Localization length at the resistivity minima of the quantum Hall effect

TL;DR

This paper calculates the localization length at resistivity minima in the quantum Hall effect for various disorder models, revealing a power-law dependence on Landau level index and aligning with experimental observations.

Contribution

It introduces a detailed calculation of localization length at quantum Hall minima across different disorder models, highlighting a power-law relationship with Landau level index.

Findings

Localization length scales as N^α with α between 1 and 10/3.

For white-noise disorder, localization length approximates the classical cyclotron radius.

Results agree with experimental data on low and moderate mobility samples.

Abstract

The resistivity minima of the quantum Hall effect arise due to the localization of the electron states at the Fermi energy, when it is positioned between adjacent Landau levels. In this paper we calculate the localization length $\xi$ of such states at even filling factors $\nu = 2 N$. The calculation is done for several models of disorder (``white-noise,'' short-range, and long-range random potentials). We find that the localization length has a power-law dependence on the Landau level index, $\xi \propto N^\alpha$ with the exponent $\alpha$ between one and ${10/3}$, depending on the model. In particular, for a ``white-noise'' random potential $\xi$ roughly coincides with the classical cyclotron radius. Our results are in reasonable agreement with experimental data on low and moderate mobility samples.