Origins of Nonstoichiometry and Vacancy Ordering in ${\mathrm{Sc}}_{1-\mathit{x}}{\square }_{\mathit{x}}S$

Whereas nearly all compounds $A_n{B}_m$ obey Dalton’s rule of integer stoichiometry ( $n$: $m$, both integer), there is a class of systems, exemplified by the rocksalt structure ${\mathrm{Sc}}_{1-x}{\square }_x\mathrm{S}$, that exhibits large deviations from stoichiometry via vacancies, even at low temperatures. By combining first-principles total energy calculations with lattice statistical mechanics, we scan an astronomical number of possible structures, identifying the stable ground states. Surprisingly, all have the same motifs: (111) planes with (112) vacancy rows arranged in (110) columns. Electronic structure calculations of the ground states (identified out of $\sim 3\times {10}^6$ structures) reveal the remarkable origins of nonstoichiometry.