Current distributions in the quantum Hall effect

The spatial distribution of the current and electron density is calculated for an ideal two-dimensional Hartree quantum Hall device in which the electron interactions are treated in the Hartree approximation. The calculations are based on an equilibrium-thermodynamics approach in which a constant current is imposed as a constraint on the system. As a consequence of the two-dimensional nature of the Coulomb interactions, there are large inhomogeneities in the electron-density and current distributions. The net current is found to be due principally to a redistribution of charges at the edges of the sample, while currents flowing in the bulk of the sample play only a minor role. The fact that a few inhomogeneously distributed states carry large currents gives a qualitative explanation for the spatial inhomogeneity and the low total current observed at the breakdown of the dissipationless quantum Hall effect.