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Starting from the Dirac-Kohn-Sham equation we derive the relativisticequation of motion of spin angular momentum in a magnetic solid under anexternal electromagnetic field. This equation of motion can be written in theform of the well-known Landau-Lifshitz-Gilbert equation for a harmonic externalmagnetic field, and leads to a more general magnetization dynamics equation fora general time-dependent magnetic field. In both cases with an electronicspin-relaxation term which stems from the spin-orbit interaction. We thusrigorously derive, from fundamental principles, a general expression for theanisotropic damping tensor which is shown to contain an isotropic Gilbertcontribution as well as an anisotropic Ising-like and a chiral,Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxationtensor comprises furthermore both electronic interband and intrabandtransitions. We also show that when the externally applied electromagneticfield possesses spin angular momentum, this will lead to an optical spin torqueexerted on the spin moment.

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics