Electrons and Phonons in Semiconductor Multilayers (Cambridge Studies in Semiconductor Physics and Microelectronic Engineering)

Contents

page xiii

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

vii

1 1 2 4

9 11 14 15 15 18 20

23

28 29 38 38 38

41 42 45 47

51

60 63 65

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Contents

3 Quasi-Continuum Theory of Lattice Vibrations 3.1 Introduction 3.2 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 3.3 The envelope function 3.4 Non-local operators 3.5 Acoustic and optical modes 3.6 Boundary conditions 3.7 Interface model 3.8 Summary Appendix: The local approximation 4 Bulk 4.1 4.2 4.3 4.4 4.5

Vibrational Modes in an Isotropic Continuum Elasticity theory Polar material Polar optical waves Energy density 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

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

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

142 142 146 152 155 156 158

7 Optical Modes in Various Structures 7.1 Introduction 7.2 Monolayers 7.2.1 Single monolayer 7.2.2 Double monolayer

161 161 161 163 167

Conren~

7.3 7.4 7.5 7.6

Metal-semiconductor structures Slab modes Quantum wires Quantum dots

ix 171 174 177 182

8 Electron-Optical Phonon Interaction in a Quantum Well 8.1 Introduction 8.2 Scattering rate 8.3 Scattering potentials for hybrids 8.4 Matrix elements for an infinitely deep well 8.5 Scattering rates for hybrids 8.6 Threshold rates 8.7 Scattering by barrier LO modes 8.8 Scattering by interface polaritons 8.9 Summary of threshold rates in an infinitely deep well 8.9.1 Intrasubband rates 8.9.2 Intersubband rates 8.10 Comparison with simple models 8.11 The interaction in a superlattice 8.12 The interaction in an alloy 8.13 Phonon resonances 8.14 Quantum wire 8.15 The sum-rule Appendix: Scalar and vector potentials

184 184 185 186 187 189 192 195 196 200 201 202 203 205 209 210 210 214 216

9 Other Scattering Mechanisms 9.1 Charged-impurity scattering 9.1.1 Introduction 9.1.2 The Coulomb scattering rate 9.1.3 Scattering by single charges 9.1.4 Scattering by fluctuations in a donor array 9.1.5 An example 9.2 Interface-roughness scattering 9.3 Alloy scattering 9.4 Electron-electron scattering 9.4.1 Basic formulae for the 2D case 9.4.2 Discussion 9.4.3 Electron-hole scattering 9.5 Phonon scattering 9.5.1 Phonon-phonon processes 9.5.2 Charged-impurity scattering

221 221 221 225 226 228 229 231 235 236 236 239 241 241 241 243

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Contents

9.5.3 Alloy fluctuations and neutral impurities 9.5.4 Interface-roughness scattering

10 Quantum Screening 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10

Introduction The density matrix The dielectric function The 3D dielectric function The quasi-2D dielectric function The quasi-ID dielectric function Lattice screening Image charges The electron-plasma/coupled-mode interaction Discussion

11 The Electron Distribution Function 11.1 11.2 11.3 11.4

11.5

11.6 11.7

The Boltzmann equation Net scattering rate by bulk polar optical phonons Optical excitation Transport 11.4.1 The 3D case 11.4.2 The 2D case 11.4.3 The ID case 11.4.4 Discussion Acoustic-phonon scattering 11.5.1 The 3D case 11.5.2 The 2D case 11.5.3 The ID case 11.5.4 Piezoelectric scattering Discussion Acoustic-phonon scattering in a degenerate gas 11.7.1 Introduction 11.7.2 Energy- and momentum-relaxation rates 11.7.3 Low-temperature approximation 11.7.4 The electron temperature 11.7.5 The high-temperature approximation

244 246 249 249 250 253 255 257 265 270 272 275 279 282 282 283 285 288 292 294 296 297 299 300 301 303 305 305 309 309 309 313 316 316

Appendix: The Polar-Optical Momentum-Relaxation Time in a 2D Degenerate Gas

321

References Index

323 329

On Seeing the Light There's some that swear by whisky, There's some that swear by rye, There's some that swear by A.p And others by ec; But those that swear by whisky Will now and then take rye, But those that swear by A.p Will never use e¢. If those that swear by whisky Would know the reason why All those who swear by A.p Would never use e¢. Then ask the e¢ people. Their answer won't be guarded: "All those who swear by A.p Are definitely retarded." In the A.p man's heaven There's a place forever bright For repentant e¢ sinners Who at last have seen the light. B. K. Ridley (1992)