Browsing by Subject "Poisson equation"
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Item Electronic States and Optical Transitions in Bulk and Quantum Well Structures of III-V Compound Semiconductors(2011-05-06) Cho, Yong Hee 1976-In this work we apply the methods of band structure calculation combined with self-consistent treatment of the light-matter interaction to a variety of problems in bulk semiconductors and semiconductor heterostructures as well as in new optoelectronic devices. In particular, we utilize the 30- and 8-band k ? p band structure calculation methods to study the electronic, magnetic, and optical properties of the diluted magnetic semiconductor, GaMnAs, in the mean-field Zener model. We calculate the anisotropic dielectric response of GaMnAs in the metallic regime and show that our model produces a good agreement with the experimental results of magneto-optical Kerr spectroscopy in the interband transition region. We also discuss the advantages of the 30-band k ? p model for spin-polarized ferromagnetic GaMnAs. We present new methods for calculating electronic states in low-dimensional semiconductor heterostructures based on the real-space Hamiltonian. The formalism provides extreme simplicity of the numerical implementation and superior accuracy of the results. They are applicable to a general n-band k ? p model and specifically tested in the 6-, 8-band k ? p models, and a simple parabolic one band model. The transparency of the new method allows us to investigate the origin and elimination of spurious solutions in the unified manner. Spurious solutions have long been a major issue in low- dimensional band structure calculations. As an application of nonlinear optical interactions in two-dimensional semiconductor heterostructures, we calculate the upper limits on the efficiency of the passive terahertz difference frequency generation based on the intersubband resonant nonlinearity. Our approach incorporates electronic states together with propagating coupled fields through the self-consistent calculation of the Poisson equation, density matrix equations, and coupled wave equations. We develop optimal device geometries and systematically study the device performance as a function of various parameters. The results are compared with a simplified analytic solution.Item Fast Algorithms for Biharmonic Problems and Applications to Fluid Dynamics(2013-10-22) Ghosh, AditiMany areas of physics, engineering and applied mathematics require solutions of inhomogeneous biharmonic problems. For example, various problems on Stokes flow and elasticity can be cast into biharmonic boundary value problems. Hence the slow viscous flow problems are generally modeled using biharmonic boundary value problems which have widespread applications in many areas of industrial problems such as flow of molten metals, flow of particulate suspensions in bio-fluid dynamics, just to mention a few. In this dissertation, we derive, implement, validate, and apply fast and high order accurate algorithms to solve Poisson problems and inhomogeneous biharmonic problems in the interior of a unit disc in the complex plane. In particular, we use two methods to solve inhomogeneous biharmonic problems: (i) the double-Poisson method which is based on transforming biharmonic problems into solving a sequence of Poisson problems (sometime also one homogeneous biharmonic problem) and then making use of the fast Poisson solver developed in this dissertation.; (ii) the direct method which uses the fast biharmoninc solver also developed in this dissertation. Both of these methods are analyzed for accuracy, complexity and efficiency. These biharmonic solvers have been compared with each other and have been applied to solve several Stokes flow problems and elasticity problems. The fast Poisson algorithm is derived here from exact analysis of the Green?s function formulation in the complex plane. This algorithm is essentially a recast of the fast Poisson algorithm of Borges and Daripa from the real plane to the complex plane. The fast biharmonic algorithms for several boundary conditions for use in the direct method mentioned above have been derived in this dissertation from exact analysis of the representation of their solutions in terms of problem specific Green?s function in the complex plane. The resulting algorithms primarily use fast Fourier transforms and recursive relations in Fourier space. The algorithms have been analyzed for their accuracy, complexity, efficiency, and subsequently tested for validity against several benchmark test problems. These algorithms have an asymptotic complexity of O(log N ) per degree of freedom with very low constant which is hidden behind the order estimate. The direct and double-Poisson methods have been applied to solving the steady, incompressible slow viscous flow problem in a cir- cular cylinder and some problems from elasticity. The numerical results from these computations agree well with existing results on these problems.Item Ionic separation in electrodialysis : analyses of boundary layer, cationic partitioning, and overlimiting current(2010-08) Kim, Younggy; Lawler, Desmond F.; Liljestrand, Howard M.; Katz, Lynn E.; Meyers, Jeremy P.; Sepehrnoori, KamyElectrodialysis performance strongly depends on the boundary layer near ion exchange membranes. The thickness of the boundary layer has not been clearly evaluated due to its substantial fluctuation around the spacer geometry. In this study, the boundary layer thickness was defined with three statistical parameters: the mean, standard deviation, and correlation coefficient between the two boundary layers facing across the spacer. The relationship between the current and potential under conditions of the competitive transport between mono- and di-valent cations was used to estimate the statistical parameters. An uncertainty model was developed for the steady-state ionic transport in a two-dimensional cell pair. Faster ionic separations were achieved with smaller means, greater standard deviations, and more positive correlation coefficients. With the increasing flow velocity from 1.06 to 4.24 cm/s in the bench-scale electrodialyzer, the best fit values for the mean thickness reduced from 40 to less than 10 μm, and the standard deviation was in the same order of magnitude as the mean. For the partitioning of mono- and di-valent cations, a CMV membrane was examined in various KCl and CaCl₂ mixtures. The equivalent fraction correlation and separation factor responded sensitively to the composition of the mixture; however, the selectivity coefficient was consistent over the range of aqueous-phase ionic contents between 5 and 100 mN and the range of equivalent fractions of each cation between 0.2 and 0.8. It was shown that small analytic errors in measuring the concentration of the mono-valent cation are amplified when estimating the selectivity coefficient. To minimize the effects of such error propagation, a novel method employing the least square fitting was proposed to determine the selectivity coefficient. Each of thermodynamic factors, such as the aqueous- and membrane-phase activity coefficients, water activity, and standard state, was found to affect the magnitude of the selectivity coefficient. The overlimiting current, occurring beyond the electroneutral limit, has not been clearly explained because of the difficulty in solving the singularly perturbed Nernst-Planck-Poisson equations. The steady-state Nernst-Planck-Poisson equations were converted into the Painlevé equation of the second kind (P[subscript II] equation). The converted model domain is explicitly divided into the space charge and electroneutral regions. Given this property, two mathematical formulae were proposed for the limiting current and the width of the space charge region. The Airy solution of the P[subscript II] equation described the ionic transport in the space charge region. By using a hybrid numerical scheme including the fixed point iteration and Newton Raphson methods, the P[subscript II] equation was successfully solved for the ionic transport in the space charge and electroneutral regions as well as their transition zone. Above the limiting current, the sum of the ionic charge in the aqueous-phase electric double layer and in the space charge region remains stationary. Thus, growth of the space charge region involves shrinkage of the aqueous-phase electric double layer. Based on this observation, a repetitive mechanism of expansion and shrinkage of the aqueous-phase electric double layer was suggested to explain additional current above the limiting current.