An isogeometric approach to flexoelectric effect in ferroelectric materials

Flexoelectricity is an electromechanical coupling effect between the polarization and strain gradient in all dielectrics regardless of point group symmetry. Due to its significant influence on material behavior at the nanoscale, the flexoelectric effect has attracted more and more attention in recent years. In this paper, a real space phase field model for the flexoelectric effect in ferroelectric materials is developed by using isogeometric analysis (IGA). The IGA employs the same smooth and high-order basis functions to describe both the geometry of material and the solution of phase field, which is able to give an accurate and efficient description of the flexoelectric effect in ferroelectrics with arbitrary geometrical shapes and boundary conditions. To this end, phase field simulations on the effect of flexoelectricity are conducted for nanoscale ferroelectrics with different geometrical shapes and boundary conditions. The simulation results show that the flexoelectric effect has significant influence on the domain structures and domain switching of ferroelectric materials at the nanoscale. For ferroelectric nanobeam under bending load, due to the flexoelectric effect, the mechanical bending can break the symmetry of hysteresis loop between electric field and polarization. As for ferroelectric nanodots, the flexoelectric effect increases the magnitude of spontaneous polarizations and results in the tilting of polarization vortex. In addition, outof-plane components appear in the polarization vortex of ferroelectric nanodots due to the flexoelectric effect, which increases the coercive field for the switching of polarization vortex and changes the switching process significantly. The present work not only presents an effective nonlocal model for the domain evolution in ferroelectric materials with the consideration of the flexoelectric effect, but also asymmetric hysteresis loop between polarization and external electric field for ferroelectric beam, and new switching behavior of the polarization vortex in ferroelectric nanodots due to the flexoelectric effect. (C) 2018 Elsevier Ltd. All rights reserved.