Abstract
The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.
Highlights
One of the most prevalent and successful measurement techniques in the applied sciences is the four-point probe method [1,2]
Two new mathematical tools turn out to be relevant to the van der Pauw problem as it pertains to holey samples: the theory of the so-called prime function on the Schottky double of a multiply connected planar domain [18] and the Fay trisecant identities associated with those same compact Riemann surfaces and which involve the prime function
Where k is an arbitrary complex number. This statement (31) of the genus-1 Fay trisecant identity expressed purely in terms of the prime function of the concentric annulus has been taken from Exercise 8.9 of Chapter 8 of the monograph [18] which asks the reader to prove it using the properties of so-called loxodromic functions
Summary
One of the most prevalent and successful measurement techniques in the applied sciences is the four-point probe method [1,2]. It is a well-known consequence of an extension of the Riemann mapping theorem due to Koebe (see [20]) that any 2D sample with a single isolated hole can be transplanted conformally into an annulus ρ < |ζ | < 1 in a complex ζ plane, say, where the radius of the inner circle of the annulus depends on the shape of sample [18,21] By conducting both numerical and actual experiments Szymanski et al showed [7] that the van der Pauw equation (1) does not hold for a sample with a hole but they conjectured that the data instead satisfies the inequality exp − Razwb + exp − Razbw ≤ 1. Using these tools we can gain insights into the two envelopes associated with the resistance measurements (4)
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