ELECTRONS AND PHONONS IN SEMICONDUCTOR MULTILAYERS

Transcription

1 ELECTRONS AND PHONONS IN SEMICONDUCTOR MULTILAYERS Second Edition B.K. RIDLEY University of Essex CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface Introduction 1 Simple Models of the Electron-Phonon Interaction 1.1 General Remarks 1.2 Early Models of Optical-Phonon Confinement The Dielectric-Continuum (DC) Model The Hydrodynamic (HD) Model The Reformulated-Mode (RM) Model Hybrid Modes 1.3 The Interaction of Electrons with Bulk Phonons The Scattering Rate The Coupling Coefficients The Overlap Integral in 2D The 2D Rates The ID Rates 1.4 The Interaction with Model Confined Phonons 2 Quantum Confinement of Carriers 2.1 The Effective-Mass Equation Introduction The Envelope-Function Equation The Local Approximation 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 v

3 Contents Quasi-Continuum Theory of Lattice Vibrations 3.1 Introduction 3.2 Linear-Chain Models Bulk Solutions Interface between Nearly Matched Media Interface between Mismatched Media Free Surface 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 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 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 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 Optical Modes in Various Structures 7.1 Introduction 7.2 Monolayers Single Monolayer Double Monolayer 7.3 Metal-Semiconductor Structures vi

4 Vll Contents 7.4 Slab Modes Quantum Wires Quantum Dots 181 Electron-Optical Phonon Interaction in a Quantum Well Introduction Scattering Rate Scattering Potentials for Hybrids Matrix Elements for an Indefinitely Deep Well Scattering Rates for Hybrids Threshold Rates Scattering by Barrier LO Modes Scattering by Interface Polaritons Summary of Threshold Rates in an Indefinitely Deep Well Intrasubband Rates Intersubband Rates Comparison with Simple Models The Interaction in a Superlattice The Interaction in an Alloy Phonon Resonances Quantum Wire The Sum-Rule 209 Appendix: Scalar and Vector Potentials 212 Other Scattering Mechanisms Charged-Impurity Scattering Introduction The Coulomb Scattering Rate Scattering by Single Charges Scattering by Fluctuations in a Donor Array An Example Interface-Roughness Scattering Alloy Scattering Electron-Electron Scattering Basic Formulae for the 2D Case Discussion Electron-Hole Scattering Phonon Scattering Phonon-Phonon Processes Charged-Impurity Scattering Alloy Fluctuations and Neutral Impurities Interface-Roughness Scattering 241

5 Contents 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 The Quasi-ID Dielectric Function 10.7 Lattice Screening 10.8 Image Charges 10.9 The Electron-Plasma/Coupled-Mode Interaction Discussion The Electron Distribution Function Spin The Boltzmann Equation Net Scattering Rate by Bulk Polar-Optical Phonons Optical Excitation Transport The 3D Case The 2D Case The ID Case Discussion Acoustic-Phonon Scattering The 3D Case The 2D Case The ID Case Piezoelectric Scattering Discussion Acoustic-Phonon Scattering in a Degenerate Gas Introduction Energy- and Momentum-Relaxation Rates Low-Temperature Approximation The Electron Temperature The High-Temperature Approximation Relaxation Introduction The Elliot-Yafet process The D'yakonov-Perel Process The DP Mechanism in a Quantum Well Quantum Wires The Rashba Mechanism The Bir-Aranov-Pikus Mechanism viii

6 IX Contents 12.6 Hyperfine Coupling 329 Appendix Appendix Appendix Electrons and Phonons in the Wurtzite Lattice The Wurtzite Lattice Energy Band Structure Eigenfunctions Optical Phonons Spontaneous Polarization 346 Appendix 1 Symmetry Nitride Heterostructures Single Heterostructures Piezoelectric Polarization Polarization Model of Passivated HFET with Field Plate The Polarization Superlattice Strain Deformation Potentials Fields The AlN/GaN Superlattice The Quantum-Cascade Laser 366 Appendix Airy Functions Terahertz Sources Introduction Bloch Oscillations Negative-Mass NDR The Esaki-Tsu Approach Lucky Drift The Hydrodynamic Model Ballistic Transport Optical-Phonon-Determined Transit-Time Oscillations Transit-Time Oscillations in a Short Diode Negative-Mass NDR Bloch Oscillations Femtosecond Generators Optical Non-Linear Rectification Surge Current Dember Diffusion Coherent Phonons Photoconductive Switch 389

7 Contents 15.6 CW Generators Photomixing Quantum-Cascade Lasers 390 Appendix 392 Appendix 1 The Polar-Optical Momentum-Relaxation Time in a 2D Degenerate Gas 393 Appendix 2 Electron/Polar Optical Phonon Scattering Rates in a Spherical Cosine Band 395 References 397 Index x