Competition and cooperation between antiferrodistortive and ferroelectric instabilities in SrTiO$_3$

We use density functional theory to explore the interplay between octahedral rotations and ferroelectricity in the model compound SrTiO$_3$. We find that over the experimentally relevant range octahedral rotations suppress ferroelectricity as is generally assumed in the literature. Somewhat surprisingly we observe that at larger angles the previously weakened ferroelectric instability strengthens significantly. By analysing geometry changes, energetics, force constants and charges, we explain the mechanisms behind this transition from competition to cooperation with increasing octahedral rotation angle.