a Phase Transition in the Low Temperature Conductance of Two Dimensional Systems.

When solid is disordered on length scales smaller than the electronic inelastic scattering length the semiclassical Boltzmann Equation for the conductivity breaks down. At low enough temperatures the conductance can be related to the elastic electronic transmittance. In this regime transport properties depend critically on the asymptotic form of the solutions of the effective single particle Schroedinger Equation, which are related to the transmittance. Disagreement exists about the asymptotic tails of states in disordered two dimensional systems. While some believe that all states are exponentially localized, others argue that a sharp transition to power law behavior, a weaker decay, can occur. Previous numerical studies have been inconclusive. A new technique for accurately calculating the transmittance of large, arbitrary systems is introduced. Called block recursion, it is a generalization of the scalar recursion method for calculating electronic densities of states. Anderson disorder is studied in one and two dimensions. In one dimension sharp, nearly transparent resonances are found to be a general feature of Anderson models at all length scales examined. Agreement with scaling and statistical predictions for the transmittance is found. In two dimensions, a sharp transition between strongly and weakly transmittive states is found near but well inside the band edge. At fixed disorder its position is stable and its strength grows with system size. This is in agreement with predictions of power law-exponential transitions, but more study is needed to demonstrate power law asymptotic behavior on the conductive side of the transition.