Stabilization of Ternary Compounds via Ordered Arrays of Defect Pairs

First-principles calculations show that the defect pair $({2V}_{\mathrm{Cu}}^{-}+{\mathrm{In}}_{\mathrm{Cu}}^{+})$ in CuInS${\mathrm{e}}_2$ has an unusually low formation energy, due both to the relative ease of forming Cu vacancies $\left(V_{\mathrm{Cu}}\right)$ and to the attractive interactions between $V_{\mathrm{Cu}}^{-}$ and ${\mathrm{In}}_{\mathrm{Cu}}^{2+}$. The defect pair is predicted to be electrically inactive. This explains the surprising electrical tolerance of CuInS${\mathrm{e}}_2$ to its huge $\left(\sim 1\%\right)$ concentration of native defects. An attractive interaction among the defect pairs is further predicted to lead to a crystallographic ordering of the pairs, explaining the observed, but hitherto surprising, structures CuI${\mathrm{n}}_5$S${\mathrm{e}}_8$, CuI${\mathrm{n}}_3$S${\mathrm{e}}_5$, C${\mathrm{u}}_2$I${\mathrm{n}}_4$S${\mathrm{e}}_7$, etc.