Determination of the Riemann modulus and sheet resistivity by a six-point generalization of the van der Pauw method

Abstract

Six point generalization of the van der Pauw method is presented. The method is applicable for two dimensional homogeneous systems with an isolated hole. A single measurement performed on the contacts located arbitrarily on the sample edge allows to determine the specific resistivity and a dimensionless parameter related to the hole, known as the Riemann modulus. The parameter is invariant under conformal mappings of the sample shape. The hole can be regarded as a high resistivity defect. Therefore the method can be applied for experimental determination of the sample inhomogeneity.

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

“…In recent years, the van der Pauw method for samples with a single hole, or even several holes, has been extensively studied [11][12][13][14][15][16][17][18][19][20]. It has been reported that the van der Pauw equation (1.1) is inaccurate for samples with several holes [21].…”

“…A sample with a single isolated hole whose boundary comprises more than a single point is the natural first case to study and has been considered by [11,16]. Any two-dimensional sample with…”

“…One objective of this paper is to show how the inequality (1.10) can be confirmed mathematically. For holey samples Szymański et al proposed some modifications to the van der Pauw setting [12,16]. Firstly, they proposed a six-point method, which uses six electrical contacts on the perimeter of a sample with a hole, and measures nine pairs of resistances [16].…”

“…For holey samples Szymański et al proposed some modifications to the van der Pauw setting [12,16]. Firstly, they proposed a six-point method, which uses six electrical contacts on the perimeter of a sample with a hole, and measures nine pairs of resistances [16]. Because the nine resistances can be expressed explicitly in terms of the coordinates of six electrical contacts on the perimeter of a unit circle, they obtained a well-conditioned equation for the unknown sample resistivity.…”

“…A sample with a single isolated hole whose boundary comprises more than a single point is the natural first case to study and has been considered by [11,16]. Any two-dimensional sample with a single hole is doubly connected and can be transplanted conformally into an annulus [9] where the radius of the inner circle of the annulus depends on the shape of sample [22].…”