  Using the self-consistent Landau-Ginzburg-Devonshire approach we simulate and
analyze the spontaneous formation of the domain structure in thin ferroelectric
films covered with the surface screening charge of the specific nature
(Bardeen-type surface states). Hence we consider the competition between the
screening and the domain formation as alternative ways to reduce the
electrostatic energy and reveal unusual peculiarities of distributions of
polarization, electric and elastic fields conditioned by the surface screening
length and the flexocoupling strength. We have established that the critical
thickness of the film and its transition temperature to a paraelectric phase
strongly depend on the Bardeen screening length, while the flexocoupling
affects the polarization rotation and closure domain structure and induces
ribbon-like nano-scale domains in the film depth far from the top open surface.
Hence the joint action of the surface screening (originating from e.g. the
adsorption of ambient ions or surface states) and flexocoupling may remarkably
modify polar and electromechanical properties of thin ferroelectric films.
