  Interface effects on the ferroelectric behavior of PbTiO$_3$ ultrathin films
deposited on SrTiO$_3$ substrate are investigated using an interatomic
potential approach with parameters fitted to first-principles calculations. We
find that the correlation of atomic displacements across the film/substrate
interface is crucial for the stabilization of the ferroelectric state in films
a few unit-cells thick. We show that the minimum film thickness for the
appearance of a spontaneous polarized domain state is not an intrinsic property
of the ferroelectric film but depends on the polarizability of the paraelectric
substrate. We also observe that the substrate displays an induced polarization
with an unusual oscillatory behavior.
