The dc voltage dependence of semiconductor grain-boundary resistance

A model is developed to describe the potential barriers which often occur at grain boundaries in polycrystalline semiconductors. The resistance of such materials is determined by thermionic emission over these barriers. The dc grain-boundary current density as a function of applied voltage is calculated using several forms for the density of defect states within the boundary region. In all cases, the currents are Ohmic at low voltages; they can attain a quasisaturated level at intermediate voltages, and they display a sharp bias dependence at high voltages. The details of the intermediate and high-voltage characteristics are found to depend strongly on the grain-doping density and on the density and energy distribution of defect states at the grain boundary. Contrary to previous assertions, we find that the large current-voltage nonlinearities found in real materials are most likely associated with defect-state densities that decrease above the zero-bias Fermi level. The results of the model are compared with previous experimental data on Si and Ge bicrystals and on polycrystalline ZnO varistors. Finally, a detailed method for determining the energy density of grain-boundary defect states from current-voltage data is developed.