Polarons and Excitons ()

POLARONS AND EXCITONS......Page 1
Half-title......Page 3
Title Page......Page 5
Copyright Page......Page 6
Preface......Page 7
Editors' Note......Page 9
Participants......Page 12
Contents......Page 15
1 Historical Introduction......Page 21
2 Qualitative Survey......Page 23
2.1 The size of the polaron......Page 25
2.2. Simple theory of polaron effective mass......Page 27
2.3. Localization......Page 30
3 Macroscopic Model......Page 31
4 Derivation of the Hamiltonian......Page 34
4.1. Example: a classical point charge at rest......Page 40
5 Solutions for Weak Coupling......Page 42
Appendix. Canonical Transformation from Coordinate and Momentum Variables to Creation and Annihilation Operators......Page 48
References......Page 52
1 Introduction......Page 53
2.1. Elimination of the electron coordinate and momentum......Page 56
2.2. The "displaced-oscillator" transformation......Page 57
3 Properties of the Intermediate-Coupling Solutions......Page 60
4 Range of Validity of Intemediate-Coupling Theory......Page 62
References......Page 64
1 Introduction......Page 65
2.1. Diagonalization of the interaction......Page 66
2.2. The eigenfunctions of H[sub(1)]......Page 68
2.3. Minimization of H[sub(1)]......Page 69
2.4. The Born–Oppenheimer approximation......Page 71
3.1. The structure of the adiabatic Hamiltonian......Page 73
3.2 Introduction of translational coordinates......Page 74
3.3. Evaluation of the kinetic energy in terms of the translational coordinates......Page 76
3.4. The Hamiltonian in the harmonic approximation......Page 81
3.5. Some properties of the harmonic Hamiltonian......Page 83
3.6. Pekar's approximate Hamiltonian......Page 85
3.7. Introduction of three extra oscillators......Page 86
Appendix. Variational Derivation of the Polariton Rest Energy and Effective Mass in the Strong-Coupling Region......Page 87
References......Page 90
1.1. Introduction......Page 91
A. Transformation functions......Page 94
B. Density matrix in the canonical ensemble......Page 95
B. Transformation function for infinitesimal times......Page 96
D. Some remarks......Page 97
1.4. Evaluation of path integrals......Page 98
B. Free harmonic oscillator in one dimension, L = 1/2 m(ẋ² – ω²x²)......Page 99
C. Forced harmonic oscillator in one dimension......Page 102
2.1. Formulation of the problem in path integrals......Page 105
2.2. The Feynman variational principle......Page 108
A. Ground-state energy......Page 110
B. Polaron effective mass......Page 113
2.4. Polaron model corresponding to the Feynman approximation......Page 115
2.5. Perturbation corrections to the Feynman approximation......Page 118
3.1. Electron-lattice partition function in path-integral formalism......Page 119
3.2. Variational principle and approximate action......Page 121
4.1. Polaron bound to a point defect......Page 124
4.2. An exciton interacting with lattice vibrations......Page 127
4.4. Polaron in a static magnetic field......Page 130
Seminar. Polaron Mobility Using the Boltzmann Equation......Page 131
Low and Pines' Results......Page 138
Feynman Mobility in Lowest Order......Page 139
References......Page 140
1 Introduction......Page 143
2 Formulation of the Mobility Problem in Terms of the Electron Coordinates Alone......Page 144
3 A Method of Approximation......Page 152
4 First Correction Term......Page 155
5 Behaviour of the Impedance......Page 158
5.2. General expression for dissipation......Page 159
5.3. Dissipation at low temperatures......Page 161
5.4. Behaviour at high temperatures......Page 166
6 Weak-Coupling Limit: The Boltzmann Equation......Page 167
7 Suggestions for Improving Accuracy......Page 170
References......Page 172
1 Introduction......Page 175
2 The One-Electron Green's Function......Page 176
2.1. Spectral representation of one-electron Green's function......Page 179
3 The Phonon Propagator......Page 182
4 Electron–Phonon Interactions: Feynman Diagrams......Page 183
References......Page 190
1 Introduction......Page 191
2 Definition and General Properties of G......Page 192
3 Equations of Motion......Page 195
4.1. Non-interacting case......Page 198
4.2. Perturbation theory......Page 199
4.3. Hartree–Fock approximation......Page 201
5 An Exactly Soluble Model......Page 204
7 Many-Electron Green's Functions and the Strong-Coupling Limit......Page 206
References......Page 210
Introduction......Page 211
Propagators on a Two-Level Model......Page 212
Dissipation......Page 216
Coherent Anomalous Dispersion and Resonant Line Shape......Page 217
Application to a Gaseous Laser......Page 219
Introduction......Page 222
The Electrical Conductivity......Page 227
Contributions from the Phonon-Exchange Graphs......Page 228
References......Page 230
1 Introduction and Continuum Model......Page 231
1.1. Elastic continuum model......Page 233
2.1. Discontinuous change in the effective mass......Page 237
2.2. The variation of the effective mass with coupling constant......Page 240
3 Adiabatic Theory of the Self-Trapped State......Page 244
3.1. The results of calculation in the case of a simple cubic lattice......Page 248
References......Page 252
G. L. Sewell......Page 253
Model A......Page 256
Model B......Page 258
The Basic Process......Page 260
Dielectric Properties......Page 262
Conduction in Periodic Lattice......Page 263
S. J. Neitel......Page 265
1. The Trial Function......Page 267
2. The Crystal Energies......Page 269
3. Electronic and Ionic Polarization......Page 272
4. Electron Spin Resonance Experiments......Page 274
1. Introduction......Page 275
2.1. Transitions between two small-polaron bands......Page 277
2.2. Transitions from a small-polaron band to a wide band......Page 283
3. Discussion......Page 284
References......Page 287
1.1 Approximations......Page 289
1.2. Formulae for optical properties......Page 291
2 Band Theory......Page 292
2.1. Band-to-band transitions in semiconductors......Page 293
3 Effective-Mass Theory for Electrons......Page 295
4.1 Direct transitions......Page 299
4.2. Indirect transitions......Page 302
5.1. Band states in a magnetic field......Page 304
5.2. Band-to-band transitions in a magnetic field......Page 305
6 Effects of Electron Spin......Page 307
8.1. Germanium......Page 309
8.2. Cuprous Oxide......Page 310
Appendix......Page 312
References......Page 313
1 Introduction......Page 315
2.2. Properties of the polaron wave functions......Page 317
2.3. Exciton variational principle......Page 318
2.4. Evaluation of the effective electron-hole potential in the intermediate-coupling region......Page 321
3.1. Historical introduction......Page 323
3.2. Treatment of the many-body Hamiltonian......Page 325
3.3. Multipole expansion of Coulomb interaction......Page 328
3.4. Diagonalization of the pair–pair interaction......Page 329
3.5. Transition to the effective two-particle Hamiltonian......Page 330
4.1. Characteristics of Feynman's polaron......Page 333
4.2. Extension of this method to the exciton......Page 334
4.3. Generalization to non-zero temperature......Page 335
4.4. On the use of time-ordered operators......Page 336
4.5. Disentangling of phonon operators and elimination of phonon coordinates......Page 337
4.6. Approximation procedure for calculating the temperature-dependent Green's function......Page 338
4.7. Use of the trial Hamiltonian to calculate polaron damping and effective mass......Page 340
References......Page 342
1.1. Introduction—Elementary concepts......Page 343
1.2. Survey of polar crystals......Page 344
A. Semiconductors......Page 346
B. Insulating photoconductors......Page 348
2.1. Theoretical formulae......Page 355
2.2. Experimental results: semiconductors and weakly polar crystals......Page 357
A. The alkali halides......Page 359
B. The silver halides......Page 360
3 Cyclotron Resonance in Silver Bromide......Page 365
4.2. Direct transitions......Page 367
4.3. Indirect transitions......Page 369
References......Page 374
G. Ascarelli......Page 377
References......Page 386
T. P. McLean......Page 387
Indirect transitions (region A)......Page 389
Direct transitions (region B)......Page 391
Theoretical binding energy of exciton......Page 392
M. Grosmann......Page 393
1. The Unperturbed Cuprous Oxide Spectrum......Page 394
(b) Interpretation......Page 396
2.2 Strains......Page 397
2.3. Neutrons and gamma rays......Page 398
2.5. Magnetic fields......Page 400
References......Page 401