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

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