Generalization of the Landau–Lifshitz–Gilbert equation by multi-body contributions to Gilbert damping for non-collinear magnets

We propose a systematic and sequential expansion of the Landau–Lifshitz–Gilbert equation utilizing the dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises from multi-body scattering processes. The tensor consists of a damping-like term and a correction to the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend on e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: ei ⋅ ej, (nij ⋅ ei)(nij ⋅ ej), nij ⋅ (ei × ej), ${({\mathbf{e}}_{i}\cdot {\mathbf{e}}_{j})}^{2}$, ei ⋅ (ej × ek) ..., where some terms are subjected to the spin–orbit field nij in first and second order. We explore the magnitude of the different contributions using both the Alexander–Anderson model and time-dependent density functional theory in magnetic adatoms and dimers deposited on Au(111) surface