ISBN: 3540433961
TITLE: Quantum Transport in Submicron Devices
AUTHOR: Magnus, Schoenmaker
TOC:

Part I. General Formalism
1.The Many Faces of Transport 3
2.Classical Mechanics 7
2.1 Generalized Coordinates and Constraints 7
2.2 d' Alembert's Principle 8
2.3 Reduction of Dynamics to Statics 10
2.4 The Lagrangian and Euler ?agrange Equations 11
2.5 Maupertuis and Maserati 11
2.6 From Lagrangian to Hamiltonian 12
2.7 Phase Space and Con guration Space 14
2.8 The Liouville Equation 15
2.9 The Micro-Canonical Ensemble 16
2.10 The Boltzmann Equation 18
2.11 Drude's Model of Carrier Transport 21
2.12 Currents in Semiconductors 23
3.Mathematical Interlude 27
3.1 Generalized Functions 27
3.1.1 The Step Function theta(x) 27
3.1.2 The Delta Function delta (x) 28
3.1.3 Some Useful Relations 28
3.2 Functions of Functions - Functionals 29
3.3 Functional Integration - Path Integrals 30
3.4 Vector Identities 31
3.5 A Few Theorems 32
Exercises 33
4.Quantum Mechanics 35
4.1 Rules of the Game 35
4.1.1 States in Fock Space 35
4.1.2 The Superposition Principle 36
4.1.3 Scalar Products and Probability Amplitudes 36
4.1.4 Observable Quantities and Operators 36
4.1.5 Measurements and Expectation Values 37
4.1.6 Dynamics - The Schrdinger Equation 38
4.2 The Book of Recipes 39
4.2.1 First Recipe - Correspondence Principle 40
4.2.2 Second Recipe - Canonical Commutation Rules 40
4.2.3 Third Recipe - Choose Your Imagination 40
4.2.4 Fourth Recipe - Choose an Appropriate Basis 40
4.3 The Position and Momentum Representation 43
4.3.1 Position Representation 43
4.3.2 Momentum Representation 44
4.3.3 Stationary Schrdinger Equation 44
4.4 Commuting Operators, 'Good' Quantum Numbers 44
4.5 Path Integrals 46
Exercises 51
5.Single-Particle Quantum Mechanics 53
5.1 Charge Density,Current and Single Particle Wave Functions 53
5.2 Constant Potential,Energy Bands and Energy Subbands 54
5.3 Potential Wells 58
5.4 Potential Barriers 62
5.5 Electromagnetic Fields 71
5.6 Spin 73
Exercises 77
6.Second Quantization 79
6.1 Identical Particles 79
6.2 Field Operators 83
6.2.1 Definition 83
6.2.2 Field Operators,Wave Functions and Topology 84
6.2.3 Field Operators in Fock Space 85
6.2.4 The Connection to First Quantization 88
6.2.5 How to Construct the Operators 90
6.3 More Creation and Annihilation Operators 94
6.3.1 The Electron Hamiltonian 96
6.3.2 The Number Operator 96
6.3.3 Charge and Current Density 96
6.3.4 Many-Particle Ground State of a Non-Interacting System 96
Summary 98
Exercise 98
7.Equilibrium Statistical Mechanics 99
7.1 The Entropy Principle 99
7.2 The Canonical and Grand-Canonical Ensembles 101
7.3 Quantum Statistical Physics 104
7.4 Quantum Ensembles 106
7.5 Photons and Phonons ?ome Partition Functions 107
7.6 Preview of Non-equilibrium Theory 111
Exercises 114
8.Non-equilibrium Statistical Mechanics 115
8.1 De nition of the Problem 115
8.2 Hydrodynamics 117
8.2.1 A First Glance at Entropy 119
8.2.2 Deriving Fourier ? Law 120
8.2.3 A Second Glance at Entropy 126
8.3 Matsubara Functions 127
Exercise 130
9.Wigner Distribution Functions 131
10.Balance Equations 137
10.1 Basic Assumptions 137
10.2 Charge and Current Density 138
10.3 Total Hamiltonian 139
10.4 Basic Equations of Motion 139
10.5 Continuity Equation 140
10.6 Energy Balance Equation 141
10.7 Linear and Angular Momentum Balance Equation 145
10.8 Calculation of the DC Current 146
10.8.1 Gedankenexperiment 147
10.8.2 Equilibrium Currents
and Broken Time Reversal Symmetry 147
10.8.3 Perturbative Solution Scheme 150
Exercise 154
Part II.Applications
11.Velocity -Field Characteristics of a Silicon MOSFET 157
11.1 Momentum and Energy Balance Equations for a MOSFET Channel 157
11.2 Calculation of the Elementary Green Functions 162
12.Gate Leakage Currents 169
12.1 Subband States and Resonances 170
12.2 Tunneling Gate Currents 177
12.3 Results of the Gate-Leakage Current Calculations 184
13.Quantum Transport in Vertical Devices 189
13.1 Quantum Transport in a Cylindrical MOSFET 190
13.2 The Hamiltonian of the System 191
13.3 The Liouville Equation 193
13.4 Electron Scattering 198
13.5 The Numerical Model 200
13.6 Numerical Results 202
Summary 207
14.An Exactly Solvable Electron -Phonon System 209
14.1 The Time Dependent Drift Velocity for the CL Model 210
14.2 Solution of the Transport Equation 213
14.3 How to Invert Laplace Transforms 215
14.4 Irreversibility and the Ohmic Case 216
14.5 Entropy Production 218
Summary 219
15.Open Versus Closed Systems 221
16.Conductance Quantization 225
16.1 Circuit Topology,Non-Conservative Fields and Dissipationless Transport 226
16.2 Quantum Rings 229
16.3 Hamiltonian and Current Response 235
16.4 Open Versus Closed Circuits 240
16.5 Energy Dissipation Versus Current Limitation 241
16.6 Flux Quantization 242
16.7 Localization of the Electric Field 243
16.8 A Quantum Lenz ?aw?.243
17.Transport in Quantum Wires 245
17.1 Balance Equations for an Imperfect Quantum Wire 245
17.2 Current-Voltage Characteristics and Local Energy Dissipation 248
18.Future Work 251
18.1 Constructing Non-equilibrium Ensembles 251
18.1.1 Covariance in Classical Physics 253
18.1.2 Canonical Quantization 255
18.1.3 Path-Integral Quantization 255
18.1.4 Guessing a Density Function 258
18.2 Quantum Circuit Theory 259
References 261
Index 265
END
