Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 15
1.1 Nanoscale systems......Page 19
How do we generate electrical currents?......Page 21
Polarization and magnetization......Page 23
Fluctuations and reservoirs......Page 24
1.2.1 Finite versus infinite systems......Page 26
1.2.2 Electron sources......Page 27
1.2.3 Intrinsic nature of the transport problem......Page 28
1.3 Measuring currents......Page 29
1.3.1 Microscopic states......Page 30
1.3.2 The current operator......Page 31
1.3.3 The measurement process......Page 34
1.3.4 Complete measurement and pure states......Page 35
An example......Page 36
1.4 The statistical operator and macro-states......Page 37
1.4.1 Pure and mixed states......Page 39
1.4.2 Quantum correlations......Page 40
1.4.3 Time evolution of the statistical operator......Page 41
1.4.4 Random or partially specified Hamiltonians......Page 42
1.4.5 Open quantum systems......Page 43
1.4.6 Equilibrium statistical operators......Page 47
1.5 Current measurement and statistical operator truncation......Page 50
1.6 One current, different viewpoints......Page 52
Exercises......Page 54
2.1 Drude model......Page 57
2.2 Resistance, coherent and incoherent transport......Page 60
2.2.1 Relaxation vs. dephasing......Page 62
2.2.2 Mean-free path......Page 66
2.2.3 The meaning of momentum relaxation time......Page 67
2.3 Kubo formalism......Page 68
2.3.1 The current-current response function......Page 73
The frequency-dependent conductivity......Page 74
2.3.2 The use of Density-Functional Theory in the Kubo approach......Page 75
2.3.3 The fluctuation-dissipation theorem......Page 78
Response to a uniform and static electric field......Page 79
2.3.3.1 Static conductivity of an ideal gas......Page 81
The interacting case......Page 83
2.3.4 Ohmic vs. ballistic regimes......Page 84
2.4 Chemical, electrochemical and electrostatic potentials......Page 86
2.5 Drift-diffusion equations......Page 90
2.5.1 Diffusion coeffcient of an ideal electron gas in the non-degenerate limit......Page 91
2.5.2 Generalization to spin-dependent transport......Page 93
2.6 Distribution functions......Page 95
2.7 Boltzmann equation......Page 97
2.7.1 Approach to local equilibrium......Page 100
2.8 Entropy, loss of information, and macroscopic irreversibility......Page 101
2.8.1 The classical statistical entropy......Page 103
Micro-canonical entropy......Page 104
Entropy as loss of information......Page 105
2.8.3 Information content of the N- and one-particle statistical operators......Page 107
2.8.4 Entropy of open quantum systems......Page 108
2.8.5 Loss of information in the Kubo formalism......Page 109
2.8.6 Loss of information with stochastic Hamiltonians......Page 110
2.8.7 Entropy associated with the measurement of currents......Page 111
Summary and open questions......Page 112
Exercises......Page 113
3 Landauer approach......Page 119
3.1 Formulation of the problem......Page 120
Loss of information in the Landauer approach......Page 125
3.2 Local resistivity dipoles and the “field response”......Page 131
3.3.1 Scattering boundary conditions......Page 133
3.3.2 Transmission and reflection probabilities......Page 137
3.3.3 Total current......Page 141
Probes......Page 146
Two-probe conductance......Page 147
Experimental verification of quantized conductance......Page 149
3.4.1 Time-dependent Lippmann-Schwinger equation......Page 150
3.4.1.1 Green’s functions......Page 151
Relation between G ± and G ±0......Page 153
3.4.1.2 Dyson’s equation and self-energy......Page 154
Incoming and outgoing states......Page 156
3.4.2 Time-independent Lippmann-Schwinger equation......Page 158
Density of states operator......Page 161
3.5 Green’s functions and self-energy......Page 163
Energy “renormalization” and state “lifetime”......Page 169
Discrete (or tight-binding) space representation......Page 170
3.5.1 Relation to scattering theory......Page 172
Transmission and reflection amplitudes......Page 175
3.6 The S matrix......Page 177
3.6.1 Relation between the total Green’s function and the S matrix......Page 180
3.7 The transfer matrix......Page 185
3.7.1.1 Resonant tunneling......Page 187
3.7.2 Incoherent scattering of two resistors in series......Page 189
3.7.3 Relation between the conductance and the transfer matrix......Page 191
3.7.4 Localization, ohmic and ballistic regimes......Page 192
3.8 Four-probe conductance in the non-invasive limit......Page 196
3.8.1 Single-channel case......Page 197
3.8.2 Geometrical “dilution”......Page 199
3.8.3 Multi-channel case......Page 200
3.9 Multi-probe conductance in the invasive limit......Page 203
3.9.1 Floating probes and dephasing......Page 205
Effect of dephasing on the current......Page 207
3.10 Generalization to spin-dependent transport......Page 208
3.10.1 Spin-dependent transmission functions......Page 212
3.10.2 Multi-probe conductance in the presence of a magnetic field......Page 213
3.10.3 Local resistivity spin dipoles and dynamical effects......Page 214
3.11 The use of Density-Functional Theory in the Landauer approach......Page 216
Non-variational properties of the current......Page 217
Fundamental limitations of ground-state DFT in transport......Page 219
Summary and open questions......Page 220
Exercises......Page 221
4 Non-equilibrium Green’s function formalism......Page 227
4.1 Formulation of the problem......Page 229
4.1.1 Contour ordering......Page 233
4.2 Equilibrium Green’s functions......Page 235
4.2.1 Time-ordered Green’s functions......Page 236
4.2.1.1 Equation of motion for the equilibrium Green’s function......Page 238
4.2.2 Dyson’s equation for interacting particles......Page 239
The single-particle non-interacting case......Page 240
4.2.3 More Green’s functions......Page 241
Density and current density from G<......Page 242
4.2.4 The spectral function......Page 243
The fluctuation-dissipation theorem......Page 244
The non-interacting limit......Page 245
Quasi-particle approximation......Page 247
Mean-field approximation......Page 248
4.3 Contour-ordered Green’s functions......Page 249
4.3.1 Equations of motion for non-equilibrium Green’s functions......Page 251
Equation of motion for G<......Page 253
Approximations 1 to 3 of the Landauer approach......Page 254
Approximation 4: Non-interacting electrons in the leads......Page 255
Proportional coupling......Page 260
The non-interacting case......Page 261
4.5 Coulomb blockade......Page 262
“Orthodox” picture of Coulomb blockade......Page 263
NEGF approach to Coulomb blockade......Page 266
4.6 Quantum kinetic equations......Page 268
Summary and open questions......Page 273
Exercises......Page 275
5 Noise......Page 276
5.1 The moments of the current......Page 279
Autocorrelation time and stationary processes......Page 280
5.2 Shot noise......Page 281
5.2.1 The classical (Poisson) limit......Page 282
5.2.2 Quantum theory of shot noise......Page 284
The single-channel case......Page 285
The average current......Page 286
The second moment of the current......Page 287
The Poisson limit......Page 289
The multi-channel case......Page 290
5.3 Counting statistics......Page 292
5.4 Thermal noise......Page 293
Exercises......Page 295
6 Electron-ion interaction......Page 298
6.1 The many-body electron-ion Hamiltonian......Page 299
6.1.1 The adiabatic approximation for a current-carrying system......Page 300
Single-particle approximation......Page 301
6.1.2 The phonon subsystem......Page 302
The propagator of non-interacting phonons......Page 305
6.1.3 Electron-phonon coupling in the presence of current......Page 306
6.2 Inelastic current......Page 308
6.2.1 Inelastic current from standard perturbation theory......Page 309
Inelastic transition probabilities......Page 310
Inelastic mean-free path......Page 312
6.2.2 Inelastic current from the NEGF......Page 314
Electron-phonon vs. elastic scattering in the leads......Page 316
6.2.2.1 Electron-phonon self-energies at equilibrium......Page 319
The Migdal theorem......Page 321
From diagrams to equations......Page 322
6.2.2.2 Electron-phonon self-energies out of equilibrium......Page 325
Total current of independent electrons interacting with phonons......Page 326
Conductance increase or decrease due to phonon scattering......Page 328
6.3 Local ionic heating......Page 330
Local ionic heating: simple considerations......Page 331
Heating from localized modes......Page 334
Quantum theory of local ionic heating......Page 335
6.3.1 Lattice heat conduction......Page 337
6.4 Thermopower......Page 341
Relation between thermopower and scattering probabilities......Page 343
Transient response and dynamical effects in thermopower......Page 345
6.5.1 Elastic vs. inelastic contribution to electro-migration......Page 346
Direct and wind forces......Page 348
Forces from Kubo linear-response theory......Page 350
6.5.4 Forces at equilibrium......Page 351
General de.nition......Page 353
An example......Page 356
Can phonons be defined in the presence of current?......Page 358
A gedanken experiment......Page 359
6.6 Local ionic heating vs. current-induced forces......Page 361
Exercises......Page 362
7 The micro-canonical picture of transport......Page 364
Quasi-steady states of open systems......Page 365
Quasi-steady states of closed systems......Page 366
7.1.2 Initial conditions and dynamics......Page 367
7.2.1 Closed and finite quantum systems in a pure state......Page 369
7.2.2 Closed quantum systems in a pure state with arbitrary boundary conditions......Page 371
7.2.3 Current in open quantum systems......Page 372
7.2.4 Closure of the BBGKY hierarchy......Page 374
7.2.5 Functional approximations and loss of information......Page 375
7.3 Transient dynamics......Page 376
7.4.1 Variational definition of quasi-steady states......Page 378
7.4.2 Dependence of quasi-steady states on initial conditions......Page 382
7.5 A non-equilibrium entropy principle......Page 383
Electron sources revisited......Page 386
Electron-electron interaction......Page 387
An elastic relaxation mechanism......Page 388
A numerical example......Page 390
7.7 Definition of conductance in the micro-canonical picture......Page 392
Summary and open questions......Page 393
8 Hydrodynamics of the electron liquid......Page 394
8.1 The Madelung equations for a single particle......Page 396
8.2 Hydrodynamic form of the Schrödinger equation......Page 398
8.2.1 Quantum Navier-Stokes equations......Page 400
8.3 Conductance quantization from hydrodynamics......Page 406
8.4 Viscosity from Time-Dependent Current Density-Functional Theory......Page 409
Loss of memory approximation......Page 411
8.4.1 Functional approximation, loss of information, and dissipative dynamics......Page 412
8.4.2 Effect of viscosity on resistance......Page 413
8.5 Turbulent transport......Page 415
Adiabatic vs. non-adiabatic potentials......Page 417
Eddy size and energy dissipation......Page 418
A numerical example......Page 419
8.6 Local electron heating......Page 421
8.6.1 Electron heat conduction......Page 423
8.6.2 Hydrodynamics of heat transfer......Page 424
Incompressible electron liquid......Page 425
8.6.3 Effect of local electron heating on ionic heating......Page 428
Summary and open questions......Page 430
Exercises......Page 431
Fermions......Page 433
Creation and annihilation operators......Page 434
Operators in second-quantized form......Page 436
Bosons......Page 437
Appendix B The quantum BBGKY hierarchy......Page 438
Appendix C The Lindblad equation......Page 441
C.1 The Lindblad theorem......Page 442
The projection method......Page 444
Energy renormalization due to the bath......Page 445
Memory kernels......Page 446
Markov approximation......Page 447
C.3 Steady-state solutions......Page 448
D.1 The Hohenberg-Kohn theorem......Page 449
D.2 The Kohn-Sham equations......Page 450
D.4 The local density approximation and beyond......Page 452
E.1 The Runge-Gross theorem......Page 454
E.3 The adiabatic local density approximation......Page 455
F.1 The current density as the main variable......Page 457
F.2 The exchange-correlation electric field......Page 458
F.3 Approximate formulas for the viscosity......Page 460
G.1 The stochastic Schrödinger equation......Page 462
G.2 Derivation of the quantum master equation......Page 464
G.3 The theorem of Stochastic TD-CDFT......Page 467
Appendix H Inelastic corrections to current and shot noise......Page 469
Appendix I Hydrodynamic form of the Schrödinger equation......Page 472
Appendix J Equation of motion for the stress tensor......Page 476
Appendix K Cut-off of the viscosity divergence......Page 479
Appendix L Bernoulli’s equation......Page 481
References......Page 482
Index......Page 488