Table of Contents
Preface vii
Acknowledgments ix
List of Available Video Lectures Quantum Transport xi
Constants Used in This Book xv
Some Symbols Used xvii
1 Overview 1
1.1 Conductance 3
1.2 Ballistic Conductance 4
1.3 What Determines the Resistance? 5
1.4 Where is the Resistance? 6
1.5 But Where is the Heat? 8
1.6 Elastic Resistors 10
1.7 Transport Theories 13
1.7.1 Why elastic resistors are conceptually simpler 14
1.8 Is Transport Essentially a Many-body Process? 16
1.9 A Different Physical Picture 17
Contact-ing Schrödinger 19
17 The Model 21
17.1 Schrödinger Equation 24
17.1.1 Spatially varying potential 25
17.2 Electron-electron Interactions and the SCF Method 28
17.3 Differential to Matrix Equation 30
17.3.1 Semi-empirical tight-binding (TB) models 31
17.3.2 Size of matrix, N = n × b 32
17.4 Choosing Matrix Parametrs 33
17.4.1 One-dimensional conductor 33
17.4.2 Two-dimensional conductor 35
17.4.3 TB parameters in B-field 37
17.4.4 Lattice with a "Basis" 38
18 NEGF Method 43
18.1 One-level Resistor 48
18.1.1 Semiclassical treatment 48
18.1.2 Quantum treatment 50
18.1.3 Quantum broadening 53
18.1.4 Do multiple sources interfere? 54
18.2 Quantum Transport Through Multiple Levels 55
18.2.1 Obtaining Eqs. (18.1) 56
18.2.2 Obtaining Eqs. (18.2) 57
18.2.3 Obtaining Eq. (18.3) 57
18.2.4 Obtaining Eq. (18.4): the current equation 58
18.3 Conductance Functions for Coherent Transport 59
18.4 Elastic Dephasing 60
19 Can Two Offer Less Resistance than One? 65
19.1 Modeling ID Conductors 65
19.1.1 1D ballistic conductor 67
19.1.2 1D conductor with one scatterer 68
19.2 Quantum Resistors in Series 70
19.3 Potential Drop Across Scatterer(s) 74
More on NEGF 79
20 Quantum of Conductance 81
20.1 2D Conductor as ID Conductors in Parallel 81
20.1.1 Modes or subbands 86
20.2 Contact Self-Energy for 2D Conductors 87
20.2.1 Method of basis transformation 87
20.2.2 General method 88
20.2.3 Graphene: ballistic conductance 90
20.3 Quantum Hall Effect 92
21 Inelastic Scattering 97
21.1 Fermi's Golden Rule 99
21.1.1 Elastic scattering 100
21.1.2 Inelastic scattering 103
21.2 Self-energy Functions 104
22 Does NEGF Include "Everything?" 107
22.1 Coulomb Blockade 108
22.1.1 Current versus voltage 110
22.2 Fock Space Description 112
22.2.1 Equilibrium in Fock space 113
22.2.2 Current in the Fock space picture 115
22.3 Entangled States 117
Spin Transport 123
23 Rotating an Electron 125
23.1 Polarizers and Analyzers 127
23.2 Spin in NEGF 130
23.3 One-level Spin Valve 131
23.4 Rotating Magnetic Contacts 134
23.5 Spin Hamiltonians 137
23.5.1 Channel with Zeeman splitting 137
23.5.2 Channel with Rashba interaction 138
23.6 Vectors and Spinors 139
23.7 Spin Precession 143
23.8 Spin-charge Coupling 146
23.9 Superconducting Contacts 150
24 Quantum to Classical 151
24.1 Matrix Electron Density 151
24.2 Matrix Potential 154
24.3 Spin Circuits 156
24.4 Pseudo-spin 158
24.5 Quantum Information 161
24.5.1 Quantum entropy 161
24.5.2 Does interaction increase the entropy? 162
24.5.3 How much information can one spin carry? 163
25 Epilogue: Probabilistic Spin Logic (PSL) 165
25.1 Spins and Magnets 166
25.1.1 Pseudospins and pseudomagnets 168
25.2 Unstable Magnets 168
25.3 Three-terminal p-bits 170
25.4 P-circuits 171
Suggested Reading 175
Appendices 187
Appendix F List of Equations and Figures Cited From Part A 189
Appendix G NEGF Equations 193
G.1 Self-energy for Contacts 194
G.2 Self-energy for Elastic Scatterers in Equilibrium 196
G.3 Self-energy for Inelastic Scatterers 196
Appendix H MATLAB Codes Used for Text Figures 199
H.1 Chapter 19 199
H.1.1 Fig. 19.2 Transmission through a single point scatterer in a ID wire 199
H.1.2 Fig. 19.4 Normalized conductance for a wire with M = 1 due to one scatterer 200
H.1.3 Fig. 19.5 Normalized conductance for a wire with M = 1 due to six scatterers 201
H.1.4 Figs. 19.6-19.7 Potential drop across a scatterer calculated from NEGF 202
H.1.5 Figs. 19.8-19.9 Potential drop across two scatterers in series calculated from NEGF 204
H.2 Chapter 20 206
H.2.1 Fig. 20.1 Numerically computed transmission as a function of energy 206
H.2.2 Fig. 20.3 Transmission calculated from NEGF for ballistic graphene sheet and CNT 209
H.2.3 Fig. 20.4 Normalized Hall resistance versus B-field for ballistic channel 211
H.2.4 Fig. 20.5 Grayscale plot of local density of states 213
H.3 Chapter 22 215
H.3.1 Fig. 22.7, n versus μ, single dot 215
H.3.2 Fig. 22.8, I versus V, single quantum dot 216
H.3.3 Fig. 22.9, n versus μ, double quantum dot 217
H.4 Chapter 23 218
H.4.1 Fig. 23.9 Voltage probe signal as the magnetization of the probe is rotated 218
H.4.2 Fig. 23.10 Voltage probe signal due to variation of gate voltage controlled Rashba coefficient 220
Appendix I Table of Contents of Part A: Basic Concepts 223
Appendix J Available Video Lectures for Part A: Basic Concepts 229
Index 231
