Theoretical Study of the Quantum Hall Effect in Systems with Finite Width.

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In order to investigate the effect of the finite system size we studied the electronic states and the current distribution in two dimensional disordered electron systems subject to perpendicular strong magnetic fields, by using computer simulations. The main results obtained in this study are as follows.1. There exist continuous transition between the bulk extended states and the edge states. When the bulk and edge states belonging to different Landau subbands are energetically degenerate, their hybridization occurs and its extent depends on the correlation length of impurity potentials.2. The delocalized states have a fractal character and extend with fractal dimensions smaller than two. As the energy decreases from the edge state region to the bulk region, the fractal dimensions vary from the anisotropic to the isotropic one.3. The sum rule about the center of mass of eigenstates, which is intimately related to the quantization of the Hall conductivity, has been confirmed numerically.4. The compensation of the Hall current has been found to be realized not only globally but also locally. Therefore, in the plateau region of the Hall conductivity, the Hall current distribution becomes uniform even in the disordered systems.5. In the region where the Hall conductivity as a function of the filling factor rises up, it shows large fluctuations and we have found large spatial fluctuations in the Hall current distribution.6. It has been found that the current control by the gate barrier is a quite useful technique in studying the current distribution inside the system not only experimentally but also theoretically.The inclusion of the Coulomb interaction has been found to reduce the apparent size effect on the Hall conductivity to some extent. However we could not succeed to clarity the theoretical limit of the accuracy of the Hall conductivity quantization.