The effect of long-range elastic interactions on the toroidal moment of polarization in a ferroelectric nanoparticle
The effect of long-range (LR) elastic interactions on the toroidal moment of polarization in a two-dimensional ferroelectric particle is investigated using a phase field model. The phase field simulations 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 LR elastic interactions, high values of spontaneous strains hinder the formation of polarization vortices in ferroelectric particles. Applied Physics Letters 88, 182904 (2006)