Eigenvalues of regular symmetric Hall-plates

TL;DR

This paper derives explicit formulas for the eigenvalues of regular symmetric Hall-plates with arbitrary contacts, enabling analysis of their electrical properties and practical approximations for common shapes.

Contribution

It provides a closed-form computation of eigenvalues for the conductance matrices of regular symmetric Hall-plates, facilitating analysis of their electrical behavior.

Findings

Eigenvalues are computable in closed form for regular symmetric Hall-plates.

Derived formulas relate eigenvalues to Hall-plate voltage and noise efficiency.

Provided practical approximations for common geometries like disks and rectangles.

Abstract

I discuss uniform, isotropic, plane, singly connected, electrically linear, regular symmetric Hall-plates with an arbitrary number of N peripheral contacts exposed to a uniform perpendicular magnetic field of arbitrary strength. In practice, the regular symmetry is the most common one. If the Hall-plates are mapped conformally to the unit disk, regular symmetry means that all contacts are equally large and all contacts spacings are equally large, yet the contacts spacings may have a different size than the contacts. Such Hall-plates do not change when they are rotated by 360{\deg}/N. Their indefinite conductance matrices are circulant matrices, whose complex eigenvalues are computable in closed form. These eigenvalues are used to discuss the Hall-output voltage, the maximum noise-efficiency, and Van-der-Pauw's method for measuring sheet resistances. For practical use, I report simple approximations for Hall-plates with four contacts and 90{\deg} symmetry with popular shapes like disks, rectangles, octagons, squares, and Greek crosses with and without rounded corners.