  We investigate two-dimensional electric dipole sheets in the superlattice
made of BaTiO$_{3}$ and BaZrO$_{3}$ using first-principles-based Monte-Carlo
simulations and density functional calculations. Electric dipole domains and
complex patterns are observed and the complex dipole structures with various
symmetries (e.g. Pma2, Cmcm and Pmc2_{1}) are further confirmed by density
functional calculations, which are found to be almost degenerate in energy with
the ferroelectric ground state of the Amm2 symmetry, therefore strongly
resembling magnetic sheets. More complex dipole patterns, including vortices
and anti-vortices, are also observed, which may constitute the intermediate
states that overcome the high energy barrier of different polarization
orientations previously predicted by Lebedev\onlinecite{Lebedev2013}. We also
show that such system possesses large electrostrictive effects that may be
technologically important.

Materials Science