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E-grāmata: Multi-Band Effective Mass Approximations: Advanced Mathematical Models and Numerical Techniques

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Multi-Band Effective Mass Approximations: Advanced Mathematical Models and Numerical TechniquesPalielināt
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This book addresses several mathematical models from the most relevant class of kp-Schrödinger systems. Both mathematical models and state-of-the-art numerical methods for adequately solving the arising systems of differential equations are presented. The operational principle of modern semiconductor nano structures, such as quantum wells, quantum wires or quantum dots, relies on quantum mechanical effects.

The goal of numerical simulations using quantum mechanical models in the development of semiconductor nano structures is threefold: First they are needed for a deeper understanding of experimental data and of the operational principle. Secondly, they allow us to predict and optimize in advance the qualitative and quantitative properties of new devices in order to minimize the number of prototypes needed. Semiconductor nano structures are embedded as an active region in semiconductor devices. Thirdly and finally, the results of quantum mechanical simulations of semiconductor nano structures can be used with upscaling methods to deliver parameters needed in semi-classical models for semiconductor devices, such as quantum well lasers. This book covers in detail all these three aspects using a variety of illustrative examples.

Readers will gain detailed insights into the status of the multiband effective mass method for semiconductor nano structures. Both users of the kp method as well as advanced researchers who want to advance the kp method further will find helpful information on how to best work with this method and use it as a tool for characterizing the physical properties of semiconductor nano structures.

The book is primarily intended for graduate and Ph.D. students in applied mathematics, mathematical physics and theoretical physics, as well as all those working in quantum mechanical research or the semiconductor / opto-electronic industry who are interested in new mathematical aspects.

Part I Physical Models
1 Kinetic and Hydrodynamic Models for Multi-Band Quantum Transport in Crystals
3(54)
Luigi Barletti
Giovanni Frosali
Omar Morandi
1.1 Introduction
4(1)
1.2 Envelope k·p Models
5(14)
1.2.1 Wannier-Slater Envelope Functions Approach
7(3)
1.2.2 Luttinger-Kohn Envelope Functions
10(2)
1.2.3 Non Uniform Materials and Generalized Wannier Functions
12(3)
1.2.4 Application of k·p Models to Heterostructures and Resonant Tunneling
15(1)
1.2.5 Some Limits: Single and Mini-Band Transport
16(3)
1.3 Wigner Approach
19(5)
1.3.1 Introduction
19(3)
1.3.2 Multi-Band Wigner Models
22(2)
1.4 Hydrodynamic Models
24(33)
1.4.1 Introduction
24(1)
1.4.2 Scalar Quantum Fluid Equations
25(8)
1.4.3 Spinorial and Multi-Band QFD
33(19)
References
52(5)
2 Electronic Properties of III-V Quantum Dots
57(30)
Andrei Schliwa
Gerald Honig
Dieter Bimberg
2.1 Introduction
57(3)
2.1.1 Role of Lattice Symmetries (Zinc Blende vs Wurtzite)
58(2)
2.2 Method of Calculation
60(11)
2.2.1 Calculation of Strain
62(1)
2.2.2 Piezoelectricity/Pyroelectricity
62(3)
2.2.3 Eight-Band k·p Method: Single Particle States
65(1)
2.2.4 Impact of Strain on Bulk Band Structure
66(4)
2.2.5 Energies of Interacting Particles
70(1)
2.3 Discussion of Selected Topics
71(9)
2.3.1 Zb(001) Versus zb(111) Substrate Orientation
71(3)
2.3.2 Type-I Versus Type-II Confinement
74(2)
2.3.3 GaN/A1N Wurtzite Quantum Dots
76(4)
2.4 Conclusion
80(7)
References
81(6)
3 Symmetries in Multiband Hamiltonians for Semiconductor Quantum Dots
87(42)
Stanko Tomic
Nenad Vukmirovic
3.1 Introduction
88(1)
3.2 Multiband Envelope Function Method
89(2)
3.3 The Effect of Interfaces
91(2)
3.4 Symmetry of the Interface Hamiltonian
93(2)
3.5 The 14-Band k·p Hamiltonian
95(4)
3.6 Symmetry of the 14-Band k·p Hamiltonian
99(6)
3.6.1 Symmetry of the 8-Band k·p Hamiltonian
99(2)
3.6.2 Symmetry of the Whole 14-Band Hamiltonian
101(4)
3.7 Plane Wave Representation
105(2)
3.8 Removal of Artificial Translational Symmetry Effects in Plane Wave Calculations
107(6)
3.9 Symmetries of Single Particle States in Quantum Dots
113(5)
3.10 Symmetries of Exciton States in Quantum Dots
118(2)
3.11 Conclusion
120(9)
Appendix
121(1)
References
122(7)
Part II Numerical Methods
4 Finite Elements for k·p Multiband Envelope Equations
129(26)
G. Ratko Veprek
Sebastian Steiger
4.1 Introduction
129(2)
4.2 Basic Principles of Finite Elements
131(3)
4.3 Nanostructure k·p Equations and Operator Ordering
134(4)
4.3.1 k·p Equations in Nanostructures
134(1)
4.3.2 Operator Ordering
135(2)
4.3.3 Zincblende Models
137(1)
4.3.4 Wurtzite Models
138(1)
4.4 FEM Discretization and Solution of the k·p Equations
138(3)
4.4.1 Weak Form
138(1)
4.4.2 Numerical Discretization
139(2)
4.4.3 Solving the Generalized Eigenvalue Problem
141(1)
4.5 Spurious Solutions and Equation Ellipticity
141(9)
4.5.1 Examples of Spurious Solutions
141(4)
4.5.2 Ellipticity Criteria
145(5)
4.6 Strain and Polarization
150(2)
4.7 Conclusion
152(3)
References
153(2)
5 Plane-Wave Approaches to the Electronic Structure of Semiconductor Nanostructures
155(38)
Eoin P. O'Reilly
Oliver Marquardt
Stefan Schulz
Aleksey D. Andreev
5.1 Plane-Wave Approaches to Real-Space Problems
155(3)
5.2 Plane-Wave Based Formulation of Elastic and Electronic Properties
158(12)
5.2.1 Semi-Analytical Plane-Wave Approaches
159(7)
5.2.2 Numerical Plane-Wave Approaches
166(4)
5.3 Strain Distribution in a Plane-Wave Formulation
170(4)
5.3.1 Analytical Approach
170(2)
5.3.2 Numerical Approach
172(2)
5.4 The Polarisation Potential in a Plane-Wave Framework
174(2)
5.5 Advantages and Disadvantages of a Plane-Wave Representation
176(3)
5.6 Plane-Wave Approach for (111)-Oriented Zincblende Dots
179(7)
5.6.1 Rotated 8-Band k·p Formalism
179(4)
5.6.2 Electronic Structure of (111)-Oriented Site-Controlled Zincblende Quantum Dots
183(1)
5.6.3 Discussion of Boundary Conditions and Plane-Wave Resolution
184(2)
5.7 Conclusion
186(7)
References
187(6)
Part III Applications
6 The Multi-Band k·p Hamiltonian for Heterostructures: Parameters and Applications
193(54)
Stefan Birner
6.1 The 8-Band k·p Hamiltonian for Bulk Materials
194(24)
6.2 Applications
218(29)
6.2.1 Spurious Solutions
218(3)
6.2.2 Spin-Orbit Coupling in Silicon Quantum Dots
221(3)
6.2.3 Type-III Broken-Gap Band Alignment: HgTe-CdTe Quantum Well
224(1)
6.2.4 Type-II Broken-Gap Band Alignment: InAs--GaSb Superlattice
225(6)
6.2.5 Valence Band Structure of Diamond
231(5)
6.2.6 Self-Consistent Calculations: Influence of Substrate Orientations on the Density of a Two-Dimensional Hole Gas in Diamond
236(5)
References
241(6)
Part IV Advanced Mathematical Topics
7 Transient Simulation of k·p-Schrodinger Systems Using Discrete Transparent Boundary Conditions
247(26)
Andrea Zisowsky
Anton Arnold
Matthias Ehrhardt
Thomas Koprucki
7.1 Introduction
248(1)
7.2 Transient k·p-Schrodinger Systems
249(5)
7.3 The Transparent Boundary Conditions
254(2)
7.4 The Discrete Transparent Boundary Conditions
256(8)
7.5 The Sum-of-Exponentials Approach and the Fast Evaluation of the Convolution
264(2)
7.6 Numerical Results
266(4)
7.7 Conclusion
270(3)
References
270(3)
8 Discrete Transparent Boundary Conditions for Multi-Band Effective Mass Approximations
273(42)
Dirk Klindworth
Matthias Ehrhardt
Thomas Koprucki
8.1 Introduction
274(1)
8.2 Single-Band Effective Mass Approximations: The Scalar Schrodinger Equation
275(20)
8.2.1 The Exterior Problem and the Quantum Mechanical Dispersion Relation
275(1)
8.2.2 Transparent Boundary Conditions
276(2)
8.2.3 The Standard Discretization
278(2)
8.2.4 Discretization of the Transparent Boundary Conditions
280(2)
8.2.5 Discrete Transparent Boundary Conditions
282(1)
8.2.6 Alternative Finite Difference Schemes
283(8)
8.2.7 Numerical Example: The Single Barrier Potential
291(4)
8.3 The General k·p-Model
295(19)
8.3.1 The Exterior Problem and the Dispersion Relation
296(3)
8.3.2 Transparent Boundary Conditions
299(4)
8.3.3 The Discretization
303(2)
8.3.4 Discrete Transparent Boundary Conditions
305(3)
8.3.5 Numerical Examples
308(6)
8.4 Conclusion
314(1)
Appendix 315(2)
References 317
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