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Professor Paul Harrison After his first degree in physics at the University of Hull, Paul decided to pursue an academic career because of his wish to share the love of his subject with students. With that in mind Paul did a a PhD in computational physics at the University of Newcastle-upon-Tyne and then returned to Hull in 1991 to work as a postdoctoral researcher assistant in the Applied Physics department. After four successful years Paul had built his CV up to the level to obtain a five-year research fellowship in 1995 in the School of Electronic & Electrical Engineering at the University of Leeds. This was a strategic move by the University of Leeds to hire early-career researchers and develop them into independent research leaders. The fellowship gave Paul the time to invest heavily in his own research portfolio and Paul built up a group of research students and postdocs, a strong publications track record and an international reputation in theory and design of semiconductor optoelectronic devices. At the end of his fellowship Paul was promoted to Reader in 2000 and subsequently Professor of Quantum Electronics in 2002. The following year Paul was nominated by the staff to be Head of the School of Electronic and Electrical Engineering, rated 5* in the 2001 Research Assessment Exercise. Under his leadership the school continued to develop its research portfolio and worked as a team to push on with its research strategy: subsequently rewarded by being ranked top of its unit of assessment in the 2008 RAE. Paul was promoted to Dean of Postgraduate Research Studies in 2011, a university-wide portfolio to develop postgraduate research as part of the overall university strategy. This included responsibility for developing research funding for postgraduate students, marketing, recruitment, admissions and progression, improving submission and completion rates, harmonising processes and improving the student experience across campus. Paul joined Sheffield Hallam University as Pro Vice-Chancellor for Research and Innovation at the start of 2014. Over his career, Paul has co-authored nearly 300 journal articles, written two books, successfully supervised 16 students to PhD, raised £3m of research funding and Google Scholar gives him a h-index of 33.
Dr Alex Valavanis (MIET) received his MEng (Hons) degree in Electronic Engineering from the University of York and his PhD degree in Electronic and Electrical Engineering from the University of Leeds in 2004 and 2009 respectively. From 2004-2005 he worked with STFC Daresbury Laboratories, Cheshire, developing X-ray detector systems. From 2005-2009, his PhD with the Quantum Electronics Group in the School of Electronic and Electrical Engineering, University of Leeds, focused on the development of numerical simulations, in C/C++, of quantum-cascade lasers (QCLs) in the silicon-germanium material system. Since 2009, he has worked in the Terahertz (THz) Photonics Laboratory within the same institute, developing new THz imaging and sensing techniques. Dr Valavanis holds full membership of the IET, and has received awards including the BNFL Peter Wilson Award (2009) for Materials Engineering, the GW Carter prize (2008) for best publication by a PhD student and the FW Carter prize (2009) for best PhD thesis within the School of Electronic and Electrical Engineering.
Dedication iii
List of Contributors xiii
Preface xv
Acknowledgements xix
Introduction xxiii
References xxiv
1 Semiconductors and heterostructures 1
1.1 The mechanics of waves 1
1.2 Crystal structure 3
1.3 The effective mass approximation 5
1.4 Band theory 5
1.5 Heterojunctions 7
1.6 Heterostructures 7
1.7 The envelope function approximation 10
1.8 Band non-parabolicity 11
1.9 The reciprocal lattice 13
Exercises 16
References 17
2 Solutions to Schrödinger's equation 19
2.1 The infinite well 19
2.2 In-plane dispersion 22
2.3 Extension to include band non-parabolicity 24
2.4 Density of states 26
2.4.1 Density-of-states effective mass 28
2.4.2 Two-dimensional systems 29
2.5 Subband populations 31
2.5.1 Populations in non-parabolic subbands 33
2.5.2 Calculation of quasi-Fermi energy 35
2.6 Thermalised distributions 36
2.7 Finite well with constant mass 37
2.7.1 Unbound states 43
2.7.2 Effective mass mismatch at heterojunctions 45
2.7.3 The infinite barrier height and mass limits 49
2.8 Extension to multiple-well systems 50
2.9 The asymmetric single quantum well 53
2.10 Addition of an electric field 54
2.11 The infinite superlattice 57
2.12 The single barrier 63
2.13 The double barrier 65
2.14 Extension to include electric field 71
2.15 Magnetic fields and Landau quantisation 72
2.16 In summary 74
Exercises 74
References 76
3 Numerical solutions 79
3.1 Bisection root-finding 79
3.2 Newton-Raphson root finding 81
3.3 Numerical differentiation 83
3.4 Discretised Schrödinger equation 84
3.5 Shooting method 84
3.6 Generalized initial conditions 86
3.7 Practical implementation of the shooting method 88
3.8 Heterojunction boundary conditions 90
3.9 Matrix solutions of the discretised Schrödinger equation 91
3.10 The parabolic potential well 94
3.11 The Pöschl-Teller potential hole 98
3.12 Convergence tests 98
3.13 Extension to variable effective mass 99
3.14 The double quantum well 103
3.15 Multiple quantum wells and finite superlattices 104
3.16 Addition of electric field 106
3.17 Extension to include variable permittivity 106
3.18 Quantum confined Stark effect 108
3.19 Field-induced anti-crossings 108
3.20 Symmetry and selection rules 110
3.21 The Heisenberg uncertainty principle 110
3.22 Extension to include band non-parabolicity 113
3.23 Poisson's equation 114
3.24 Matrix solution of Poisson's equation 118
3.25 Self-consistent Schrödinger-Poisson solution 119
3.26 Modulation doping 121
3.27 The high-electron-mobility transistor 122
3.28 Band filling 123
Exercises 124
References 125
4 Diffusion 127
4.1 Introduction 127
4.2 Theory 129
4.3 Boundary conditions 130
4.4 Convergence tests 131
4.5 Numerical stability 133
4.6 Constant diffusion coefficients 133
4.7 Concentration dependent diffusion coefficient 135
4.8 Depth dependent diffusion coefficient 136
4.9 Time dependent diffusion coefficient 138
4.10 d-doped quantum wells 138
4.11 Extension to higher dimensions 141
Exercises 142
References 142
5 Impurities 145
5.1 Donors and acceptors in bulk material 145
5.2 Binding energy in a heterostructure 147
5.3 Two-dimensional trial wave function 152
5.4 Three-dimensional trial wave function 158
5.5 Variable-symmetry trial wave function 164
5.6 Inclusion of a central cell correction 170
5.7 Special considerations for acceptors 171
5.8 Effective mass and dielectric mismatch 172
5.9 Band non-parabolicity 173
5.10 Excited states 173
5.11 Application to spin-flip Raman spectroscopy 174
5.11.1 Diluted magnetic semiconductors 174
5.11.2 Spin-flip Raman spectroscopy 176
5.12 Alternative approach to excited impurity states 178
5.13 The ground state 180
5.14 Position dependence 181
5.15 Excited states 181
5.16 Impurity occupancy statistics 184
Exercises 188
References 189
6 Excitons 191
6.1 Excitons in bulk 191
6.2 Excitons in heterostructures 193
6.3 Exciton binding energies 193
6.4 1s exciton 198
6.5 The two-dimensional and three-dimensional limits 202
6.6 Excitons in single quantum wells 206
6.7 Excitons in multiple quantum wells 208
6.8 Stark ladders 210
6.9 Self-consistent effects 211
6.10 2s exciton 212
Exercises 214
References 215
7 Strained quantum wells 217
7.1 Stress and strain in bulk crystals 217
7.2 Strain in quantum wells 221
7.3 Critical thickness of layers 224
7.4 Strain balancing 226
7.5 Effect on the band profile of quantum wells 228
7.6 The piezoelectric effect 231
7.7 Induced piezoelectric fields in quantum wells 234
7.8 Effect of piezoelectric fields on quantum wells 236
Exercises 239
References 240
8 Simple models of quantum wires and dots 241
8.1 Further confinement 241
8.2 Schrödinger's equation in quantum wires 243
8.3 Infinitely deep rectangular wires 245
8.4 Simple approximation to a finite rectangular wire 247
8.5 Circular cross-section wire 251
8.6 Quantum boxes 255
8.7 Spherical quantum dots 256
8.8 Non-zero angular momentum states 259
8.9 Approaches to pyramidal dots 262
8.10 Matrix approaches 263
8.11 Finite difference expansions 263
8.12 Density of states 265
Exercises 267
References 268
9 Quantum dots 269
9.1 0-dimensional systems and their experimental realization 269
9.2 Cuboidal dots 271
9.3 Dots of arbitrary shape 272
9.3.1 Convergence tests 277
9.3.2 Efficiency 279
9.3.3 Optimization 281
9.4 Application to real problems 282
9.4.1 InAs/GaAs self-assembled quantum dots 282
9.4.2 Working assumptions 282
9.4.3 Results 283
9.4.4 Concluding remarks 286
9.5 A more complex model is not always a better model 288
Exercises 289
References 290
10 Carrier scattering 293
10.1 Introduction 293
10.2 Fermi's Golden Rule 294
10.3 Extension to sinusoidal perturbations 296
10.4 Averaging over two-dimensional carrier distributions 296
10.5 Phonons 298
10.6 Longitudinal optic phonon scattering of two-dimensional carriers 301
10.7 Application to conduction subbands 313
10.8 Mean intersubband LO phonon scattering rate 315
10.9 Ratio of emission to absorption 316
10.10 Screening of the LO phonon interaction 318
10.11 Acoustic deformation potential scattering 319
10.12 Application to conduction subbands 324
10.13 Optical deformation potential scattering 326
10.14 Confined and interface phonon modes 328
10.15 Carrier-carrier scattering 328
10.16 Addition of screening 336
10.17 Mean intersubband carrier-carrier scattering rate 337
10.18 Computational implementation 339
10.19 Intrasubband versus intersubband 340
10.20 Thermalized distributions 341
10.21 Auger-type intersubband processes 342
10.22 Asymmetric intrasubband processes 343
10.23 Empirical relationships 344
10.24 A generalised expression for scattering of two-dimensional carriers 345
10.25 Impurity scattering 346
10.26 Alloy disorder scattering 351
10.27 Alloy disorder scattering in quantum wells 354
10.28 Interface roughness scattering 355
10.29 Interface roughness scattering in quantum wells 359
10.30 Carrier scattering in quantum wires and dots 362
Exercises 362
References 364
11 Optical properties of quantum wells 367
11.1 Carrier-photon scattering 367
11.2 Spontaneous emission lifetime 372
11.3 Intersubband absorption in quantum wells 374
11.4 Bound-bound transitions 376
11.5 Bound-free transitions 377
11.6 Rectangular quantum well 379
11.7 Intersubband optical non-linearities 382
11.8 Electric polarization 383
11.9 Intersubband second harmonic generation 384
11.10 Maximization of resonant susceptibility 387
Exercises 390
References 391
12 Carrier transport 393
12.1 Introduction 393
12.2 Quantum cascade lasers 393
12.3 Realistic quantum cascade laser 398
12.4 Rate equations 400
12.5 Self-consistent solution of the rate equations 402
12.6 Calculation of the current density 404
12.7 Phonon and carrier-carrier scattering transport 404
12.8 Electron temperature 405
12.9 Calculation of the gain 408
12.10 QCLs, QWIPs, QDIPs and other methods 411
12.11 Density matrix approaches 412
12.11.1 Time evolution of the density matrix 415
12.11.2 Density matrix modelling of terahertz QCLs 416
Exercises 418
References 420
13 Optical waveguides 423
13.1 Introduction to optical waveguides 423
13.2 Optical waveguide analysis 425
13.2.1 The wave equation 425
13.2.2 The transfer matrix method 428
13.2.3 Guided modes in multi-layer waveguides 431
13.3 Optical properties of materials 434
13.3.1 Semiconductors 434
13.3.2 Influence of free-carriers 436
13.3.3 Carrier mobility model 438
13.3.4 Influence of doping 439
13.4 Application to waveguides of laser devices 440
13.4.1 Double heterostructure laser waveguide 441
13.4.2 Quantum cascade laser waveguides 443
13.5 Thermal properties of waveguides 447
13.6 The heat equation 449
13.7 Material properties 450
13.7.1 Thermal conductivity 450
13.7.2 Specific heat capacity 451
13.8 Finite difference approximation to the heat equation 453
13.9 Steady-state solution of the heat equation 454
13.10 Time-resolved solution 457
13.11 Simplified RC thermal models 458
Exercises 461
References 462
14 Multiband envelope function (k.p) method 465
14.1 Symmetry, basis states and band structure 465
14.2 Valence band structure and the 6 × 6 Hamiltonian 466
14.3 4 × 4 valence band Hamiltonian 470
14.4 Complex band structure 471
14.5 Block-diagonalization of the Hamiltonian 472
14.6 The valence band in strained cubic semiconductors 474
14.7 Hole subbands in heterostructures 476
14.8 Valence band offset 478
14.9 The layer (transfer matrix) method 479
14.10 Quantum well subbands 483
14.11 The influence of strain 484
14.12 Strained quantum well subbands 484
14.13 Direct numerical methods 485
Exercises 486
References 486
15 Empirical pseudo-potential bandstructure 487
15.1 Principles and approximations 487
15.2 Elemental band structure calculation 488
15.3 Spin-orbit coupling 496
15.4 Compound semiconductors 498
15.5 Charge densities 501
15.6 Calculating the effective mass 504
15.7 Alloys 504
15.8 Atomic form factors 506
15.9 Generalization to a large basis 507
15.10 Spin-orbit coupling within the large basis approach 510
15.11 Computational implementation 511
15.12 Deducing the parameters and application 512
15.13 Isoelectronic impurities in bulk 515
15.14 The electronic structure around point defects 520
Exercises 520
References 521
16 Pseudo-potential calculations of nanostructures 523
16.1 The superlattice unit cell 523
16.2 Application of large basis method to superlattices 526
16.3 Comparison with envelope function approximation 530
16.4 In-plane dispersion 531
16.5 Interface coordination 532
16.6 Strain-layered superlattices 533
16.7 The superlattice as a perturbation 534
16.8 Application to GaAs/AlAs superlattices 539
16.9 Inclusion of remote bands 541
16.10 The valence band 542
16.11 Computational effort 542
16.12 Superlattice dispersion and the interminiband laser 543
16.13 Addition of electric field 545
16.14 Application of the large basis method to quantum wires 549
16.15 Confined states 552
16.16 Application of the large basis method to tiny quantum dots 552
16.17 Pyramidal quantum dots 554
16.18 Transport through dot arrays 555
16.19 Recent progress 556
Exercises 556
References 557
Concluding remarks 559
A Materials parameters 561
B Introduction to the simulation tools 563
B.1 Documentation and support 564
B.2 Installation and dependencies 564
B.3 Simulation programs 565
B.4 Introduction to scripting 566
B.5 Example calculations 567
De Broglie (see reference [1]) stated that a particle of momentum p has an associated wave of wavelength ? given by:
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.
where t is the time, ? the angular frequency and the modulus of the wave vector is given by:
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.
where
which, when operating on the electron vacuum wave function in equation (1.2), would give the following:
and therefore
Thus the eigenvalue
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:
Therefore it may be expected that the quantum mechanical analogy can also be represented by an eigenvalue equation with an operator:
i.e.
where T is the kinetic energy eigenvalue, and, given the form of ? in equation (1.5), then:
When acting upon the electron vacuum wave function, i.e.
then
Thus the kinetic energy eigenvalue is given by:
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:
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.
or
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:
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:
Figure 1.2 The face-centred cubic Bravais lattice
The lattice vectors then follow as the set of vectors:
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
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:
and the energy solutions follow as:
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
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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