# Solid State and Quantum Theory for Optoelectronics

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## Book Description

While applications rapidly change one to the next in our commercialized world, fundamental principles behind those applications remain constant. So if one understands those principles well enough and has ample experience in applying them, he or she will be able to develop a capacity for reaching results via conceptual thinking rather than having to always rely on models to test various conditions.

In **Solid State and Quantum Theory for Optoelectronics**, Michael Parker provides a general conceptual framework for matter that leads to the matter-light interaction explored in the author’s Physics of Optoelectronics (CRC Press). Instead of overburdening readers with the definition–theorem– proof format often expected in mathematics texts, this book instructs readers through the development of conceptual pictures. Employing a proven pedagogic approach, as rigorous as it is intuitive, Professor Parker –

- Provides several lead-ins to the quantum theory including a brief review of Lagrange and Hamilton’s approach to classical mechanics and the fundamental quantum link with Hilbert space
- Demonstrates the Schrödinger wave equation from the Feynman path integral
- Discusses standard topics such as the quantum well, harmonic oscillator, representations, perturbation theory, and spin
- Expands discussion from the density operator and its applications to quantum computing and teleportation
- Provides the concepts for ensembles and microstates in detail with emphasis on the derivation of particle population distributions across energy levels

Professors Parker includes problems to help readers understand and internalize the material. But just as important, the working-through of these problems will help readers develop the sort of approach that, instead of wholly relying on models, enables them to extrapolate solutions guided by informed intuition developed over the course of formal study and laboratory experiment. It is the kind of conceptual thinking that will allow readers to move with deeper understanding from optical applications to more theoretical topics in physics.

## Table of Contents

**Chapter 1 Introduction to the Solid State **1.1 Brief Preview

1.2 Introduction to Matter and Bonds

1.2.1 Gasses and Liquids

1.2.2 Solids

1.2.3 Bonding and the Periodic Table

1.2.4 Dopant Atoms

1.3 Introduction to Bands and Transitions

1.3.1 Intuitive Origin of Bands

1.3.2 Indirect Bands and Light- and Heavy-Hole Bands

1.3.3 Introduction to Transitions

1.3.4 Introduction to Band-Edge Diagrams

1.3.5 Bandgap States and Defects

1.4 Introduction to the pn Junction

1.4.1 Junction Technology

1.4.2 Band-Edge Diagrams and the pn Junction

1.4.3 Nonequilibrium Statistics

1.5 Device Trends

1.5.1 Monolithic Integration of Device Types

1.5.2 Year 2000 Benchmarks

1.5.3 Small Optical Signals

1.5.4 Fabrication Challenges

1.6 Vacuum Tubes and Transistors

1.6.1 Vacuum Tube

1.6.2 Bipolar Transistor

1.6.3 Field-Effect Transistor

1.7 Brief Summary of Some Early Nanometer-Scale Devices

1.7.1 Resonant-Tunnel Device

1.7.2 Resonant-Tunneling Transistor

1.7.2.1 Single-Electron Transistors

1.7.2.2 Quantum Cellular Automation (QCA)

1.7.2.3 Aharanov–Bohm Effect Device

1.7.2.4 Quantum Interference Devices

1.7.2.5 Josephson Junction

1.8 Review Exercises

References and Further Readings

**Chapter 2 Vector and Hilbert Spaces**

2.1 Vector and Hilbert Spaces

2.1.1 Motivation for Linear Algebra in Quantum Theory

2.1.2 Definition of Vector Space

2.1.3 Hilbert Space

2.1.4 Comment on the Length of a Vector for Quantum Theory

2.1.5 Linear Isomorphism

2.1.6 Antilinear Isomorphism

2.2 Dirac Notation and Euclidean Vector Spaces

2.2.1 Kets, Bras, and Brackets for Euclidean Space

2.2.2 Basis and Completeness for Euclidean Space

2.2.3 Closure Relation for the Euclidean Vector Space

2.2.4 Euclidean Dual Vector Space

2.2.5 Inner Product and Norm

2.3 Introduction to Coordinate and Vector Representation of Functions

2.3.1 Initial View of the Coordinate Representation of Functions

2.3.2 Coordinate Basis Set

2.3.3 Introduction to the Inner Product for Functions

2.3.4 Representations of Functions

2.4 Function Space with Discrete Basis Sets

2.4.1 Introduction to Hilbert Space

2.4.2 Hilbert Space of Functions with Discrete Basis Vectors

2.4.3 Closure Relation for Functions with a Discrete Basis

2.4.4 Norms and Inner Products for Function Spaces with Discrete Basis Sets

2.4.5 Discussion of Weight Functions

2.4.6 Some Miscellaneous Notes on Notation

2.5 Function Spaces with Continuous Basis Sets

2.5.1 Continuous Basis Set of Functions

2.5.2 Coordinate Space

2.5.3 Representations of the Dirac Delta Using Basis Vectors

2.6 Graham–Schmidt Orthonormalization Procedure

2.6.1 Simplest Case of Two Vectors

2.6.2 More than Two Vectors

2.7 Fourier Basis Sets

2.7.1 Fourier Cosine Series

2.7.2 Fourier Sine Series

2.7.3 Fourier Series

2.7.4 Alternate Basis for the Fourier Series

2.7.5 Fourier Transform

2.8 Closure Relations, Kronecker Delta, and Dirac Delta Functions

2.8.1 Alternate Closure Relations and Representations of the Kronecker Delta Function for Euclidean Space

2.8.2 Cosine Basis Functions

2.8.3 Sine Basis Functions

2.8.4 Fourier Series Basis Functions

2.8.5 Some Notes

2.9 Introduction to Direct Product Spaces

2.9.1 Overview of Direct Product Spaces

2.9.2 Introduction to Dyadic Notation for the Tensor Product of Two Euclidean Vectors

2.9.3 Direct Product Space from the Fourier Series

2.9.4 Components and Closure Relation for the Direct Product of Functions with Discrete Basis Sets

2.9.5 Notes on the Direct Products of Continuous Basis Sets

2.10 Introduction to Minkowski Space

2.10.1 Coordinates and Pseudo-Inner Product

2.10.2 Pseudo-Orthogonal Vector Notation

2.10.3 Tensor Notation

2.10.4 Derivatives

2.11 Brief Discussion of Probability and Vector Components

2.11.1 Simple 2-D Space for Starters

2.11.2 Introduction to Applications of the Probability

2.11.3 Discrete and Continuous Hilbert Spaces

2.11.4 Contrast with Random Vectors

2.12 Review Exercises

References and Further Readings

**Chapter 3 Operators and Hilbert Space**

3.1 Introduction to Operators and Groups

3.1.1 Linear Operator

3.1.2 Transformations of the Basis Vectors Determine the Linear Operator

3.1.3 Introduction to Isomorphisms

3.1.4 Comments on Groups and Operators

3.1.5 Permutation Group and a Matrix Representation: An Example

3.2 Matrix Representations

3.2.1 Definition of Matrix for an Operator with Identical Domain and Range Spaces

3.2.2 Matrix of an Operator with Distinct Domain and Range Spaces

3.2.3 Dirac Notation for Matrices

3.2.4 Operating on an Arbitrary Vector

3.2.5 Matrix Equation

3.2.6 Matrices for Function Spaces

3.2.7 Introduction to Operator Expectation Values

3.2.8 Matrix Notation for Averages

3.3 Common Matrix Operations

3.3.1 Composition of Operators

3.3.2 Isomorphism between Operators and Matrices

3.3.3 Determinant

3.3.4 Introduction to the Inverse of an Operator

3.3.5 Trace

3.3.6 Transpose and Hermitian Conjugate of a Matrix

3.4 Operator Space

3.4.1 Concepts and Section Summary

3.4.2 Basis Expansion of a Linear Operator

3.4.3 Introduction to the Inner Product for a Hilbert Space of Operators

3.4.4 Proof of the Inner Product

3.4.5 Basis for Matrices

3.5 Operators and Matrices in Direct Product Space

3.5.1 Review of Direct Product Spaces

3.5.2 Operators

3.5.3 Matrices of Direct Product Operators

3.5.4 Matrix Representation of Basis Vectors for Direct Product Space

3.6 Commutators and Algebra of Operators

3.6.1 Initial Discussion of Operator Algebra

3.6.2 Introduction to Commutators

3.6.3 Some Commutator Theorems

3.7 Unitary Operators and Similarity Transformations

3.7.1 Orthogonal Rotation Matrices

3.7.2 Unitary Transformations

3.7.3 Visualizing Unitary Transformations

3.7.4 Trace and Determinant

3.7.5 Similarity Transformations

3.7.6 Equivalent and Reducible Representations of Groups

3.8 Hermitian Operators and the Eigenvector Equation

3.8.1 Adjoint, Self-Adjoint, and Hermitian Operators

3.8.2 Adjoint and Self-Adjoint Matrices

3.9 Relation between Unitary and Hermitian Operators

3.9.1 Relation between Hermitian and Unitary Operators

3.10 Eigenvectors and Eigenvalues for Hermitian Operators

3.10.1 Basic Theorems for Hermitian Operators

3.10.2 Direct Product Space

3.11 Eigenvectors, Eigenvalues, and Diagonal Matrices

3.11.1 Motivation for Diagonal Matrices

3.11.2 Eigenvectors and Eigenvalues

3.11.3 Diagonalize a Matrix

3.11.4 Relation between a Diagonal Operator and the Change-of-Basis Operator

3.12 Theorems for Hermitian Operators

3.12.1 Common Theorems

3.12.2 Bounded Hermitian Operators Have Complete Sets of Eigenvectors

3.12.3 Derivation of the Heisenberg Uncertainty Relation

3.13 Raising–Lowering and Creation–Annihilation Operators

3.13.1 Definition of the Ladder Operators

3.13.2 Matrix and Basis-Vector Representations of the Raising and Lowering Operators

3.13.3 Raising and Lowering Operators for Direct Product Space

3.14 Translation Operators

3.14.1 Exponential Form of the Translation Operator

3.14.2 Translation of the Position Operator

3.14.3 Translation of the Position-Coordinate Ket

3.14.4 Example Using the Dirac Delta Function

3.14.5 Relation among Hilbert Space and the 1-D Translation, and Lie Group

3.14.6 Translation Operators in Three Dimensions

3.15 Functions in Rotated Coordinates

3.15.1 Rotating Functions

3.15.2 Rotation Operator

3.15.3 Rectangular Coordinates for the Generator of Rotations about z

3.15.4 Rotation of the Position Operator

3.15.5 Structure Constants and Lie Groups

3.15.6 Structure Constants for the Rotation Lie Group

3.16 Dyadic Notation

3.16.1 Notation

3.16.2 Equivalence between the Dyad and the Matrix

3.17 Review Exercises

References and Further Reading

**Chapter 4 Fundamentals of Classical Mechanics**

4.1 Constraints and Generalized Coordinates

4.1.1 Constraints

4.1.2 Generalized Coordinates

4.1.3 Phase Space Coordinates

4.2 Action, Lagrangian, and Lagrange’s Equation

4.2.1 Origin of the Lagrangian in Newton’s Equations

4.2.2 Lagrange’s Equation from a Variational Principle

4.3 Hamiltonian

4.3.1 Hamiltonian from the Lagrangian

4.3.2 Hamilton’s Canonical Equations

4.4 Poisson Brackets

4.4.1 Definition of the Poisson Bracket and Relation to the Commutator

4.4.2 Basic Properties for the Poisson Bracket

4.4.3 Constants of the Motion and Conserved Quantities

4.5 Lagrangian and Normal Coordinates for a Discrete Array of Particles

4.5.1 Lagrangian and Equations of Motion

4.5.2 Transformation to Normal Coordinates

4.5.3 Lagrangian and the Normal Modes

4.6 Classical Field Theory

4.6.1 Lagrangian and Hamiltonian Density

4.6.2 Lagrange Density for 1-D Wave Motion

4.7 Lagrangian and the Schrödinger Equation

4.7.1 Schrödinger Wave Equation

4.7.2 Hamiltonian Density

4.8 Brief Summary of the Structure of Space-Time

4.8.1 Introduction to Space-Time Warping

4.8.2 Minkowski Space

4.8.3 Lorentz Transformation

4.8.4 Some Examples

4.9 Review Exercises

References and Further Readings

**Chapter 5 Quantum Mechanics**

5.1 Relation between Quantum Mechanics and Linear Algebra

5.1.1 Observables and Hermitian Operators

5.1.2 Eigenstates

5.1.3 Meaning of Superposition of Basis States and the Probability Interpretation

5.1.4 Probability Interpretation

5.1.5 Averages

5.1.6 Motion of the Wave Function

5.1.7 Collapse of the Wave Function

5.1.8 Interpretations of the Collapse

5.1.9 Noncommuting Operators and the Heisenberg Uncertainty Relation

5.1.10 Complete Sets of Observables

5.2 Fundamental Operators and Procedures for Quantum Mechanics

5.2.1 Summary of Elementary Facts

5.2.2 Momentum Operator

5.2.3 Hamiltonian Operator and the Schrödinger Wave Equation

5.2.4 Introduction to Commutation Relations and Heisenberg Uncertainty Relations

5.2.5 Derivation of the Heisenberg Uncertainty Relation

5.2.6 Program

5.3 Examples for Schrödinger’s Wave Equation

5.3.1 Discussion of Quantum Wells

5.3.2 Solutions to Schrödinger’s Equation for the Infinitely Deep Well

5.3.3 Finitely Deep Square Well

5.4 Harmonic Oscillator

5.4.1 Introduction to Classical and Quantum Harmonic Oscillators

5.4.2 Hamiltonian for the Quantum Harmonic Oscillator

5.4.3 Introduction to the Ladder Operators for the Harmonic Oscillator

5.4.4 Ladder Operators in the Hamiltonian

5.4.5 Properties of the Raising and Lowering Operators

5.4.6 Energy Eigenvalues

5.4.7 Energy Eigenfunctions

5.5 Introduction to Angular Momentum

5.5.1 Classical Definition of Angular Momentum

5.5.2 Origin of Angular Momentum in Quantum Mechanics

5.5.3 Angular Momentum Operators

5.5.4 Pictures for Angular Momentum in Quantum Mechanics

5.5.5 Rotational Symmetry and Conservation of Angular Momentum

5.5.6 Eigenvalues and Eigenvectors

5.5.7 Eigenvectors as Spherical Harmonics

5.6 Introduction to Spin and Spinors

5.6.1 Basic Idea of Spin

5.6.2 Link between Physical Space and Hilbert Space

5.6.3 Pauli Spin Matrices

5.6.4 Rotations

5.6.5 Direct Product Space for a Single Electron

5.6.6 Spin Hamiltonian

5.7 Angular Momentum for Multiple Systems

5.7.1 Adding Angular Momentum

5.7.2 Clebsch–Gordon Coefficients

5.8 Quantum Mechanical Representations

5.8.1 Discussion of the Schrödinger, Heisenberg, and Interaction Representations

5.8.2 Schrödinger Representation

5.8.3 Rate of Change of the Average of an Operator in the Schrödinger Picture

5.8.4 Ehrenfest’s Theorem for the Schrödinger Representation

5.8.5 Heisenberg Representation

5.8.6 Heisenberg Equation

5.8.7 Newton’s Second Law from the Heisenberg Representation

5.8.8 Interaction Representation

5.9 Time-Independent Perturbation Theory

5.9.1 Initial Discussion of Perturbations

5.9.2 Nondegenerate Perturbation Theory

5.9.3 Unitary Operator for Time-Independent Perturbation Theory

5.10 Time-Dependent Perturbation Theory

5.10.1 Physical Concept

5.10.2 Time-Dependent Perturbation Theory Formalism in the Schrödinger Picture

5.10.3 Example for Further Thought and Questions

5.10.4 Time-Dependent Perturbation Theory in the Interaction Representation

5.10.5 Evolution Operator in the Interaction Representation

5.11 Introduction to Optical Transitions

5.11.1 EM Interaction Potential

5.11.2 Integral for the Probability Amplitude

5.11.3 Rotating Wave Approximation

5.11.4 Absorption

5.11.5 Emission

5.11.6 Discussion of the Results

5.12 Fermi’s Golden Rule

5.12.1 Introductory Concepts on Probability

5.12.2 Definition of the Density of States

5.12.3 Equations for Fermi’s Golden Rule

5.13 Density Operator

5.13.1 Introduction to the Density Operator

5.13.2 Density Operator and the Basis Expansion

5.13.3 Ensemble and Quantum Mechanical Averages

5.13.4 Loss of Coherence

5.13.5 Some Properties

5.14 Introduction to Multiparticle Systems

5.14.1 Introduction

5.14.2 Permutation Operator

5.14.3 Simultaneous Eigenvectors of the Hamiltonian and the Interchange Operator

5.14.4 Introduction to Fock States

5.14.5 Origin of Fock States

5.14.5.1 Bosons

5.14.5.2 Fermions

5.15 Introduction to Second Quantization

5.15.1 Field Commutators

5.15.2 Creation and Annihilation Operators

5.15.3 Introduction to Fock States

5.15.4 Interpretation of the Amplitude and Field Operators

5.15.5 Fermion–Boson Occupation and Interchange Symmetry

5.15.6 Second Quantized Operators

5.15.7 Operator Dynamics

5.15.8 Origin of Boson Creation and Annihilation Operators

5.16 Propagator

5.16.1 Idea of the Green Function

5.16.2 Propagator for a Conservative System

5.16.3 Alternate Formulation

5.16.4 Propagator and the Path Integral

5.16.5 Free-Particle Propagator

5.17 Feynman Path Integral

5.17.1 Derivation of the Feynman Path Integral

5.17.2 Classical Limit

5.17.3 Schrödinger Equation from the Propagator

5.18 Introduction to Quantum Computing

5.18.1 Turing Machines

5.18.2 Block Diagrams for the Quantum Computer

5.18.3 Memory Register with Multiple Spins

5.18.4 Feynman Computer for Negation without a Program Counter

5.18.5 Example Physical Realizations of Quantum Computers

5.19 Introduction to Quantum Teleportation

5.19.1 Local versus Nonlocal

5.19.2 EPR Paradox

5.19.3 Bell’s Theorem

5.19.4 Quantum Teleportation

5.20 Review Exercises

References and Further Reading

**Chapter 6 Solid-State: Structure and Phonons**

6.1 Origin of Crystals

6.1.1 Orbitals and Spherical Harmonics

6.1.2 Hybrid Orbital

6.2 Crystal, Lattice, Atomic Basis, and Miller Notation

6.2.1 Lattice

6.2.2 Translation Operator

6.2.3 Atomic Basis

6.2.4 Unit Cells

6.2.5 Miller Indices

6.3 Special Unit Cells

6.3.1 Body-Centered Cubic Lattice

6.3.2 Face-Centered Cubic Lattice

6.3.3 Wigner–Seitz Primitive Cell

6.3.4 Diamond and Zinc Blende Lattice

6.3.5 Tetrahedral Bonding and the Diamond Structure

6.4 Reciprocal Lattice

6.4.1 Primitive Reciprocal Lattice Vectors

6.4.2 Discussion of Reciprocal Lattice Vector in the Fourier Series

6.4.3 Fourier Series and General Lattice Translations

6.4.4 Application to X-Ray Diffraction

6.4.5 Comment on Band Diagrams and Dispersion Curves

6.5 Comments on Crystal Symmetries

6.5.1 Space and Point Groups

6.5.2 Rotations

6.5.3 Defects

6.5.4 Introduction to Symmetries in Quantum Mechanics

6.6 Phonon Dispersion Curves for Monatomic Crystal

6.6.1 Introduction to Normal Modes for Monatomic Linear Crystal

6.6.2 Equations of Motion

6.6.3 Phonon Group Velocity for Monatomic Crystal

6.6.4 Three-Dimensional Monatomic Crystals

6.6.5 Longitudinal Vibration of a Rod and Young’s Modulus

6.7 Classical Phonons in Diatomic Linear Crystal

6.7.1 The Dispersion Curves

6.7.2 Approximation for Small Wave Vector

6.7.3 Discussion

6.8 Phonons and Modes

6.8.1 Modes in Monatomic 1-D Finite Crystal with 1-D Motion and Fixed-Endpoint Boundary Conditions

6.8.2 Periodic Boundary Conditions

6.8.3 Modes for 2-D and 3-D Waves on Linear Monatomic Array

6.8.4 Modes for the 2-D and 3-D Crystal

6.8.5 Amplitude and Phonons

6.9 The Phonon Density of States

6.9.1 Introductory Discussion

6.9.2 The Density of States in ~k-Space

6.9.3 Density of States for 2-D Crystal Near k¼0 for the Acoustic Branch

6.9.4 Summary of Technique

6.9.5 3-D Crystal in Long-Wavelength Limit

6.10 Comments on Phonon Crystal Momentum

6.10.1 Anticipations for Momentum

6.10.2 Conservation of Momentum in Crystals

6.11 The Phonon Bose–Einstein Probability Distribution

6.11.1 Discussion of Reservoirs and Equilibrium

6.11.2 Equilibrium Requires Equal Temperatures

6.11.3 Discussion of Boltzmann Factor

6.11.4 Bose–Einstein Probability Distribution for Phonons

6.11.5 Statistical Moments for Phonon Bose–Einstein Distribution

6.12 Introduction to Specific Heat

6.12.1 Discussion of Specific Heat

6.12.2 Einstein Model for Specific Heat

6.12.3 Debye Model for Specific Heat

6.13 Quantum Mechanical Development of Phonon Fields

6.13.1 Basis States for Fourier Series with Periodic Boundary Conditions

6.13.2 Lagrangian for Line of Atoms

6.13.3 Classical Hamiltonian

6.13.4 Introduction to Quantizing Phonon Field and Hamiltonian

6.13.5 Introduction to Phonon Fock States

6.14 Phonons and Continuous Media

6.14.1 Wave Equation and Speed

6.14.2 Hamiltonian for One-Dimensional Wave Motion

Review Exercises

References and Further Readings

**Chapter 7 Solid-State: Conduction, States, and Bands**

7.1 Equation of Continuity

7.1.1 Classical DC Conduction

7.1.2 Collisions and Drift Mobility

7.1.3 Classical Equation of Continuity

7.1.4 Equation of Continuity for Quantum Particles

7.2 Scattering Matrices

7.2.1 Introduction to Scattering Theory

7.2.2 Amplitudes

7.2.3 Reflectivity and Transmissivity

7.2.4 Modifications for Heterostructure

7.2.5 Reflectance and Transmittance

7.2.6 Current-Density Amplitudes

7.3 The Transfer Matrix

7.3.1 Simple Interface

7.3.2 Simple Electronic Waveguide

7.3.3 Transfer Matrix for Electron-Resonant Device

7.3.4 Resonance Conditions for Electron Resonance Device

7.3.5 Quantum Tunneling

7.3.6 Tunneling and Electrical Contacts

7.4 Introduction to Free and Nearly Free Quantum Models

7.4.1 Potential in Cubic Monatomic Crystal

7.4.2 Free Electron Model

7.4.3 Nearly Free Electron Model

7.4.4 Bragg Diffraction and Group Velocity

7.4.5 Brief Discussion of Electron Density and Bandgaps

7.5 Bloch Function

7.5.1 Introduction to Bloch Wave Function

7.5.2 Proof of Bloch Wave Function

7.5.3 Orthonormality Relation for Bloch Wave Functions

7.6 Introduction to Effective Mass and Band Current

7.6.1 Mass, Momentum, and Newton’s Second Law

7.6.2 Electron and Hole Current

7.7 3-D Band Diagrams and Tensor Effective Mass

7.7.1 E–k Diagrams for 3-D Crystals

7.7.2 Effective Mass for Three-Dimensional Band Structure

7.7.3 Introduction to Band-Edge Diagrams

7.8 The Kronig–Penney Model for Nearly Free Electrons

7.8.1 Model

7.8.2 Bands

7.8.3 Bandwidth and Periodic Potential

7.9 Tight Binding Approximation

7.9.1 Introduction

7.9.2 Bloch Wave Functions

7.9.3 Dispersion Relation and Bands

7.10 Introduction to Effective Mass Equation

7.10.1 Thesis

7.10.2 Discussion of the Single-Band Effective-Mass Equation

7.10.3 Envelope Approximation

7.10.4 Diagonal Matrix Elements of VE

7.10.5 Summary

7.11 Introduction to k∙p Band Theory

7.11.1 Brief Reminder on Bloch Wave Function

7.11.2 k∙p Equation for Periodic Bloch Function

7.11.3 Nondegenerate Bands

7.11.4 k∙p Theory for Two Nondegenerate Bands

7.12 Introduction to k∙p Theory for Degenerate Bands

7.12.1 Summary of Concepts and Procedure

7.12.2 Hamiltonian for Kane’s Model

7.12.3 Eigenequation for Periodic Bloch States

7.12.4 Initial Basis Set

7.12.5 Matrix of Hamiltonian

7.12.6 Eigenvalues

7.12.7 Effective Mass

7.12.8 Wave Functions

7.13 Introduction to Density of States

7.13.1 Introduction to Localized and Extended States

7.13.2 Definition of Density of States

7.13.3 Relation between Density of Extended States and Boundary Conditions

7.13.4 Fixed-Endpoint Boundary Conditions

7.13.5 Periodic Boundary Condition

7.13.6 Density of k-States

7.13.7 Electron Density of Energy States for Two-Dimensional Crystal

7.13.8 Electron Density of Energy States for Three-Dimensional Crystal

7.13.9 General Relation between k and E Mode Density

7.13.10 Tensor Effective Mass and Density of States

7.13.11 Overlapping Bands

7.13.12 Density of States from Periodic and Fixed-Endpoint

Boundary Conditions

7.13.13 Changing Summations to Integrals

7.13.14 Comment on Probability

7.14 Infinitely Deep Quantum Well in a Semiconductor

7.14.1 Envelope Function Approximation for Infinitely Deep Well

7.14.2 Solutions for Infinitely Deep Quantum Well in 3-D Crystal

7.14.3 Introduction to the Density of States

7.15 Density of States for Reduced Dimensional Structures

7.15.1 Envelope Function Approximation

7.15.2 Density of Energy States for Quantum Well

7.15.3 Density of Energy States for Quantum Wire

7.16 Review Exercises

References and Further Readings

**Chapter 8 Statistical Mechanics**

8.1 Introduction to Reservoirs

8.1.1 Definition of Reservoir

8.1.2 Example of the Fluctuation-Dissipation Theorem

8.1.3 Reservoirs for Optical Emitter

8.1.4 Comment

8.2 Statistical Ensembles and Introduction to Statistical Mechanics

8.2.1 Microcanonical Ensemble, Entropy, and States

8.2.2 Canonical Ensemble

8.2.3 Grand Canonical Ensemble

8.3 The Boltzmann Distribution

8.3.1 Preliminary Discussion of States and Probability

8.3.2 Derivation of Boltzmann Distribution Using

a Thermal Reservoir

8.3.3 Derivation of Boltzmann DistributionUsing an Ensemble

8.3.4 Counting Degenerate States

8.3.5 Boltzmann Distribution for DistinguishableBoson-Like Particles

8.3.6 Independent, Distinguishable Subsystems

8.4 Introduction to Fermi–Dirac Distribution

8.4.1 Fermi–Dirac Distribution

8.4.2 Density of Carriers

8.4.3 Comments

8.5 Derivation of Fermi–Dirac Distribution

8.5.1 Pauli Exclusion Principle

8.5.2 Brief Review of Maxwell–Boltzmann Distribution

8.5.3 Fermi–Dirac and Bose–Einstein Distributions

8.6 Effective Density of States, Doping, and Mass Action

8.6.1 Carrier Concentrations

8.6.2 Law of Mass Action

8.6.3 Electric Fields

8.6.4 Some Comments

8.7 Dopant Ionization Statistics

8.7.1 Dopant Fermi Function

8.7.2 Derivation

8.8 pn Junction at Equilibrium

8.8.1 Introductory Concepts

8.8.2 Quick Calculation of Built-in Voltage of pn Junction

8.8.3 Junction Fields

8.9 Review Exercises

References and Further Readings

**Appendices**

Appendix A Growth and Fabrication Methods

Appendix B Dirac Delta Function

Appendix C Fourier Transform from the Fourier Series

Appendix D Brief Review of Probability

Appendix E Review of Integrating Factors

Appendix F Group Velocity

Appendix G Note on Combinatorials

Appendix H Lagrange Multipliers

Appendix I Comments on System Return to Equilibrium

Appendix J Bose–Einstein Distribution

Appendix K Density Operator and the Boltzmann Distribution

Appendix L Coordinate Representations of Schrödinger Wave Equation

**Index**