Classical theory of the Hall-effect in an inhomogeneous magnetic field

TL;DR

This paper develops a classical theoretical framework for understanding the Hall effect in a two-dimensional electron gas subjected to an inhomogeneous magnetic field created by flux tubes, aligning with recent experimental observations.

Contribution

It introduces a new approach to modeling the Hall effect in inhomogeneous fields, avoiding the failure of vortex scatterer models and matching experimental results.

Findings

Vortex scatterer approach predicts zero Hall constant at T=0.

Smooth field treatment yields a finite Hall constant consistent with experiments.

Classical theory aligns with experimental data when Fermi wavelength is short.

Abstract

Inspired by recent experiments by Geim et al. we discuss the classical theory of the Hall effect of a 2 dimensional electron gas in an inhomogeneous magnetic field. The field modulation is in the form of flux tubes created by a superconductor overlayer. We find that an approach, where the vortices are treated as individual scatterers contributing to the collision term in the Boltzman equation will not work --- it leads to a vanishing Hall constant at $T=0$. If the field is treated as a smooth contribution to the driving term in the Boltzman equation, the classical Hall constant emerges, in agreement with experiments when the Fermi wavelength is short in comparison with all other lengths in the problem.