Theory of magnon-skyrmion scattering in chiral magnets

We study theoretically the dynamics of magnons in the presence of a single skyrmion in chiral magnets featuring Dzyaloshinskii-Moriya interaction. We show by micromagnetic simulations that the scattering process of magnons by a skyrmion can be clearly defined although both originate in the common spins. We find that (i) the magnons are deflected by a skyrmion, with the angle strongly dependent on the magnon wave number due to the effective magnetic field of the topological texture, and (ii) the skyrmion motion is driven by magnon scattering through exchange of the momenta between the magnons and a skyrmion: the total momentum is conserved. This demonstrates that the skyrmion has a well-defined, though highly non-Newtonian, momentum.

Figure 1

Snapshots of scattering processes with three different wave numbers. (a1)–(a4): $\overline{k}\xi \simeq 1.87\pi$; (b1)–(b4): $\overline{k}\xi \simeq 0.83\pi$; (c1)–(c4): $\overline{k}\xi \simeq 0.52\pi$; time steps of the snapshots as indicated. The inset of (a1) shows the color representation of the in-plane spin component in $\left(xy\right)$ spin space. In (a1), (b1), and (c1), the wavelengths $\lambda \equiv 2\pi /\overline{k}$ of the incident waves are compared with the size of the skyrmion $\xi$. The vertical blue line in (a4), (b4), and (c4) denotes the incoming magnon direction. For the higher wave numbers we can clearly identify the skew scattering of the magnons. In (a4) the white dashed lines indicate the equal phase contour of the scattered magnons, and blue line perpendicular to those defines the scattering skew angle $\overline{\phi }$. The yellow lines represent the path traversed by the skyrmion, also clearly showing skew scattering over an angle $\Phi$. Hence we see that the skyrmion skew angle is nearly half of the magnon skew angle as expected from the conservation of the momentum.

Figure 2

The scattering properties obtained by numerical and analytical calculations. (a) The Hall angles $\Phi$ (red line) and velocities $v$ (blue line) of skyrmion motion are estimated from the numerical results for different wave numbers $\overline{k}$. To obtain these values, we traced the center-of-mass coordinate $\mathbf{R}$ of a skyrmion between $Y=51$ and $Y=31$. The coordinate $\mathbf{R}$ is defined as $\mathbf{R}\equiv \int d^{2}r{\rho }_{\mathrm{top}}\left(\mathbf{r}\right)\mathbf{r}/\int d^{2}r{\rho }_{\mathrm{top}}\left(\mathbf{r}\right)$, where ${\rho }_{\mathrm{top}}\left(\mathbf{r}\right)\equiv \mathbf{m}\left(\mathbf{r}\right)·[{\partial }_x\mathbf{m}\left(\mathbf{r}\right)\times {\partial }_y\mathbf{m}\left(\mathbf{r}\right)]$. There is a strong nonmonotonic wave-number-dependent behavior in both quantities. We compare these observations to the idealized cases of magnons scattering off a uniform flux tube by an Aharonov-Bohm-type calculation: (b) Expectation of the magnon Hall angle $\overline{\phi }$ as a function of wave number $k$. It is strongly peaked around $k\xi \approx 1$, and vanishes for both low and high wave number, in the latter case as $\sim$$1/k$. The relation $\Phi =\overline{\phi }/2$ derived by momentum conservation seems to be well obeyed by comparing images (a) and (b). (c) Magnon scattering amplitude of several wave numbers $k$. The asymmetry in left or right scattering can be clearly seen and is due to the effective Lorentz force induced by the Berry phase of the skyrmion. For low wave numbers, the scattering amplitude is almost flat, indicating the wave “missing” or “ignoring” the skyrmion; for high wave numbers it is strongly peaked, indicating mostly forward scattering, which is well known in the Aharonov-Bohm effect.