Semiconductor Nanostructures: Quantum states and electronic transport

Semiconductor Nanostructures (February 2010) (ATTiCA)......Page 1
Contents......Page 10
1.1 A short survey......Page 18
1.2 What is a semiconductor?......Page 22
1.3 Semiconducting materials......Page 25
Exercises......Page 26
2.2.1 Silicon......Page 28
2.2.2 Germanium......Page 30
2.3.1 Molecular beam epitaxy – MBE......Page 32
2.3.2 Other methods......Page 34
Exercises......Page 35
3.1 Spinless and noninteracting electrons......Page 36
3.2 Electron spin and the Zeeman hamiltonian......Page 44
3.3 Spin–orbit interaction......Page 46
3.4 Band structure of some semiconductors......Page 48
3.5 Band structure near band extrema: k·p-theory......Page 50
3.6 Spin–orbit interaction within k·p-theory......Page 59
3.7 Thermal occupation of states......Page 64
3.8 Measurements of the band structure......Page 66
Exercises......Page 68
4.1 Quantum mechanical motion in a parabolic band......Page 70
4.2 Semiclassical equations of motion, electrons and holes......Page 76
Further reading......Page 77
Exercises......Page 78
5.1 Band engineering......Page 80
5.2 Doping, remote doping......Page 89
5.3 Semiconductor surfaces......Page 93
5.4 Metal electrodes on semiconductor surfaces......Page 94
Exercises......Page 99
6.1 Growth methods......Page 100
6.2 Lateral patterning......Page 105
Further reading......Page 110
7.1 The electrostatic problem......Page 112
7.2 Formal solution using Green’s function......Page 113
7.3 Induced charges on gate electrodes......Page 115
7.4 Total electrostatic energy......Page 116
7.5 Simple model of a split-gate structure......Page 117
Exercises......Page 119
8.1 General hamiltonian......Page 120
8.2 Single-particle approximations for the many-particle problem......Page 123
Further reading......Page 129
Exercises......Page 130
9.1 Electrostatics of a GaAs/AlGaAs heterostructure......Page 132
9.2 Electrochemical potentials and applied gate voltage......Page 134
9.4 Fang–Howard variational approach......Page 135
9.5.1 Spatial potential fluctuations......Page 139
9.5.2 Linear static polarizability of the electron gas......Page 140
9.5.3 Linear screening......Page 142
9.5.4 Screening a single point charge......Page 145
9.5.5 Mean amplitude of potential fluctuations......Page 149
9.5.6 Nonlinear screening......Page 151
9.6 Spin–orbit interaction......Page 152
9.7 Summary of characteristic quantities......Page 155
Further reading......Page 157
Exercises......Page 158
10.1 Ohm’s law and current density......Page 160
10.2 Hall effect......Page 162
10.3 Drude model with magnetic field......Page 163
10.4 Sample geometries......Page 167
10.5 Conductivity from Boltzmann’s equation......Page 174
10.6 Scattering mechanisms......Page 178
10.7 Quantum treatment of ionized impurity scattering......Page 182
10.8 Einstein relation: conductivity and diffusion constant......Page 186
10.9 Scattering time and cross-section......Page 187
10.10 Conductivity and field effect in graphene......Page 188
Further reading......Page 190
Exercises......Page 191
11.1 Experimental observation of conductance quantization......Page 192
11.2 Current and conductance in an ideal quantum wire......Page 194
11.3 Current and transmission: adiabatic approximation......Page 199
11.4 Saddle point model for the quantum point contact......Page 202
11.5 Conductance in the nonadiabatic case......Page 203
11.6 Nonideal quantum point contact conductance......Page 205
11.8 Diffusive limit: recovering the Drude conductivity......Page 206
Exercises......Page 209
12.1 Tunneling through a single delta-barrier......Page 210
12.2 Perturbative treatment of the tunneling coupling......Page 212
12.3 Tunneling current in a noninteracting system......Page 215
Exercises......Page 217
13.1 Generalization of conductance: conductance matrix......Page 218
13.2 Conductance and transmission: Landauer–Büttiker approach......Page 219
13.3 Linear response: conductance and transmission......Page 220
13.4 The transmission matrix......Page 221
13.5 S-matrix and T-matrix......Page 222
13.6 Time-reversal invariance and magnetic field......Page 225
13.7 Four-terminal resistance......Page 226
13.8 Ballistic transport experiments in open systems......Page 229
Exercises......Page 240
14.1 Double-slit interference......Page 242
14.2 The Aharonov–Bohm phase......Page 243
14.3 Aharonov–Bohm experiments......Page 246
14.4 Berry’s phase and the adiabatic limit......Page 252
14.5 Aharonov–Casher phase and spin–orbit interaction induced phase effects......Page 260
14.6 Experiments on spin–orbit interaction induced phase effects in rings......Page 266
14.7.1 Decoherence by entanglement with the environment......Page 267
14.7.2 Decoherence by motion in a fluctuating environment......Page 270
14.8 Conductance fluctuations in mesoscopic samples......Page 273
Exercises......Page 279
15.1 Weak localization effect......Page 282
15.2 Decoherence in two dimensions at low temperatures......Page 284
15.3 Temperature-dependence of the conductivity......Page 285
15.4 Suppression of weak localization in a magnetic field......Page 286
15.5 Validity range of the Drude–Boltzmann theory......Page 289
15.6 Thouless energy......Page 290
15.7 Scaling theory of localization......Page 292
15.8 Length scales and their significance......Page 296
15.9 Weak antilocalization and spin–orbit interaction......Page 297
Exercises......Page 303
16.1 Shubnikov–de Haas effect......Page 304
16.1.1 Electron in a perpendicular magnetic field......Page 305
16.1.2 Quantum treatment of E X B-drift......Page 309
16.1.3 Landau level broadening by scattering......Page 310
16.1.4 Magnetocapacitance measurements......Page 314
16.1.5 Oscillatory magnetoresistance and Hall resistance......Page 315
16.2 Electron localization at high magnetic fields......Page 318
16.3 The integer quantum Hall effect......Page 322
16.3.1 Phenomenology of the quantum Hall effect......Page 323
16.3.2 Bulk models for the quantum Hall effect......Page 326
16.3.3 Models considering the sample edges......Page 327
16.3.4 Landauer–Büttiker picture......Page 328
16.3.5 Self-consistent screening in edge channels......Page 335
16.3.6 Quantum Hall effect in graphene......Page 337
16.4.1 Experimental observation......Page 339
16.4.2 Laughlin’s theory......Page 341
16.4.3 New quasiparticles: composite fermions......Page 342
16.4.4 Composite fermions in higher Landau levels......Page 344
16.4.5 Even denominator fractional quantum Hall states......Page 345
16.4.6 Edge channel picture......Page 346
16.5 The electronic Mach–Zehnder interferometer......Page 347
Further reading......Page 349
Exercises......Page 350
17.1 Influence of screening on the Drude conductivity......Page 352
17.2 Quantum corrections of the Drude conductivity......Page 355
Exercises......Page 356
18.1.1 Phenomenology......Page 358
18.1.2 Experiments demonstrating the quantization of charge on the quantum dot......Page 361
18.1.3 Energy scales......Page 362
18.1.4 Qualitative description......Page 366
18.2.1 Overview......Page 371
18.2.2 Capacitance model......Page 372
18.2.3 Approximations for the single-particle spectrum......Page 376
18.2.4 Energy level spectroscopy in a perpendicular magnetic field......Page 377
18.2.5 Spectroscopy of states using gate-induced electric fields......Page 381
18.2.6 Spectroscopy of spin states in a parallel magnetic field......Page 382
18.2.7 Two electrons in a parabolic confinement: quantum dot helium......Page 383
18.2.8 Hartree and Hartree–Fock approximations......Page 389
18.2.9 Constant interaction model......Page 392
18.2.10 Configuration interaction, exact diagonalization......Page 393
18.3.1 Resonant tunneling......Page 394
18.3.2 Sequential tunneling......Page 404
18.3.3 Higher order tunneling processes: cotunneling......Page 415
18.3.4 Tunneling with spin-flip: the Kondo effect in quantum dots......Page 420
Further reading......Page 423
Exercises......Page 424
19 Coupled quantum dots......Page 426
19.1 Capacitance model......Page 427
19.2 Finite tunneling coupling......Page 432
19.3.1 The effect of the tunneling coupling......Page 434
19.3.2 The effect of the hyperfine interaction......Page 435
19.4.2 Two quantum dots connected in series......Page 437
Exercises......Page 442
20.1 Classification of noise......Page 444
20.2 Characterization of noise......Page 445
20.3 Filtering and bandwidth limitation......Page 448
20.4 Thermal noise......Page 451
20.5.1 Shot noise of a vacuum tube......Page 453
20.5.2 Landauer’s wave packet approach......Page 455
20.5.3 Noise of a partially occupied monoenergetic stream of fermions......Page 457
20.5.4 Zero temperature shot noise with binomial distribution......Page 458
20.6 General expression for the noise in mesoscopic systems......Page 459
20.7.1 Shot noise in open mesoscopic systems......Page 462
20.7.2 Shot noise and full counting statistics in quantum dots......Page 464
Further reading......Page 467
Exercises......Page 468
21.1 The Fano effect......Page 470
21.2 Measurements of the transmission phase......Page 475
21.3 Controlled decoherence experiments......Page 478
Further reading......Page 484
Exercises......Page 485
22 Quantum information processing......Page 486
22.1.1 Uncertainty and information......Page 487
22.1.2 What is a classical bit?......Page 490
22.1.4 Information processing: loss of information and noise......Page 492
22.1.5 Sampling theorem......Page 501
22.1.6 Capacitance of a noisy communication channel......Page 503
22.2.1 Information entropy and physical entropy......Page 505
22.2.2 Energy dissipation during bit erasure: Landauer’s principle......Page 509
22.2.3 Boolean logic......Page 510
22.2.4 Reversible logic operations......Page 512
22.3.1 Quantum information theory: the basic idea......Page 513
22.3.2 Qubits......Page 515
22.3.3 Qubit operations......Page 522
22.4 Implementing qubits and qubit operations......Page 523
22.4.1 Free oscillations of a double quantum dot charge qubit......Page 524
22.4.2 Rabi oscillations of an excitonic qubit......Page 526
22.4.3 Quantum dot spin-qubits......Page 529
Further reading......Page 536
Exercises......Page 537
A.3 Fourier transform in two dimensions......Page 538
B.2 Proof of the symmetry of Green’s functions......Page 540
C: The delta-function......Page 542
References......Page 544
C......Page 562
D......Page 563
G......Page 564
L......Page 565
Q......Page 566
R......Page 567
S......Page 568
Z......Page 569