Solustions of the Landau problem and some uses of the Density functional theory in materials science
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The ordinary time-independent Landau problem of a charged particle moving in an external perpendicular homogeneous magnetic field and the case of when a linear potential is introduced are reviewed for motion occurring in an infinite plane, on an infinite strip and in the half-plane. The first objective of this part of the study is to obtain a semi-numerical solution of the modified Landau Hamiltonian for the case of motion in the half-plane; this is done using the discretization method. The structure of the energy eigenvalue spectrum is presented and an analysis of the results reveals that it is of the linear harmonic operator form with slight variations which are as a result of the addition of the linear perturbation. In an actual physical system such as a quantum Hall slab, the linear perturbation could be an electric field or a gravitational quantum well. The second objective of this part of the study is to investigate Landau quantization in a two-dimensional electron gas by working in the Landau gauge. Such a system can actually be physically created at semiconductor heterojunctions. Our results reveal that the degree of degeneracy in the energy levels becomes finite if the motion is restricted to an area and the differential equation that governs this motion is the same as that presented in the first part of the study for the case of motion in the half-plane.Within the Density Functional Theory (DFT), an introduction to the self consistent field method for solving the Kohn-Sham system is also presented. The objective of this part of the study is to investigate the stability and formation sequence of the NiGe, Ni2Ge, Ni3Ge2 and NisGcs thin film phases; this is accomplished by using an open-source computer program called QUANTUM ESPRESSO. A comparison of our results with those obtained from experiments reported in the literature proves that significantly accurate predictions of the properties of materials can be made by using DFT methods. Ultimately our results show that the Ni2Ge, NisGea and NisGes phases are more stable and hence must form in the first phase while NiGe is less stable and should form in the second phase.
- Natural Sciences