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

Abstract

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: $\vec{e}_i\cdot\vec{e}_j$, $(\vec{n}_{ij}\cdot\vec{e}_i)(\vec{n}_{ij}\cdot\vec{e}_j)$, $\vec{n}_{ij}\cdot(\vec{e}_i\times\vec{e}_j)$, $(\vec{e}_i\cdot\vec{e}_j)^2$, $\vec{e}_i\cdot(\vec{e}_j\times\vec{e}_k)$..., where some terms are subjected to the spin-orbit field $\vec{n}_{ij}$ 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.

First-principles calculation of the parameters used by atomistic magnetic simulations

“…Alternatively, a real-space extension for classical Gilbert damping tensor was proposed recently by Brinker et al [112], by introducing two-site Gilbert damping tensor G ij entering the site-resolved LLG equation…”

“…with the Green function G(E ± iη) = (E − H ± iη) −1 corresponding to the Hamiltonian H. Moreover, this approach allows a multisite expansion of the GD accounting for higher-order non-local contributions for non-collinear structures [112]. For this purpose, the Hamiltonian H is split into the on-site contribution H 0 and the intersite hopping term t ij , which is spin dependent in the general case.…”

First-principles calculation of the parameters used by atomistic magnetic simulations

“…Trace T 0µ τ 0n T nµ τ n0 c (86) with the g-factor 2(1 + µ orb /µ spin ) in terms of the spin and orbital moments, µ spin and µ orb , respectively, the total magnetic moment µ tot = µ spin + µ orb , and τ 0n ΛΛ ′ = 1 2i (τ 0n ΛΛ ′ − τ 0n Λ ′ Λ ) and with the energy argument E F omitted. The matrix elements T nµ are identical to those occurring in the context of exchange coupling 6 and can be expressed in terms of the spin-dependent part B of the electronic potential with matrix elements:…”

“…Alternatively, a real-space extension for classical Gilbert damping tensor was proposed recently by Brinker et al 86 , by introducing two-site Gilbert damping tensor G ij entering the site-resolved LLG equation…”

“…with the Green function G(E ± iη) = (E − H ± iη) −1 corresponding to the Hamiltonian H. Moreover, this approach allows a multisite expansion of the GD accounting for higher-order non-local contributions for non-collinear structures 86 . For this purpose, the Hamiltonian H is split into the on-site contribution H ′ and the intersite hopping term t ij , which is spin dependent in the general case.…”