Condensed Matter Physics

Transcription

1 Condensed Matter Physics Michael P. Marder University of Texas at Austin A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 v Preface References xix xxii I ATOMICSTRUCTURE 1 1 The Idea of Crystals Introduction Why are Solids Crystalline? Two-Dimensional Lattices Bravais Lattices Enumeration of Two-Dimensional Bravais Lattices Lattices with Bases Primitive Cells Wigner-Seitz Cells Symmetries The Space Group Translation and Point Groups 11 Problems 13 References 15 2 Three-Dimensional Lattices Introduction Distribution Among Elements Monatomic Lattices The Simple Cubic Lattice The Face-Centered Cubic Lattice The Body-Centered Cubic Lattice The Hexagonal Lattice The Hexagonal Close-Packed Lattice The Diamond Lattice Compounds Rocksalt Sodium Chloride Cesium Chloride Fluorite Calcium Fluoride 26

3 2.3.4 Zincblende Zinc Sulfide Wurtzite Zinc Oxide Perovskite Calcium Titanate Classification of Lattices by Symmetry Fourteen Bravais Lattices and Seven Crystal Systems Symmetries of Lattices with Bases Thirty-Two Crystallographic Point Groups Two Hundred Thirty Distinct Lattices Some Macroscopic Implications of Microscopic Symmetries Pyroelectricity Piezoelectricity Optical Activity 38 Problems 38 References 41 Experimental Determination of Crystal Structures Introduction Theory of Scattering from Crystals Lattice Sums Reciprocal Lattice Miller Indices Scattering from a Lattice with a Basis Experimental Methods LaueMethod Rotating Crystal Method Powder Method Further Features of Scattering Experiments Interaction of X-Rays with Matter Productionof X-Rays Neutrons Electrons Deciphering Complex Structures Accuracy of Structure Determinations 64 Problems 65 References 67 Surfaces and Interfaces Introduction Geometry of Interfaces Coherent and Commensurate Interfaces Stacking Period and Interplanar Spacing Other Topics in Surface Structure Experimental Observation and Creation of Surfaces Low-Energy Electron Diffraction (LEED) 74

4 vii Reflection High-Energy Electron Diffraction (RHEED) Molecular Beam Epitaxy (MBE) Field Ion Microscopy (FIM) Scanning Tunneling Microscopy (STM) Atomic Force Microscopy (AFM) High Resolution Electron Microscopy (HREM) 82 Problems 82 References 85 5 Complex Structures Introduction Alloys Equilibrium Structures Phase Diagrams Superlattices Phase Separation Nonequilibrium Structures in Alloys Dynamics of Phase Separation Simulations Monte Carlo Molecular Dynamics Liquids Correlation Functions Extended X-Ray Absorption Fine Structure (EXAFS) Calculating Correlation Functions Glasses Liquid Crystals Nematics, Cholesterics, and Smectics Liquid Crystal Order Parameter Polymers Ideal Radius of Gyration Quasicrystals One-Dimensional Quasicrystal Two-Dimensional Quasicrystals Penrose Tiles Experimental Observations Fullerenes 124 Problems 125 References 129 II ELECTRONIC STRUCTURE The Single-Electron Model Introduction 135

5 viii 6.2 The Basic Hamiltonian Densities of States Definition of Density of States D Results for Free Electrons Statistical Mechanics of Noninteracting Electrons Sommerfeld Expansion Specific Heat of Noninteracting Electrons at Low Temperatures 149 Problems 150 References The Schrödinger Equation and Symmetry Introduction Translational Symmetry Bloch's Theorem Van Hove Singularities Fourier Analysis of Bloch's Theorem Kronig-Penney Model Rotational Symmetry Group Representations Classes and Characters Consequences of point group symmetries for Schrödinger's equation 178 Problems 181 References Nearly Free and Tightly Bound Electrons Introduction Nearly Free Electrons Degenerate Perturbation Theory Brillouin Zones Nearly Free Electron Fermi Surfaces Tightly Bound Electrons Wannier Functions Tight Binding Model Problems 199 References Electron-Electron Interactions Introduction Hartree and Hartree-Fock Equations Variational Principle Hartree-Fock Equations Numerical Implementation Hartree-Fock Equations for Jellium Density Functional Theory 214

6 ix Thomas-Fermi Theory Kohn-Sham Equations Stabilityof Matter 220 Problems 223 References CalculationofBandStructures Introduction Numerical Methods Pseudopotentials and Orthogonalized Planes Waves (OPW) Linear Combination of Atomic Orbitals (LCAO) Plane Waves Linear Augmented Plane Waves (LAPW) Linearized Muffin Tin Orbitals (LMTO) Definition of Metals, Insulators, and Semiconductors Brief Surveyofthe Periodic Table Noble Gases Nearly Free Electron Metals Semiconductors Transition Metals RareEarths 252 Problems 254 References 258 III MECHANICAL PROPERTIES CohesionofSolids Introduction Radiiof Atoms Noble Gases Ionic Crystals Ewald Sums Metals Use of Pseudopotentials Band Structure Energy Peierls Distortion Structural Phase Transitions Hydrogen-Bonded Solids Cohesive Energy from Band Calculations Classical Potentials 282 Problems 283 References 285

7 x 12 Elasticity Introduction General Theory of Linear Elasticity SolidsofCubicSymmetry Isotropie Solids Other Constitutive Laws Liquid Crystals Rubber Composite and Granulär Materials 301 Problems 301 References Phonons Introduction Vibrations of a Classical Lattice Normal Modes Lattice with a Basis Vibrations of a Quantum-Mechanical Lattice Phonon Specific Heat Einstein and Debye Models Thermal Expansion Inelastic Scattering from Phonons Neutron Scattering Formal Theory of Neutron Scattering Averaging Exponentials Evaluation of Structure Factor Kohn Anomalies The Mössbauer Effect 336 Problems 339 References Dislocations and Cracks Introduction Dislocations Experimental Observations of Dislocations Force to Move a Dislocation One-Dimensional Dislocations: Frenkel-Kontorova Model Two-Dimensional Dislocations and Hexatic Phases Impossibility of Crystalline Order in Two Dimensions Orientational Order Kosterlitz-Thouless-Berezinskii Transition Cracks Fracture of a Strip Stresses Around an Elliptical Hole 366

8 XI Stress Intensity Factor Atomic Aspects of Fracture 368 Problems 370 References Fluid Mechanics Introduction Newtonian Fluids Euler's Equation Navier-Stokes Equation Polymerie Solutions Plasticity Superfluid 4 He Two-Fluid Hydrodynamics Second Sound Origin of Superfluidity Lagrangian Theory of Wave Function Superfluid 3 He 403 Problems 404 References 408 IV ELECTRON TRANSPORT Dynamics of Bloch Electrons Introduction Drude Model Semiclassical Electron Dynamics Bloch Oscillations PMethod Effective Mass Noninteracting Electrons in an Electric Field Zener Tunneling Semiclassical Equations from Wave Packets Formal Dynamics of Wave Packets Quantizing Semiclassical Dynamics Wannier-Stark Ladders de Haas-van Alphen Effect Experimental Measurements of Fermi Surfaces 434 Problems 437 References 440

9 xii 17 Transport Phenomena and Fermi Liquid Theory Introduction Boltzmann Equation Boltzmann Equation Relaxation Time Approximation Relation to Rate of Production of Entropy Transport Symmetries Onsager Relations Thermoelectric Phenomena Electrical Current Effective Mass and Holes Mixed Thermal and Electrical Gradients Wiedemann-Franz Law Thermopower Seebeck Effect Peltier Effect Thomson Effect Hall Effect Magnetoresistance Giant Magnetoresistance Fermi Liquid Theory Basic Ideas Statistical Mechanics of Quasi-Particles Effective Mass Specific Heat Fermi Liquid Parameters Traveling Waves Comparison with Experiment in 3 He 473 Problems 474 References Microscopic Theories of Conduction Introduction Weak Scattering Theory of Conductivity General Formula for Relaxation Time Matthiessen's Rule Fluctuations Metal-Insulator Transitions Typesoflmpurities Impurity Scattering and Green's Functions Green's Functions Single Impurity Coherent Potential Approximation Localization Exact Results in One Dimension 501

10 xiii Scaling Theory of Localization Comparison with Experiment 509 Problems 510 References Electronics Introduction Metal Interfaces Work Functions Schottky Barrier Contact Potentials Semiconductors Pure Semiconductors Semiconductor in Equilibrium Intrinsic Semiconductor Extrinsic Semiconductor Diodes and Transistors Surface States Semiconductor Junctions Boltzmann Equation for Semiconductors Detailed Theory of Rectification Transistor Inversion Layers Heterostructures Quantum Point Contact Quantum Dot 553 Problems 556 References 557 V OPTICAL PROPERTIES Phenomenological Theory Introduction Maxwell's Equations Traveling Waves Mechanical Oscillators as Dielectric Function Kramers-Kronig Relations Application to Optical Experiments The Kubo-Greenwood Formula Born Approximation Susceptibility Many-Body Green Functions 578 Problems 578

11 xiv References Optical Properties of Semiconductors Introduction Cyclotron Resonance Electron Energy Surfaces Semiconductor Band Gaps Direct Transitions Indirect Transitions Excitons Mott-Wannier Excitons Frenkel Excitons Electron-Hole Liquid Optoelectronics Solar Cells Lasers 596 Problems 602 References Optical Properties of Insuiators Introduction Polarization Ferroelectrics Clausius-Mossotti Relation Optical Modes in Ionic Crystals Polaritons Polarons Experimental Observations of Polarons Point Defects and Color Centers Vacancies F Centers Electron Spin Resonance and Electron Nuclear Double Resonance Other Centers Franck-Condon Effect UrbachTails 632 Problems 633 References Optical Properties of Metals and Inelastic Scattering Introduction Plasma Frequency Metals at Low Frequencies Anomalous Skin Effect 642

12 xv 23.3 Plasmons Experimental Observation of Plasmons Interband Transitions Brillouin and Raman Scattering Brillouin Scattering Raman Scattering Inelastic X-Ray Scattering Photoemission Measurement of Work Functions Angle-Resolved Photoemission Core-Level Photoemission and Charge-Transfer Insulators 658 Problems 664 References 667 VI MAGNETISM Classical Theories of Magnetism and Ordering Introduction Three Views of Magnetism From Magnetic Moments From Conductivity From a Free Energy Magnetic Dipole Moments Spontaneous Magnetization of Ferromagnets Ferrimagnets Antiferromagnets Mean Field Theory and the Ising Model Domains Hysteresis Other Order-Disorder Transitions Alloy Superlattices SpinGlasses Critical Phenomena Landau Free Energy Scaling Theory 698 Problems 702 References Magnetism of Ions and Electrons Introduction Atomic Magnetism Hund'sRules Curie's Law 714

13 xvi 25.3 Magnetism of the Free-Electron Gas Pauli Paramagnetism Landau Diamagnetism Aharonov-Bohm Effect Tightly Bound Electrons in Magnetic Fields Quantum Hall Effect Integer Quantum Hall Effect Fractional Quantum Hall Effect 733 Problems 739 References Quantum Mechanics of Interacting Magnetic Moments Introduction Origin of Ferromagnetism Heitler-London Calculation Spin Hamiltonian Heisenberg Model Indirect Exchange and Superexchange Ground State SpinWaves Spin Waves in Antiferromagnets Comparison with Experiment Ferromagnetism in Transition Metals Stoner Model Calculations Within Band Theory Kondo Effect Scaling Theory Hubbard Model Mean-Field Solution 773 Problems 776 References Superconductivity Introduction Phenomenology of Superconductivity Phenomenological Free Energy Thermodynamics of Superconductors Landau-Ginzburg Free Energy Type I and Type II Superconductors Flux Quantization The Josephson Effect Circuits with Josephson Junction Elements SQUIDS Origin of Josephson's Equations 800

14 xvii 27.3 Microscopic Theory of Superconductivity Electron-Ion Interaction Formal Derivation Instability of the Normal State: Cooper Problem Self-Consistent Ground State Thermodynamics of Superconductors Superconductor in External Magnetic Field Derivation of Meissner Effect Comparison with Experiment High-Temperature Superconductors 828 Problems 833 References 837 APPENDICES 841 A Lattice Sums and Fourier Transforms 843 A.l One-Dimensional Sum 843 A.2 Area Under Peaks 843 A.3 Three-Dimensional Sum 844 A.4 Discrete Case 845 A.5 Convolution 846 A.6 Using the Fast Fourier Transform 846 References 848 B Variational Techniques 849 B.l Functionals and Functional Derivatives 849 B.2 Time-Independent Schrödinger Equation 850 B.3 Time-Dependent Schrödinger Equation 851 B.4 Method of Steepest Descent 852 References 852 C Second Quantization 853 C.l Rules 853 C.l.l States 853 C.1.2 Operators 853 C.1.3 Hamiltonians 854 C.2 Derivations 855 C.2.1 Bosons 855 C.2.2 Fermions 856 Index 859