Gilbert damping in non-collinear magnetic system

The modification of the magnetization dissipation or Gilbert damping caused by an inhomogeneous magnetic structure and expressed in terms of a wave vector dependent tensor $\underset{‾}{\alpha }(\vec{q})$ is investigated by means of linear response theory. A corresponding expression for $\underset{‾}{\alpha }(\vec{q})$ in terms of the electronic Green function has been developed giving in particular the leading contributions to the Gilbert damping linear and quadratic in $q$. Numerical results for realistic systems are presented that have been obtained by implementing the scheme within the framework of the fully relativistic KKR (Korringa-Kohn-Rostoker) band structure method. Using the multilayered system (Cu/Fe${}_{1-x}$Co${}_x$/Pt)${}_n$ as an example for systems without inversion symmetry we demonstrate the occurrence of non-vanishing linear contributions. For the alloy system bcc Fe${}_{1-x}$Co${}_x$ having inversion symmetry, on the other hand, only the quadratic contribution is non-zero. As it is shown, this quadratic contribution does not vanish even if the spin-orbit coupling is suppressed, i.e.\ it is a direct consequence of the non-collinear spin configuration.