We have studied the behavior of micro four-point probe (M4PP) measurements on two-dimensional (2D) sheets composed of grains of varying size and grain boundary resistivity by Monte Carlo based finite element (FE) modelling. The 2D sheet of the FE model was constructed using Voronoi tessellation to emulate a polycrystalline sheet, and a square sample was cut from the tessellated surface. Four-point resistances and Hall effect signals were calculated for a probe placed in the center of the square sample as a function of grain density n and grain boundary resistivity ρGB. We find that the dual configuration sheet resistance as well as the resistance measured between opposing edges of the square sample have a simple unique dependency on the dimension-less parameter √nρGBG0, where G0 is the sheet conductance of a grain. The value of the ratio RA/RB between resistances measured in A- and B-configurations depends on the dimensionality of the current transport (i.e., one- or two-dimensional). At low grain density or low grain boundary resistivity, two-dimensional transport is observed. In contrast, at moderate grain density and high grain resistivity, one-dimensional transport is seen. Ultimately, this affects how measurements on defective systems should be interpreted in order to extract relevant sample parameters. The Hall effect response in all M4PP configurations was only significant for moderate grain densities and fairly large grain boundary resistivity.
Lotz, M. R., Boll, M., Østerberg, F. W., Hansen, O., & Petersen, D. H. (2016). Mesoscopic current transport in two-dimensional materials with grain boundaries: Four-point probe resistance and Hall effect. Journal of Applied Physics, 120(13), . https://doi.org/10.1063/1.4963719