Study of Periodic Domain Patterns in Tetragonal Ferroelectrics Using Phase-Field Methods

Abstract

A sharp-interface model based on the linear constrained theory of laminates identifies eight distinct rank-2 periodic patterns in tetragonal ferroelectrics. While some of the periodic solutions, such as the herringbone and stripe patterns are commonly observed, others such as the checkerboard pattern consisting of repeating polarization vortices are rarely seen in experiments. The linear constrained theory predicts compatible domain arrangements, but neglects gradient effects at domain walls and misfit stresses due to junctions of domains. Here, we employ a phase-field model to test the stability of the periodic domain patterns with in-plane polarizations, under periodic boundary conditions which impose zero average stress and electric field. The results indicate that domain patterns containing strong disclinations are of high energy and typically unstable in the absence of external stresses or electric fields. The study also provides insight into the internal stresses developed in the various domain patterns.

Nanoscale domain patterns and a concept for an energy harvester

“…The stripe domain pattern, figure 2(a) and the herringbone domain pattern, figure 2(b) are found to be stable at equilibrium, while all the other patterns shown dissolved into either single domain states or a simple stripe pattern. The energy of each polarization pattern, and the associated strain and stress fields have been discussed in previous work [35] where it was found that the stripe and herringbone pattern are stable due to the absence of stressed domains. Each of the other patterns are stressed due to incompatible domain junctions that increase their elastic energy.…”

“…where u i is the displacement, f is the electric potential and P i is the polarization at each node. A similar set of periodic boundary condition is applied on the representative node 'S' [35]. These constraints enforce periodicity of the strain fields with zero average stress, and periodicity of the electric potential values with zero average electric field.…”

“…The volume average free energy (normalized[35]) of the stripe domain pattern as a function of domain wall separation, lw. Inset schematics denote polarization patterns.…”