Numerical study of disorder effects in two dimensional electron gases and three dimensional diluted magnetic semiconductors

Numerical studies of the integer quantum Hall effect in mesoscopic samples and magnetic properties of diluted magnetic semiconductors are the two main topics studied in this dissertation.; The mesoscopic integer quantum Hall effect is studied in light of two recent experiments, both of which show reproducible fluctuations in the measured resistances of mesoscale (∼mum) samples of two dimensional electron gas in semiconductor heterostructures. Based on existing theories, a model of the real experiment is developed and solved by computational methods. This makes it possible to detect the physical causes for the new features found in the resistances curves. For the first experiment, the simulations provide qualitative explanations for the appearance and asymmetric distribution of the fluctuations. For the second experiment, the simulations capture all the features observed in the experiment, and a unified understanding of the phenomena is obtained, which reduces the correlations and symmetries of the longitudinal and Hall resistances between various terminals of the device to a set of algebraic identities.; The study of the diluted magnetic semiconductor (Ga,Mn)As was motivated by the recent suggestion that anisotropic exchange may cause the incomplete magnetization of (Ga,Mn)As at low temperatures. (Ga,Mn)As has been receiving much attention recently because of its potential applications in spintronics. Here, the interplay and effects of anisotropy, disorder, and antiferromagnetic exchange are studied by Monte Carlo simulations. Small and moderate anisotropy are found to be ineffective in suppressing the low temperature magnetization, whereas antiferromagnetic exchange, such as those introduced by RKKY interaction, contributes considerably to this effect. Spin-glass behavior is also found in models with anisotropy at low temperatures.; In addition, this dissertation includes some studies of the Wang-Landau algorithm, which focus on the computational aspects of the Monte Carlo simulation.