  We consider theoretically the influence of the flexoelectric coupling on the
spatial distribution and temperature behavior of spontaneous polarization for
several types of stable domain structure in thin ferroelectric films, such as
stripe domains and vortices.Finite element modelling (FEM) for BaTiO3 films and
analytical calculations within the Landau-Ginzburg-Devonshire approach reveals
that an out-of-plane polarization component can be very sensitive to the
flexoelectric coupling for periodic quasi-2D stripe domains and 3D
vortex-antivortex structures.However, the influence is rather different for
these structures. The flexoelectric coupling increases significantly the
amplitude of a small out-of-plane polarization component in the stripe domains,
and the "up" or "down" direction of the component is defined by the sign of the
flexoelectric coefficients.Concerning the vortex-antivortex pairs, their
antivortices with in-plane anti-circulation have smooth wide dipolar cores
through the entire film, whose shape and other features are almost insensitive
to the coupling.The vortices with in-plane vorticity have spike-like cores with
an out-of-plane quadrupolar moment induced by the flexoelectric coupling. The
cores are located near the film-dead layer interfaces.FEM results corroborated
by analytical calculations prove that a change of the flexoelectric coefficient
sign leads to a reorientation of the core axial polarization, making the
flexo-sensitive 3D vortices similar to the recently introduced "flexons" in
cylindrical nanoparticles.The relatively wide temperature range (from 200 to
400 K) of the flexo-sensitive vortices' existence gives us the hope that they
can be observed experimentally in thin ferroelectric films by scanning probe
and nonlinear optical microscopy methods.
