Contacts and Edge State Equilibration in the Fractional Quantum Hall Effect

TL;DR

This paper develops a kinetic equation model to analyze edge state equilibration in the fractional quantum Hall effect, revealing how impurity scattering and contact models influence conductance quantization.

Contribution

It introduces a simple kinetic framework to study edge mode equilibration in FQHE, highlighting the roles of impurity scattering and contact models in conductance quantization.

Findings

Electrical contact models do not equilibrate counter-propagating edge modes at ν=2/3.

Impurity scattering restores quantized conductance in certain FQHE states.

Equilibration length diverges at specific filling factors, affecting quantization at low temperatures.

Abstract

We develop a simple kinetic equation description of edge state dynamics in the fractional quantum Hall effect (FQHE), which allows us to examine in detail equilibration processes between multiple edge modes. As in the integer quantum Hall effect (IQHE), inter-mode equilibration is a prerequisite for quantization of the Hall conductance. Two sources for such equilibration are considered: Edge impurity scattering and equilibration by the electrical contacts. Several specific models for electrical contacts are introduced and analyzed. For FQHE states in which edge channels move in both directions, such as $\nu=2/3$, these models for the electrical contacts {\it do not} equilibrate the edge modes, resulting in a non-quantized Hall conductance, even in a four-terminal measurement. Inclusion of edge-impurity scattering, which {\it directly} transfers charge between channels, is shown to restore the four-terminal quantized conductance. For specific filling factors, notably $\nu =4/5$ and $\nu=4/3$, the equilibration length due to impurity scattering diverges in the zero temperature limit, which should lead to a breakdown of quantization for small samples at low temperatures. Experimental implications are discussed.