Spin Dynamics and Multiple Reflections in Ferromagnetic Film in Contact with Normal Metal Layers

Spin dynamics of a metallic ferromagnetic film imbedded between normal metal layers is studied using the spin-pumping theory of Tserkovnyak et al. [Phys. Rev. Lett. 88, 117601 (2002)]. The scattering matrix for this structure is obtained using a spin-dependent potential with quantum well in the ferromagnetic region. Owing to multiple reflections in the well, the excess Gilbert damping and the gyromagnetic ratio exhibit quantum oscillations as a function of the thickness of the ferromagnetic film. The wavelength of the oscillations is given by the depth of the quantum well. For iron film imbedded between gold layers, the amplitude of the oscillations of the Gilbert damping is in an order of magnitude agreement with the damping observed by Urban et al. [Phys. Rev. Lett. 87, 217204 (2001)]. The results are compared with the linear response theory of Mills [Phys. Rev. B 68, 0144419 (2003)].