Periodic boundary conditions for the simulation of 3D domain patterns in tetragonal ferroelectric material

I Muench,A Renuka Balakrishna,J E Huber

doi:10.1007/s00419-018-1411-9

Abstract

Periodic domain patterns in tetragonal ferroelectrics are explored using a phase field model calibrated for barium titanate. In this context, we discuss the standard periodic boundary condition and introduce the concept of reverse periodic boundary conditions. Both concepts allow the assembly of cubic cells in accordance with mechanical and electrical conditions. However, application of the reverse periodic boundary condition is due to an increased size of the RVE and enforces more complex structures compared to the standard condition. This may be of particular interest for other multiphysics simulations. Additionally, we formulate mechanical side conditions with minimal spherical (hydrostatic) stress, or conditions with controlled average strain. It is found that in sufficiently small periodic cells, only a uniform single domain, or the simplest stripe domains constitute equilibrium states. However, once the periodic cells are of order 20 domain wall widths in size, more complex, 3-dimensional patterns emerge. Some of these patterns are known from prior studies, but we also identify other domain patterns with long, ribbon-like domains threaded through them and some vortex-like structures.

Highlights

Tetragonal ferroelectrics such as barium titanate, BaTiO3, are characterized by a spontaneous electric polarization lying parallel to the pseudo-cubic axes of the Tc ∼= 120 ◦C, the material gains a spontaneous crystal
Domain topologies resulting from the use of periodic boundary conditions in 3-dimensions, combined with each of the four strain conditions (IFPT, Adaptive Spherical Stress Free (ASSF), Isotropic Spherical Stress Free (ISSF), Transversely Isotropic Deformation (TID)), are presented
For the well-known standard conditions, our solutions tend to evolve patterns with 90◦ domain walls only, if the representative volume elements (RVE) is allowed to adopt the inherent volumetric growth resulting from the materials phase transition from the cubic to the tetragonal crystal state

Summary

Introduction

Tetragonal ferroelectrics such as barium titanate, BaTiO3, are characterized by a spontaneous electric polarization lying parallel to the pseudo-cubic axes of the Tc ∼= 120 ◦C, the material gains a spontaneous crystal. On cooling BaTiO3 through the polarization and transforms its crystal. Structure from the cubic, paraelectric to the tetragonal, ferroelectric state. This phase transformation induces lattice strains and, depending on the surrounding constraints, it causes deformation or internal mechanical stress [1,2]. Regions with uniform electric polarization group together, forming domains separated by narrow domain walls.

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