Compatible domain structures and the poling of single crystal ferroelectrics

Ferroelectric crystals are known to adopt low energy, compatible configurations of microstructure. Many models have been proposed to describe these low-energy domain configurations, mainly based on the idea of fine phase mixtures and average compatibility. The present work develops criteria for the exact compatibility of domain structure in the form of a periodic multi-rank laminate. Exactly compatible structure is expected to be energetically favourable and does not require the concept of a fine mixture. A hierarchical binary tree representation is used to describe the relations among the domains in a multi-rank laminate, and this enables a concise statement of the conditions for exact compatibility. The disclinations associated with domain junctions are also taken into account. The resulting method is a rapid and systematic procedure for finding exactly compatible domain arrangements. This is used to explore minimum rank compatible microstructure in the tetragonal and rhombohedral crystal systems. The results reveal routes in polarization and strain space along which the microstructure can continuously evolve, including low-rank poling paths for single crystals. The method generates several well known domain configurations and many new structures.