The construction of problem-tailored basis functions for modeling anisotropic quantum wire structures

Smith, A 2011, The construction of problem-tailored basis functions for modeling anisotropic quantum wire structures, Doctor of Philosophy (PhD), Electrical and Computer Engineering, RMIT University.


Document type: Thesis
Collection: Theses

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Title The construction of problem-tailored basis functions for modeling anisotropic quantum wire structures
Author(s) Smith, A
Year 2011
Abstract Due to the continuous reduction in feature size of electronic devices, quantum mechanical effects play a pivotal role in device characteristics today. In modeling the quantum mechanical phenomena in small-scale electronic devices, the fundamental building blocks that makeup the model for these devices are quantum wires and quantum dots. By solving the Schroedinger equation, subject to device specifications and constraints, the underlying physical properties can be calculated numerically to a desired accuracy. Only a few canonical problems exist that can be solved in closed-form, rendering the determination of the solutions to the Schroedinger equation a most challenging task for any realistic model. In this thesis, a new modeling technique has been introduced to solve the Schroedinger equation for quantum wire structures, including the inhomogeneous and anisotropic effective mass phenomenon. The proposed technique is based on Galerkin's method. In contrast to existing techniques, the proposed method utilizes the eigensolutions associated with a hierarchy of auxiliary problems, as the analysing and synthesizing functions, to tackle the original Schroedinger equation. By utilizing problem-specific basis functions, the effectiveness of the proposed technique is shown by significantly reducing the number of required basis functions for achieving convergent solutions. The proposed technique has been applied to common isotropic and anisotropic quantum wire structures. It is shown how the decomposition of the potential function, defining the Hamiltonian in the Schroedinger equation, allows the design of a number of independent lower-dimensional auxiliary problems, which can be solved with comparatively less computational resources. The obtained solutions are then employed to solve the problem of interest. Using this method, a laterally confined periodic quantum wire model is considered, and the influence of the effective mass tensor on the Brillouin energy dispersion diagram is investigated. To validate the proposed method, several carefully chosen examples have been analyzed for determining the quantum mechanical effects in quantum wire structures. The energy eigenvalues for GaAs/Ga0.63Al0.37As and Ga0.47In0.53As/InP single quantum wire structures have been calculated and compared with existing calculated eigenvalues in literature. An energy dispersion diagram for a periodic array of GaAs/Ga0.63Al0.37As quantum wires has also been calculated and compared against an energy dispersion diagram given in literature. Excellent agreement has been achieved.
Degree Doctor of Philosophy (PhD)
Institution RMIT University
School, Department or Centre Electrical and Computer Engineering
Keyword(s) Numerical solution of time-independent Schroedinger equation
periodic and non-periodic boundary conditions
construction of problem-specific auxiliary problems
physics-based basis functions
Wannier functions
Model Order Reduction
Galerkin’s Method
Quantum Wire
Effective Mass
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