A continuum theory of deformable, semiconducting ferroelectrics

Ferroelectric solids, especially ferroelectric perovskites, are widely used as sensors, actuators, filters, memory devices, and optical components. While these have traditionally been treated as insulators, they are in reality wide-band-gap semiconductors. This semiconducting behavior affects the microstructures or domain patterns of the ferroelectric material and the interaction of ferroelectrics with electrodes, and is affected significantly by defects and dopants. In this paper, we develop a continuum theory of deformable, semiconducting ferroelectrics. A key idea is to introduce space charges and dopant density as field (state) variables in addition to polarization and deformation. We demonstrate the theory by studying oxygen vacancies in barium titanate. We find the formation of depletion layers, regions of depleted electrons, and a large electric field at the ferroelectric-electrode boundary. We also find the formation of a charge double layer and a large electric field across 90 degrees domain walls but not across 180 degrees domain walls. We show that these internal electric fields can give rise to a redistribution or forced diffusion of oxygen vacancies, which provides a mechanism for aging of ferroelectric materials.