Surface reconstruction and ferroelectricity in PbTiO$_3$ thin films

Surface and ferroelectric properties of PbTiO$_3$ thin films are investigated using an interatomic potential approach with parameters computed from first-principles calculations. We show that a model developed for the bulk describes properly the surface properties of PbTiO$_3$. In particular, the antiferrodistortive surface reconstruction, recently observed from X-ray scattering, is correctly reproduced as a result of the change in the balance of long-range Coulombic and short-range interactions at the surface. The effects of the surface reconstruction on the ferroelectric properties of ultrathin films are investigated. Under the imposed open-circuit electrical boundary conditions, the model gives a critical thickness for ferroelectricity of 4 unit cells. The surface layer, which forms the antiferrodistortive reconstruction, participates in the ferroelectricity. A decrease in the tetragonality of the films leads to the stabilization of a phase with non-vanishing in-plane polarization. A peculiar effect of the surface reconstruction on the in-plane polarization profile is found.