Quantum Wells, Wires and Dots

Theoretical and Computational Physics of Semiconductor Nanostructures
 
 
Wiley (Verlag)
  • 4. Auflage
  • |
  • erschienen am 29. April 2016
  • |
  • 624 Seiten
 
E-Book | ePUB mit Adobe DRM | Systemvoraussetzungen
978-1-118-92334-4 (ISBN)
 
Quantum Wells, Wires and Dots provides all the essential information, both theoretical and computational, to develop an understanding of the electronic, optical and transport properties of these semiconductor nanostructures. The book will lead the reader through comprehensive explanations and mathematical derivations to the point where they can design semiconductor nanostructures with the required electronic and optical properties for exploitation in these technologies.
This fully revised and updated 4th edition features new sections that incorporate modern techniques and extensive new material including:
* Properties of non-parabolic energy bands
* Matrix solutions of the Poisson and Schrödinger equations
* Critical thickness of strained materials
* Carrier scattering by interface roughness, alloy disorder and impurities
* Density matrix transport modelling
* Thermal modelling
Written by well-known authors in the field of semiconductor nanostructures and quantum optoelectronics, this user-friendly guide is presented in a lucid style with easy to follow steps, illustrative examples and questions and computational problems in each chapter to help the reader build solid foundations of understanding to a level where they can initiate their own theoretical investigations. Suitable for postgraduate students of semiconductor and condensed matter physics, the book is essential to all those researching in academic and industrial laboratories worldwide.
Instructors can contact the authors directly (p.harrison@shu.ac.uk / a.valavanis@leeds.ac.uk) for Solutions to the problems.
4. Auflage
  • Englisch
  • Chicester
  • |
  • Großbritannien
John Wiley & Sons
  • 9,11 MB
978-1-118-92334-4 (9781118923344)
1118923340 (1118923340)
weitere Ausgaben werden ermittelt
  • Title Page
  • Copyright
  • Dedication
  • Table of Contents
  • List of contributors
  • Principal authors
  • Contributing authors
  • Preface
  • Acknowledgements
  • Introduction
  • References
  • Chapter 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 Band non-parabolicity
  • 1.9 The reciprocal lattice
  • Exercises
  • References
  • Chapter 2: Solutions to Schrödinger's equation
  • 2.1 The infinite well
  • 2.2 In-plane dispersion
  • 2.3 Extension to include band non-parabolicity
  • 2.4 Density of states
  • 2.5 Subband populations
  • 2.6 Thermalised distributions
  • 2.7 Finite well with constant mass
  • 2.8 Extension to multiple-well systems
  • 2.9 The asymmetric single quantum well
  • 2.10 Addition of an electric field
  • 2.11 The infinite superlattice
  • 2.12 The single barrier
  • 2.13 The double barrier
  • 2.14 Extension to include electric field
  • 2.15 Magnetic fields and Landau quantisation
  • 2.16 In summary
  • Exercises
  • References
  • Chapter 3: Numerical solutions
  • 3.1 Bisection root-finding
  • 3.2 Newton-Raphson root finding
  • 3.3 Numerical differentiation
  • 3.4 Discretised Schrödinger equation
  • 3.5 Shooting method
  • 3.6 Generalised initial conditions
  • 3.7 Practical implementation of the shooting method
  • 3.8 Heterojunction boundary conditions
  • 3.9 Matrix solutions of the discretised Schrödinger equation
  • 3.10 The parabolic potential well
  • 3.11 The Pöschl-Teller potential hole
  • 3.12 Convergence tests
  • 3.13 Extension to variable effective mass
  • 3.14 The double quantum well
  • 3.15 Multiple quantum wells and finite superlattices
  • 3.16 Addition of electric field
  • 3.17 Extension to include variable permittivity
  • 3.18 Quantum-confined Stark effect
  • 3.19 Field-induced anti-crossings
  • 3.20 Symmetry and selection rules
  • 3.21 The Heisenberg uncertainty principle
  • 3.22 Extension to include band non-parabolicity
  • 3.23 Poisson's equation
  • 3.24 Matrix solution of Poisson's equation
  • 3.25 Self-consistent Schrödinger-Poisson solution
  • 3.26 Modulation doping
  • 3.27 The high-electron-mobility transistor
  • 3.28 Band filling
  • Exercises
  • References
  • Chapter 4: Diffusion
  • 4.1 Introduction
  • 4.2 Theory
  • 4.3 Boundary conditions
  • 4.4 Convergence tests
  • 4.5 Numerical stability
  • 4.6 Constant diffusion coefficients
  • 4.7 Concentration-dependent diffusion coefficient
  • 4.8 Depth-dependent diffusion coefficient
  • 4.9 Time-dependent diffusion coefficient
  • 4.10 d-doped quantum wells
  • 4.11 Extension to higher dimensions
  • Exercises
  • References
  • Chapter 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 in diluted magnetic semiconductors
  • 5.12 Alternative approach to excited impurity states
  • 5.13 Direct evaluation of the expectation value of the Hamiltonian for the ground state
  • 5.14 Validation of the model for the position dependence of the impurity
  • 5.15 Excited states
  • 5.16 Impurity occupancy statistics
  • Exercises
  • References
  • Chapter 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 2s exciton
  • Exercises
  • References
  • Chapter 7: Strained quantum wells
  • 7.1 Stress and strain in bulk crystals
  • 7.2 Strain in quantum wells
  • 7.3 Critical thickness of layers
  • 7.4 Strain balancing
  • 7.5 Effect on the band profile of quantum wells
  • 7.6 The piezoelectric effect
  • 7.7 Induced piezoelectric fields in quantum wells
  • 7.8 Effect of piezoelectric fields on quantum wells
  • Exercises
  • References
  • Chapter 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
  • Exercises
  • References
  • Chapter 9: Quantum dots
  • 9.1 Zero-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
  • Exercises
  • References
  • Chapter 10: Carrier scattering
  • 10.1 Introduction
  • 10.2 Fermi's golden rule
  • 10.3 Extension to sinusoidal perturbations
  • 10.4 Averaging over two-dimensional carrier distributions
  • 10.5 Phonons
  • 10.6 Longitudinal optic phonon scattering of two-dimensional carriers
  • 10.7 Application to conduction subbands
  • 10.8 Mean intersubband longitudinal optic phonon scattering rate
  • 10.9 Ratio of emission to absorption
  • 10.10 Screening of the longitudinal optical phonon interaction
  • 10.11 Acoustic deformation potential scattering
  • 10.12 Application to conduction subbands
  • 10.13 Optical deformation potential scattering
  • 10.14 Confined and interface phonon modes
  • 10.15 Carrier-carrier scattering
  • 10.16 Addition of screening
  • 10.17 Mean intersubband carrier-carrier scattering rate
  • 10.18 Computational implementation
  • 10.19 Intrasubband versus intersubband
  • 10.20 Thermalised distributions
  • 10.21 Auger-type intersubband processes
  • 10.22 Asymmetric intrasubband processes
  • 10.23 Empirical relationships
  • 10.24 A generalised expression for scattering of two-dimensional carriers
  • 10.25 Impurity scattering
  • 10.26 Alloy disorder scattering
  • 10.27 Alloy disorder scattering in quantum wells
  • 10.28 Interface roughness scattering
  • 10.29 Interface roughness scattering in quantum wells
  • 10.30 Carrier scattering in quantum wires and dots
  • Exercises
  • References
  • Chapter 11: Optical properties of quantum wells
  • 11.1 Carrier-photon scattering
  • 11.2 Spontaneous emission lifetime
  • 11.3 Intersubband absorption in quantum wells
  • 11.4 Bound-bound transitions
  • 11.5 Bound-free transitions
  • 11.6 Rectangular quantum well
  • 11.7 Intersubband optical nonlinearities
  • 11.8 Electric polarisation
  • 11.9 Intersubband second harmonic generation
  • 11.10 Maximisation of resonant susceptibility
  • Exercises
  • References
  • Chapter 12: Carrier transport
  • 12.1 Introduction
  • 12.2 Quantum cascade lasers
  • 12.3 Realistic quantum cascade laser
  • 12.4 Rate equations
  • 12.5 Self-consistent solution of the rate equations
  • 12.6 Calculation of the current density
  • 12.7 Phonon and carrier-carrier scattering transport
  • 12.8 Electron temperature
  • 12.9 Calculation of the gain
  • 12.10 QCLs, QWIPs, QDIPs and other methods
  • 12.11 Density matrix approaches
  • Exercises
  • References
  • Chapter 13: Optical waveguides
  • 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
  • 13.5 Thermal properties of waveguides
  • 13.6 The heat equation
  • 13.7 Material properties
  • 13.8 Finite-difference approximation to the heat equation
  • 13.9 Steady-state solution of the heat equation
  • 13.10 Time-resolved solution
  • 13.11 Simplified RC thermal models
  • Exercises
  • References
  • Chapter 14: Multiband envelope function (k.p) method
  • 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
  • Exercises
  • References
  • Chapter 15: Empirical pseudo-potential band structure
  • 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
  • Exercises
  • References
  • Chapter 16: Pseudo-potential calculations of nanostructures
  • 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
  • 16.14 Application of the large-basis method to quantum wires
  • 16.15 Confined states
  • 16.16 Application of the large-basis method to tiny quantum dots
  • 16.17 Pyramidal quantum dots
  • 16.18 Transport through dot arrays
  • 16.19 Recent progress
  • Exercises
  • References
  • Concluding remarks
  • Appendix A: Materials parameters
  • Appendix B: Introduction to the simulation tools
  • B.1 Documentation and support
  • B.2 Installation and dependencies
  • B.3 Simulation programs
  • B.4 Introduction to scripting
  • B.5 Example calculations
  • Index
  • End User License Agreement

Chapter 1
Semiconductors and heterostructures


1.1 The mechanics of waves


De Broglie (see reference [1]) stated that a particle of momentum p has an associated wave of wavelength ? given by:

1.1

Thus, an electron in a vacuum at a position r and away from the influence of any electromagnetic potentials could be described by a state function, which is of the form of a wave, i.e.

1.2

where t is the time, ? the angular frequency and the modulus of the wave vector is given by:

1.3

The quantum mechanical momentum has been deduced to be a linear operator [2] acting upon the wave function ?, with the momentum p arising as an eigenvalue, i.e.

1.4

where

1.5

which, when operating on the electron vacuum wave function in equation (1.2), would give the following:

1.6

and therefore

1.7 1.8

Thus the eigenvalue

1.9

which, not surprisingly, can be simply manipulated (p = hk = (h/2p)(2p/?)) to reproduce de Broglie's relationship in equation (1.1).

Following on from this, classical mechanics gives the kinetic energy of a particle of mass m as:

1.10

Therefore it may be expected that the quantum mechanical analogy can also be represented by an eigenvalue equation with an operator:

1.11

i.e.

1.12

where T is the kinetic energy eigenvalue, and, given the form of ? in equation (1.5), then:

1.13

When acting upon the electron vacuum wave function, i.e.

1.14

then

1.15

Thus the kinetic energy eigenvalue is given by:

1.16

For an electron in a vacuum away from the influence of electromagnetic fields, the total energy E is just the kinetic energy T. Thus the dispersion or energy versus momentum (which is proportional to the wave vector k) curves are parabolic, just as for classical free particles, as illustrated in Fig. 1.1.

Figure 1.1 The energy versus wave vector (proportional to momentum) curve for an electron in a vacuum

The equation describing the total energy of a particle in this wave description is called the time-independent Schrödinger equation and, for this case with only a kinetic energy contribution, can be summarised as follows:

1.17

A corresponding equation also exists that includes the time dependency explicitly; this is obtained by operating on the wave function by the linear operator ih?/?t, i.e.

1.18

or

1.19

Clearly, this eigenvalue h? is also the total energy but in a form usually associated with waves, e.g. a photon. These two operations on the wave function represent the two complementary descriptions associated with wave-particle duality. Thus the second, i.e. time-dependent, Schrödinger equation is given by:

1.20

1.2 Crystal structure


The vast majority of the mainstream semiconductors have a face-centred cubic Bravais lattice, as illustrated in Fig 1.2. The lattice points are defined in terms of linear combinations of a set of primitive lattice vectors, one choice for which is:

1.21

Figure 1.2 The face-centred cubic Bravais lattice

The lattice vectors then follow as the set of vectors:

1.22

where a1, a2, and a3 are integers.

The complete crystal structure is obtained by placing the atomic basis at each Bravais lattice point. For materials such as Si, Ge, GaAs, AlAs and InP, this consists of two atoms, one at and the other at , in units of A0.

For the group IV materials, such as Si and Ge, as the atoms within the basis are the same, the crystal structure is equivalent to diamond (see Fig. 1.3 (left)). For III-V and II-VI compound semiconductors such as GaAs, AlAs, InP, HgTe and CdTe, the cation sits on the site and the anion on ; this type of crystal is called the zinc blende structure, after ZnS (see Fig. 1.3 (right)). The only exception to this rule is GaN, and its important InxGa1-xN alloys, which have risen to prominence in recent years due to their use in green and blue light emitting diodes and lasers (see, for example, [3]); these materials have the wurtzite structure (see [4], p. 47).

Figure 1.3 The diamond (left) and zinc blende (right) crystal structures

From an electrostatics viewpoint, the crystal potential consists of a three-dimensional lattice of spherically symmetric ionic core potentials screened by the inner shell electrons (see Fig. 1.4), which are further surrounded by the covalent bond charge distributions that hold everything together.

Figure 1.4 Schematic illustration of the ionic core component of the crystal potential across the {001} planes-a three-dimensional array of spherically symmetric potentials

1.3 The effective mass approximation


Therefore, the crystal potential is complicated; however, using the principle of simplicity,1 imagine that it can be approximated by a constant! Then the Schrödinger equation derived for an electron in a vacuum would be applicable. Clearly, though, a crystal is not a vacuum so allow the introduction of an empirical fitting parameter called the effective mass, m*. Thus the time-independent Schrödinger equation becomes:

1.23

and the energy solutions follow as:

1.24

This is known as the effective mass approximation and has been found to be very suitable for relatively low electron momenta as occur with low electric fields. Indeed, it is the most widely used parameterisation in semiconductor physics (see any good solid state physics book, e.g. ).[4, 5, 6] Experimental measurements of the effective mass have revealed it to be anisotropic-as might be expected since the crystal potential along, say, the [001] axis is different than along the [111] axis. Adachi [7] collates reported values for GaAs and its alloys; the effective mass in other materials can be found in Landolt and Börnstein [8].

In GaAs, the reported effective mass is around 0.067 m0, where m0 is the rest mass of an electron. Figure 1.5 plots the dispersion curve for this effective mass, in comparison with that of an electron in a vacuum.

Figure 1.5 The energy versus wave vector (proportional to momentum) curves for an electron in GaAs compared to that in a vacuum

1.4 Band theory


It has also been found from experiment that there are two distinct energy bands within semiconductors. The lower band is almost full of electrons and can conduct by the movement of the empty states. This band originates from the valence electron states which constitute the covalent bonds holding the atoms together in the crystal. In many ways, electric charge in a solid resembles a fluid, and the analogy for this band, labelled the valence band, is that the empty states behave like bubbles within the fluid-hence their name, holes.

In particular, the holes rise to the uppermost point of the valence band, and just as it is possible to consider the release of carbon dioxide through the motion of beer in a glass, it is actually easier to study the motion of the bubble (the absence of beer), or in this case the motion of the hole.

In a semiconductor, the upper band is almost devoid of electrons. It represents excited electron states which are occupied by electrons promoted from localised covalent bonds into extended states in the body of the crystal. Such electrons are readily accelerated by an applied electric field and contribute to current flow. This band is therefore known as the conduction band.

Figure 1.6 illustrates these two bands. Notice how the valence band is inverted-this is a reflection of the fact that the 'bubbles' rise to the top, i.e. their lowest-energy states are at the top of the band. The energy difference between the two bands is known as the band gap, labelled as Egap on the figure. The particular curvatures used in both bands are indicative of those measured experimentally for GaAs, namely effective masses of around 0.067 m0 for an electron in the conduction band, and 0.6 m0 for a (heavy) hole in the valence band. The convention is to put the zero of the energy at the top of the valence band. Note the extra qualifier 'heavy'. In fact, there is more than one valence band, and they are distinguished by their different effective masses. Chapter 15 will discuss band structure in more detail; this will be in the context of a microscopic...

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