Electrostatic derivation for the van der Pauw formula and simulation using arbitrarily shaped resistive materials

The van der Pauw method is an important technique in measuring the resistivity of flat materials by means of the van der Pauw formula. The formula is derived analytically from a rectangular material by solving the two-dimensional Laplace equation of electric potential. The input and output current densities are two Neumann boundary conditions. The use of the δ function as the current boundary condition, in comparison with the rect function in the literature, makes the deduction simple. The van der Pauw measurement is simulated with the finite element method by using two arbitrarily shaped materials of different resistivities and four point contacts around the periphery. The input current is used as the boundary condition at one contact and electric potential as the variable to be solved. The potential differences between two voltage contacts are employed to calculate the two resistances for the formula. With the reduction of mesh size, the simulated resistivity approaches the actual value progressively. The minimum relative error is on the order of ppm. In addition, one circular hole is produced in the irregular material, and the relative error is investigated in response to the hole position and radius. The equipotential curves and current pattern are displayed for demonstration. This confirms that the van der Pauw method is valid for measurement of arbitrarily shaped materials without an inner hole.