The longitudinal resistance of a quantum Hall system with a density gradient

TL;DR

This paper analyzes how a density gradient in a quantum Hall system causes non-uniform current flow and voltage distribution, revealing that the voltage on one side depends solely on Hall resistivity changes, independent of microscopic longitudinal resistivity.

Contribution

It introduces a model for current flow in quantum Hall systems with density gradients, showing how voltage distributions relate to resistivity variations and proposing methods to extract microscopic resistivity.

Findings

Current is pushed towards one side of the sample due to density gradient.

Voltage on the side with large Hall resistivity change is independent of microscopic longitudinal resistivity.

Proposed measurement techniques to determine microscopic longitudinal resistivity.

Abstract

Following recent experiments, we consider current flow in two dimensional electronic systems in the quantum Hall regime where a gradient in the electron density induces a spatial variation in the Hall resistivity. Describing the system in terms of a spatially varying local resistivity tensor, we show that in such a system the current density is generically non-uniform, with the current being pushed towards one side of the sample. We show that, for sufficiently large density gradient, the voltage along that side is determined by the change of the Hall resistivity, and is independent of the microscopic longitudinal resistivity, while the voltage on the opposite side is small and determined by both longitudinal and Hall resistivities. We solve some particular models in detail, and propose ways by which the microscopic longitudinal resistivity may be extracted from measurements of the longitudinal voltage on both sides of the sample.