Assessing different approaches to ab initio calculations of spin wave stiffness

Ab initio calculations of the spin-wave stiffness constant $D$ for elemental Fe and Ni performed by different groups in the past have led to values with a considerable spread of 50–100%. We present results for the stiffness constant $D$ of Fe, Ni, and permalloy ${\mathrm{Fe}}_{0.19}{\mathrm{Ni}}_{0.81}$ obtained by three different approaches: (i) by finding the quadratic term coefficient of the power expansion of the spin-wave energy dispersion, (ii) by a damped real-space summation of weighted exchange coupling constants, and (iii) by integrating the appropriate expression in reciprocal space. All approaches are implemented by means of the same Korringa-Kohn-Rostoker (KKR) Green's function formalism. We demonstrate that if properly converged, all procedures yield comparable values, with uncertainties of 5–10% remaining. By a careful analysis of the influence of various technical parameters, we estimate the margin of errors for the stiffness constants evaluated by different approaches and suggest procedures to minimize the risk of getting incorrect results.

Figure 1

Magnetic moment per atom (lower panel) and energy dispersion $[E\left(\mathit{q}\right)-E_0]/{sin}^2\theta$ (upper panel) for spin spiral waves propagated along the [001] direction in Fe, obtained by means of self-consistent calculations (markers) and by means of magnetic force theorem (lines). The spiral cone angle $\theta$ is specified in the legend.

Figure 2

As Fig. 1 but for Ni. Note that the energy dispersion curves for $\theta =5^{\circ }$ and ${20}^{\circ }$ almost coincide.

Figure 3

The energy dispersion $[E\left(\mathit{q}\right)-E_0]/{sin}^2\theta$ for spin spiral waves in Fe obtained when the $\mathit{k}$-space integrals were evaluated using a regular mesh corresponding to ${89}^3$ points, ${106}^3$ points, and ${165}^3$ points in the full BZ. The $\mathit{q}$-vector was varied along the [001] direction and the cone angle is $\theta ={20}^{\circ }$. The energies were calculated employing self-consistent potentials.

Figure 4

Dependence of the spin-wave stiffness constant $D$ on the maximum distance $R_{\text{max}}$ up to which the terms in Eq. (10) are included, for different damping parameters $\eta$. Lower panel shows data for for Ni obtained using the mesh of ${112}^3$ points in the full BZ, upper panel shows data for Py obtained using the mesh of ${75}^3$ points in the full BZ. The distance $R_{\text{max}}$ is in units of the lattice constant $a_0$.

Figure 5

Dependence of the stiffness constant $D$ for Ni on the maximum distance $R_{\text{max}}$ up to which the terms in Eq. (10) are included, for different numbers of $\mathit{k}$ points in the full BZ (shown in the legend). The data were obtained for damping parameter $\eta$ = 0.30.

Figure 6

Dependence of the spin-wave stiffness constant $D$ for Ni (lower panel) and for Py (upper panel) on the damping parameter $\eta$. Data are shown for different $\mathit{k}$-space grids. The maximum distance $R_{\text{max}}$ up to which the individual terms in Eq. (10) were evaluated is $20.5a_0$.

Figure 7

Dependence of the spin-wave stiffness constant $D$ for Ni on the damping parameter $\eta$, together with fits of $D\left(\eta \right)$ by polynomials of the second, third, and fifth degree in $\eta$ (as indicated by the legend). To obtain the fitting polynomials, only data for $\eta \in [0.4,1.0]$ were used. The $\mathit{k}$-space integrals were evaluated using the grid of ${112}^3$ points in the full BZ, the maximum distance $R_{\text{max}}$ up to which the terms in Eq. (10) were included is $20.5a_0$.