Anderson localization in anisotropic random media

Anderson localization in systems where the randomness is anisotropic and intermediate between the different dimensions is studied both analytically and numerically. As a function of the anisotropy parameter θ, which interpolates between a one-dimensional randomly layered system and three-dimensional isotropic randomness, our results indicate the existence of a Fermi-energy-dependent critical ${\mathrm{\theta }}_{\mathit{c}}$ below which the wave function is localized for arbitrarily small randomness. An interesting reentry phenomenon is found in the localization phase diagram near the band edge, where the density of states is small. Expressions for the localization lengths are obtained analytically in the localized regime when the randomness is small. The behavior of the localization length in the layering direction is found to follow the simple one-dimensional result, while in the lateral direction the localization length behaves differently from the standard two-dimensional result. Physical arguments are presented to make plausible the above behaviors. Our model is contrasted with the anisotropic hopping model. Significant differences are noted.