Generalization of the van der Pauw relationship derived from electrostatics

► A van der Pauw result is presented for an electrode array smaller than the sample. ► The consequences of this modified arrangement are presented. ► Its disadvantages relative to the original van der Pauw arrangement are discussed. ► As a consequence of this work, a new mathematical relationship has been uncovered. ► This electrostatics technique can be applied to the calculation of the Hall voltage. In an earlier paper, this author, along with two others Weiss et al. (2008) [1], demonstrated that the original van der Pauw relationship could be derived from three-dimensional electrostatics, as opposed to van der Pauw’s use of conformal mapping. The earlier derivation was done for a conducting material of rectangular cross section with contacts placed at the corners. Presented here is a generalization of the previous work involving a square sample and a square array of electrodes that are not confined to the corners, since this measurement configuration could be a more convenient one. As in the previous work, the effects of non-zero sample thickness and contact size have been investigated. Buehler and Thurber derived a similar relationship using an infinite series of current images on a large and thin conducting sheet to satisfy the conditions at the boundary of the sample. The results presented here agree with theirs numerically, but analytic agreement could not be shown using any of the perused mathematical literature. By simply equating the two solutions, it appears that, as a byproduct of this work, a new mathematical relationship has been uncovered. Finally, the application of this methodology to the Hall Effect is discussed.