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Strain-induced ferroelectricity in CaTiO$_3$ from first principles

First principles calculations are used to investigate the effects of epitaxial strain on the structure of the perovskite oxide CaTiO$_3$, with particular focus on the stabilization of a ferroelectric phase related to a polar instability hidden in the orthorhombic equilibrium bulk $Pbnm$ structure but found in previous first-principles studies of the ideal cubic perovskite high-symmetry reference structure. At 1.5% strain, we find an epitaxial orientation transition between the $ab$-$ePbnm$ phase, favored for compressive strains, and the $c$-$ePbnm$ phase. For larger tensile strains, a polar instability develops in the $c$-$ePbnm$ phase and an epitaxial-strain-induced ferroelectric phase is obtained with polarization along a $<$110$>$ direction with respect to the primitive perovskite lattice vectors of the square substrate.