Resonant scattering on impurities in the Quantum Hall Effect

TL;DR

This paper introduces a novel approach to model carrier transport in the Quantum Hall Effect via resonant scattering on impurities, revealing how phase effects cause Aharonov-Bohm oscillations and identifying extended states within Landau bands.

Contribution

It develops a new theoretical framework for resonant impurity scattering in Quantum Hall systems, connecting complex phase effects to conductance oscillations and localization properties.

Findings

Total conductance can be derived from an effective Schrödinger equation.

Aharonov-Bohm oscillations arise due to complex phase in impurity hopping.

Each Landau band contains an extended state with localized others.

Abstract

We develop a new approach to carrier transport between the edge states via resonant scattering on impurities, which is applicable both for short and long range impurities. A detailed analysis of resonant scattering on a single impurity is performed. The results are used for study of the inter-edge transport by multiple resonant hopping via different impurities' sites. It is shown that the total conductance can be found from an effective Schroedinger equation with constant diagonal matrix elements in the Hamiltonian, where the complex non-diagonal matrix elements are the amplitudes of a carrier hopping between different impurities. It is explicitly demonstrated how the complex phase leads to Aharonov-Bohm oscillations in the total conductance. Neglecting the contribution of self-crossing resonant-percolation trajectories, one finds that the inter-edge carrier transport is similar to propagation in one-dimensional system with off-diagonal disorder. We demonstrated that each Landau band has an extended state $\bar E_N$, while all other states are localized. The localization length behaves as $L_N^{-1}(E)\sim (E-\bar E_N)^2$.