  We use density-functional theory with the local-density approximation to
study the structural and ferroelectric properties of SrTiO$_3$ under misfit
strains. Both the antiferrodistortive (AFD) and ferroelectric (FE)
instabilities are considered. The rotation of the oxygen octahedra and the
movement of the atoms are fully relaxed within the constraint of a fixed
in-plane lattice constant. We find a rich misfit strain-induced phase
transition sequence and is obtained only when the AFD distortion is taken into
account. We also find that compressive misfit strains induce ferroelectricity
in the tetragonal low temperature phase only whilst tensile strains induce
ferroelectricity in the orthorhombic phases only. The calculated FE
polarization for both the tetragonal and orthorhombic phases increases
monotonically with the magnitude of the strains. The AFD rotation angle of the
oxygen octahedra in the tetragonal phase increases dramatically as the misfit
strain goes from the tensile to compressive strain region whilst it decreases
slightly in the orthorhombic (FO4) phase. This reveals why the polarization in
the epitaxially strained SrTiO$_3$ would be larger when the tensile strain is
applied, since the AFD distortion is found to reduce the FE instability and
even to completely suppress it in the small strain region. Finally, our
analysis of the average polar distortion and the charge density distribution
suggests that both the Ti-O and Sr-O layers contribute significantly to the FE
polarization.
