An extended grain boundary barrier height model including the impact of internal electric field

An extended phenomenological model is proposed to rationalize the potential barriers of the varistor like boundaries of piezoelectric semiconductors. This model takes self-consistently into account the inverse piezoelectric effect of the internal electric field associated with the barrier height at grain boundaries. A double Schottky barrier at a grain boundary gives rise to a strong internal electric field in the depletion layer. Due to the inverse piezoelectric effect, this internal electric field leads to mechanical strain and consequently piezoelectric charge at a grain boundary and also an additional space charge associated with the polarization inhomogeneity. The piezoelectric charges in return impacts the barrier height. Results show that the piezoelectric charge induced by the internal field tends to adjust the grain boundary charge and lowers the barrier height. Furthermore, the barrier height becomes less sensitive to mechanical stress and applied voltage if the influence of the internal field is taken into account. The extended model with the inverse piezoelectric effect of the internal field allows to further elucidate their piezotronic response. The work improves the grain boundary barrier height theory of piezoelectric ceramics and is expected to have general impact on piezotronics and other junction devices.