Effective damping enhancement in noncollinear spin structures

Damping mechanisms in magnetic systems determine the lifetime, diffusion, and transport properties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological Landau-Lifshitz-Gilbert equation, here the effective damping parameter in noncollinear magnetic systems is determined describing the linewidth in resonance experiments or the decay parameter in time-resolved measurements. It is shown how the effective damping can be calculated from the elliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded that the effective damping is larger than the Gilbert damping, and it may significantly differ between excitation modes. Numerical results for the effective damping are presented for the localized magnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model–type system.

Figure 1

Effective damping parameter ${\alpha }_{k,\mathrm{eff}}$ as a function of inverse aspect ratio $b_k/a_k$ of the polarization ellipse, assuming constant $a_k$ and $b_k$ functions in Eq. (6). Insets illustrate the precession for different values of $b_k/a_k$.

Figure 2

Localized magnons in the isolated skyrmion, with the interaction parameters corresponding to the Pd/Fe/Ir(111) system [58]: $\mathcal{A}=2.0\mathrm{pJ}/\mathrm{m},\mathcal{D}=3.9\mathrm{mJ}/{\mathrm{m}}^2,\mathcal{K}=2.5\mathrm{MJ}/{\mathrm{m}}^3,\mathcal{M}=1.1\mathrm{MA}/\text{m}$. (a) Magnon frequencies $f=\omega /2\pi$ for $\alpha =0$. Illustrations display the shapes of the excitation modes visualized on the triangular lattice of Fe magnetic moments, with red and blue colors corresponding to positive and negative out-of-plane spin components, respectively. (b) Effective damping coefficients ${\alpha }_{m,\mathrm{eff}}$, calculated from Eq. (6).

Figure 3

Precession of a single spin in the skyrmion in the Pd/Fe/Ir(111) system in the $m=0$ and $m=-3$ modes at $B=0.75\mathrm{T}$, from numerical simulations performed at $\alpha =0.1$. Inset shows the elliptic precession paths. From fitting the oscillations with Eq. (4), we obtained $\left|\mathrm{Re}{\omega }_{m=0}\right|/2\pi =39.22\mathrm{GHz}, \left|\mathrm{Im}{\omega }_{m=0}\right|=0.0608{\mathrm{ps}}^{-1}, {\alpha }_{m=0,\mathrm{eff}}=0.25$ and $\left|\mathrm{Re}{\omega }_{m=-3}\right|/2\pi =40.31\mathrm{GHz}, \left|\mathrm{Im}{\omega }_{m=-3}\right|=0.0276{\mathrm{ps}}^{-1}, {\alpha }_{m=-3,\mathrm{eff}}=0.11$.