Existence conditions for single-vertex structure of polarization in ferroelectric nanoparticles

The distribution of polarization in low-dimensional ferroelectric nanostructures appears to form vertex structures under open-circuit boundary conditions. A thermodynamics-based continuum phase field approach is employed to investigate the conditions for existence of single-vertex structure in two-dimensional (2D) square-shaped nanodots. It has been observed that, for a given material, such structure exists only in a certain range in dimension, which is associated to an exchange energy parameter in the model. It has been further noticed that, instead of a double-vertex structure, a single-vertex distribution remains in a 2D rectangular-shaped nanodot with initial conditions of polarization in favor of single-toroid shape.