$(\mathrm{Sm},\mathrm{Zr}){\mathrm{Fe}}_{12-x}{M}_x$ ($M=\mathrm{Zr},\mathrm{Ti},\mathrm{Co})$ for Permanent-Magnet Applications: Ab Initio Material Design Integrated with Experimental Characterization

In rare-earth permanent magnets (REPMs), trade-offs between intrinsic magnetic properties are often encountered. A recent example is ${\mathrm{Sm}\mathrm{Fe}}_{12}$ where excellent magnetic properties can be achieved at the sacrifice of bulk structure stability. Bulk structure stability is sustained by the presence of a third substitute element as is the case with ${\mathrm{Sm}\mathrm{Fe}}_{11}\mathrm{Ti}$, where $\mathrm{Ti}$ degrades magnetic properties. It is now in high demand to find out with which chemical composition a good compromise in the trade-off between structure stability and strong ferromagnetism is reached. We inspect the effects of representative substitute elements, $\mathrm{Zr}$, $\mathrm{Ti}$, and $\mathrm{Co}$ in ${\mathrm{Sm}\mathrm{Fe}}_{12}$ by combining ab initio data with experimental data from neutron diffraction. The trend in the intrinsic properties with respect to the concentration of substitute elements are monitored and a systematic way to search the best compromise is constructed. A certain minimum amount of $\mathrm{Ti}$ is identified with respect to the added amount of $\mathrm{Co}$ and $\mathrm{Zr}$. It is found that $\mathrm{Zr}$ brings about a positive effect on magnetization, in line with recent experimental developments, and we argue that this can be understood as an effective doping of extra electrons.

Figure 1

Crystal structure of ${\mathrm{Sm}\mathrm{Fe}}_{12}$. Large and red balls represent $\mathrm{Sm}(2a)$ sites and green, cyan, blue, and small balls represent $\mathrm{Fe}(8i),\mathrm{Fe}(8j),\mathrm{and}\mathrm{Fe}(8f)$ sites, respectively. In the tetragonal box shown here, two formula units are included.

Figure 2

Calculated formation energy, $\mathrm{\Delta }E$, of ${\mathrm{Sm}\mathrm{Fe}}_{12}$ with substitute elements $\mathrm{Ti}$, $\mathrm{Co}$, or $\mathrm{Zr}$. In the tetragonal unit shown in Fig. 1 containing two formula units of ${\mathrm{Sm}\mathrm{Fe}}_{12}$, one substitute atom, either $\mathrm{Ti}$, $\mathrm{Co}$, or $\mathrm{Zr}$, replaces one host atom and ab initio structure optimization is done to extract the formation energy per formula unit (f.u.).

Figure 3

Scheme of the “$\mathrm{LDA}+\mathrm{Rietveld}$” self-consistent iterations.

Figure 4

Initial shot of the Rietveld analysis of which output makes an initial input to ab initio Korringa-Kohn-Rostoker (KKR) coherent potential approximation (CPA) in the overall “$\mathrm{LDA}+\mathrm{Rietveld}$” iteration.

Figure 5

Calculated formation energy of ${\mathrm{Sm}\mathrm{Fe}}_{12}$ per f.u. as a function of the concentration of substitute elements. In these calculations, the substitute elements are selectively put into the most energetically favorable sublattice, unless otherwise stated.

Figure 6

Utility of the target material ${\mathrm{Sm}}_{1-n}{\mathrm{Zr}}_n{\mathrm{Fe}}_{12-m-l-l^{\prime }}{\mathrm{Co}}_m{\mathrm{Ti}}_l{\mathrm{Zr}}_{l^{\prime }}$ as a function of $l$, with $n=0.2$ and $l^{\prime }=0$ fixed.

Figure 7

Compromise between structure stability and magnetization inspected with the partial utility functions of the target material ${\mathrm{Sm}}_{1-n}{\mathrm{Zr}}_n{\mathrm{Fe}}_{12-m-l-l^{\prime }}{\mathrm{Co}}_m{\mathrm{Ti}}_l{\mathrm{Zr}}_{l^{\prime }}$ as a function of $l$, with $n=0.2$ and $l^{\prime }=0$ fixed.

Figure 8

Compromise between structure stability, magnetization, Curie temperature, and room-temperature anisotropy field of which leading-order trend may be well captured by $\mathrm{Sm}$-$\mathrm{Fe}$ exchange couplings. Overall product of the partial utility functions for the target material ${\mathrm{Sm}}_{1-n}{\mathrm{Zr}}_n{\mathrm{Fe}}_{12-m-l-l^{\prime }}{\mathrm{Co}}_m{\mathrm{Ti}}_l{\mathrm{Zr}}_{l^{\prime }}$ as a function of $l$, with $n=0.2$ and $l^{\prime }=0$ fixed.

Figure 9

Calculated magnetization of doped ${\mathrm{Sm}\mathrm{Fe}}_{12}$ per f.u. on a fixed lattice of ${\mathrm{Sm}\mathrm{Fe}}_{12}$ [22]. It is seen that one extra electron from $\mathrm{Zr}(2a)$ or $\mathrm{Ce}(2a)$, with a proper rescaling of the concentration on the horizontal axis, works like one more electron in the minority-spin band from $\mathrm{Co}$ and an analogue of the Slater-Pauling curve can be realized in the doped $2a$ site.

Figure 10

Calculated formation energy of $\mathrm{Co}$-substituted ${\mathrm{Sm}\mathrm{Fe}}_{12}$, ${\mathrm{Sm}\mathrm{Fe}}_{11.5}{\mathrm{Ti}}_{0.5}$, and ${\mathrm{Sm}\mathrm{Fe}}_{11}\mathrm{Ti}$ per f.u. The reference systems are the ingredient elements.

Figure 11

Calculated formation energy of $M$-substituted ${\mathrm{Sm}\mathrm{Fe}}_{12}$ ($M=\mathrm{Ti}$, $\mathrm{Co}$, and $\mathrm{Zr}$) per f.u. The reference system is taken to be ${\mathrm{Sm}}_2{\mathrm{Fe}}_{17}$ in the same spirit as is written in Eq. (A2). The same data as those shown in Fig. 2 in the main text are plotted with the reference systems set to be ${\mathrm{Sm}}_2{\mathrm{Fe}}_{17}$.

Figure 12

Interpolated formation energy of ${\mathrm{SmT}}_{12}$ per f.u. with the reference systems being ${\mathrm{Sm}}_2{\mathrm{Fe}}_{17}$ and ${\mathrm{Sm}}_2{\mathrm{Co}}_{17}$ following Eqs. (A2), (A3), (A4), and (A5).

Figure 13

Deriving the derivative matrix of the intrinsic properties for the target alloy, ${\mathrm{Sm}}_{1-n}{\mathrm{Zr}}_n({\mathrm{Fe}}_{12-m-l-l^{\prime }}{\mathrm{Co}}_m{\mathrm{Ti}}_l{\mathrm{Zr}}_{l^{\prime }}$) around the pristine limit, ${\mathrm{Sm}\mathrm{Fe}}_{12}$, from the calculated data via KKR CPA using AkaiKKR. Dependence of magnetization $M$ per f.u., Curie temperature $T_{\mathrm{Curie}}$, and Sm-Fe indirect exchange couplings $J_{\mathrm{RT}}$ for $\mathrm{Fe}(8i)$, $\mathrm{Fe}(8j)$, $\mathrm{Fe}(8f)$ that are crucial for anisotropy field in the operation temperature range, are plotted with respect to the concentration of substitute elements, Zr, Co, and Ti. For Zr two possible substitution sublattices, $\mathrm{Sm}(2a)$ and $\mathrm{Fe}(8i)$, are explored.

Figure 14

Calculated density of states (DOS) projected onto Fe atoms for the pristine ${\mathrm{Sm}\mathrm{Fe}}_{12}$ on an optimized lattice [22]. The horizontal axis shows the energy $E$ as measured from the Fermi level, $E_F$. The arrow in the plot points to the extra density of states in the majority spin state projected onto $\mathrm{Fe}(8f)$. Additional Co reduces this region leading to the Slater-Pauling curve.

Figure 15

(a) Calculated density of states projected onto $\mathrm{Fe}(8f)$ atoms for ${\mathrm{Sm}\mathrm{Fe}}_{12}$, ${\mathrm{Zr}\mathrm{Fe}}_{12}$, and ${\mathrm{Ce}\mathrm{Fe}}_{12}$ on the same fixed lattice. The full-line arrow in the plot points to the majority-spin density of states on the Fermi level, which is seen significantly mostly for ${\mathrm{Sm}\mathrm{Fe}}_{12}$. Dotted arrow points to the relatively sparse contribution from the majority spin of $\mathrm{Fe}(8f)$ in ${\mathrm{Ce}\mathrm{Fe}}_{12}$ and ${\mathrm{Zr}\mathrm{Fe}}_{12}$, which means that magnetic moments on $\mathrm{Fe}(8f)$ are close to saturation for those materials. (b),(c) Analogous data to (a) for $\mathrm{Fe}(8i)$ and $\mathrm{Fe}(8j)$, respectively. The axis labels follow those in Fig. 14.