Statistical theory of relaxation of high energy electrons in quantum Hall edge states

TL;DR

This paper develops a theoretical framework to describe how high-energy electrons relax and exchange energy in quantum Hall edge states, using coupled Fokker-Planck equations to model the process as a drift-diffusion in energy space.

Contribution

It introduces a novel analytical model for electron relaxation in quantum Hall edge states considering non-momentum conserving collisions due to disorder.

Findings

Relaxation described by coupled Fokker-Planck equations.

Electron relaxation modeled as a generalized drift-diffusion process.

Edge state heating results from injected electron relaxation.

Abstract

We investigate theoretically the energy exchange between electrons of two co-propagating, out-of-equilibrium edge states with opposite spin polarization in the integer quantum Hall regime. A quantum dot tunnel-coupled to one of the edge states locally injects electrons at high energy. Thereby a narrow peak in the energy distribution is created at high energy above the Fermi level. A second downstream quantum dot performs an energy resolved measurement of the electronic distribution function. By varying the distance between the two dots, we are able to follow every step of the energy exchange and relaxation between the edge states - even analytically under certain conditions. In the absence of translational invariance along the edge, e.g. due to the presence of disorder, energy can be exchanged by non-momentum conserving two-particle collisions. For weakly broken translational invariance, we show that the relaxation is described by coupled Fokker-Planck equations. From these we find that relaxation of the injected electrons can be understood statistically as a generalized drift-diffusion process in energy space for which we determine the drift-velocity and the dynamical diffusion parameter. Finally, we provide a physically appealing picture in terms of individual edge state heating as a result of the relaxation of the injected electrons.