Energy-polarization behaviors of AA$'$BB$'$O$_6$ perovskites with double rock-salt order

Using first-principles methods, we study the energy-polarization relation of double perovskites AA$'$BB$'$O$_6$ where atoms in both A and B sites are arranged in rock-salt order. The high-symmetry structure in this case is the {\it tetrahedral} $F\bar{4}3m$ space group. If a ferroelectric instability occurs, the energy-vs.-polarization landscape $E({\bf P})$ will tend to have minima for {$\bf P$} along tetrahedral directions leading to a rhombohedral space group $R3m$, with two different values of spontaneous polarization and associated energy along opposite body-diagonal directions; or along Cartesian directions, leading to orthorhombic space group $Imm2$. We search for polar {\it soft modes} at the $\Gamma$ point of the high-symmetry $F\bar{4}3m$ structure and analyze its eigenvectors to identify ferroelectric instabilities, which we find in CaBaTiZrO$_6$, KCaZrNbO$_6$ and PbSnTiZrO$_6$. The results of the first-principle calculations are modeled with a Landau-Devonshire expansion that is truncated at either 4th or 5th order in $\bf P$, and its predictions are found to agree favorably with our calculation. The 5th-order calculation improves the agreement further except in PSTZ. Recently, synthesis of SrCaTiMnO$_6$ in rock-salt order has been reported.\footnote{J.L Blok, G. Rijnders and D.H.A. Blank, private communication.} Unfortunately, preliminary results do not seem to indicate any polarized structure.