The Classical Hall Effect in Multiply-Connected Plane Regions Part I: Topologies with Stream Function

Udo Ausserlechner

doi:10.4236/jamp.2019.79136

Abstract

Typical Hall plates for practical magnetic field sensing purposes are plane, simply-connected regions with peripheral contacts. Their output voltage is the sum of even and odd functions of the applied magnetic field. They are commonly called offset and Hall voltage. Contemporary smart Hall sensor circuits extract the Hall voltage via spinning current Hall probe schemes, thereby cancelling out the offset very efficiently. The magnetic field response of such Hall plates can be computed via the electric potential or via the stream function. Conversely, Hall plates with holes show new phenomena: 1) the stream function exists only for a limited class of multiply-connected domains, and 2) a sub-class of 1) behaves like a Hall/Anti-Hall bar configuration, i.e., no Hall voltage appears between any two points on the hole boundary if current contacts are on their outer boundary. The paper studies the requirements under which these effects occur. Canonical cases of simply and doubly connected domains are computed analytically. The focus is on 2D multiply-connected Hall plates where all boundaries are insulating and where all current contacts are point sized.

Highlights

IntroductionThe stunning result is that the voltage between C and F depends on whether the current is supplied via points on the inner or outer boundary, the current streamlines near C and F are perfectly horizontal in both cases
Hall plates with holes show new phenomena: 1) the stream function exists only for a limited class of multiply-connected domains, and 2) a sub-class of 1) behaves like a Hall/Anti-Hall bar configuration, i.e., no Hall voltage appears between any two points on the hole boundary if current contacts are on their outer boundary
The Hall effect in double boundary geometries with small contacts was studied in [10] with the goal of reducing the zero point error of Hall plates. The authors called their rectangular ring “anti-Hall bar within a Hall bar” and focused on the fact that current flowing through points on the outer boundary gives no Hall signal on the inner boundary, and vice versa, whereas the offsets measured on both boundaries are affected by both currents

Summary

Introduction

The stunning result is that the voltage between C and F depends on whether the current is supplied via points on the inner or outer boundary, the current streamlines near C and F are perfectly horizontal in both cases This holds even if the aspect ratio of the device becomes infinite and the points of current injection move to left and right infinity— at applied magnetic field the voltage between C and F changes markedly if we swap the current contacts between inner and outer boundaries. These numerical findings are in perfect agreement to experimental observations [21].

Assumptions and Basic Definitions

Doubly-Connected Asymmetric Hall Plates with Large Contacts

Plane Hall Plates Where All Boundaries Are Insulating

Hall Plates without Holes and with Point Current Contacts on the Boundary

Hall Plates with Extended Contacts at Large Hall Angle

10. Discussion