Anisotropic damping of the magnetization dynamics in Ni, Co, and Fe

The Gilbert parameter α describing the damping of magnetization dynamics is commonly taken to be an isotropic scalar. We argue that it is a tensor ̱̱α that is anisotropic, leading to a dependence of the damping on both the instantaneous direction of the magnetization M(t) (orientational anisotropy) and on the direction of rotation of the magnetization (rotational anisotropy). For small-angle precession of M around a prescribed axis in the crystal, the rotational anisotropy of Ni, Co, and Fe is calculated as a function of the electronic scattering rate. For circular precession, the rotational anisotropy of M is averaged out and the damping is determined by an effective damping scalar αeff which depends on the axis of rotation. The quantity αeff of Ni, Co, and Fe is calculated for various crystallographic orientations. All calculations are performed by the ab initio density-functional electron theory within the framework of the torque-correlation model. The intraband contribution of this model (breathing Fermi-surface contribution) maintains both orientational and rotational anisotropy for all scattering rates. In contrast, the interband contribution (bubbling Fermi-surface contribution) exhibits these anisotropies only at small scattering rates (τ^-1) and becomes increasingly isotropic (both orientationally and rotationally) as τ^-1 increases. Because the interband contribution dominates at high τ^-1, each material should exhibit isotropic damping at sufficiently high τ^-1 (i.e., sufficiently high temperatures).