Thin films of single crystal ferroelectric material are constrained by their substrate to a fixed state of macroscopic in-plane strain. Domain structures can form to minimize the overall energy of the system by matching the imposed strain conditions in an average sense. We study low-energy equilibrium states of ferroelectric thin films using the theory of domain compatibility. In a thin film with a given state of average strain, periodic laminate microstructures can be predicted or designed using compatibility theory and the condition that domain wall orientations parallel to the substrate are prohibited by the reduction to 2-dimensions. The theory is applied to [001], [011] and [111] oriented single crystal films of tetragonal ferroelectric materials. Compatible configurations are generated and the ability of such films to be poled by electric field is then explored.

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