  The effect of layer-by-layer heterostructuring and epitaxial strain on
lattice instabilities and related ferroelectric properties is investigated from
first principles for the [001]-(PbTiO$_3$)$_1$(PbZrO$_3$)$_1$ superlattice and
pure PbTiO$_3$ on a cubic substrate. The results for the superlattice show an
enhancement of the stability of the monoclinic r-phase with respect to pure
PbTiO$_3$. Analysis of the lattice instabilities of the relaxed centrosymmetric
reference structure computed within density functional perturbation theory
suggests that this results from the presence of two unstable zone-center modes,
one confined in the PbTiO$_3$ layer and one in the PbZrO$_3$ layer, which
produce in-plane and normal components of the polarization, respectively. The
zero-temperature dielectric response is computed and shown to be enhanced not
only near the phase boundaries, but throughout the r-phase. Analysis of the
analogous calculation for pure PbTiO$_3$ is consistent with this
interpretation, and suggests useful approaches to engineering the dielectric
properties of artificially structured perovskite oxides.
