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Edge effects in four-point direct current potential drop measurements on metal plates

Four-point direct current potential drop (DCPD) measurements are commonly used to measure the conductivity (or resistivity) of semiconductors and ferrous or non-ferrous metals. The measured electrical potential difference is often interpreted in terms of analytic expressions developed for large plates that are either ‘thin’ or ‘thick’ relative to the probe length. It is well known that the presence of the back surface of a plate leads to a solution expressed in terms of an infinite series representing the current source and its images. This approach can be generalized to account for multiple surfaces in order to obtain a solution for a finite plate, but convergence of the series is poor when the plate dimensions are similar to or smaller than the separation of the current injection and extraction points. Here, Fourier series representations of the infinite sums are obtained. It is shown that the Fourier series converge with many fewer terms than the series obtained from image theory, for plates with dimensions similar to or smaller than the separation of the current injection and extraction points. Comparing calculated results for the potential drop obtained by a four-point probe centred on finite plates of varying dimension, with those for a probe in contact with a large (laterally infinite) plate, estimates are given of the uncertainty due to edge effects in measurements on small plates interpreted using analytic formulae developed for large plates. It is also shown that these uncertainties due to edge effects are reduced, for a given plate size, if the probe pick-up points are moved closer to the current injection points, rather than adopting the common arrangement in which the four probe points are equally spaced. Calculated values of DCPD are compared with experimental data taken on aluminium and spring-steel plates of various sizes and excellent agreement is obtained.