Hall transport in nonuniform two-dimensional conductors

Hall transport in a nonuniform system is considered in the phenomenological approach of the local conductivity tensor σ_αβ(r) varying in the plane. The simplest example of a two-terminal rectangular device with σ_αβ depending on only one coordinate is studied in detail. In the case of weak scattering σₓₓ{}σxy, the current across the sample is shown to concentrate in a very narrow region along the line of maximum Hall conductivity σxy. The potential distribution in this region is evaluated in a wide range of aspect ratios of the sample. The physical picture obtained is in striking contrast to that in a uniform conductor, where the current is known to occupy the whole area of the sample. Generalization of this result for the case of an arbitrary analytic distribution σxy(r) is given. The current flow across the sample is shown to collapse, in the limit σₓₓ{→}0, to a cluster formed by a finite number of distinct lines of zero width.