Phase field simulations of ferroelectric materials

Ferroelectric materials have become preferred materials in a wide variety of electronic and mechatronic devices due to their pronounced dielectric, piezoelectric, and pyroelectric properties. The material macroscopic properties are related to the microscopic domain structure of the materials. To understand and predict the relation between the macroscopic properties and the domain structure, a continuum phase field model incorporating with the long-range elastic and electrostatic interactions is employed to simulate the ferroelectric/paraelectric phase transition, polarization switching under external loading, influence of applied strains on ferroelectric and dielectric properties, size effect of ferroelectric epitaxial islands and thin films, effect of long-range elastic interactions on toroidal moment in ferroelectric nanoparticles, and fracture toughness variation induced by domain switching near an electrically permeable crack tip in ferroelectrics.;Phase field simulations were conducted to understand polarization switching under external electric or/and mechanical loading. The temporal evolution of the polarization switching shows that the switching is a process of nucleation, if needed, and growth of energy-favorite domains. The simulation results successfully reveal the hysteresis loop of macroscopic polarization versus the applied electric field, the butterfly curve of macroscopic strain versus the applied electric field, and the macroscopically nonlinear strain response to applied compressive stresses. The simulation results show that the pyroelectric/ferroelectric phase transition temperature linearly increases with the applied mechanical strain under mechanical clamping conditions. Analogous to the classical Ehrenfest equation, a thermodynamics equation was derived to describe the relationship between the transition temperature and the applied strain. The change in the domain structure with temperature under applied inequiaxial strains is different from that under applied equiaxial strains. The simulations also illustrate the changes in the coercive field, the remanent polarization and the nonlinear dielectric constant with the applied strain.;Phase field simulations were also conducted in real space by using different numerical methods, such as finite element analysis and finite difference method to study size effects of epitaxial ferroelectric islands, thin films, and free-standing ferroelectric nanoparticle. The simulations exhibit spatial polarization distributions with different types of domain walls in the epitaxial ferroelectric islands and find two critical thicknesses, at which the simulated material changes from a multi-domain state to a single-domain state and from ferroelectric phase to paraelectric phase, respectively. The two critical thicknesses and the domain wall types vary with the length-to-thickness ratio. The remanent polarization and the coercive field of the simulated ferroelectric films both decrease with decreasing film thickness. The phase field simulations on free-standing ferroelectric nanoparticles exhibit vortex patterns with purely toroidal moments of polarization and negligible macroscopic polarization when the spontaneous strains are low and the simulated ferroelectric size is small. However, a monodomain structure with a zero toroidal moment of polarization is formed when the spontaneous strains are high in small simulated ferroelectrics, indicating that, because of the long-range elastic interactions, high values of spontaneous strains hinder the formation of polarization vortices in ferroelectric particles.;Phase field simulations were conducted to understand domain-switching-induced shielding or anti-shielding of an electrically permeable crack in a mono-domain ferroelectric material. Phase field simulations give not only the switching zone but also the explicit polarization distribution without any prior assumed switching criterion. The simulation results show that the switching-induced internal stresses shield the crack tip from the applied mechanical loads, resulting in switching-toughening. Although a mechanical load plays a predominant role in the domain switching near the electrically permeable crack, an applied electric field plays also an important role there. When the electric field is parallel to the original polarization direction, it reduces the shielding, whereas an anti-parallel electric field enhances the shielding. The simulated phenomenon is consistent with other theoretical predictions and some experimental observations. Moreover, the phase field simulations exhibit new switching features, such as head-to-tail arrangements of polarizations and appearance of domain walls.