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

The stability of $(R,\mathrm{Zr}){(\mathrm{Fe}, \mathrm{Co}, \mathrm{Ti})}_{12}$ with a ${\mathrm{ThMn}}_{12}$ structure is investigated using first-principles calculations. We consider energetic competition with multiple phases that have the ${\mathrm{Th}}_2{\mathrm{Zn}}_{17}$ 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 ${\mathrm{ThMn}}_{12}$ phases.

Figure 1

Phase diagram on the $(\mathrm{Y},\mathrm{Z}r){(\mathrm{Fe},\mathrm{Co})}_{12}$ plane. (1) $(\mathrm{Y},\mathrm{Zr}){\mathrm{Fe}}_{12}$, $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Zr}}_2{\mathrm{Fe}}_{17}$, and unary Fe; (2) $(\mathrm{Y},\mathrm{Zr}){\mathrm{Fe}}_{12}$, ${\mathrm{Y}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (3) $(\mathrm{Y},\mathrm{Zr}){\mathrm{Fe}}_{12}$, $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Y}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (4) $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Y}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (5) ${\mathrm{Y}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (6) ${\mathrm{Y}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe and Co; (7) ${\mathrm{Y}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Co.

Figure 2

Phase diagram on the $(\mathrm{Nd},\mathrm{Zr}){(\mathrm{Fe},\mathrm{Co})}_{12}$ plane. (1) $(\mathrm{Nd},\mathrm{Zr}){\mathrm{Fe}}_{12}$, Zr(${\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Zr}}_2{\mathrm{Fe}}_{17}$, and unary Fe; (2) $(\mathrm{Nd},\mathrm{Zr}){\mathrm{Fe}}_{12}$, $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, and unary Nd and Fe; (3) $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Nd}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Nd and Fe; (4) $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Nd}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (5) $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Nd}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe; (6) ${\mathrm{Nd}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe; (7) ${\mathrm{Nd}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe and Co.

Figure 3

Phase diagram on the (Sm,Zr)(${\mathrm{Fe},\mathrm{Co})}_{12}$ plane. (1) ($\mathrm{Sm},\mathrm{Zr}){\mathrm{Fe}}_{12}$, $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Zr}}_2{\mathrm{Fe}}_{17}$, and unary Fe; (2) $(\mathrm{Sm},\mathrm{Zr}){\mathrm{Fe}}_{12}$, ${\mathrm{Sm}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Sm and Fe; (3) $(\mathrm{Sm},\mathrm{Zr}){\mathrm{Fe}}_{12}$, ${\mathrm{Sm}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (4) $(\mathrm{Sm},\mathrm{Zr}){\mathrm{Fe}}_{12}$, $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Sm}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (5) $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Sm}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, and unary Fe; (6) $\mathrm{Zr}{(\mathrm{Fe},\mathrm{Co})}_{12}$, ${\mathrm{Sm}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe; (7) ${\mathrm{Sm}}_2{(\mathrm{Fe},\mathrm{Co})}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe; (8) ${\mathrm{Sm}}_2{\mathrm{Co}}_{17}$, ${\mathrm{Zr}}_2{\mathrm{Co}}_{17}$, and unary Fe and Co.

Figure 4

Total DOS of (top) ${\mathrm{YFe}}_{12}$ and ${\mathrm{ZrFe}}_{12}$ and (bottom) ${\mathrm{Y}}_2{\mathrm{Fe}}_{17}$ and ${\mathrm{Zr}}_2{\mathrm{Fe}}_{17}$.

Figure 5

Hull distance in electronvolts for (${\mathrm{Y}}_{1-\alpha }{\mathrm{Zr}}_{\alpha }){\left({\mathrm{Fe}}_{1-\beta }{\mathrm{Co}}_{\beta }\right)}_{12-\gamma }{\mathrm{Ti}}_{\gamma }$ with the ${\mathrm{ThMn}}_{12}$ structure with (top) $\gamma =0$, (middle) $\gamma =0.5$, and (bottom) $\gamma =1$.

Figure 6

Hull distance in electronvolts for (${\mathrm{Nd}}_{1-\alpha }{\mathrm{Zr}}_{\alpha }){\left({\mathrm{Fe}}_{1-\beta }{\mathrm{Co}}_{\beta }\right)}_{12-\gamma }{\mathrm{Ti}}_{\gamma }$ with the ${\mathrm{ThMn}}_{12}$ structure with (top) $\gamma =0$, (middle) $\gamma =0.5$, and (bottom) $\gamma =1$.

Figure 7

Hull distance in electronvolts for (${\mathrm{Sm}}_{1-\alpha }{\mathrm{Zr}}_{\alpha }){\left({\mathrm{Fe}}_{1-\beta }{\mathrm{Co}}_{\beta }\right)}_{12-\gamma }{\mathrm{Ti}}_{\gamma }$ with the ${\mathrm{ThMn}}_{12}$ structure with (top) $\gamma =0$, (middle) $\gamma =0.5$, and (bottom) $\gamma =1$.

Figure 8

Phonon dispersion for ${\mathrm{ZrFe}}_{12}$.