bound Excitons) 509 13-21 Fundamental Absorption Above it 520 13-22 UrbachEdge(S) 521

Notes 524 Problems 526

Ferroelectricity and Structural Phase Transitions 14-1 Introduction 531 14-2 The Free Energy 536 14-3 Soft Modes 542 14-4 Microscopic Model of Soft Modes 550 14-5 Renormalization Group 552 14-6 Optical Properties of Ferroelectrics (S) 554 14-7 Other Related Properties

Notes 559 Problems 562

Magnetism PART A DIAMAGNETISM AND PARAMAGNETISM 565

15-1 Introduction 565 15-2 Diamagnetism 567 15-3 Paramagnetism 569

PART В FERROMAGNETISM, ANTIFERRO-MAGNETISM, AND RELATED TOPICS 584

15-4 Introduction 584 15-5 Molecular Field Theory 584 15-6 The Heisenberg Exchange Interaction 588 15-7 Magnetic Structures 590 15-8 Special Techniques Used to Study Magnetic

Structures 596

CONTENTS

PART С OTHER TOPICS 604 15-9 Spin Waves (S, A) 604 15-10 Anisotropy, Hysteresis, Domains, and Bloch Walls 15-11 Metals and Magnetism (S, A) 619 15-12 Spin Glasses (S) 625

Notes 627 Problems 629

Superconductivity 16-1 Introduction (dc Conductivity) 633 16-2 The Occurrence of Superconductivity 634 16-3 Effects that Destroy Superconductivity 635 16-4 Magnetic Properties 637 16-5 The BCS Theory 642 16-6 BCS Predictions 648 16-7 BCS Related Measurements 653 16-8 The Josephson Effect 659

Notes 666 Problems 668

Surface Science 17-1 Introduction — The Need for UH V 674 17-2 Crystal Shape 675 17-3 Preparation of Clean Surfaces and LEED 677 17-4 The Structure of Surfaces 679 17-5 Interaction of Gases with Surfaces 686 17-6 Surface Related Techniques 692 17-7 Electronic Surface Structure 702

Notes 710 Problems 712 Appendix to Chapter 17 712

Artificial Structures PART A SEMICONDUCTORS 716 18-1 Introduction 716 18-2 A Particle in a 1 -D Rectangular Well 717 18-3 3-D Motion with a 1 -D Rectangular Well 719 18-4 Experimental Aspects 724 18-5 Semiconductor Superlattices 726 18-6 Inversion Layers 737

xvi CONTENTS

PART В METALS 747 18-7 Introduction 747 18-8 Sample Preparation 748 18-9 Properties of Layered Metal Structures 749 18-10 Other Artificial Structures (S) 752

Notes 753 Problems 755

Appendix 757

Appendix — Units 792

Bibliography 796 ,

Index 800