Unusual polarization patterns in flat epitaxial ferroelectric nanoparticles

We investigate the effects of a lattice misfit strain on a ground state and polarization patterns in flat perovskite nanoparticles (nanoislands of BaTiO$_{3}$ and PbZr$_{0.5}$Ti$_{0.5}$O$_{3}$) with the use of an \textit{ab-initio} derived effective Hamiltonian. We show that the strain strongly controls the balance between the depolarizing field and the polarization anizotropy in determining the equilibrium polarization patterns. Compressive strain favors 180 $^0$ stripe/tweed domains while a tensile strain leads to in-plane vortex formation, with the unusual intermediate phase (s) where both ordering motifs coexist [1]. The results may allow to explain contradictions in recent experimental data for ferroelectric nanoparticles. [1] Ivan Naumov and Alexander M. Bratkovsky, Phys. Rev. Lett. 101, 107601 (2008).