Inverted band gap trend through octahedral ordering in Cs$_2$Au$_2$X$_6$ (X=Cl, Br, I)

Double perovskites Cs$_2$Au$_2$X$_6$ (X=Cl, Br, I) are prototypical materials that exhibit charge disproportionation of gold into 1+ and 3+ states. It is known that the disproportionation is resolved under high pressures, and this has stimulated many studies into the pressurization of these materials. At present, the phase changes in these materials are still strongly contested. Here, we use density functional theory to study the pressure-dependent behavior of Cs$_2$Au$_2$X$_6$. We find that a tetragonal--cubic transition occurs directly from the ground state $I4/mmm$ structure. Even so, we also found an intermediate tetragonal $P4/mmm$ structure to be very close in energy, suggesting it to be observable. We also find several other competing metastable phases, which explains some of the controversies in the literature. Focusing on one of the metastable phases, we suggest that Cs$_2$Au$_2$X$_6$ can be prepared in a $P4_2/mnm$ structure, analogous to that of KCuF$_3$. The band gap in the $P4_2/mnm$ structure widened as atomic number of the halide was increased, which is the inverse trend compared to the ground state structure. We explain this by the different octahedral distortion ordering in the two structural phases. Furthermore, we show that the conduction band in $P4_2/mnm$ is three dimensionally connected, which is favorable for opto-electronic applications. We submit that this work demonstrates that octahedral distortion ordering is a promising avenue for developing new double perovskites and suggests it to be particular effective in tuning the electronic structure properties.