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 {\em et al.} 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.