Improvement of magnetic hardness at finite temperatures: Ab initio disordered local-moment approach for ${\mathrm{YCo}}_5$

Temperature dependence of the magnetocrystalline anisotropy energy and magnetization of the prototypical rare-earth magnet ${\mathrm{YCo}}_5$ is calculated from first principles, utilizing the relativistic disordered local-moment approach. We discuss a strategy to enhance the finite-temperature anisotropy field by hole doping, paving the way for an improvement of the coercivity near room temperature or higher.

Figure 1

Calculated temperature dependence of the anisotropy field $H_{\mathrm{A}}$ for pure and hole-doped ${\mathrm{YCo}}_5$, where doping-induced enhancement of $H_{\mathrm{A}}$ is observed in the temperature range $T\gtrsim 450$–550 K with the number of doped holes being $N_h=0.24$ and $0.36$, respectively.

Figure 2

Calculated magnetic anisotropy energy plotted as the function of the valence electron number for ${\mathrm{YCo}}_5$ near the ground state $T=0$ K. The numbers on the left-hand side of the vertical axis indicate the results per the formula unit of ${\mathrm{YCo}}_5$ and those on the right-hand side of the vertical axis give the same quantities converted to per the unit volume using the experimental lattice constants in Ref. [26]. The previous theoretical result for MAE at $T=0$ is taken from Ref. [27]. The number of valence electrons per formula unit of ${\mathrm{YCo}}_5$ is 54 ($\text{a},K_{\mathrm{u}1}=7.38\mathrm{MJ}/{\mathrm{m}}^3$ at $T=4.2$ K from Ref. [23]; $\text{b},K_{\mathrm{u}1}=6.3\mathrm{MJ}/{\mathrm{m}}^3$ at $T=77$ K from Ref. [24]; $\text{c},K=6.03\mathrm{MJ}/{\mathrm{m}}^3$ from Ref. [25] at $T=293$ K. We assume $K_{\mathrm{u}1}$ dominates in $K$ as was shown in Ref. [23], and $\text{d},K_{\mathrm{u}1}=5.5\mathrm{MJ}/{\mathrm{m}}^3$ at room temperature from Ref. [8]).

Figure 3

Crystal structure of ${\mathrm{YCo}}_5$: (a) bird's-eye view and (b) top view seen along the $c$ axis. Nearest-neighbor pairs in the $\mathrm{Co}\left(2c\right)\text{−}\mathrm{Co}\left(3g\right)$ network are only partially drawn for the illustration. Labels for atoms in (b) follow Table 1.

Figure 4

Calculated temperature dependence of the total (including both spin and orbital) local moment for each magnetic sublattice in ${\mathrm{YCo}}_5$.

Figure 5

Calculated magnetization of ${\mathrm{YCo}}_5$ plotted as a function of the number of electrons near the ground state at $m=0.967$. Our calculated result $8.03{\mu }_{\mathrm{B}}$ at around $T=100$ K falls onto the experimental numbers within two digits. For the closeness of such data to the ground state, see Fig. 7 ($\text{a},M=8.33{\mu }_{\mathrm{B}}$ at liquid-helium temperature from Ref. [23] and $\text{b,} M=7.99{\mu }_{\mathrm{B}}$ at room temperature from Ref. [26]).

Figure 6

Calculated temperature dependence of magnetic anisotropy energy for ${\mathrm{YCo}}_5$. Previous calculation result at $T=0$ is taken from Ref. [27] (a, Ref. [24] for $K_{\mathrm{u}1}$; b, Ref. [23] for $K_{\mathrm{u}1}$; and c, Ref. [25] for $K$, with $K_{\mathrm{u}1}$ dominating).

Figure 7

Calculated temperature dependence of magnetization for ${\mathrm{YCo}}_5$. Experimental Curie temperature, 920 K, is taken from Ref. [26] (a, Ref. [23] and b, Ref. [26]).

Figure 8

Temperature dependence of the anisotropy field for ${\mathrm{YCo}}_5$ as deduced from the numbers presented in Figs. 6 and 7, following Eq. (16). The experimental data at 293 K is taken from Ref. [25].

Figure 9

Calculated temperature dependence of local-moment-resolved MAE.

Figure 10

Calculated MAE plotted as a function of the order parameter $m\equiv M\left(T\right)/M\left(0\right)$ for each local moment $i$. The inset is a zoomed-in plot in region $M\left(T\right)/M\left(0\right)\lesssim 1$.

Figure 11

Scaling of the bulk MAE, $K_{\mathrm{u}1,\mathrm{bulk}}\left(T\right)$, with respect to the order parameter $m=M\left(T\right)/M\left(0\right)$. The scaling exponent $\alpha$ in $K_{\mathrm{u}1,\mathrm{bulk}}\left(T\right)=A{m}^{\alpha }$ depends on the temperature-dependent scaling region: (i) $\alpha =3.5$ for $m>0.975$ which corresponds to $41\mathrm{K}<T<75\mathrm{K}$, (ii) $\alpha =5.6$ for $0.83<m<0.95$ which corresponds to $186\mathrm{K}<T<477\mathrm{K}$, and (iii) $\alpha =6.5$ for $0.75<m<0.86$ which corresponds to $428\mathrm{K}<T<610\mathrm{K}$.

Figure 12

Calculated temperature dependence of MAE for the hole-doped ${\mathrm{YCo}}_5$.

Figure 13

Calculated Fermi-level dependence of the number of electrons for pure ${\mathrm{YCo}}_5$ near the ground state at $m=0.967$ [$m=L\left(\beta h\right)$ with $\beta h=30$ as in Eq. (8)]. Equation of state is solved with the artificially given Fermi level to adjust the valence electron number exactly to the number as set up by the calculation.