HALL TRANSPORT IN NONUNIFORM 2-DIMENSIONAL CONDUCTORS

Hall transport in a nonuniform system is considered in the phenomenolo gical approach of the local conductivity tensor sigma(alphabeta)(r) va rying in the plane. The simplest example of a two-terminal rectangular device with sigma(alphabeta) depending on only one coordinate is stud ied 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 pot ential distribution in this region is evaluated in a wide range of asp ect ratios of the sample. The physical picture obtained is in striking contrast to that in a uniform conductor, where the current is known t o occupy the whole area of the sample. Generalization of this result f or the case of an arbitrary analytic distribution sigma(xy)(r) is give n. The current flow across the sample is shown to collapse, in the lim it sigma(xx) --> 0, to a cluster formed by a finite number of distinct lines of zero width.