Phase field simulations of stress-free ferroelectric nanoparticles with different long-range electrostatic interactions

Two-dimensional phase field simulations of stress-free ferroelectric nanoparticles with different long-range (LR) electrostatic interactions are conducted based on the time-dependent Ginzburg-Landau equation. Polarization patterns and the toroidal moment of polarization are found to be dependent on the LR electrostatic interaction and the size of the simulated nanoparticle. Phase field simulations exhibit vortex patterns with purely toroidal moments of polarization and negligible macroscopic polarization in the stress-free ferroelectric nanoparticles when the LR electrostatic interaction is fully taken into account. However, a single-domain structure without any toroidal moment of polarization is formed in small particles if the LR electrostatic interaction is completed ignored. The result indicates that the LR electrostatic interaction and the particle size play crucial roles in the formation of polarization vortices in the ferroelectric nanoparticles.