Hall transport in nonuniform two-dimensional conductors

Hall transport in a nonuniform system is considered in the phenomenological approach of the local conductivity tensor ${\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}$(r) varying in the plane. The simplest example of a two-terminal rectangular device with ${\mathrm{\sigma }}_{\mathrm{\alpha }\mathrm{\beta }}$ depending on only one coordinate is studied in detail. In the case of weak scattering ${\mathrm{\sigma }}_{\mathit{x}\mathit{x}}$≪${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$, the current across the sample is shown to concentrate in a very narrow region along the line of maximum Hall conductivity ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$. 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 ${\mathrm{\sigma }}_{\mathit{x}\mathit{y}}$(r) is given. The current flow across the sample is shown to collapse, in the limit ${\mathrm{\sigma }}_{\mathit{x}\mathit{x}}$→0, to a cluster formed by a finite number of distinct lines of zero width.