  We present a phenomenological model of periodic ferroelectric-superconductor
(FE-S) heterostructures containing two alternating ferroelectric and
superconducting layers. The interaction at the FE-S contacts is described as a
coupling of the local carrier density of the superconductor with the
spontaneous ferroelectric polarization near the FE-S interface. We obtain a
stable symmetric domain-type phase exhibiting a contact-induced polarization
and the ferroelectric domain structure at temperatures above the bulk
ferroelectric transition temperature. With an increasing coupling energy, we
find the appearance of the ferroelectric phase coexisting with the suppressed
superconductivity in the S-film. The system is analyzed for different
thicknesses of the FE- and S-films demonstrating the dramatic change of the
topology of the phase diagrams with a variation of the layers thickness. The
results are expected to shed light on the processes occurring in
high-temperature superconducting films grown on perovskite alloy-substrates
exhibiting ferroelectric properties at lower temperatures.
