Dynamics and scaling properties of localization in energy space in two-dimensional mesoscopic systems

Energy-space localization in two-dimensional (2D) mesoscopic systems is studied. The geometry considered is that of a cylinder threaded by a linearly time-dependent magnetic flux. A formalism for a single Zener transition is described, and applied to transport in energy space when there are multiple Zener transitions. The degree of localization in energy space, measured by the appropriate participation ratio ρ, is evaluated as a function of the disorder in the system (W) and the driving rate α. It is argued that, similarly to the 1D case, when α is very large, ρ depends on W only, while for sufficiently small values of α, ρ=ρ(${\mathit{W}}^2$/α). The degree of localization is, however, nonmonotonic in the disorder and the driving field, in clear contradistinction to the 1D results. These results can be understood qualitatively on the basis of the properties of a single Zener transition and the structure of the disorder free adiabatic spectrum.