Spin wave stiffness and exchange stiffness of doped permalloy via ab initio calculations

The way doping affects the spin wave stiffness and the exchange stiffness of permalloy (Py) is investigated via ab initio calculations, using the Korringa-Kohn-Rostoker (KKR) Green function formalism. By considering various types of dopants of different nature (V, Gd, and Pt), we are able to draw general conclusions. To describe the trends of the stiffness with doping it is sufficient to account for the exchange coupling between nearest neighbors. The polarizability of the impurities is not important for the spin wave stiffness. Rather, the decisive factor is the hybridization between the impurity and the host states as reflected by changes in the Bloch spectral function. Our theoretical results agree well with earlier experiments.

Figure 1

Theoretical spin wave stiffness $D$ (upper panel) and exchange stiffness $A_{\text{ex}}$ (central panel) for Py doped with V, Gd, and Pt. Experimental values for $A_{\text{ex}}$ are shown in the lowermost panel, with the data from Ref. [5] for V-doped Py, from Ref. [6] for Gd-doped Py, and from Ref. [7] (circles) and Ref. [3] (circles with crosses) for Pt-doped Py.

Figure 2

Spin wave stiffness $D$ for Py doped with V and Pt calculated by considering the coupling constants $J_{ij}^{\left(\alpha \beta \right)}$ in Eqs. (7) and (8) for all atoms on the same footing (full lines), by completely ignoring the coupling constants for the dopants (dashed lines), and by additionally suppressing the exchange field $B$ for the dopant atoms (circles).

Figure 3

Dependence of the stiffness constant $D$ on the maximum distance $R_{0j}^{\text{(max)}}$ accounted for when evaluating Eq. (8) for undoped permalloy and for permalloy doped by 2.5% and 10% of V (left) and Pt (right).

Figure 4

Exchange coupling constants for Fe-Fe, Fe-Ni, and Ni-Ni atomic pairs if they are first nearest neighbors (upper panels) or second nearest neighbors (lower panels). Data for V-doped Py are shown by triangles, for Gd-doped Py by squares, and for Pt-doped Py by circles.

Figure 5

Upper panels: Dependence of the theoretical spin wave stiffness $D$ on the effective exchange coupling parameter $J_0$ (left) and on the nearest-neighbor coupling $J_1$ (right). The markers denote calculated values of $D, J_0$, and $J_1$ for different dopants and concentrations, the straight lines show the linear fits. Lower panels: Calculated spin wave stiffness $D$ (markers) together with $D$ obtained from respective $J_0$ values relying on the linear $D\left(J_0\right)$ fit (left) and from respective $J_1$ values relying on the $D\left(J_1\right)$ fit (right).

Figure 6

Upper panels: Comparison of the DOS for Fe atoms in undoped Py and in Py doped by 10% of V (left), Gd (center), and Pt (right). Lower panels: As in the upper panels but for Ni atoms.

Figure 7

Comparison of Bloch spectral functions for undoped Py and for Py doped by 10% of V, Gd, and Pt.

Figure 8

Spin wave stiffness constant $D\left(\eta \right)$ for undoped Py evaluated by performing the summation in Eq. (8) up to $R_{0j}^{\text{(max)}}=20.5a_0$ ($a_0$ is the lattice constant) for several values of the damping parameter $\eta$ together with the fifth-degree polynomial used to extrapolate the data to zero damping.