Spin-wave stiffness and micromagnetic exchange interactions expressed by means of the KKR Green function approach

We represent an approach to calculate micromagnetic model parameters such as the tensor of exchange stiffness, Dzyaloshinskii-Moriya interaction (DMI), as well as spin-wave stiffness. The scheme is based on the fully relativistic Korringa-Kohn-Rostoker Green function (KKR-GF) technique and can be seen as a relativistic extension of the work of Lichtenstein et al. [J. Magn. Magn. Mater. 67, 65 (1987)]. The expression for $D^{z\alpha }$ elements of DMI differ from the expressions for $D^{x\alpha }$ and $D^{y\alpha }$ elements as the former are derived via second-order perturbation term of the energy caused by spin-spiral while the latter are associated with the first-order term. Corresponding numerical results are compared with those obtained using other schemes reported in the literature.

Figure 1

Calculated spin-wave stiffness ${\mathfrak{D}}_{xx}$ (see text) of ${\mathrm{Fe}}_{1-x}{\mathrm{Ni}}_x$ alloys, with fcc (left) and bcc (right) crystal structures, represented as a function of the concentration in comparison with experiment: Nakai [19] (filled circles), Hatherly et al. [20] (open circles), and Rusov [21] (filled squares).

Figure 2

The elements $D^{xx}$, $D^{yy}$, and $D^{zz}$ of the micromagnetic DMI tensor calculated for ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ (left) and ${\mathrm{Co}}_{1-x}{\mathrm{Fe}}_x\mathrm{Ge}$ represented as a function of Fe concentration. The results for $D^{xx}$ and $D^{yy}$ based on Eq. (33) were obtained calculating $D_{ij}^x$ and $D_{ij}^y$ as described in Ref. [16]. The corresponding result for $D^{zz}$, however, is based on $D_{ij}^z$ calculated as described in Ref. [5].

Figure 3

The elements $D^{xy}$ of the micromagnetic DMI tensor for multilayer ${(\mathrm{Cu}/{\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x/\mathrm{Pt})}_n$ system as function of Co concentration.

Figure 4

Spin- and element-resolved $d$-DOS for the $d$ states in the multilayer system ${(\mathrm{Cu}/{\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x/\mathrm{Pt})}_n$ for $x=0.1$ (a), $x=0.5$ (b) and $x=0.9$ (c): Pt (black line), Co (red line), and Fe (blue line). The $d$-DOS for the Cu atoms in the systems are shown in (d) for three different concentrations.