Stochastic description of the stationary Hall effect

TL;DR

This paper models the stationary Hall effect using Langevin equations to describe Brownian motion of charges, revealing how boundary layers and charge accumulation influence the effect in various materials.

Contribution

It introduces a stochastic Langevin equation approach to derive properties of the stationary Hall effect, emphasizing boundary layer effects and charge accumulation.

Findings

Non-uniform current density forms near edges

Boundary layer confined within Debye-Fermi length

Theory aligns with established results

Abstract

The properties which characterize the stationary Hall effect in a Hall bar are derived from the Langevin equations describing the Brownian motion of an ensemble of interacting moving charges in a constant externally applied electromagnetic field. It is demonstrated that a non-uniform current density a) superimposes on the injected one, b) is confined in a boundary layer located near the edges over the Debye-Fermi length scale c) results from the coupling between diffusion and conduction and d) arises because of charge accumulation at the edges. The theory can easily be transposed to describe the Hall effect in metals, semi-conductors and plasmas and agrees with standard and previously published results.