Influence of space charge on domain patterns and susceptibility in a rhombohedral ferroelectric film

The presence of a space charge region induces an internal electric field within the charged region that, in a ferroelectric material, would rotate the polarisations to align with the field. The strength of the induced field would therefore determine the domain patterns and polarisation switching properties of the material. Using a phase-field model, we investigate the effect of charged layers in fully and partially depleted BiFeO$_3$ thin films in the rhombohedral phase. While the domain pattern in a charge-free BiFeO$_3$ film consists of only two polarisation variants, we observe complex patterns with four coexisting variants that form within the charged layers at sufficiently high induced fields. These variants form a head-to-head configuration with an interface that is either wavy or planar depending on the internal field strength, which is determined by the charge density as well as the thickness of the charged layer. For depletion layers with sufficient thickness, there exists a range of charge density values for which the interface is wavy, while at high densities the interface becomes planar. We find that films with wavy interfaces exhibit enhanced susceptibilities with reduced hystereses compared to the charge-free film. The results of our work suggest that introducing space charge regions by careful selection of dopant density and electrode materials can engineer domain patterns that yield a higher response with a smaller hysteresis.