Spin-wave dispersion and exchange stiffness in ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ and $R{\mathrm{Fe}}_{11}\mathrm{Ti}$ ($R=\mathrm{Y},\mathrm{Nd},\mathrm{Sm}$) from first-principles calculations

We theoretically investigate spin-wave dispersion in rare-earth magnet compounds by using first-principles calculations and a method we call the reciprocal-space algorithm. The value of the calculated exchange stiffness for ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ is within the range of reported experimental values. We find that the exchange stiffness is considerably anisotropic when only short-range exchange couplings are considered, whereas inclusion of long-range couplings weakens the anisotropy. In contrast, $R{\mathrm{Fe}}_{11}\mathrm{Ti}$ ($R=\mathrm{Y},\mathrm{Nd},\mathrm{Sm}$) shows large anisotropy in the exchange stiffness.

Figure 1

Spin-wave dispersion for ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$.

Figure 2

Anisotropy in the exchange stiffness for ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ calculated by Eq. (8). The horizontal axis shows the spatial cutoff for $J_{ij}$. The horizontal dotted line shows the anisotropy ratio obtained by the RSA calculation.

Figure 3

Angular-integrated spin-wave dispersion for $R{\mathrm{Fe}}_{11}\mathrm{Ti}$ and ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$. These color maps show the density of states $I(q,E)$ of Eq. (B13) where the horizontal axis denotes the absolute value of a wave vector, $q$, and the vertical axis is the energy, $E$.

Figure 4

Spin-wave dispersion in (top) bcc-Fe and (bottom) hcp Co. The solid lines are from the calculations with the largest number of $k$ points ($nk$): $18\times 18\times 18$ for Fe; $15\times 15\times 15$ for Co. The dashed lines and the dash-dotted lines are for smaller $nk$.

Figure 5

Convergence of the spin-wave stiffness with respect to the number of $k$ points ($nk$) where $nk$ for SCF is the same as the spin-wave calculation (filled circle), and where $nk$ is fixed to 1000 (open circle). Note that the horizontal axis is in log scale.

Figure 6

Outlines of $I(q,E)$ in Eq. (B13) where (left) $q$ is fixed and (right) $E$ is fixed.