Influence of flexoelectric coupling on domain patterns in ferroelectrics

Using Ginzburg-Landau theory and two-dimensional (2D) phase field simulations, we analyze the influence of flexoelectric coupling on the domain patterns in ferroelectrics. The phase field simulations predict that a high strength of the flexoelectric coupling leads to formation of a fine structure in domain patterns in ferroelectrics. The fine structure forms when the coupling strength exceeds a critical value and is related to local transition into an incommensurate phase. Depending on the parameters, a structure with stripe patterns with antiparallel polarizations or another one, not seen before, with two-dimensional arrays of alternating vortices is found. Complex domain configurations with coexisting phases and unusual domain walls between them are observed. Although the incommensurate phase does not form for weaker couplings, the influence of flexoelectricity on bulk domain patterns can still be significant. The results of the calculations are rationalized using an analytical model. Directions for the modulation wave vectors in the fine structure are found in the framework of a linear analysis, while the type of the structure-stripes or vortices-is determined by anharmonicity.