Dimensional Crossover of Localisation and Delocalisation in a Quantum Hall Bar

TL;DR

This paper investigates how electron localization properties change from two to one dimension in a disordered quantum wire, deriving an analytical formula for the localization length and analyzing implications for the quantum Hall effect.

Contribution

It provides a new analytical formula for the localization length during the dimensional crossover in quantum wires, linking it to conductance and magnetic field effects.

Findings

Derived an analytical formula for localization length as a function of width, conductance, and magnetic field.

Reconsidered scaling analysis of the quantum Hall effect in high Landau levels.

Analyzed delocalization transition in quantum Hall wires.

Abstract

The 2-- to 1--dimensional crossover of the localisation length of electrons confined to a disordered quantum wire of finite width $L_y$ is studied in a model of electrons moving in the potential of uncorrelated impurities. An analytical formula for the localisation length is derived, describing the dimensional crossover as function of width $L_y$, conductance $g$ and perpendicular magnetic field $B$ . On the basis of these results, the scaling analysis of the quantum Hall effect in high Landau levels, and the delocalisation transition in a quantum Hall wire are reconsidered.