Title Details

Acknowledgements

About the author(s)

About the book

Introduction

1 Semiconductors and heterostructures

1.1 The mechanics of waves

1.2 Crystal structure

1.3 The effective mass approximation

1.4 Band theory

1.5 Heterojunctions

1.6 Heterostructures

1.7 The envelope function approximation

1.8 The reciprocal lattice

2 Solutions to Schrödinger’s equation

2.1 The infinite well

2.2 In-plane dispersion

2.3 Density of states

2.4 Subband populations

2.5 Finite well with constant mass

2.6 Effective mass mismatch at heterojunctions

2.7 The infinite barrier height and mass limits

2.8 Hermiticity and the kinetic energy operator

2.9 Alternative kinetic energy operators

2.10 Extension to multiple-well systems

2.11 The asymmetric single quantum well

2.12 Addition of an electric field

2.13 The infinite superlattice

2.14 The single barrier

2.15 The double barrier

2.16 Extension to include electric field

2.17 Magnetic fields and Landau quantisation

2.18 In summary

3 Numerical solutions

3.1 Shooting method

3.2 Generalised initial conditions

3.3 Practical implementation of the shooting method

3.4 Heterojunction boundary conditions

3.5 The parabolic potential well

3.6 The Pöschl–Teller potential hole

3.7 Convergence tests

3.8 Extension to variable effective mass

3.9 The double quantum well

3.10 Multiple quantum wells and finite superlattices

3.11 Addition of electric field

3.12 Quantum confined Stark effect

3.13 Field–induced anti-crossings

3.14 Symmetry and selection rules

3.15 The Heisenberg uncertainty principle

3.16 Extension to include band non-parabolicity

3.17 Poisson’s equation

3.18 Self-consistent Schrödinger–Poisson solution

3.19 Computational implementation

3.20 Modulation doping

3.21 The high-electron-mobility transistor

3.22 Band filling

4 Diffusion

4.1 Introduction

4.2 Theory

4.3 Boundary conditions

4.4 Convergence tests

4.5 Constant diffusion coefficients

4.6 Concentration dependent diffusion coefficient

4.7 Depth dependent diffusion coefficient

4.8 Time dependent diffusion coefficient

4.9 !-doped quantum wells

4.10 Extension to higher dimensions

5 Impurities

5.1 Donors and acceptors in bulk material

5.2 Binding energy in a heterostructure

5.3 Two-dimensional trial wave function

5.4 Three-dimensional trial wave function

5.5 Variable-symmetry trial wave function

5.6 Inclusion of a central cell correction

5.7 Special considerations for acceptors

5.8 Effective mass and dielectric mismatch

5.9 Band non-parabolicity

5.10 Excited states

5.11 Application to spin-flip Raman spectroscopy

5.12 Alternative approach to excited impurity states

5.13 The ground state

5.14 Position dependence

5.15 Excited States

5.16 Impurity occupancy statistics

6 Excitons

6.1 Excitons in bulk

6.2 Excitons in heterostructures

6.3 Exciton binding energies

6.4 1s exciton

6.5 The two-dimensional and three-dimensional limits

6.6 Excitons in single quantum wells

6.7 Excitons in multiple quantum wells

6.8 Stark Ladders

6.9 Self-consistent effects

6.10 Spontaneous symmetry breaking

6.11 2s exciton

7 Strained quantum wells, V. D. Jovanovíc

7.1 Stress and strain in bulk crystals

7.2 Strain in quantum wells

7.3 Strain balancing

7.4 Effect on the band profile of quantum wells

7.5 The piezoelectric effect

7.6 Induced piezoelectric fields in quantum wells

7.7 Effect of piezoelectric fields on quantum wells

8 Simple models of quantum wires and dots

8.1 Further confinement

8.2 Schrödinger’s equation in quantum wires

8.3 Infinitely deep rectangular wires

8.4 Simple approximation to a finite rectangular wire

8.5 Circular cross-section wire

8.6 Quantum boxes

8.7 Spherical quantum dots

8.8 Non-zero angular momentum states

8.9 Approaches to pyramidal dots

8.10 Matrix approaches

8.11 Finite difference expansions

8.12 Density of states

9 Quantum dots, M. Califano

9.1 0-dimensional systems and their experimental realisation

9.2 Cuboidal dots

9.3 Dots of arbitrary shape

9.4 Application to real problems

9.5 A more complex model is not always a better model

10 Carrier scattering

10.1 Fermi’s Golden Rule

10.2 Phonons

10.3 Longitudinal optic phonon scattering of bulk carriers

10.4 LO phonon scattering of two-dimensional carriers

10.5 Application to conduction subbands

10.6 Averaging over carrier distributions

10.7 Ratio of emission to absorption

10.8 Screening of the LO phonon interaction

10.9 Acoustic deformation potential scattering

10.10 Application to conduction subbands

10.11 Optical deformation potential scattering

10.12 Confined and interface phonon modes

10.13 Carrier–carrier scattering

10.14 Addition of screening

10.15 Averaging over an initial state population

10.16 Intrasubband versus intersubband

10.17 Thermalised distributions

10.18 Auger-type intersubband processes

10.19 Asymmetric intrasubband processes

10.20 Empirical relationships

10.21 Carrier–photon scattering

10.22 Carrier scattering in quantum wires and dots

11 Electron transport

11.1 Introduction

11.2 Mid-infrared quantum cascade lasers

11.3 Realistic quantum cascade laser

11.4 Rate equations

11.5 Self-consistent solution of the rate equations

11.6 Calculation of the current density

11.7 Phonon and carrier-carrier scattering transport

11.8 Electron temperature

11.9 Calculation of the gain

11.10 QCLs, QWIPs, QDIPs and other methods

12 Optical properties of quantum wells, D. Indjin

12.1 Intersubband absorption in quantum wells

12.2 Bound-bound transitions

12.3 Bound-free transitions

12.4 Fermi level

12.5 Rectangular quantum well

12.6 Intersubband optical non-linearities

12.7 Electric polarisation

12.8 Intersubband second harmonic generation

12.9 Maximization of resonant susceptibility

13 Optical waveguides, C. A. Evans

13.1 Introduction to optical waveguides

13.2 Optical waveguide analysis

13.3 Optical properties of materials

13.4 Application to waveguides of laser devices

14 Multiband envelope function (k.p) method, Z. Ikoníc

14.1 Symmetry, basis states and band structure

14.2 Valence band structure and the 6 × 6 Hamiltonian

14.3 4 × 4 valence band Hamiltonian

14.4 Complex band structure

14.5 Block-diagonalisation of the Hamiltonian

14.6 The valence band in strained cubic semiconductors

14.7 Hole subbands in heterostructures

14.8 Valence band offset

14.9 The layer (transfer matrix) method

14.10 Quantum well subbands

14.11 The influence of strain

14.12 Strained quantum well subbands

14.13 Direct numerical methods

15 Empirical pseudopotential theory

15.1 Principles and Approximations

15.2 Elemental Band Structure Calculation

15.3 Spin–orbit coupling

15.4 Compound Semiconductors

15.5 Charge densities

15.6 Calculating the effective mass

15.7 Alloys

15.8 Atomic form factors

15.9 Generalisation to a large basis

15.10 Spin–orbit coupling within the large basis approach

15.11 Computational implementation

15.12 Deducing the parameters and application

15.13 Isoelectronic impurities in bulk

15.14 The electronic structure around point defects

16 Microscopic electronic properties of heterostructures

16.1 The superlattice unit cell

16.2 Application of large basis method to superlattices

16.3 Comparison with envelope–function approximation

16.4 In-plane dispersion

16.5 Interface coordination

16.6 Strain-layered superlattices

16.7 The superlattice as a perturbation

16.8 Application to GaAs/AlAs superlattices

16.9 Inclusion of remote bands

16.10 The valence band

16.11 Computational effort

16.12 Superlattice dispersion and the interminiband laser

16.13 Addition of electric field

17 Application to quantum wires and dots

17.1 Recent progress

17.2 The quantum-wire unit cell

17.3 Confined states

17.4 V-grooved quantum wires

17.5 Along-axis dispersion

17.6 Tiny quantum dots

17.7 Pyramidal quantum dots

17.8 Transport through dot arrays

17.9 Anti-wires and anti-dots

Concluding Remarks

Appendix A: Materials parameters

References

Topic Index