Microscopic theory on the Gilbert damping due to spin pumping effects in the magnetic multi-layer system

We have calculated the Gilbert damping coefficient α in the magnetic multi-layer structures consisted of ferromagnetic (FM) and non-magnetic (NM) layers which are FM/NM and FM/NM/FM from microscopic theory. In this study, our model Hamiltonian doesn't include the spin-orbit interaction term, but the kinetic electron hopping term. For obtaining α in theory, we used Kubo formula and Projection method. We found α is dependent on structure of systems and both thickness of the FM and NM layer. Spin pumping effect which is predicted by experimental results is confirmed by this calculation in theory. We conclude that transferring conduction electrons carrying angular momentum to adjacent layers are the origin of the Gilbert damping for multi-layer systems.