A Boundary Element Method Coupled to Phase Field to Compute Ferroelectric Domains in Complex Geometries

Abstract

Ferroelectrics display couplings between mechanics, electromagnetism and optics. Hence they are in use and have been proposed as elements of MEMS and nanotechnological devices. In these applications, ferroelectrics are machined to have complex geometries and electrode arrangements, and are subjected to microstructural rearrangements. Current mesoscale calculations of ferroelectric microstructure with the phase-field method are typically limited to completely periodic geometries or other special boundary conditions. This is due to the infeasible computational expense in accurately accounting for the stray electric fields caused by charges and electrodes; it requires solution of the electrostatic equations over all of space. We present a computational technique to predict ferroelectric microstructure in complex geometries. The key issue is to resolve both the internal electric fields in the ferroelectric, as well as the stray electric fields that exist outside the material. In particular, we apply a Boundary Element Method (BEM) or Dirichlet-to-Neumann map to calculate the stray electrostatic fields that are generated by a ferroelectric specimen with no a priori restrictions on geometry. This allows us to move beyond periodic boundary conditions, yet retain accuracy and numerical efficiency in computing the stray electric fields. We couple this implementation to a real-space phase-field method and elasticity solver to obtain domain patterns under electromechanical loads. We apply this to calculate the microstructural rearrangements at the tip of a notch.