Electronic band structure trends of perovskite halides: Beyond Pb and Sn to Ge and Si

The trends in electronic band structure are studied in the cubic $AB{X}_{3}$ halide perovskites for $A=\text{Cs}$; $B=\text{Pb}$, Sn, Ge, Si; and $X=\text{I}$, Br, Cl. The gaps are found to decrease from Pb to Sn and from Ge to Si, but increase from Sn to Ge. The trend is explained in terms of the atom $s$ levels of the group-IV element and the atomic sizes which changes the amount of hybridization with $X\text{\ensuremath{-}}p$ and hence the valence bandwidth. Along the same series spin-orbit coupling also decreases and this tends to increase the gap because of the smaller splitting of the conduction band minimum. Both effects compensate each other to a certain degree. The trend with halogens is to reduce the gap from Cl to I, i.e., with decreasing electronegativity. The role of the tolerance factor in avoiding octahedron rotations and octahedron edge sharing is discussed. The Ge containing compounds have tolerance factor $t>1$ and hence do not show the series of octahedral rotation distortions and the existence of edge-sharing octahedral phases known for Pb and Sn-based compounds, but rather a rhombohedral distortion. ${\mathrm{CsGeI}}_{3}$ is found to have a suitable gap for photovoltaics both in its cubic (high-temperature) and rhombohedral (low-temperature) phases. The structural stability of the materials in the different phases is also discussed. We find the rhombohedral phase to have lower total energy and slightly larger gaps but to present a less significant distortion of the band structure than the edge-sharing octahedral phases, such as the yellow phase in ${\mathrm{CsSnI}}_{3}$. The corresponding silicon based compounds have not yet been synthesized and therefore our estimates are less certain but indicate a small gap for cubic ${\mathrm{CsSiI}}_{3}$ and ${\mathrm{CsSiBr}}_{3}$ of about $0.2\ifmmode\pm\else\textpm\fi{}0.2$ eV and $0.8\ifmmode\pm\else\textpm\fi{}0.6$ eV for ${\mathrm{CsSiCl}}_{3}$. The intrinsic stability of the Si compounds is discussed.