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We present a density functional theory investigation of strainedCa$_{0.5}$Sr$_{0.5}$TiO$_3$. We have determined the structure and polarizationfor a number of arrangements of Ca and Sr in a 2$\times$2$\times$2 supercell.The a and b lattice vectors are strained to match the lattice constants of therotated Si(001) face. To set the context for the CSTO study, we also includesimulations of the Si(001) constrained structures for CaTiO$_3$ and SrTiO$_3$.Our primary findings are that all Ca$_{0.5}$Sr$_{0.5}$TiO$_3$ structuresexamined except one are ferroelectric, exhibiting polarizations ranging from0.08 C/m$^2$ for the lowest energy configuration to about 0.26 C/m$^2$ for thehigher energy configurations. We find that the configurations with largerpolarizations have lower c/a ratios. The net polarization of the cell is theresult of Ti-O ferroelectric displacements regulated by A-site cations.

Ferroelectricity in strained Ca$_{0.5}$Sr$_{0.5}$TiO$_3$ from first principles