Localized systems

We have found the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of $\ln g$ grow with lateral size as $L^{1/3}$ and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.

Many layered materials are more anisotropic than it is predicted by the band theory: the ratio of the in-plane and out-of-plane conductivities exceeds by orders of magnitudes the (inverse) ratio of the effective masses. To explain this anomaly, we propose a simple model of randomly spaced potential barriers (mimicking stacking faults) with isotropic impurities in between the barriers. In the absence of bulk disorder, electron motion in the out-of-plane direction is localized. Bulk disorder destroys 1D localization. As a result, the out-of-plane conductivity is finite and scales linearly with the scattering rate by bulk impurities until planar and bulk disorder become comparable. The out-of-plane conductivity is of a manifestly non-Drude form, with a maximum at the frequency corresponding to the scattering rate by potential barriers. We are interested in several other problems of localization in anystropic systems.