Correction factors for 4-probe electrical measurements with finite size electrodes and material anisotropy: a finite element study

In four-probe (4-probe) electrical measurements, especially on highly resistive materials, it is not always possible to configure the electrodes such that the current density is uniform throughout the sample. Under such circumstances, simply considering the material's electrical resistivity to be proportional to the measured resistance with the proportionality constant given by the sample geometry can give an incorrect result. In this paper, a numerical finite element model is presented which can extract a material's true resistivity from co-linear 4-probe electrical measurements on highly resistive samples with large electrodes that extend across the sample width. The finite element model is used to investigate the influence of material anisotropy, the resistance of the sample-electrode interfaces and the relative electrode-to-sample size on the potential and current density distributions in the sample. A correction factor is introduced to account for the impact of these effects on the measured resistivity. In the limit of large interface resistance, excellent agreement is found with an analytical expression derived elsewhere (Esposito et al 2000 J. Appl. Phys. 88 2724-9). The approach presented here can be used to evaluate a variety of effects on co-linear 4-probe electrical measurements, can be extended to complex specimen geometries with arbitrary electrode arrangements and, additionally, could find use in the evaluation of data from 4-probe thermal conductivity measurements.