Magnetization dynamics and damping due to electron-phonon scattering in a ferrimagnetic exchange model

We present a microscopic calculation of magnetization damping for a magnetic ``toy model.'' The magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless band of localized spins, and the magnetization damping is due to coupling of the itinerant carriers to a phonon bath in the presence of spin-orbit coupling. Using a mean-field approximation for the kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we derive Boltzmann scattering integrals for the distributions and spin conherences in the case of an antiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime broadening and the magnetic gap. For incoherent scattering of itinerant carriers with the phonon bath, i.e., the Elliott-Yafet mechanism, we extract dephasing and magnetization times T₁ and T₂ from initial conditions corresponding to a tilt of the magnetization vector and draw a comparison to phenomenological equations such as the Landau-Lifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for this system including an anisotropy field and find a carrier mediated dephasing of the localized spin via the mean-field coupling.