Two dimensional disordered electron systems

Nahm, In Hyun (1989) Two dimensional disordered electron systems. University of Southampton, Doctoral Thesis.

Abstract

Two dimensional disordered electron systems have recently been the focus of intensive theoretical and experimental interest since they exhibit several remarkable features such as two dimensional localisation, quantisation of Hall conductivity under strong magnetic field, and the Wigner crystal or charge density wave state at low temperature. In this thesis we investigate two intimately related subjects, two dimensional localisation and the quantum Hall effect, using various simplified models. Our main concern is the behaviour of the diffusion constant near the mobility edge under the variation of the parameter q²/ω, i.e., the ratio of wavevector squared versus frequency. For this purpose we employ both analytical and numerical approaches. In the analytic work we choose the n-orbital local gauge invariant model of Oppermann and Wegner to calculate the diffusion constant to second order for the three different universality classes which were originally proposed by Dyson. In all of these cases we obtained a dependence of diffusion constant on the parameter, q^2/ω, which had been missing from previous work. In the numerical work we set up a tight binding Hamiltonian with spin-orbit coupling and examine how the diffusion constant varies as the ratio q²/ω increases. We obtain a dependence of the diffusion constant on the scaling variable q^2/ω consistent with our analytical results. Finally using a model for the quantum Hall effect we examined the distribution of zeros of the wavefunction in the system, and obtained a formula fitting the numerical result with the conclusion that the distribution is highly correlated.

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