Lambert meets van der Pauw: Analytical expressions for fast numerical computation of dual configuration sheet resistance

Abstract

The van der Pauw theorem [van der Pauw, L.J. 1958; Philips Res. Rep 13 no 1, 1–9] enables accurate determination of sheet resistance irrespective of either sample or probing geometry. While van der Pauw's identities form the theoretical cornerstone of electrical four-point probe metrology, the formulae are implicit with respect to sheet resistance, enabling to date only numerical solutions or approximations. Here we briefly review former approaches of solving the van der Pauw identities, recognize the problem as root finding of a trinomial, introduce four alternative calculation schemes, and evaluate both the legacy and the proposed approaches in terms of both their accuracy and time complexity. We demonstrate that an iterative solution based on Lambert's transcendental equation yields a thousand-fold acceleration with respect to a numerical solution of van der Pauw's original formula, with no loss of numerical accuracy. We demonstrate that this acceleration remains significant within the scope of current-in-plane tunnelling measurements of magnetic tunnel junctions, where ∼10³ individual solutions of the van der Pauw identity are typically required during the acquisition of a single measurement point.