HALL TRANSPORT IN NONUNIFORM 2-DIMENSIONAL CONDUCTORS

Hall transport in a nonuniform system is considered in the phenomenological approach of the local conductivity tensor sigma(alphabeta)(r) varying in the plane. The simplest example of a two-terminal rectangular device with sigma(alphabeta) depending on only one coordinate is studied in detail. In the case of weak scattering sigma(xx) << sigma(xy). the current across the sample is shown to concentrate in a very narrow region along the line of maximum Hall conductivity sigma(xy). 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 sigma(xy)(r) is given. The current flow across the sample is shown to collapse, in the limit sigma(xx) --> 0, to a cluster formed by a finite number of distinct lines of zero width.