Magnetic anisotropy of $R_2{\text{Fe}}_{14}\text{B}$ ($R=\text{Nd}$, Gd, Y): Density functional calculation by using the linear combination of pseudo-atomic-orbital method

The magnetocrystalline anisotropy energy (MAE) of rare-earth transition-metal intermetallics $R_2{\text{Fe}}_{14}\text{B}$ ($R=\text{Nd}$, Gd, Y) was calculated by using the linear combination of pseudo-atomic-orbital method. Electronic structures were calculated by using relativistic density functional theory with Hubbard-type on-site Coulomb potential $U$ for $f$ electrons in the $R$ sites. The calculated magnetic moments of ${\text{Y}}_2{\text{Fe}}_{14}\text{B}$ and ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$ are in agreement with the experimentally measured saturation moments of these compounds. The calculated MAEs of ${\text{Y}}_2{\text{Fe}}_{14}\text{B}$ and ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$ are also in agreement with the value measured at low temperature. Moreover, it was found that the shape of the charge density at the Nd sites is aspherical, which causes larger MAE of ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$ than that of ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$. However, the orbital moment of the $4f$ electrons at the Nd sites and the MAE of ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$ were underestimated within the scope of a relativistic local-density approximation plus $U$ with spherical average potential.

Figure 1

Spin-resolved density of states of ${\text{Y}}_2{\text{Fe}}_{14}\text{B}$, ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$, and ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$. The directions of the spin moments are parallel to $c$ axis. The majority (minority) spin states are above (below) the zero axis.

Figure 2

Spin-resolved density of states on the Fe sites of ${\text{Y}}_2{\text{Fe}}_{14}\text{B}$, ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$, and ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$. The directions of the spin moments are parallel to $c$ axis. The majority- (minority-) spin states are above (below) the zero axis.

Figure 3

Spin-resolved density of states on the rare-earth sites of ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$, and ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$. The directions of the spin moments are parallel to $c$ axis. The majority- (minority-) spin states are above (below) the zero axis.

Figure 4

Convergence of MAE for ${\text{Y}}_2{\text{Fe}}_{14}\text{B}$ and ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$.

Figure 5

Local magnetic moments of Fe sites in ${\text{Y}}_2{\text{Fe}}_{14}\text{B}$, ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$, and ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$. The ferri indicates the moment in ferrimagnetic configuration.

Figure 6

(Color online) Valence-electron-density map of ${\text{Gd}}_2{\text{Fe}}_{14}\text{B}$ (a) map in (001) plane and (b) map in (110) plane. The unit of density is $e/{\text{Å}}^3$.

Figure 7

(Color online) Valence-electron-density map of ${\text{Nd}}_2{\text{Fe}}_{14}\text{B}$ (a) map in (001) plane and (b) map in (110) plane. The unit of density is $e/{\text{Å}}^3$.