# Charge equilibration in integer and fractional quantum Hall edge channels in a generalized Hall-bar device

## TL;DR

This study investigates charge equilibration in quantum Hall edge channels, revealing differences in equilibration lengths between various regimes and states, and demonstrating a new method using generalized Hall bars to analyze edge structures in IQH and FQH systems.

## Contribution

The paper introduces a systematic approach using generalized Hall bars to evaluate charge equilibration in both integer and fractional quantum Hall regimes, highlighting differences in edge channel structures.

## Key findings

- Equilibration length is shorter for counter-propagating channels at ν=2/3.
- Equilibration length is longer for co-propagating channels at ν=4/3.
- Edge channel geometry influences charge equilibration behavior.

## Abstract

Charge equilibration between quantum-Hall edge states can be studied to reveal geometric structure of edge channels not only in the integer quantum Hall (IQH) regime but also in the fractional quantum Hall (FQH) regime particularly for hole-conjugate states. Here we report on a systematic study of charge equilibration in both IQH and FQH regimes by using a generalized Hall bar, in which a quantum Hall state is nested in another quantum Hall state with different Landau filling factors. This provides a feasible way to evaluate equilibration in various conditions even in the presence of scattering in the bulk region. The validity of the analysis is tested in the IQH regime by confirming consistency with previous works. In the FQH regime, we find that the equilibration length for counter-propagating $\delta \nu $ = 1 and $\delta \nu $ = -1/3 channels along a hole-conjugate state at Landau filling factor $\nu $ = 2/3 is much shorter than that for co-propagating $\delta \nu $ = 1 and $\delta \nu $ = 1/3 channels along a particle state at $\nu $ = 4/3. The difference can be associated to the distinct geometric structures of the edge channels. Our analysis with generalized Hall bar devices would be useful in studying edge equilibration and edge structures.