Gilbert damping in noncollinear magnetic systems

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 $\underline{\ensuremath{\alpha}}(\stackrel{P\vec}{q})$ is investigated by means of linear response theory. A corresponding expression for $\underline{\ensuremath{\alpha}}(\stackrel{P\vec}{q})$ 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 ($\mathrm{Cu}/{\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Co}}_x{/\mathrm{Pt})}_n$ as an example for systems without inversion symmetry we demonstrate the occurrence of nonvanishing linear contributions. For the alloy system bcc ${\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Co}}_x$ having inversion symmetry, on the other hand, only the quadratic contribution is nonzero. 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 noncollinear spin configuration.