Surface currents in Hall devices

M Creff,J M Rubì,F Faisant,J.-E Wegrowe

doi:10.1063/5.0013182

Abstract

One hundred and forty years after his discovery, the Hall effect still deserves attention. If it is well-known that the Hall voltage measured in Hall bar devices is due to the electric charges accumulated at the edges in response to the magnetic field, the nature of the corresponding boundary conditions is still problematic. In order to study this out-of-equilibrium stationary state, the Onsager’s least-dissipation principle is applied. It is shown that, beside the well-known expression of the charge accumulation and the corresponding Hall voltage, a longitudinal surface current proportional to the charge accumulation is generated. An expression of the surface current is given. The surface currents allow the Hall voltage to be stabilized at a stationary state, despite, e.g., the presence of leakage of charges at the edges.

Highlights

The classical Hall effect1 is usually described by the local transport equations for the charge carriers that takes into account the effect of the Laplace–Lorentz force generated by a static magnetic field
We have shown that the stationary state of the ideal Hall bar with small charge leakage at the edges can be derived from the principle of least dissipation and the global boundary conditions
The stationary state is characterized by the accumulation of electric charges at the edges—that generates the Hall voltage like in a simple capacitor at equilibrium—and by surface currents confined at the edges

Summary

INTRODUCTION

The classical Hall effect is usually described by the local transport equations for the charge carriers that takes into account the effect of the Laplace–Lorentz force generated by a static magnetic field. In contrast to usual non-equilibrium stationary sates, the presence of the static magnetic field leads to specific transverse boundary conditions— the charge accumulation at the edges—that are not imposed directly by external constraints but by the system itself, according to the Le Chatelier–Braun principle.. In contrast to usual non-equilibrium stationary sates, the presence of the static magnetic field leads to specific transverse boundary conditions— the charge accumulation at the edges—that are not imposed directly by external constraints but by the system itself, according to the Le Chatelier–Braun principle.2,3 This specificity of the boundary conditions for the Hall effect has been discussed within a large variety of theoretical models, but the nature of the charge accumulation at the edges still remains rather mysterious.

JOULE DISSIPATION

ÀθH θH 1

LEAST-DISSIPATION PRINCIPLE

EXPRESSION OF THE SURFACE CURRENTS

STABILITY

CONCLUSION