The Physics of Low-dimensional Semiconductors: An Introduction

by John H. Davies

• ISBN13: 9780521484916
• ISBN10: 052148491X
• Format: Paperback
• Copyright: 1997-12-13
• Publisher: Cambridge University Press

Summary

The composition of modern semiconductor heterostructures can be controlled precisely on the atomic scale to create low-dimensional systems. These systems have revolutionised semiconductor physics, and their impact on technology, particularly for semiconductor lasers and ultrafast transistors, is widespread and burgeoning. This book provides an introduction to the general principles that underlie low-dimensional semiconductors. As far as possible, simple physical explanations are used, with reference to examples from actual devices. The author shows how, beginning with fundamental results from quantum mechanics and solid-state physics, a formalism can be developed that describes the properties of low-dimensional semiconductor systems. Among numerous examples, two key systems are studied in detail: the two-dimensional electron gas, employed in field-effect transistors, and the quantum well, whose optical properties find application in lasers and other opto-electronic devices. The book includes many exercises and will be invaluable to undergraduate and first-year graduate physics or electrical engineering students taking courses in low-dimensional systems or heterostructure device physics.

Table of Contents

Preface	xiii
Introduction	xv
Foundations	1	(44)
Wave Mechanics and the Schrodinger Equation	1	(2)
Free Particles	3	(1)
Bound Particles: Quantum Well	4	(5)
Charge and Current Densities	9	(4)
Operators and Measurement	13	(7)
Mathematical Properties of Eigenstates	20	(2)
Counting States	22	(8)
Filling States: The Occupation Function	30	(15)
Further Reading	40	(1)
Exercises	41	(4)
Electrons and Phonons in Crystals	45	(35)
Band Structure in One Dimension	45	(5)
Motion of Electrons in Bands	50	(4)
Density of States	54	(1)
Band Structure in Two and Three Dimensions	55	(2)
Crystal Structure of the Common Semiconductors	57	(4)
Band Structure of the Common Semiconductors	61	(8)
Optical Measurement of Band Gaps	69	(1)
Phonons	70	(10)
Further Reading	76	(1)
Exercises	76	(4)
Heterostructures	80	(38)
General Properties of Heterostructures	80	(2)
Growth of Heterostructures	82	(3)
Band Engineering	85	(3)
Layered Structures: Quantum Wells and Barriers	88	(4)
Doped Heterostructures	92	(4)
Strained Layers	96	(4)
Silicon-Germanium Heterostructures	100	(2)
Wires and Dots	102	(3)
Optical Confinement	105	(2)
Effective-Mass Approximation	107	(4)
Effective-Mass Theory in Heterostructures	111	(7)
Further Reading	114	(1)
Exercises	114	(4)
Quantum Wells and Low-Dimensional Systems	118	(32)
Infinitely Deep Square Well	118	(1)
Square Well of Finite Depth	119	(6)
Parabolic Well	125	(3)
Triangular Well	128	(2)
Low-Dimensional Systems	130	(3)
Occupation of Subbands	133	(2)
Two- and Three-Dimensional Potential Wells	135	(5)
Further Confinement Beyond Two Dimensions	140	(2)
Quantum Wells in Heterostructures	142	(8)
Further Reading	146	(1)
Exercises	146	(4)
Tunnelling Transport	150	(56)
Potential Step	150	(3)
T-Matrices	153	(5)
More on T-Matrices	158	(4)
Current and Conductance	162	(5)
Resonant Tunnelling	167	(10)
Superlattices and Minibands	177	(6)
Coherent Transport with Many Channels	183	(12)
Tunnelling in Heterostructures	195	(4)
What Has Been Brushed Under the Carpet?	199	(7)
Further Reading	200	(1)
Exercises	201	(5)
Electric and Magnetic Fields	206	(43)
The Schrodinger Equation with Electric and Magnetic Fields	206	(2)
Uniform Electric Field	208	(8)
Conductivity and Resistivity Tensors	216	(3)
Uniform Magnetic Field	219	(14)
Magnetic Field in a Narrow Channel	233	(5)
The Quantum Hall Effect	238	(11)
Further Reading	245	(1)
Exercises	246	(3)
Approximate Methods	249	(41)
The Matrix Formulation of Quantum Mechanics	249	(3)
Time-Independent Perturbation Theory	252	(9)
k.p Theory	261	(2)
WKB Theory	263	(7)
Variational Method	270	(3)
Degenerate Perturbation Theory	273	(2)
Band Structure: Tight Binding	275	(5)
Band Structure: Nearly Free Electrons	280	(10)
Further Reading	284	(1)
Exercises	284	(6)
Scattering Rates: The Golden Rule	290	(39)
Golden Rule for Static Potentials	290	(5)
Impurity Scattering	295	(6)
Golden Rule for Oscillating Potentials	301	(1)
Phonon Scattering	302	(6)
Optical Absorption	308	(5)
Interband Absorption	313	(3)
Absorption in a Quantum Well	316	(5)
Diagrams and the Self-Energy	321	(8)
Further Reading	324	(1)
Exercises	324	(5)
The Two-Dimensional Electron Gas	329	(42)
Band Diagram of Modulation-Doped Layers	329	(7)
Beyond the Simplest Model	336	(6)
Electronic Structure of a 2DEG	342	(7)
Screening by an Electron Gas	349	(7)
Scattering by Remote Impurities	356	(6)
Other Scattering Mechanisms	362	(9)
Further Reading	365	(1)
Exercises	366	(5)
Optical Properties of Quantum Wells	371	(38)
General Theory	371	(6)
Valence-Band Structure: The Kane Model	377	(7)
Bands in a Quantum Well	384	(3)
Interband Transitions in a Quantum Well	387	(6)
Intersubband Transitions in a Quantum Well	393	(2)
Optical Gain and Lasers	395	(2)
Excitons	397	(12)
Further Reading	406	(1)
Exercises	406	(3)
A1 TABLE OF PHYSICAL CONSTANTS	409	(1)
A2 PROPERTIES OF IMPORTANT SEMICONDUCTORS	410	(2)
A3 PROPERTIES OF GAAS-AIAS ALLOYS AT ROOM TEMPERATURE	412	(1)
A4 HERMITE'S EQUATION: HARMONIC OSCILLATOR	413	(2)
A5 AIRY FUNCTIONS: TRIANGULAR WELL	415	(2)
A6 KRAMERS-KRONIG RELATIONS AND RESPONSE FUNCTIONS	417	(6)
A6.1 Derivation of the Kramers-Kronig Relations	417	(2)
A6.2 Model Response Functions	419	(4)
Bibliography	423	(4)
Index	427