Fay meets van der Pauw: the trisecant identity and the resistivity of holey samples

Abstract

The van der Pauw method is commonly used in the applied sciences to find the resistivity of a simply connected, two-dimensional conducting laminate. Given the usefulness of this ‘4-point probe’ method there has been much recent interest in trying to extend it to holey, that is, multiply connected, samples. This paper introduces two new mathematical tools to this area of investigation—the prime function on the Schottky double of a planar domain and the Fay trisecant identity—and uses them to show how the van der Pauw method can be extended to find the resistivity of a sample with a hole. We show that an integrated form of the Fay trisecant identity provides valuable information concerning the appearance of ‘envelopes’ observed in the case of holey samples by previous authors. We find explicit formulae for these envelopes, as well as an approximate formula relating two pairs of resistance measurements to the sample resistivity that is expected to be valid when the hole is sufficiently small and not too close to the outer boundary. We describe how these new mathematical tools have enabled us to prove certain conjectures recently made in the engineering literature.

The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

“…By introducing the prime function on the Schottky double of a conformally equivalent concentric annulus to any given doubly connected sample with a single uncharged hole, and by using an integrated form of an associated Fay trisecant identity expressed in terms of that prime function, we have shown how to prove some conjectures made in the literature on an observed envelope structure associated with pairs of 4-point resistance measurements in a van der Pauw procedure. More practical implications of our results are discussed elsewhere [19].…”

“…This article focusses on the function theory aspects. A companion article [19] explores some of the practical implications of our results.…”

“…To explain the derivation of (65) we note that the integrand G θ,η (ζ ) defined in (61) has two simple poles at ζ = ν and ζ = 1/ν and is readily confirmed, on use of (19), to be loxodromic. The function G θ,η (ζ ) can therefore also be written as…”

“…The logarithmic derivative of the prime function K (ζ ) and its derivative L(ζ ) are also important functions in the general function theory on multiply connected domains [18]. For the concentric annulus it is easily shown, on use of (19), that these functions satisfy the functional relations…”

Mathematical analysis of Van der Pauw’s method for measuring resistivity

“…For example, F.S. Oliveira et al evaluated Van der Pauw's method for measuring resistivity as shown in Figure 1 and thin-layer resistance and proposed an improved measurement method [2]; Jonas Kleiza popularized Van der Pauw's method in anisotropic media, showing that the solution of Van der Pauw's equation in anisotropic media happened to be the value of the conductivity tensor determinant [3]; Hiroyuki Miyoshi et al used the trisecant theorem to obtain a measurement method for porous materials [4].…”