Nonlinear extrinsic permittivity and piezoelectricity in lead titanate due to 90° domain walls pinning

It is known that the restoring force acting on a disequilibrated domain wall is the key factor, which controls macroscopic values of extrinsic (i.e. domain wall) contributions to permittivity and piezoelectricity. Considering the bulk properties of perovskites, the main contribution to the restoring force comes from the interaction of domain walls with so called pinning centers (PnCs) such as crystal lattice point defects or dislocations. It is clear that the number of PnCs can be essentially controlled by the material composition and the processing technology. In addition, there exist two types of domain wall movement between PnCs: First, the planar wall passes through the PnC and, second, the domain wall bends between PnCs. It has been shown by analytical modeling that each physical phenomenon that controls the restoring force on the domain wall introduces qualitatively different macroscopic features to the nonlinear dielectric and piezoelectric response. These features can be identified in experimental data. In the presented work, we develop an analytical model for extrinsic contribution due to reversible movements of pinned planar (e.g. ferroelectric-ferroelastic) domain walls. Qualitative features of the analytical model are verified using phase-field simulations.