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The Physics of Low-dimensional Semiconductors

Book Description

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

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Introduction
  9. 1. Foundations
    1. 1.1 Wave Mechanics and the Schrödinger Equation
    2. 1.2 Free Particles
    3. 1.3 Bound Particles: Quantum Well
    4. 1.4 Charge and Current Densities
    5. 1.5 Operators and Measurement
    6. 1.6 Mathematical Properties of Eigenstates
    7. 1.7 Counting States
    8. 1.8 Filling States: The Occupation Function
    9. Further Reading
    10. Exercises
  10. 2. Electrons and Phonons in Crystals
    1. 2.1 Band Structure in One Dimension
    2. 2.2 Motion of Electrons in Bands
    3. 2.3 Density of States
    4. 2.4 Band Structure in Two and Three Dimensions
    5. 2.5 Crystal Structure of the Common Semiconductors
    6. 2.6 Band Structure of the Common Semiconductors
    7. 2.7 Optical Measurement of Band Gaps
    8. 2.8 Phonons
    9. Further Reading
    10. Exercises
  11. 3. Heterostructures
    1. 3.1 General Properties of Heterostructures
    2. 3.2 Growth of Heterostructures
    3. 3.3 Band Engineering
    4. 3.4 Layered Structures: Quantum Wells and Barriers
    5. 3.5 Doped Heterostructures
    6. 3.6 Strained Layers
    7. 3.7 Silicon–Germanium Heterostructures
    8. 3.8 Wires and Dots
    9. 3.9 Optical Confinement
    10. 3.10 Effective-Mass Approximation
    11. 3.11 Effective-Mass Theory in Heterostructures
    12. Further Reading
    13. Exercises
  12. 4. Quantum Wells and Low-Dimensional Systems
    1. 4.1 Infinitely Deep Square Well
    2. 4.2 Square Well of Finite Depth
    3. 4.3 Parabolic Well
    4. 4.4 Triangular Well
    5. 4.5 Low-Dimensional Systems
    6. 4.6 Occupation of Subbands
    7. 4.7 Two- and Three-Dimensional Potential Wells
    8. 4.8 Further Confinement Beyond Two Dimensions
    9. 4.9 Quantum Wells in Heterostructures
    10. Further Reading
    11. Exercises
  13. 5. Tunnelling Transport
    1. 5.1 Potential Step
    2. 5.2 T-Matrices
    3. 5.3 More on T-Matrices
    4. 5.4 Current and Conductance
    5. 5.5 Resonant Tunnelling
    6. 5.6 Superlattices and Minibands
    7. 5.7 Coherent Transport with Many Channels
    8. 5.8 Tunnelling in Heterostructures
    9. 5.9 What Has Been Brushed Under the Carpet?
    10. Further Reading
    11. Exercises
  14. 6. Electric and Magnetic Fields
    1. 6.1 The Schrödinger Equation with Electric and Magnetic Fields
    2. 6.2 Uniform Electric Field
    3. 6.3 Conductivity and Resistivity Tensors
    4. 6.4 Uniform Magnetic Field
    5. 6.5 Magnetic Field in a Narrow Channel
    6. 6.6 The Quantum Hall Effect
    7. Further Reading
    8. Exercises
  15. 7. Approximate Methods
    1. 7.1 The Matrix Formulation of Quantum Mechanics
    2. 7.2 Time-Independent Perturbation Theory
    3. 7.3 k · p Theory
    4. 7.4 WKB Theory
    5. 7.5 Variational Method
    6. 7.6 Degenerate Perturbation Theory
    7. 7.7 Band Structure: Tight Binding
    8. 7.8 Band Structure: Nearly Free Electrons
    9. Further Reading
    10. Exercises
  16. 8. Scattering Rates: The Golden Rule
    1. 8.1 Golden Rule for Static Potentials
    2. 8.2 Impurity Scattering
    3. 8.3 Golden Rule for Oscillating Potentials
    4. 8.4 Phonon Scattering
    5. 8.5 Optical Absorption
    6. 8.6 Interband Absorption
    7. 8.7 Absorption in a Quantum Well
    8. 8.8 Diagrams and the Self-Energy
    9. Further Reading
    10. Exercises
  17. 9. The Two-Dimensional Electron Gas
    1. 9.1 Band Diagram of Modulation-Doped Layers
    2. 9.2 Beyond the Simplest Model
    3. 9.3 Electronic Structure of a 2DEG
    4. 9.4 Screening by an Electron Gas
    5. 9.5 Scattering by Remote Impurities
    6. 9.6 Other Scattering Mechanisms
    7. Further Reading
    8. Exercises
  18. 10. Optical Properties of Quantum Wells
    1. 10.1 General Theory
    2. 10.2 Valence-Band Structure: The Kane Model
    3. 10.3 Bands in a Quantum Well
    4. 10.4 Interband Transitions in a Quantum Well
    5. 10.5 Intersubband Transitions in a Quantum Well
    6. 10.6 Optical Gain and Lasers
    7. 10.7 Excitons
    8. Further Reading
    9. Exercises
  19. A1. Table of Physical Constants
  20. A2. Properties of Important Semiconductors
  21. A3. Properties of GaAs–AIAs Alloys at Room Temperature
  22. A4. Hermite’s Equation: Harmonic Oscillator
  23. A5. Airy Functions: Triangular Well
  24. A6. Kramers–Kronig Relations and Response Functions
    1. A6.1 Derivation of the Kramers–Kronig Relations
    2. A6.2 Model Response Functions
  25. Bibliography
  26. Index