Phase field simulations of low-dimensional ferroelectrics

This paper reviews the previous work on phase field simulations of low-dimensional ferroelectrics of two-dimensional epitaxial ferroelectric islands, thin films, and nanoparticles. The simulations are conducted in real space with exact boundary conditions of a low-dimensional ferroelectric, but consuming a much longer simulation time in comparison with that conducted in Fourier space. For ferroelectric islands and thin films, the simulations exhibit spatial polarization distributions with different types of domain walls and find two critical thicknesses, at which the simulated material changes from a multidomain state to a single-domain state and from ferroelectric phase to paraelectric phase, respectively. The remanent polarization and the coercive field of the simulated ferroelectric films both decrease with decreasing film thickness. The simulations exhibit vortex patterns of polarizations, which have purely toroidal moments of polarizations and macroscopically negligible averaged polarizations, in stress-free nanoparticles when long-range electrostatic interactions are fully taken into account. However, a single-domain structure without any toroidal moment of polarizations is formed in small nanoparticles with strong long-range elastic interactions.