Springer Solid-State Sciences : A Theoretical Introduction - Wim Magnus

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Springer Solid-State Sciences

A Theoretical Introduction

By: Wim Magnus (Editor), Wim Schoenmaker (Editor)

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Hardcover | 11 November 2003

At a Glance

Hardcover
292 Pages

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24.77 x 16.51 x 1.27

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$169.00

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In this book, the problem of electron and hole transport is approached from the point of view that a coherent and consistent physical theory can be constructed for transport phenomena. Along the road readers will visit some exciting citadels in theoretical physics as the authors guide them through the strong and weak aspects of the various theoretical constructions. Our goal is to make clear the mutual coherence and to put each theoretical model in an appropriate perspective. The mere fact that so many partial solutions have been proposed to describe transport, be it in condensed matter, fluids, or gases, illustrates that we are entering a world of physics with a rich variety of phenomena. Theoretical physics always seeks to provide a unifying picture. By presenting this tour of many very inventive attempts to build such a picture, it is hoped that the reader will be inspired and encouraged to help find the unifying principle behind the many faces of transport.

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General Formalism
The Many Faces of Transport	p. 3
Classical Mechanics	p. 7
Generalized Coordinates and Constraints	p. 7
d'Alembert's Principle	p. 8
Reduction of Dynamics to Statics	p. 10
The Lagrangian and Euler-Lagrange Equations	p. 11
Maupertuis and Maserati	p. 11
From Lagrangian to Hamiltonian	p. 12
Phase Space and Configuration Space	p. 14
The Liouville Equation	p. 15
The Micro-Canonical Ensemble	p. 16
The Boltzmann Equation	p. 18
Drude's Model of Carrier Transport	p. 21
Currents in Semiconductors	p. 23
Mathematical Interlude	p. 27
Generalized Functions	p. 27
The Step Function (x)	p. 27
The Delta Function (x)	p. 28
Some Useful Relations	p. 28
Functions of Functions - Functionals	p. 29
Functional Integration - Path Integrals	p. 30
Vector Identities	p. 31
A Few Theorems	p. 32
Exercises	p. 33
Quantum Mechanics	p. 35
Rules of the Game	p. 35
States in Fock Space	p. 35
The Superposition Principle	p. 36
Scalar Products and Probability Amplitudes	p. 36
Observable Quantities and Operators	p. 36
Measurements and Expectation Values	p. 37
Dynamics - The Schrodinger Equation	p. 38
The Book of Recipes	p. 39
First Recipe - Correspondence Principle	p. 40
Second Recipe - Canonical Commutation Rules	p. 40
Third Recipe - Use Your Imagination	p. 40
Fourth Recipe - Choose an Appropriate Basis	p. 40
The Position and Momentum Representation	p. 43
Position Representation	p. 43
Momentum Representation	p. 44
Stationary Schrödinger Equation	p. 44
Commuting Operators, `Good' Quantum Numbers	p. 44
Path Integrals	p. 46
Exercises	p. 51
Single-Particle Quantum Mechanics	p. 53
Charge Density,Current and Single Particle Wave Functions	p. 53
Constant Potential,Energy Bands and Energy Subbands	p. 54
Potential Wells	p. 58
Potential Barriers	p. 62
Electromagnetic Fields	p. 71
Spin	p. 73
Exercises	p. 77
Second Quantization	p. 79
Identical Particles	p. 79
Field Operators	p. 83
Definition	p. 83
Field Operators, Wave Functions and Topology	p. 84
Field Operators in Fock Space	p. 85
The Connection to First Quantization	p. 88
How to Construct the Operators	p. 90
More Creation and Annihilation Operators	p. 94
The Electron Hamiltonian	p. 96
The Number Operator	p. 96
Charge and Current Density	p. 96
Many-Particle Ground State of a Non-Interacting System	p. 96
Summary	p. 98
Exercise	p. 98
Equilibrium Statistical Mechanics	p. 99
The Entropy Principle	p. 99
The Canonical and Grand-Canonical Ensembles	p. 101
Quantum Statistical Physics	p. 104
Quantum Ensembles	p. 106
Photons and Phonons - Some Partition Functions	p. 107
Preview of Non-equilibrium Theory	p. 111
Exercises	p. 114
Non-equilibrium Statistical Mechanics	p. 115
Definition of the Problem	p. 115
Hydrodynamics	p. 117
A First Glance at Entropy	p. 119
Deriving Fourier's Law	p. 120
A Second Glance at Entropy	p. 126
Matsubara Functions	p. 127
Exercise	p. 130
Wigner Distribution Functions	p. 131
Balance Equations	p. 137
Basic Assumptions	p. 137
Charge and Current Density	p. 138
Total Hamiltonian	p. 139
Basic Equations of Motion	p. 139
Continuity Equation	p. 140
Energy Balance Equation	p. 141
Linear and Angular Momentum Balance Equation	p. 145
Calculation of the DC Current	p. 146
Gedankenexperiment	p. 147
Equilibrium Currents and Broken Time Reversal Symmetry	p. 147
Perturbative Solution Scheme	p. 150
Exercise	p. 154
Applications
Velocity-Field Characteristics of a Silicon MOSFET	p. 157
Momentum and Energy Balance Equations for a MOSFET Channel	p. 157
Calculation of the Elementary Green Functions	p. 162
Gate Leakage Currents	p. 169
Subband States and Resonances	p. 170
Tunneling Gate Currents	p. 177
Results of the Gate-Leakage Current Calculations	p. 184
Quantum Transport in Vertical Devices	p. 189
Quantum Transport in a Cylindrical MOSFET	p. 190
The Hamiltonian of the System	p. 191
The Liouville Equation	p. 193
Electron Scattering	p. 198
The Numerical Model	p. 200
Numerical Results	p. 202
Summary	p. 207
An Exactly Solvable Electron-Phonon System	p. 209
The Time Dependent Drift Velocity for the CL Model	p. 210
Solution of the Transport Equation	p. 213
How to Invert Laplace Transforms	p. 215
Irreversibility and the Ohmic Case	p. 216
Entropy Production	p. 218
Summary	p. 219
Open Versus Closed Systems	p. 221
Conductance Quantization	p. 225
Circuit Topology, Non-Conservative Fields and Dissipationless Transport	p. 226
Quantum Rings	p. 229
Hamiltonian and Current Response	p. 235
Open Versus Closed Circuits	p. 240
Energy Dissipation Versus Current Limitation	p. 241
Flux Quantization	p. 242
Localization of the Electric Field	p. 243
A Quantum Lenz' Law?	p. 243
Transport in Quantum Wires	p. 245
Balance Equations for an Imperfect Quantum Wire	p. 245
Current-Voltage Characteristics and Local Energy Dissipation	p. 248
Future Work	p. 251
Constructing Non-equilibrium Ensembles	p. 251
Covariance in Classical Physics	p. 253
Canonical Quantization	p. 255
Path-Integral Quantization	p. 255
Guessing a Density Function	p. 258
Quantum Circuit Theory	p. 259
References	p. 261
Index	p. 265
Table of Contents provided by Publisher. All Rights Reserved.

Springer Solid-State Sciences

A Theoretical Introduction

At a Glance

Hardcover

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