Nonlocal Gilbert damping tensor within the torque-torque correlation model

An essential property of magnetic devices is the relaxation rate in magnetic switching, which depends strongly on the damping in the magnetization dynamics. It was recently measured that damping depends on the magnetic texture and, consequently, is a nonlocal quantity. The damping enters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter a, which does not, in a straightforward formulation, account for nonlocality. Efforts were spent recently to obtain Gilbert damping from first principles for magnons of wave vector q. However, to the best of our knowledge, there is no report about real-space nonlocal Gilbert damping aij. Here, a torque-torque correlation model based on a tight-binding approach is applied to the bulk elemental itinerant magnets and it predicts significant off-site Gilbert damping contributions, which could be also negative. Supported by atomistic magnetization dynamics simulations, we reveal the importance of the nonlocal Gilbert damping in atomistic magnetization dynamics. This study gives a deeper understanding of the dynamics of the magnetic moments and dissipation processes in real magnetic materials. Ways of manipulating nonlocal damping are explored, either by temperature, materials doping, or strain.