THE GUNN-HILSUM EFFECT
CONTENTS PART I BASIC CONCEPTS IN SEMICONDUCTOR PHYSICS SURVEY OF CLASSICAL PHYSICS NEWTON DYNAMICS WORK AND ENERGY HAMILTONIAN FORMULATION OF DYNAMICS CANONICAL TRANSFORMATIONS SMALL OSCILLATIONS MAXWELL EQUATIONS ELECTROMAGNETIC POTENTIALS AND GAUGE TRANSFORMATIONS HAMILTONIAN OF A CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD FUNDAMENTALS OF QUANTUM MECHANICS THE FIRST POSTULATES EQUATIONS OF MOTION PICTURES AND REPRESENTATIONS EVOLUTION OPERATOR AND ITS EQUATION OF MOTION EQUATION OF MOTION IN SCHR¨ODINGER AND HEISENBERG PICTURES INTERACTION PICTURE HEISENBERG UNCERTAINTY RELATIONS HOW TO DEAL WITH A GENERAL QUANTUM-MECHANICAL PROBLEM IN A SYSTEM WITH A CONSTANT HAMILTONIAN THE {Q} REPRESENTATION: WAVE MECHANICS IDENTICAL PARTICLES AND PAULI EXCLUSION PRINCIPLE FUNDAMENTALS OF STATISTICAL PHYSICS INTRODUCTION LIOUVILLE THEOREM THE FUNDAMENTAL HYPOTHESES OF STATISTICAL MECHANICS MAIN DE?NITIONS AND RESULTS OF STATISTICAL MECHANICS THERMAL BATH THE THREE FUNDAMENTAL STATISTICAL ENSEMBLES MICROCANONICAL ENSEMBLE CANONICAL ENSEMBLE GRAND CANONICAL ENSEMBLE EQUILIBRIUM PARTICLE DISTRIBUTIONS IN IDEAL GASES CLASSICAL GAS: MAXWELL–BOLTZMANN DISTRIBUTION BOSE DISTRIBUTIONS FERMI DISTRIBUTION CLASSICAL LIMIT CRYSTAL STRUCTURES CRYSTALS LATTICES CRYSTAL BONDING RECIPROCAL LATTICE PHONONS THE VIBRATING STRING THE SIMPLEST LINEAR CHAIN MONATOMIC LINEAR CHAIN WITH MULTIPLE COUPLING DIATOMIC LINEAR CHAIN THREE-DIMENSIONAL LATTICE VIBRATIONS NORMAL COORDINATES AND QUANTIZATION – PHONONS PHONON MOMENTUM AND CRYSTAL MOMENTUM EXPERIMENTAL DETERMINATION OF PHONON DISPERSIONS BLOCH STATES AND BAND THEORY BLOCH THEOREM DENSITY OF STATES TIGHT-BINDING APPROACH BAND-STRUCTURE CALCULATIONS LCAO METHOD K P METHOD PSEUDOPOTENTIAL METHOD BAND STRUCTURES OF MOST IMPORTANT SEMICONDUCTORS E?ECTIVE-MASS APPROXIMATION BLOCH WAVEPACKETS GROUP VELOCITY E?ECTIVE-MASS THEOREMS, ENVELOPE FUNCTION, AND SEMICLASSICAL DYNAMICS E?ECTIVE-MASS THEOREM FOR BLOCH STATES E?ECTIVE-MASS THEOREM IN PRESENCE OF A SCALAR POTENTIAL ACCELERATED WAVES ACCELERATED CLASSICAL ELECTRONS IN FREE SPACE ACCELERATED QUANTUM ELECTRONS IN FREE SPACE ACCELERATED BLOCH STATES ENVELOPE FUNCTION FOR STEADY STATES E?ECTIVE-MASS THEOREM FOR A WAVEPACKET IN SLOW-VARYING ELECTRIC AND MAGNETIC FIELDS TIME-DEPENDENT ENVELOPE FUNCTION SEMICLASSICAL DYNAMICS SEMICONDUCTORS FREE DYNAMICS OF BLOCH ELECTRONS A FULLY OCCUPIED BAND CANNOT CARRY CURRENT HOLES INSULATORS, CONDUCTORS, SEMICONDUCTORS INTRINSIC AND DOPED SEMICONDUCTORS DONORS AND ACCEPTORS N-DOPED AND P-DOPED SEMICONDUCTORS CHARGE-CARRIER STATISTICS METALS GENERAL MODELS OF BANDS FOR CUBIC SEMICONDUCTORS DI?ERENT TYPES OF E?ECTIVE MASSES HERRING–VOGT TRANSFORMATION NONPARABOLICITY PART II SEMICLASSICAL TRANSPORT IN BULK SEMICONDUCTORS ELECTRONIC INTERACTIONS CLASSI?CATION FUNDAMENTALS OF SCATTERING – CRYSTAL-MOMENTUM CONSERVATION ELECTRON–PHONON SCATTERING RATES – DEFORMATION POTENTIAL ELECTRON INTRAVALLEY SCATTERING BY ACOUSTIC PHONONS ELECTRON INTRAVALLEY SCATTERING BY OPTICAL PHONONS ELECTRON INTERVALLEY SCATTERING HOLE INTRABAND SCATTERING BY ACOUSTIC PHONONS HOLE INTRABAND SCATTERING BY OPTICAL PHONONS HOLE INTERBAND SCATTERING ELECTRON–PHONON SCATTERING RATES – ELECTROSTATIC INTERACTION ACOUSTIC PHONONS – PIEZOELECTRIC INTERACTION OPTICAL PHONONS – POLAR INTERACTION SELECTION RULES IMPURITY SCATTERING IONIZED IMPURITIES NEUTRAL IMPURITIES ALLOY SCATTERING CARRIER–CARRIER INTERACTION RELATIVE IMPORTANCE OF THE DI?ERENT SCATTERING MECHANISMS BOLTZMANN EQUATION 10 1 THE DISTRIBUTION FUNCTION 10 1 1 MEAN QUANTITIES 10 2 ELEMENTARY DERIVATION OF THE BOLTZMANN EQUATION 10 3 THE COLLISION INTEGRAL – DETAILED BALANCE 10 4 MOMENT METHOD 10 4 1 ZERO-ORDER MOMENT: CONTINUITY EQUATION 10 4 2 FIRST-ORDER MOMENT 10 4 3 DRIFT-DI?USION EQUATION 10 4 4 HIGHER-ORDER MOMENTS: HYDRODYNAMIC EQUATIONS 10 5 CHAMBERS’ INTEGRAL EQUATION 10 5 1 PATH VARIABLES 10 5 2 CHAMBERS’ INTEGRAL EQUATION LINEAR TRANSPORT 11 1 LINEARIZATION OF BOLTZMANN EQUATION 11 2 RELAXATION-TIME APPROXIMATION 11 3 LINEAR TRANSPORT PROPERTIES IN A “SIMPLE SEMICONDUCTOR” 11 3 1 OHMIC MOBILITY 11 3 2 MATTHIESSEN RULE 11 3 3 MAGNETOTRANSPORT 11 3 4 HALL E?ECT 11 4 HIGH-MAGNETIC-FIELD E?ECTS 11 5 EVALUATION OF THE MOMENTUM RELAXATION TIMES 11 5 1 RELAXATION TIME FOR VELOCITY-RANDOMIZING COLLISIONS 11 5 2 RELAXATION TIME FOR ELASTIC COLLISIONS 11 6 MOBILITIES 11 6 1 ACOUSTIC–PHONON SCATTERING, DEFORMATION POTENTIAL, ELASTIC 11 6 2 OPTICAL–PHONON SCATTERING, DEFORMATION POTENTIAL 11 6 3 IONIZED–IMPURITY SCATTERING 11 6 4 ACOUSTIC–PHONON SCATTERING, PIEZOELECTRIC, ELASTIC 11 6 5 OPTICAL–PHONON SCATTERING, POLAR INTERACTION DI?USION, FLUCTUATIONS, AND NOISE 12 1 FICK LAWS 12 2 EINSTEIN RELATION 12 3 DRIFT-DI?USION EQUATION AND THE GAUSSIAN SOLUTION 12 4 MOMENTS AND CORRELATIONS 12 5 SPECTRAL DENSITY AND WIENER–KINTCHINE THEOREM 12 6 NYQUIST THEOREM NONLINEAR TRANSPORT 13 1 HOT ELECTRONS 13 1 1 THE WARM ELECTRON REGION 13 2 ELECTRON–ELECTRON COLLISIONS AND THE HEATED AND DRIFTED MAXWELL DISTRIBUTION 13 3 ANISOTROPY OF TRANSPORT COE?CIENTS 13 4 NEGATIVE DI?ERENTIAL MOBILITY AND GUNN E?ECT 13 5 HIGH-FIELD DI?USIVITY 13 5 1 INTERVALLEY DI?USION 13 6 TRANSIENT TRANSPORT 13 7 HOT PHONONS 13 8 ULTRAFAST SPECTROSCOPY MONTE CARLO SIMULATION OF BULK ELECTRON TRANSPORT 14 1 THE MONTE CARLO METHOD 14 2 DIRECT MONTE CARLO SIMULATION 14 2 1 A TYPICAL MONTE CARLO PROGRAM FOR HOMOGENEOUS, STATIONARY TRANSPORT 14 2 2 TIME- AND SPACE-DEPENDENT PHENOMENA – ENSEMBLE MC 14 2 3 DI?USION 14 2 4 OHMIC MOBILITY 14 2 5 ELECTRON–ELECTRON INTERACTION AND DEGENERATE STATISTICS 14 2 6 IMPACT IONIZATION 14 2 7 VARIANCE-REDUCING TECHNIQUES 14 2 8 FULL-BAND MONTE CARLO 14 3 FORMAL MONTE CARLO SOLUTION OF THE BE – WEIGHTED MONTE CARLO 14 3 1 MONTE CARLO EVALUATION OF SUMS AND INTEGRALS 14 3 2 THE INTEGRAL BOLTZMANN EQUATION WITH APPROXIMATE TOTAL SCATTERING RATE 14 3 3 THE NEUMANN EXPANSION 14 3 4 SAMPLING BULK TRANSPORT PROPERTIES OF MAIN SEMICONDUCTORS 15 1 ELECTRONS IN SILICON 15 2 HOLES IN SILICON 15 3 ELECTRONS IN GALLIUM ARSENIDE 15 4 HOLES IN GALLIUM ARSENIDE 15 5 ORGANIC SEMICONDUCTORS PART III QUANTUM TRANSPORT IN BULK SEMICONDUCTORS QUANTUM TRANSPORT IN HOMOGENEOUS SYSTEMS 16 1 INTRODUCTION TO QUANTUM TRANSPORT 16 1 1 SEMICLASSICAL TRANSPORT AND QUANTUM PHYSICS 16 1 2 FROM REVERSIBLE DYNAMICS TO IRREVERSIBLE BOLTZMANN EQUATION 16 1 3 COHERENCE, DEPHASING, AND ENTANGLEMENT 16 1 4 WHEN IS QUANTUM TRANSPORT NECESSARY? 16 2 THE DENSITY MATRIX 16 3 REDUCED DENSITY MATRIX 16 4 KUBO FORMULA 16 5 THE PATH-INTEGRAL APPROACH THE WIGNER-FUNCTION APPROACH TO QUANTUM TRANSPORT 17 1 INTRODUCTION 17 2 DE?NITION AND MAIN PROPERTIES 17 2 1 WEYL–WIGNER TRANSFORMATION 17 2 2 TRANSFORMATION BETWEEN THE MATRIX ELEMENTS OF AN OPERATOR AND ITS WEYL–WIGNER TRANSFORM 17 2 3 DE?NITION OF THE WIGNER FUNCTION 17 2 4 MAIN PROPERTIES 17 3 COHERENT EVOLUTION OF THE WIGNER FUNCTION 17 4 DYNAMICAL EQUATIONS OF THE WIGNER FUNCTION 17 4 1 MOYAL EXPANSION 17 5 ELECTRON–PHONON INTERACTION 17 6 WIGNER PATHS AND MC SIMULATION 17 6 1 INTEGRAL EQUATION 17 6 2 NEUMANN EXPANSION AND WIGNER PATHS 17 6 3 MONTE CARLO SIMULATION 17 7 TWO-TIME WIGNER FUNCTION 17 8 MANY-PARTICLE WIGNER FUNCTION PART IV TRANSPORT IN SEMICONDUCTOR STRUCTURES INHOMOGENEOUS AND OPEN SYSTEMS: ELECTRONIC DEVICES 18 1 INHOMOGENEOUS, OPEN SYSTEMS 18 2 SELF-AVERAGING TRANSPORT, COHERENT TRANSPORT, AND INTERMEDIATE CASES 18 3 PN JUNCTIONS 18 3 1 PN JUNCTION AT EQUILIBRIUM 18 3 2 PN DIODE 18 3 3 SOLAR CELLS 18 3 4 LIGHT-EMITTING DIODES 18 4 THE BIPOLAR JUNCTION TRANSISTOR 18 5 METAL–SEMICONDUCTOR JUNCTIONS, SCHOTTKY BARRIER DIODE 18 6 FIELD-E?ECT TRANSISTORS 18 7 DEVICE SIMULATION 18 7 1 DRIFT-DI?USION MODELS 18 7 2 HYDRODYNAMIC MODELS 18 7 3 MONTE CARLO SIMULATIONS LOW-DIMENSIONAL STRUCTURES 19 1 EPITAXIAL HETEROSTRUCTURES 19 2 QUANTUM WELLS 19 2 1 ELECTRON STATES 19 2 2 TRANSPORT 19 2 3 MULTIPLE QUANTUM WELLS 19 3 QUANTUM WIRES 19 4 QUANTUM DOTS 19 4 1 TRANSPORT: COULOMB BLOCKADE 19 5 SUPERLATTICES 19 5 1 MINIBANDS 19 5 2 TRANSPORT: BLOCH OSCILLATIONS 19 5 3 WANNIER–STARK LADDER 19 5 4 NEGATIVE DI?ERENTIAL CONDUCTIVITY 19 6 APPLICATIONS CARBON NANOTUBES 20 1 INTRODUCTION 20 2 STRUCTURE 20 3 ELECTRON STATES: BANDS 20 4 ELECTRON TRANSPORT COHERENT TRANSPORT IN MESOSCOPIC STRUCTURES 21 1 LANDAUER–B¨UTTIKER THEORY OF TRANSPORT 21 2 POINT CONTACTS 21 3 QUANTUM HALL E?ECT 21 4 AHARONOV–BOHM OSCILLATIONS 21 5 LOCALIZATION 21 6 WEAK LOCALIZATION – QUANTUM CORRECTIONS 21 7 UNIVERSAL CONDUCTION FLUCTUATIONS 21 8 RESONANT TUNNELING DIODE SEMICONDUCTOR PHOTO GALLERY PART V QUANTUM TRANSPORT WITH NONEQUILIBRIUM GREEN FUNCTIONS SECOND-QUANTIZATION FORMALISM 23 1 MANY-PARTICLE WAVEFUNCTIONS 23 1 1 EXPANSION IN SYMMETRIC WAVEFUNCTIONS 23 2 VECTOR SPACE OF MANY-PARTICLE STATES 23 2 1 CREATION AND ANNIHILATION OPERATORS 23 2 2 FIELD OPERATORS 23 3 FROM FIRST TO SECOND QUANTIZATION 23 4 DYNAMICS 23 5 COMMUTATIONS AT DI?ERENT TIMES FOR NON-INTERACTING PARTICLES 23 6 FIELD OPERATORS IN MOMENTUM AND ENERGY SPACE INTRODUCTION TO GREEN FUNCTIONS 24 1 GFS FROM DI?ERENTIAL EQUATIONS TO MANY-BODY THEORY 24 1 1 GF OF SCHR¨ODINGER EQUATION 24 1 2 THE EVOLUTION OPERATOR AS GREENIAN 24 1 3 GREEN FUNCTIONS FOR A ONE-PARTICLE SYSTEM 24 1 4 SINGLE-PARTICLE GREEN FUNCTIONS IN MANY-PARTICLE SYSTEMS 24 2 GREEN FUNCTIONS IN MOMENTUM AND ENERGY SPACE 24 3 EQUILIBRIUM GFS FOR NONINTERACTING PARTICLES 24 4 GREEN FUNCTIONS AND MEAN QUANTITIES 24 5 SPECTRAL DENSITY 24 5 1 RELATION BETWEEN G< AND G> AND THE SPECTRAL DENSITY AT EQUILIBRIUM WICK–MATSUBARA THEOREMS 25 1 TIME-ORDERED PRODUCTS, NORMAL PRODUCTS, AND CONTRACTIONS 25 2 WICK THEOREM 25 2 1 LEMMA 25 2 2 WICK THEOREM 25 3 WICK–MATSUBARA THEOREM PERTURBATION EXPANSION OF GREEN FUNCTIONS: FEYNMAN DIAGRAMS AND DYSON EQUATION 26 1 THE INTERACTION PICTURE 26 2 CONTOUR INTEGRATION 26 3 PERTURBATION EXPANSION AND FEYNMAN DIAGRAMS, POTENTIAL INTERACTION 26 3 1 CANCELLATION OF DISCONNECTED DIAGRAMS 26 3 2 TERM MULTIPLICITY 26 4 PARTICLE–PARTICLE INTERACTION 26 5 ELECTRON–PHONON INTERACTION 26 6 SELF-ENERGY AND DYSON EQUATION 26 6 1 MATRIX FORMULATION OF G AND OF DYSON EQUATION 26 6 2 DYSON EQUATIONS FOR SEPARATE GFS 26 6 3 (K, ?) REPRESENTATION 26 7 ELECTRON–PHONON SELF-ENERGY NONEQUILIBRIUM GREEN FUNCTIONS APPLIED TO TRANSPORT: QUANTUM BOLTZMANN EQUATION 27 1 THE EQUATIONS FOR G<(R, T, R?, T?) AND GR(R, T, R?, T?) 27 2 THE EQUATIONS FOR G<(R, T, K, ?) AND GR(R, T, K, ?) 27 3 GRADIENT-EXPANSION APPROXIMATION 27 4 EQUATIONS FOR LINEAR RESPONSE IN HOMOGENEOUS SYSTEMS IN STEADY STATE NONEQUILIBRIUM GREEN FUNCTIONS APPLIED TO TRANSPORT: MESOSCOPIC SYSTEMS 28 1 GFS FOR THE TIME-INDEPENDENT SCHR¨ODINGER EQUATION 28 2 GF FOR A PERFECT, IN?NITE, TWO-DIMENSIONAL WIRE 28 3 FROM GREEN FUNCTION TO S MATRIX 28 4 FINITE-DI?ERENCE SCHEME FOR THE CONDUCTOR GF 28 5 THE E?ECT OF THE LEADS 28 6 CONDUCTANCE