Low Energy Periodic Microstructure in Ferroelectric Single Crystals
Two distinct modelling approaches are used to find minimum energy (equilibrium) microstructural states in tetragonal ferroelectric single crystals. The first approach treats domain walls as sharp interfaces and uses analytical solutions of the compatibility conditions at domain walls to identify multi-rank laminate microstructures that are free of residual stress and electric field. The second method treats domain walls as diffuse interfaces, using a phase-field model in 3-dimensions. This is computationally intensive, but takes the full field equations into account and allows a more general class of periodic microstructure to be explored. By searching for minimum energy configurations of a cube of tetragonal material, candidate unit cells of a periodic microstructure are identified. Adding periodic boundary conditions allows the assembly of the unit cells into a macro-structure of low energy. A noteworthy structure identified in this way is a “hexadomain” vortex consisting of six tetragonal domains meeting along the major diagonal of a cube. Several of the structures identified by the phase-field model are found to be special cases of multi-rank laminate structure. Thus the analytical approach offers a fast method for finding equilibrium microstructures, while the phase-field model provides a validation of these solutions.
KeywordsDomain Wall Barium Titanate Polarization Vortex Diffuse Interface Periodic Assembly
J.E.H. Acknowledges support of EPSRC project No. EP/E026095/1. I.M. gratefully acknowledges the Karlsruhe House of Young Scientists for awarding a research fellowship to support his stay at the University of Oxford.