# Surface currents in Hall devices

## TL;DR

This paper uses a variational approach to analyze stationary states in Hall devices, revealing boundary layer effects and a minimal power dissipation state with zero transverse current.

## Contribution

It introduces a variational method to study charge, potential, and current distributions in Hall devices, deriving a simple expression for minimal power dissipation states.

## Key findings

- Identifies a boundary layer near device edges with charge accumulation and surface currents.
- Derives a state of minimum power dissipation with zero transverse current.
- Shows surface currents are proportional to charge accumulation near edges.

## Abstract

A variational approach is used in order to study the stationary states of Hall devices. Charge accumulation, electric potentials and electric currents are investigated on the basis of the Kirchhoff-Helmholtz principle of least heat dissipation. A simple expression for the state of minimum power dissipated -- that corresponds to zero transverse current and harmonic chemical potential -- is derived. It is shown that a longitudinal surface current proportional to the charge accumulation is flowing near the edges of the device. Charge accumulation and surface currents define a boundary layer over a distance of the order of the Debye-Fermi length.