Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films

The stability and evolution of ferroelectric domain structures in thin films are studied. Elastic solutions are derived for both elastically anisotropic and isotropic thin films with arbitrary domain structures, subject to the mixed stress-free and constraint boundary conditions. These solutions are employed in a three-dimensional phase-field model to investigate simultaneously the effect of substrate constraint and temperature on the volume fractions of domain variants, domain-wall orientations, surface topology, domain shapes, and their temporal evolution for a cubic-to-tetragonal ferroelectric phase transition. A specific example of a [001] orientated film heteroepitaxially grown on a [001] cubic substrate is considered. It is shown that the shapes of a-domains with tetragonal axes parallel to the film surface are significantly different from those of c-domains with tetragonal axes perpendicular to the film surface. For the substrate constraints and temperatures under which both a- and c-domains coexist, both types of a-domains are present with their tetragonal axes perpendicular to each other, and the domain wall orientations deviate from the 45 orientation generally assumed in thermodynamic analyses. It is demonstrated that a substrate constraint results in sequential nucleation and growth of different tetragonal domains during a ferroelectric phase transition.