Intrinsic magnetic properties in $R{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_{11}\mathrm{Ti}Z(R=\mathrm{Y}\text{and Ce};Z=\mathrm{H},\text{C},\text{and N})$

To guide improved properties coincident with reduction of critical materials in permanent magnets, we investigate via density functional theory (DFT) the intrinsic magnetic properties of a promising system, $R{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_{11}\mathrm{Ti}Z$ with $R=\text{Y}$, Ce and interstitial doping $(Z=\text{H},\text{C},\text{N})$. The magnetization $M$, Curie temperature $T_{\text{C}}$, and magnetocrystalline anisotropy energy $K$ calculated in local density approximation to DFT agree well with measurements. Site-resolved contributions to $K$ reveal that all three Fe sublattices promote uniaxial anisotropy in ${\mathrm{YFe}}_{11}\mathrm{Ti}$, while competing anisotropy contributions exist in ${\mathrm{YCo}}_{11}\mathrm{Ti}$. As observed in experiments on $R{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_{11}\mathrm{Ti}$, we find a complex nonmonotonic dependence of $K$ on Co content and show that anisotropy variations are a collective effect of MAE contributions from all sites and cannot be solely explained by preferential site occupancy. With interstitial doping, calculated $T_{\text{C}}$ enhancements are in the sequence of $\mathrm{N}>\mathrm{C}>\mathrm{H}$, with volume and chemical effects contributing to the enhancement. The uniaxial anisotropy of $R{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_{11}\mathrm{Ti}Z$ generally decreases with C and N; although, for $R=\text{Ce}$, C doping is found to greatly enhance it for a small range of $0.7<x<0.9$.

Figure 1

Schematic representation of the crystal structures of $R{\mathrm{Fe}}_{11}\mathrm{Ti}$. $R\left(2a\right)$ atoms are indicated with larger spheres in yellow color. Three transition metal sublattices $8i,8j$, and $8f$ are in red, blue, and green, respectively. Each $R$ atom has four nearest $8i$, eight nearest $8j$, and $8f$ neighbor atoms. Among three $TM$ sites, the $8i$ site has the shortest distance from the $R$ atom. Interstitial sites $2b$ (not shown) are halfway between two $R$ atoms along the $c$ axis and coordinated by an octahedron of two $R$ and four $\text{Fe}\left(8j\right)$ sites.

Figure 2

Atom- and spin-projected partial densities of states (DOS) in (a) ${\mathrm{YFe}}_{11}\mathrm{Ti}$, (b) ${\mathrm{YFe}}_{11}\mathrm{TiH}$, (c) ${\mathrm{YFe}}_{11}\mathrm{TiC}$, and (d) ${\mathrm{YFe}}_{11}\mathrm{TiN}$ within the LDA and no SOC. For ${\mathrm{YFe}}_{11}\mathrm{Ti}$, the Fe DOS are further resolved by averaging states projected on $8i,8j$, and $8f$ sites. Majority spin (positive values) and minority spin (negative values) DOS are shown separately. Fermi energy $E_F$ is at 0 eV.

Figure 3

Comparison of measured and calculated (squares) $M$ versus Co content in ${\mathrm{Y}(\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_{11}\mathrm{Ti}$. Experimental data are from Wang et al. [19] at 5 K (circles) and Yang et al. [8] at 1.5 K (triangles).

Figure 4

Total and sublattice-resolved $K_{\text{so}}$ in ${\mathrm{YFe}}_{11}\mathrm{Ti},{\mathrm{YFe}}_7{\mathrm{Co}}_4\mathrm{Ti},{\mathrm{YFe}}_3{\mathrm{Co}}_8\mathrm{Ti}$, and ${\mathrm{YCo}}_{11}\mathrm{Ti}$. Calculated $K$ and measured $K_{\text{expt}}$ values are also compared. Experimental values were from Refs. [29] and [18], measured at 4.2 K for ${\mathrm{YFe}}_{11}\mathrm{Ti}$ and 293 K for ${\mathrm{YCo}}_{11}\mathrm{Ti}$, respectively. In calculations, we assume that all four Co occupy the $8j$ sites in ${\mathrm{YFe}}_7{\mathrm{Co}}_4\mathrm{Ti}$ while all eight Co occupy the $8j$ and $8f$ sites in ${\mathrm{YFe}}_3{\mathrm{Co}}_8\mathrm{Ti}$.

Figure 5

$K$ and sublattice-resolved $K_{\text{so}}$ in Y-based (top) and Ce-based (bottom) $R\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right){}_{11}\mathrm{Ti}$.

Figure 6

$K$ versus Co content in $R\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right){}_{11}\mathrm{Ti}Z$ with $R=\text{Y}$ (top) and $R=\text{Ce}$ (bottom), with and without $Z=\text{H}$, C, and N.