TITLE:
Simple Proofs of Upper and Lower Envelopes of Van Der Pauw’s Equation for Hall-Plates with an Insulated Hole and Four Peripheral Point-Contacts
AUTHORS:
Udo Ausserlechner
KEYWORDS:
Conformal Mapping, Contraction Process, Doubly Connected Domains, Envelopes, Hall Plate, Large Hole Angle, Sheet Resistance, Small Hole Angle, Van Der Pauw
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.10 No.3,
March
31,
2022
ABSTRACT: For plane singly-connected domains with insulating boundary and four point-sized contacts, C0 … C3, van der Pauw derived a famous equation relating the two trans-resistances R01,23, R12,30 with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.