Gilbert damping in ferromagnetic films due to adjacent normal-metal layers

Enhancement of the spin relaxation rate in a monolayer ferromagnetic sheet due to adjacent normal metal layers is studied using a dynamic version of the Ruderman-Kittel-Kasuya-Yosida (RKKY) theory. We follow the previously published approach [E. Simanek and B. Heinrich, Phys. Rev. B 67, 144418 (2003)] in which the damping comes from the time lag of the RKKY oscillations induced by s-d exchange interaction with the precessing magnetization. The present work extends this approach by incorporating into the transverse dynamic susceptibility the effect of the static inhomogeneous spin polarization due to the longitudinal component of the magnetization. The calculated excess Gilbert damping constant is found to agree with the spin pumping approach of Tserkovnyak [Phys. Rev. Lett. 88, 117601 (2002)]. Comparison with this approach allows predictions to be made about the role of electron-electron interaction in the present theory.