The Gilbert equation revisited: anisotropic and nonlocal damping of magnetization dynamics

Abstract

An equation of motion for the magnetization dynamics of systems with collinear or noncollinear magnetization is derived by a combination of the breathing Fermi surface model with a variant of the ab initio density functional electron theory given by the magnetic force theorem. The equation corresponds to a Gilbert equation with the constant Gilbert damping scalar α replaced by a nonlocal damping matrix , which depends on the momentary orientation of all atomic magnetic moments in the system. For collinear situations this corresponds to an anisotropy of the damping because it depends on the orientation of the magnetization in the crystal, and for systems with atomic-scale noncollinearity such as extremely narrow domain walls or vortices the nonlocality is essential. The range of validity of the theory is discussed, and the predictions are compared with experimental observations. In particular, it is outlined how the prediction of anisotropic damping can be tested by ferromagnetic resonance experiments.