Stability of Vortex Phases in Ferroelectric Easy-Plane nano-Cylinders

Bounded charges induced by the polarization gradient in finite-size ferroelectrics are known to produce the unfavorable depolarization electric field that suppresses the uniform ferroelectric state. To reduce the depolarization energy the non-uniform vortex (toroidal) state is formed inside ferroelectric nano-particles, nano-disks, and nano-rods. Based on self-consistent solution of Ginzburg-Landau equations coupled with electrostatic equations, we study the multivortex toroidal states appearing in the nanometric easy-plane ferroelectric cylinder. The geometrical textures, critical temperatures and stability regions for these states are calculated.