First-principles study on the stability of ($R$, Zr)(Fe, Co, Ti)$_{12}$ against 2-17 and unary phases ($R$ = Y, Nd, Sm)

The stability of ($R$, Zr)(Fe, Co, Ti)$_{12}$ with a ThMn$_{12}$ structure is investigated using first-principles calculations. We consider energetic competition with multiple phases that have the Th$_2$Zn$_{17}$ structure and the unary phases of $R$, Zr, Fe, Co, and Ti simultaneously by constructing a quinary energy convex hull. From the analysis, we list the stable phases at zero temperature, and show possible stable and metastable ThMn$_{12}$ phases.