Magnon-skyrmion scattering in chiral magnets

Chiral magnets support topological skyrmion textures due to the Dzyaloshinskii-Moriya spin-orbit interaction. In the presence of a sufficiently large applied magnetic field, such skyrmions are large-amplitude excitations of the field-polarized magnetic state. We investigate analytically the interaction between such a skyrmion excitation and its small-amplitude fluctuations, i.e., the magnons in a clean two-dimensional chiral magnet. The magnon spectrum is found to include two magnon-skyrmion bound states corresponding to a breathing mode and, for intermediate fields, a quadrupolar mode, which will give rise to subgap magnetic and electric resonances. Due to the skyrmion topology, the magnons scatter from an Aharonov-Bohm flux density that leads to skew and rainbow scattering, characterized by an asymmetric differential cross section with, in general, multiple peaks. As a consequence of the skew scattering, a finite density of skyrmions will generate a topological magnon Hall effect. Using the conservation law for the energy-momentum tensor, we demonstrate that the magnons also transfer momentum to the skyrmion. As a consequence, a magnon current leads to magnon pressure reflected in a momentum-transfer force in the Thiele equation of motion for the skyrmion. This force is reactive and governed by the scattering cross sections of the skyrmion; it causes not only a finite skyrmion velocity but also a large skyrmion Hall effect. Our results provide, in particular, the basis for a theory of skyrmion caloritronics for a dilute skyrmion gas in clean insulating chiral magnets.

Figure 1

Skyrmion texture of a chiral magnet.

Figure 2

Energy dependence of the single skyrmion solution as a function of the dimensionless parameter ${\kappa }^2/Q^2$. The energy of the skyrmion is positive for ${\kappa }^2>{\kappa }_{\mathrm{cr}}^2\approx 0.8Q^2$. The negative skyrmion energy for smaller $\kappa$ signals an instability of the magnetic systems toward a proliferation of skyrmions. The inset shows two skyrmion profiles $\theta \left(\rho \right)$ for values of ${\kappa }^2/Q^2$ marked as dots in the main panel.

Figure 3

The potentials $v_0,v_x$, and $v_z$ of Eqs. (42) in units of ${\varepsilon }_{\mathrm{gap}}$ plotted as functions of the dimensionless parameter $\kappa \rho$ for a value of $\kappa =Q$. All three potentials vanish exponentially for $\kappa \rho \gg 1$.

Figure 4

Energy spectrum of magnons in the presence of a single skyrmion excitation of the field-polarized ground state. The latter becomes thermodynamically unstable for ${\kappa }^2<{\kappa }_{\mathrm{cr}}^2\approx 0.8Q^2$ (dashed-dotted vertical line); see Eq. (12). In addition to the continuous magnon spectrum for $\varepsilon >{\varepsilon }_{\mathrm{gap}}={\varepsilon }_{\mathrm{DM}}{\kappa }^2/Q^2$, one obtains in-gap bound states for smaller energies. The dashed vertical line signals the local bimeron instability identified by the vanishing of the quadrupolar eigenfrequency; see the text. The images show snapshots of the corresponding excitation modes; see also Fig. 5.

Figure 5

Time dependence of the skyrmion profile for the bound magnon modes with oscillation period $T=\hslash /\varepsilon$, where $\varepsilon$ is the corresponding eigenenergy. The color code reflects the $z$ component of the local magnetization. Corresponding movies can be found in Ref. [51].

Figure 6

Phase shifts of the scattering states for various angular momenta $m$ for $\kappa =Q$; numerically exact values are shown as solid lines, and the dashed lines are obtained within the WKB approximation. For large angular momenta or high energies, the WKB approximation provides satisfying results.

Figure 7

Effective WKB potential $U_{\mathrm{WKB}}$, Eq. (57), of the magnon scattering states for various values of angular momentum $m$ for $\kappa =Q$. In the classically allowed regimes, the energy of the scattering states obeys $\varepsilon >U_{\mathrm{WKB}}$. In the left panel, classical turning points, ${\rho }_0$, for the energy $\varepsilon =2{\varepsilon }_{\mathrm{gap}}$ are indicated by dots. For certain values of $m$, the potential develops a local maximum giving rise to resonances that are reflected in a pronounced energy dependence of the phase shift.

Figure 8

Differential cross section (61) evaluated within the WKB approximation for various energies $\varepsilon$ at magnetic fields (a) ${\kappa }^2=Q^2$ and (b) ${\kappa }^2=2Q^2$. Skew scattering results in a pronounced asymmetry with respect to forward scattering $\chi =0$ with characteristic multiple peaks at high energies.

Figure 9

Scattering wave function in the WKB approximation for the energy $\varepsilon =20{\varepsilon }_{\mathrm{gap}}$ at $\kappa =Q$ for an incoming wave with $\widehat{k}=\widehat{x}$. The arrows indicate the position of peaks in the differential scattering cross section; see Fig. 8. The skyrmion at the origin is represented by the circle with radius $1/\kappa$. Panel (a) shows only the scattered wave $\mathrm{Re}\{(1,0){\vec{\psi }}_{\mathrm{lab}}^{\mathrm{scatter}}\left(\mathbf{r}\right)-e^{i\mathbf{k}\mathbf{r}}\}$ and panel (b) displays $\mathrm{Re}\left\{(1,0){\vec{\psi }}_{\mathrm{lab}}^{\mathrm{scatter}}\left(\mathbf{r}\right)\right\}$ of Eq. (59) that exhibits the interference between the incoming and the scattered wave.

Figure 10

Smooth part of the effective magnetic flux density (65) for various values of $\kappa$.

Figure 11

Classical deflection function ${\Theta }_m=2{\delta }_m^{\prime }$ of Eq. (70) for the energy $\varepsilon =20{\varepsilon }_{\mathrm{gap}}$ for various values of $\kappa$. As a function of increasing $\kappa$, the three stationary points merge into a single maximum. The inset shows the corresponding phase shifts evaluated within the WKB approximation.

Figure 12

Illustration of the magnon pressure and the resulting skyrmion velocity, $d_t\mathbf{R}$. The magnons are emitted from a source on the left-hand side, see also Fig. 9, and scatter from the skyrmion which is represented by the circle with radius $1/\kappa$. The skyrmion experiences the force $\mathbf{F}$ given in Eq. (83), which results in a finite skyrmion velocity $d_t\mathbf{R}$ of Eq. (86), with a skyrmion Hall angle $\Phi$ given in Eq. (87). The arrows only illustrate the orientation but not the length of the corresponding vectors.

Figure 13

The skyrmion Hall angle $\Phi$ as defined in Eq. (87) as a function of energy for various values of $\kappa$; see the text for details. The inset shows the ratio $-\Phi /{\langle \chi \rangle }_{\varepsilon }$. The Hall angle is maximal $\Phi =\pi /2$ in the limit $\varepsilon \to {\varepsilon }_{\mathrm{gap}}$ where magnon $s$-wave scattering prevails.