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

Abstract

The Gilbert parameter $\ensuremath{\alpha}$ describing the damping of magnetization dynamics is commonly taken to be an isotropic scalar. We argue that it is a tensor $\underset{̱}{\underset{̱}{\ensuremath{\alpha}}}$ that is anisotropic, leading to a dependence of the damping on both the instantaneous direction of the magnetization $\mathbf{M}(t)$ (orientational anisotropy) and on the direction of rotation of the magnetization (rotational anisotropy). For small-angle precession of $\mathbf{M}$ 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 $\mathbf{M}$ is averaged out and the damping is determined by an effective damping scalar ${\ensuremath{\alpha}}_{\text{eff}}$ which depends on the axis of rotation. The quantity ${\ensuremath{\alpha}}_{\text{eff}}$ 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 $({\ensuremath{\tau}}^{\ensuremath{-}1})$ and becomes increasingly isotropic (both orientationally and rotationally) as ${\ensuremath{\tau}}^{\ensuremath{-}1}$ increases. Because the interband contribution dominates at high ${\ensuremath{\tau}}^{\ensuremath{-}1}$, each material should exhibit isotropic damping at sufficiently high ${\ensuremath{\tau}}^{\ensuremath{-}1}$ (i.e., sufficiently high temperatures).