ELECTRONS AND PHONONS IN SEMICONDUCTOR MULTILAYERS В. К. RIDLEY University of Essex CAMBRIDGE UNIVERSITY PRESS
Contents Introduction 1 Simple Models of the Electron-Phonon Interaction 1.1 General remarks 1.2 Early models of optical-phonon confinement 1.2.1 The dielectric-continuum (DC) model 1.2.2 The hydrodynamic (HD) model 1.2.3 The reformulated mode (RM) model 1.2.4 Hybrid modes 1.3 The interaction of electrons with bulk phonons 1.3.1 The scattering rate 1.3.2 The coupling coefficients 1.3.3 The overlap integral in 2D 1.3.4 The 2D rates 1.3.5 The ID rates 1.4 The interaction with model confined phonons 2 Quantum Confinement of Carriers 2.1 The effective-mass equation 2.1.1 Introduction 2.1.2 The envelope-function equation 2.1.3 The local approximation 2.1.4 The effective-mass approximation 2.2 The confinement of electrons 2.3 The confinement of holes 2.4 Angular dependence of matrix elements 2.5 Non-parabolicity 2.6 Band-mixing Vll
viii Contents 3 Quasi-Continuum Theory of Lattice Vibrations 3.1 3.2 Introduction ^ 3.3 3.4 3.5 3.6 3.7 3.8 Linear-chain models 3.2.1 Bulk solutions 3.2.2 Interface between nearly matched media 3.2.3 Interface between mismatched media 3.2.4 Free surface 3.2.5 Summary The envelope function Non-local operators Acoustic and optical modes Boundary conditions Interface model Summary Appendix: The local approximation 4 Bulk Vibrational Modes in an Isotropic Continuum 4.1 Elasticity theory 4.2 Polar material 4.3 Polar optical waves 4.4 Energy density 4.5 Two-mode alloys 5 Optical Modes in a Quantum Well 5.1 Non-polar material 5.2 Polar material 5.3 Barrier modes: optical phonon tunnelling 5.4 The effect of dispersion 5.5 Quantization of hybrid modes 6 Superlattice Modes 6.1 Superlattice hybrids 6.2 Superlattice dispersion 6.3 General features 6.4 Interface polaritons in a superlattice 6.5 The role of LO and TO dispersion 6.6 Acoustic phonons 7 Optical Modes in Various Structures 7.1 Introduction 7.2 Monolayers 7.2.1 Single monolayer 7.2.2 Double monolayer 66 66 68 68 71 74 75 75 76 78 80 84 86 93 96 99 99 104 106 108 115 120 120 123 129 138 139 142 142 146 152 155 156 158 163 167
Contents ix 7.3 Metal-semiconductor structures 171 7.4 Slab modes 174 7.5 Quantum wires 177 7.6 Quantum dots 182 8 Electron-Optical Phonon Interaction in a Quantum Well 184 8.1 Introduction 184 8.2 Scattering rate 185 8.3 Scattering potentials for hybrids 186 8.4 Matrix elements for an infinitely deep well 187 8.5 Scattering rates for hybrids 189 8.6 Threshold rates 192 8.7 Scattering by barrier LO modes 195 8.8 Scattering by interface polaritons 196 8.9 Summary of threshold rates in an infinitely deep well 200 8.9.1 Intrasubband rates 201 8.9.2 Intersubband rates 202 8.10 Comparison with simple models 203 8.11 The interaction in a superlattice 205 8.12 The interaction in an alloy 209 8.13 Phonon resonances 210 8.14 Quantum wire 210 8.15 The sum-rule 214 Appendix: Scalar and vector potentials 216 9 Other Scattering Mechanisms 221 9.1 Charged-impurity scattering 221 9.1.1 Introduction 221 9.1.2 The Coulomb scattering rate 225 9.1.3 Scattering by single charges 226 9.1.4 Scattering by fluctuations in a donor array 228 9.1.5 An example 229 9.2 Interface-roughness scattering 231 9.3 Alloy scattering 235 9.4 Electron-electron scattering 236 9.4.1 Basic formulae for the 2D case 236 9.4.2 Discussion 239 9.4.3 Electron-hole scattering 241 9.5 Phonon scattering 241 9.5.1 Phonon-phonon processes 241 9.5.2 Charged-impurity scattering 243
X Contents 9.5.3 Alloy fluctuations and neutral impurities 244 9.5.4 Interface-roughness scattering 246 10 Quantum Screening 10.1 Introduction 10.2 The density matrix 10.3 The dielectric function 10.4 The 3D dielectric function 10.5 The quasi-2d dielectric function 10.6 The quasi-id dielectric function 10.7 Lattice screening 10.8 Image charges 10.9 The electron-plasma/coupled-mode interaction 10.10 Discussion 249 249 250 253 255 257 265 270 272 275 279 11 The Electron Distribution Function 282 11.1 The Boltzmann equation 282 11.2 Net scattering rate by bulk polar optical phonons 283 11.3 Optical excitation 285 11.4 Transport 288 11.4.1 The 3D case 292 11.4.2 The 2D case 294 11.4.3 The ID case 296 11.4.4 Discussion 297 11.5 Acoustic-phonon scattering 299 11.5.1 The 3D case 300 11.5.2 The 2D case 301 11.5.3 The ID case 303 11.5.4 Piezoelectric scattering 305 11.6 Discussion 305 11.7 Acoustic-phonon scattering in a degenerate gas 309 11.7.1 Introduction 309 11.7.2 Energy- and momentum-relaxation rates 309 11.7.3 Low-temperature approximation 313 11.7.4 The electron temperature 316 11.7.5 The high-temperature approximation 316 Appendix: The Polar-Optical Momentum-Relaxation Time in a 2D Degenerate Gas 321 References 323 Index 329