Gilbert Damping in Noncollinear Ferromagnets

The precession and damping of a collinear magnetization displaced from its equilibrium are well described by the Landau-Lifshitz-Gilbert equation. The theoretical and experimental complexity of noncollinear magnetizations is such that it is not known how the damping is modified by the noncollinearity. We use first-principles scattering theory to investigate transverse domain walls (DWs) of the important ferromagnetic alloy ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ and show that the damping depends not only on the magnetization texture but also on the specific dynamic modes of Bloch and Néel DWs in ways that were not theoretically predicted. Even in the highly disordered ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ alloy, the damping is found to be remarkably nonlocal.

Figure 1

Sketch of Néel (a) and Bloch (b) DWs. (c) Calculated effective Gilbert damping parameters for Permalloy DWs (Néel, black lines; Bloch, red lines) as a function of the inverse of the DW width ${\lambda }_w$. Without spin-orbit coupling, calculations for the two DW types yield the same results (blue lines). The green dot represents the value of Gilbert damping calculated for collinear Permalloy. For each value of ${\lambda }_w$, we typically consider eight different disorder configurations and the error bars are a measure of the spread of the results.

Figure 2

(a) Sketch of the magnetization gradient for two SSSs separated by collinear magnetization with (green, dashed line) and without (red, solid line) a broadening of the magnetization gradient at the ends of the SSSs. The length of each segment is $L$. (b) Calculated energy pumping $E_r$ as a function of $L$ for a single Permalloy Bloch-DW-type SSS without SOC. The upper horizontal axis shows the total winding angle of the SSS. (c) Calculated energy pumping $E_r$ without SOC as a function of the number of SSSs that are separated by a stretch of collinear magnetization.

Figure 3

(a) Illustration of electronic transport in a two-band, free-electron DW. The global quantization axis of the system is defined by the majority- and minority-spin states in the left domain. (b) Calculated ${\alpha }_{\mathrm{o}}^{\text{eff}}$ for two-band free-electron DWs as a function of $1/(\pi {\lambda }_w)$ on a log-log scale. The black circles show the calculated results for the clean DWs, which are in perfect agreement with the analytical model Eq. (2), shown as a dashed violet line. When disorder (characterized by the resistivity $\rho$ calculated for the corresponding collinear magnetization) is introduced, ${\alpha }_{\mathrm{o}}^{\text{eff}}$ shows a transition from a linear dependence on $1/{\lambda }_w$ for narrow DWs to a quadratic behaviour for wide DWs. The solid lines are fits using Eq. (3). The dashed orange lines illustrate quadratic behavior.

Figure 4

Calculated out-of-plane damping ${\alpha }_{\mathrm{o}}^{\text{eff}}-{\alpha }_{\text{coll}}$ from Fig. 1 plotted as a function of $1/(\pi {\lambda }_w)$ on a log-log scale. The solid lines are fitted using Eq. (3). The dashed violet and orange lines illustrate linear and quadratic behavior, respectively.