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

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

Figure 1

(Color online) A sketch of the equilibrium Fermi surface $S$ for time $t-dt$ and time $t$ [(a), left part]. In a strict equilibrium situation the yellow (light gray) states are occupied only at time $t-dt$, the red (gray) states are occupied only at time $t$, whereas the blue (dark gray) states are occupied at both times. Panel (b) (right part) shows a sketch of the equilibrium band structure ${\epsilon }_{j\mathbf{k}}\left(t\right)$ along the direction in $k$ space indicated by the horizontal dashed line in (a). For a realistic situation where the $n_{j\mathbf{k}}$ lag behind the $f_{j\mathbf{k}}$ there may be some yellow (light gray) states which should be empty in a strict equilibrium situation but which are occupied in a realistic situation. Furthermore, there may be some red (gray) states which should be occupied but which are still empty. In (b) there is also a sketch for the interband transitions.

Figure 2

(Color online) The effective damping parameter ${\alpha }_{\text{eff}}$ for circular precession for Ni (top), Co (middle), and Fe (bottom) vs electron scattering rate ${\tau }^{-1}$, for various orientations of the prescribed axis around which the small-angle precession of the magnetization vector takes place. The convention for the real-space coordinate directions for Co are described in the text. Full lines: total damping, dashed lines: conductivitylike contribution and dotted lines: resistivitylike contribution.

Figure 3

(Color online) The effective damping parameter ${\alpha }_{\text{eff}}$ for Ni (top), Co (middle), and Fe (bottom) vs electron scattering rate ${\tau }^{-1}$ for various ellipticities of orbit about the prescribed axis, which is $⟨011⟩$ for Ni and Fe, and the real-space $\left(1,0,1\right)/\sqrt{2}$ direction for Co. Ellipticities and real-space axes convention are as described in the text. Full lines: total damping, dashed lines: conductivitylike contribution and dotted lines: resistivitylike contribution.