Breakdown of the adiabatic approach for magnetization damping in metallic ferromagnets

We present a microscopic theory for magnetization relaxation in metallic ferromagnets of nanoscopic dimensions that is based on the calculation of the dynamic spin response matrix in the presence of spin-orbit coupling. Our approach takes into account the collective character of spin excitations in ferromagnetic metals. We show that calculations of the Gilbert damping based on a mean-field, adiabatic approximation fail to describe the relaxation of collective excitations in the ballistic regime. Namely, we show that collective spin excitations such as the ferromagnetic resonance mode have finite damping rates even for perfectly crystalline systems, whereas existing microscopic approaches, based on adiabatic and mean-field approximations, predict infinite damping rates. Moreover, we demonstrate that within a finite frequency approach the relaxation properties are not completely determined by the transverse susceptibility alone, and that the damping rate has a non-negligible frequency dependence in experimentally relevant situations. Our results indicate that the widely used adiabatic approach for spin dynamics breaks down for metallic nanostructures in the presence of strong spin-orbit coupling.