Quantum Kinetics in Transport and Optics of Semiconductors

by Haug, Hartmut; Jauho, Antti-Pekka

• ISBN13: 9783540735618
• ISBN10: 3540735615
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-12-04
• Publisher: Springer Verlag

Summary

Nanoscale miniaturization and femtosecond laser-pulse spectroscopy require a quantum mechanical description of the carrier kinetics that goes beyond the conventional Boltzmann theory. On these extremely short length and time scales, the electrons behave as do partially coherent waves. This monograph deals with quantum kinetics for transport in low-dimensional microstructures and for ultra-short laser pulse spectroscopy. The nonequilibrium Green function theory is described and used for the derivation of the quantum kinetic equations. Numerical methods for the solution of the retarded quantum kinetic equations are discussed and results are presented for high-field transport and for mesoscopic transport phenomena. Quantum beats, polarization decay, and non-Markovian behaviour are treated for femtosecond spectroscopy on a microscopic basis. Since the publishing of the first edition in 1996, the nonequilibrium Green function technique has been applied to a large number of new research topics, and the revised edition introduces the reader to many of these areas, such as molecular electronics, noise calculations, build-up of screening and polaron correlations, and non-Markovian relaxation, among others. Connection to recent experiments is made, and it is emphasized how the quantum kinetic theory is essential in their interpretation.

Author Biography

Hartmut Haug obtained his Ph. D. (Dr. rer. nat. 1966) in Physics at the University of Stuttgart. From 1967 to 1969 he was a faculty member at the Department of Electrical Engeneering, University of Wisconsin in Madiason. After working as a scientific staff member at the Philips Research Laboratories in Eindhoven from 1969 to 1973, he joined the Institute of Theoretical Physics of the J.W.Goethe-University Frankfurt, where he was a full professor from 1975 to 2001 and currently is an emeritus. He has been a visiting scientist at many international research centers and universities.Antti-Pekka Jauho obtained his Ph.D in Theoretical Condensed Matter Physics at Cornell University, USA, in 1982. He has been a faculty member at University of Copenhagen, Nordita (Copenhagen), and, since 1993, at Technical University of Denmark, where he has been Professor of Theoretical Nanotechnology at MIC, Department of Micro and Nanotechnology, since 2003. He is also a Distinguished Professor of the Finnish Academy since 2007, and spends half of his time at the Technical University of Helskinki, Finland.

Table of Contents

Introduction to Kinetics and Many-Body Theory
Boltzmann Equation	p. 3
Heuristic Derivation of the Semiclassical Boltzmann Equation	p. 3
Approach to Equilibrium: H-Theorem	p. 5
Linearization: Eigenfunction Expansion	p. 8
Numerical Solutions of the Boltzmann Equation	p. 11
Introduction	p. 11
Linearized Coulomb Boltzmann Kinetics of a 2D Electron Gas	p. 12
Ensemble Monte Carlo Simulation	p. 20
General Theory	p. 20
Simulation of the Relaxation Kinetics of a 2D Electron Gas	p. 23
N[superscript +]N[superscript -]N[superscript +] Structure: Boltzmann Equation Analysis	p. 29
Equilibrium Green Function Theory	p. 35
Second Quantization	p. 35
Density Matrix Equations: An Elementary Derivation of a Non-Markovian Quantum Kinetic Equation	p. 38
Green Functions	p. 41
Examples of Measurable Quantities	p. 43
Fluctuation-Dissipation Theorem	p. 45
Perturbation Expansion of the Green Function	p. 47
Examples of Simple Solvable Models	p. 50
Free-Particle Green Function	p. 50
Resonant-Level Model	p. 50
Self-Energy	p. 52
Electron-Phonon Interaction	p. 52
Elastic Impurity System: Impurity Averaging	p. 54
Finite Temperatures	p. 58
Nonequilibrium Many-Body Theory
Contour-Ordered Green Functions	p. 63
General Remarks	p. 63
Two Transformations	p. 64
Analytic Continuation	p. 69
Basic Quantum Kinetic Equations	p. 75
Introductory Remarks	p. 75
The Kadanoff-Baym Formulation	p. 75
Keldysh Formulation	p. 77
Boltzmann Limit	p. 79
Gradient Expansion	p. 79
Quasiparticle Approximation	p. 81
Recovery of the Boltzmann Equation	p. 82
Gauge Invariance	p. 85
Choice of Variables	p. 85
Gauge Invariant Quantum Kinetic Equation	p. 87
Driving Term	p. 87
Collision Term	p. 90
Retarded Green Function	p. 91
Quantum Distribution Functions	p. 93
Relation to Observables, and the Wigner Function	p. 93
Generalized Kadanoff-Baym Ansatz	p. 94
Summary of the Main Formal Results	p. 97
Quantum Transport in Semiconductors
Linear Transport	p. 101
Quantum Boltzmann Equation	p. 101
Linear Conductivity of Electron-Elastic Impurity Systems	p. 104
Kubo Formula	p. 105
Quantum Kinetic Formulation	p. 109
Weak Localization Corrections to Electrical Conductivity	p. 111
Field-Dependent Green Functions	p. 115
Free Green Functions and Spectral Functions in an Electric Field	p. 115
A Model for Dynamical Disorder: The Gaussian White Noise Model	p. 121
Introduction	p. 121
Determination of the Retarded Green Function	p. 121
Kinetic Equation for the GWN	p. 123
Other Transport Properties	p. 127
Introduction to High-Field Transport in Semiconductors	p. 129
Resonant-Level Model in High Electric Fields	p. 131
Introduction	p. 131
Retarded Green Function: Single Impurity Problem	p. 131
Retarded Green Function: Dilute Concentration of Impurities	p. 133
Analytic Continuation	p. 140
Quantum Kinetic Equation	p. 141
Quantum Kinetic Equation for Electron-Phonon Systems	p. 144
An Application: Collision Broadening for a Model Semiconductor	p. 148
Analytical Considerations	p. 148
A Simple Model: Optical Phonon Emission at T = 0	p. 150
Spatially Inhomogeneous Systems	p. 151
Optical Absorption in Intense THz Fields	p. 157
Introductory Remarks	p. 157
Optical Absorption as a Response Function	p. 158
Absorption Coefficient	p. 162
Static Electric Field	p. 163
Harmonically Varying External Electric Fields	p. 164
Joint Density of States, 2D	p. 167
Joint Density of States, 3D	p. 169
Dynamical Franz-Keldysh Effect: Excitonic Effects	p. 171
Matrix Truncation	p. 172
Floquet Space Formulation	p. 175
Transport in Mesoscopic Semiconductor Structures	p. 181
Introduction	p. 181
Nonequilibrium Techniques in Mesoscopic Tunneling Structures	p. 184
Model Hamiltonian	p. 185
General Expression for the Current	p. 186
Current Conservation	p. 191
Noninteracting Resonant-Level Model	p. 192
Density Functional Theory and Modeling of Molecular Electronics	p. 195
Resonant Tunneling with Electron-Phonon Interactions	p. 196
Transport in a Semiconductor Superlattice	p. 198
Transport in Atomic Gold Wires: Signature of Coupling to Vibrational Modes	p. 202
Transport Through a Coulomb Island	p. 205
Time-Dependent Phenomena	p. 213
Introduction	p. 213
Applicability to Experiments	p. 214
Mathematical Formulation	p. 215
Average Current	p. 217
Time-Dependent Resonant-Level Model	p. 218
Response to harmonic Modulation	p. 221
Response to Step-Like Modulation	p. 224
Linear-Response	p. 227
Fluctuating Energy Levels	p. 229
Noise	p. 230
The Disconnected Terms	p. 235
The Connected Terms	p. 236
Theory of Ultrafast Kinetics in Laser-Excited Semiconductors
Optical Free-Carrier Interband Kinetics in Semiconductors	p. 243
Interband Transitions in Direct-Gap Semiconductors	p. 243
Reduced Density Matrices	p. 243
Nonequilibrium Green Functions	p. 245
Calculations of the Two-Time-Dependent Nonequilibrium Green Function	p. 245
Replacement of Two-Time Propagators by a One-Time Density Matrix and a Two-Time Spectral Function	p. 246
Free-Carrier Kinetics Under Laser-Pulse Excitation	p. 251
The Optical Free-Carrier Bloch Equations	p. 255
Interband Quantum Kinetics with LO-Phonon Scattering	p. 259
Derivation of the Interband Quantum Kinetic Equations	p. 259
The Spectral Green Functions G[superscript r subscript mu nu] and G[superscript a subscript mu nu]	p. 266
Free-Particle Retarded Green Function	p. 266
Retarded GF in the Mean-Field Approximation	p. 267
Spectra of Retarded Gfs	p. 269
Dephasing of Retarded Green Functions	p. 274
Intraband Relaxation	p. 279
Interband-Polarization Dephasing	p. 281
Numerical Strategies	p. 283
Two-Pulse Spectroscopy	p. 287
Introductory Remarks	p. 287
Thin Samples	p. 289
Low-Intensity Two-Beam Experiments	p. 290
LO-Phonon Relaxation Cascades	p. 291
LO-Phonon Quantum Beats in FWM	p. 292
Two-Time Electron-Phonon Quantum Kinetics	p. 294
Coulomb Quantum Kinetics in a Dense Electron-Hole Plasma	p. 301
Introduction	p. 301
Screening in the Nonequilibrium GF Theory	p. 302
Coulomb Quantum Kinetics	p. 306
Plasmon-Pole Approximation for the Two-Time-Dependent Potential	p. 309
Parametric Plasma Oscillations	p. 311
Instantaneous Static Potential Approximation	p. 313
The Buildup of Screening	p. 317
Screening of the Coulomb Interaction	p. 317
Calculations of the Two-Time-Dependent Screened Potential	p. 318
Femtosecond Optical Pump and THz Probe Spectroscopy	p. 320
Time-Dependent Screening of Phonon-Mediated and Coulomb Interactions	p. 320
Buildup of the Phonon-Plasmon Mixed Modes	p. 324
Femtosecond Four-Wave Mixing with Dense Plasmas	p. 331
Time-Resolved Four-Wave Mixing	p. 331
Time-Integrated Four-Wave Mixing	p. 334
Four-Wave Mixing with Coherent Control	p. 335
References	p. 341
Index	p. 353
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