Ferroelectric Phase Transitions in Small Particles and Local Regions

Phase transitions in spherical particles of a cubic ferroelectric are considered within Landau-Ginzburg-Devonshire theory. Concentrating on effects of the depolarizing field, we study competition between states with homogeneous polarization and vortex structures. For large radii of the sphere ($R>{R}_{c}$), the phase transition is into a vortex state while for $R<{R}_{c}$ it might be into an homogeneous state. ${R}_{c}$ is proportional to the square root of the dielectric constant of the environment. If this constant is of the order of unity, a transition into homogeneous state is practically impossible. The obtained results are applied to a discussion of the formation of ``polar nanoclusters'' in relaxors.