Gilbert damping in non-collinear magnetic system

TL;DR

This paper investigates how non-uniform magnetic structures influence Gilbert damping in non-collinear systems, deriving a wave vector dependent tensor expression and analyzing its effects through relativistic band structure calculations.

Contribution

It develops a Green function-based expression for wave vector dependent Gilbert damping and demonstrates non-zero linear contributions in systems lacking inversion symmetry.

Findings

Non-zero linear damping contributions in asymmetric multilayers.

Quadratic damping contributions in symmetric systems.

Quadratic damping persists without spin-orbit coupling.

Abstract

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 $\underline{\alpha}(\vec{q})$ is investigated by means of linear response theory. A corresponding expression for $\underline{\alpha}(\vec{q})$ 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.