QUANTUM STATES & SCATTERING IN SEMICONDUCTOR NANOSTRUCTURES

by Gerald Bastard, Francesca Carosella, Camille Ndebeka-bandou

Overview

This book is an introduction to quantum states and of their scattering in semiconductor nanostructures. Written with exercises and detailed solutions, it is designed to enable readers to start modelling actual electron states and scattering in nanostructures. It first looks at practical aspects of quantum states and emphasises the variational and perturbation approaches. Following this there is analysis of quasi two-dimensional materials, including discussion of the eigenstates of nanostructures, scattering mechanisms and their numerical results.Focussing on practical applications, this book moves away from standard discourse on theory and provides students of physics, nanotechnology and materials science with the opportunity to fully understand the electronic properties of nanostructures.

Product Details

ISBN-13:	9781786343048
Publisher:	World Scientific Publishing Europe Ltd
Publication date:	08/03/2017
Series:	ADVANCED TEXTBOOKS IN PHYSICS
Sold by:	Barnes & Noble
Format:	eBook
Pages:	448
File size:	32 MB
Note:	This product may take a few minutes to download.

Table of Contents

Foreword v

About the Authors ix

Part I Practical Quantum Mechanics 1

I.1 Schrödinger Equation 3

I.1.1 Eigenvalues, eigenvector of linear operators 6

I.1.2 Adjoint operator 7

I.1.3 Hermitian operators: observables 8

I.1.4 Unitary operators 8

I.1.5 Projectors 9

I.1.6 Commuting operators 9

I.1.7 Two important examples of non-commuting operators 13

I.1.8 Heisenberg inequalities 14

I.1.9 Spin 15

I.1.10 Spin-orbit coupling 16

I.1.11 Density of states 17

I.1.12 Identical particles and Pauli principle 20

I.1.13 Tensorial products 24

I.2 Bound and Extended States 27

I.2.1 Propagating and evanescent states 27

I.2.2 Probability current 30

I.2.3 Boundary conditions 31

I.2.4 Bound states 39

I.2.5 The problem of plane waves 48

I.2.6 Schrödinger equation, time-dependent aspects 52

I.3 Approximate Methods 57

I.3.1 Variational method 57

I.3.2 Perturbation theory 60

I.3.2.1 Non-degenerate perturbation theory 62

I.3.2.2 Degenerate perturbation theory 67

I.3.3 Time-dependent perturbation theory 70

I.3.3.1 Static scatterers 71

I.3.3.2 Time-dependent scattering 77

I.4 Landau Quantisation of Electron Motion in Ideal Semiconductor Bulks and Heterostructures 83

I.4.1 Landau level degeneracy 88

I.4.2 Perturbative estimates of δH₁ and δH₁ 89

I.4.3 Magnetic field-dependent density of states 91

I.4.4 A tractable case of lifting of the k_y degeneracy: the crossed E, B fields 93

Part II The Physics of Heterostructures 95

II.1 Background on Heterostructures 97

II.2 Electrons States in Nanostructures 103

II.2.1 The envelope function approximation 103

II.2.1.1 Introduction 103

II.2.1.2 Electronic states in bulk semiconductors 108

II.2.1.3 Heterostructure states 115

II.2.2 Multiple quantum wells: transfer matrix method 120

II.2.2.1 Multiple quantum wells and superlattices 120

II.2.2.2 Transfer matrix method 122

II.2.3 Double quantum wells 126

II.2.3.1 Tight binding analysis 131

II.2.3.2 Symmetrical double quantum well 134

II.2.4 Holes 135

II.3 Beyond the Ideal World 139

II.3.1 Population, velocity, energy relaxation times through rate equations 147

II.3.2 Rate equations with elastic and inelastic processes 151

II.3.3 Analysis of the relaxation times in rate equations 154

II.3.3.1 Impurity form factor 155

II.3.3.2 Phonon form factors and transition rates 158

II.3.4 Consequence of the Born approximation on the additivity of scattering frequencies 162

II.4 Screening at the Semi-classical Approximation 165

II.4.1 Case of a single subband occupation 166

II.4.2 Case of many subbands occupation 171

II.4.3 Screening of inter-subband matrix elements 173

II.5 Results for Static Scatterers 177

II.5.1 Scattering by static disorder 177

II.5.2 Scattering of composite particles/excitons at the Born approximation 184

II.5.3 Scattering on magnetic impurities 193

II.5.3.1 The "spin"-flip scattering of electrons 193

II.5.3.2 The "spin"-flip scattering of holes 201

II.5.4 Three-body collisions 205

II.5.4.1 FCA in imperfect bulks and heterostructures 206

II.5.4.2 Phonon scattering in the presence of static scatterers 218

II.6 Results for Electron-Phonon Interaction 223

II.6.1 Optical phonon scattering 224

II.6.2 Acoustical phonon scattering 229

II.6.3 Energy loss irate 231

II.7 Beyond the Born Approximation 235

II.7.1 Scattering between Landau levels 238

Part III Exercises 255

1 Average position and velocity 257

2 Average velocity in a bound state 257

3 Density of states 258

4 Density of states of a camel back shaped dispersion relation 260

5 Heisenberg inequality in a quantum well with infinitely high barriers 263

6 Manipulating Slater determinants 266

7 Pauli principle for two weakly interacting electrons in ID 267

8 Calculation with Pauli matrices 270

9 Moss-Burstein shift of inter band absorption 271

10 Virial theorem 273

11 Abseuce of degeneracy for the ID bound states 276

12 Variational method: hydrogen atom 278

13 Variational method: electron in a triangular-potential 279

14 Variational method: anharmonic oscillator 280

15 Screened coulombic bound states 281

16 A two-dimensional coulombic problem 287

17 Inter-subband transitions in cubic GaN/AlN quantum wells: information on the conduction hand offset 290

18 Asymmetrical square quantum well 294

19 Spherical quantum dots 296

20 Delta quantum well 298

21 Wavefunction amplitude at the interfaces 302

22 Interface state in HgTe/CdTe heterojunctions 304

23 Step quantum well 306

24 Application of the Bohr-Sommerfeld quantisation rule to ID confining potential: digital alloying 313

25 Transmission/reflection in a delta quantum well 319

26 Static perturbation of a harmonic oscillator 323

27 Static perturbation (degenerate case) 324

28 Degenerate perturbation calculus applied to quantum dots with cylindrical symmetry 325

29 Quantum well and a delta potential: perturbative estimate 327

30 Quantum dot anisotropy 332

31 Defect in a superlattice: tight binding approach 339

32 Bound states created by two delta scatterers in a Landau level 345

33 Time-dependent evolution in an infinitely deep quantum well 351

34 Time-dependent problem: evolution 352

35 A touch of interaction representation 356

36 Time evolution if A and H commute 357

37 Oscillator: time evolution of averages 358

38 Time evolution of a system where one level is coupled to N degenerate levels 359

39 Time-dependent Hamiltonian: an exactly solvable model 363

40 Time evolution of superlattice states 365

41 Wavepackets 370

42 Average velocity of a wavepacket 372

43 Time-dependent perturbation in a 2-level system 373

44 Universal absorption probability for interband transitions in graphene 375

45 Scattering by N random impurity dimmers 383

46 A tractable example of selective doping by delta scatterers 388

47 Comparison between Born and self-cons is tent Born approximations 391

48 Influence of a fast emptying of the final subband on the equilibrium between two subbands 396

49 Phonon-mediated equilibration of the electronic temperature to the lattice temperature 398

50 Inter-subband scattering by unscreened coulombic impurities 402

51 Evaluation of a double sum appearing in the free carrier absorption 407

52 Energy loss rate for the in-plane polarization T = 0 K 410

53 Inter-subband absorption versus carrier concentration in an ideal heterostructure 413

54 Electron-LO phonon interaction: dimensionality dependence 416

Bibliography 423

Index 429