Solitary wave excitations of skyrmion strings in chiral magnets

Chiral magnets possess topological line excitations where the magnetization within each cross section forms a skyrmion texture. We study analytically and numerically the low-energy, nonlinear dynamics of such a skyrmion string in a field-polarized cubic chiral magnet, and we demonstrate that it supports solitary waves. These waves are in general nonreciprocal, i.e., their properties depend on the sign of their velocity $v$, but this nonreciprocity diminishes with decreasing $\left|v\right|$. An effective field-theoretical description of the solitary waves is derived that is valid in the limit $v\to 0$ and gives access to their profiles and their existence regime. Our analytical results are quantitatively confirmed with micromagnetic simulations for parameters appropriate for the chiral magnet FeGe. Similarities with solitary waves found in vortex filaments of fluids are pointed out.

Figure 1

Solitary wave excitation of an isolated skyrmion string, that is aligned with the magnetic field $\mathit{H}=\widehat{\mathit{z}}H$, propagating in a direction parallel ($v>0$) and antiparallel ($v<0$) to $\mathit{H}$. The figure is produced by micromagnetic simulation with parameters ${\omega }_{c2}/\left(2\pi \right)=10.4$ GHz and $2\pi /Q=70$ nm typical for the chiral magnet FeGe [11, 12]. The dimensionless field is $h=2.16$ corresponding to ${\mu }_{0}H\approx 0.8$ T for $g\approx 2$ resulting in a skyrmion string radius of approximately 5 nm. Solitary waves with amplitude $R_0\approx 5.8$ nm are created at time $t=0$ and propagate with velocity $\left|v\right|\approx 1$ km/s. Periodic boundary conditions are employed, and the red line is defined according to Eq. (2). Lengths in the $(x,y)$ plane are upscaled by a factor of 5 for visualization.

Figure 2

(a) Linear spin-wave spectrum of a skyrmion string for wave vectors $k_z$ along the string and an applied magnetic field $h=2.16$. The spectrum of the low-energy translational mode has the form $\omega \left(k_z\right)/{\omega }_{c2}\approx a_1{(k_z/Q)}^2+a_2{(k_z/Q)}^3+\cdots$ for small $k_z$. The nonreciprocity due to $a_2$ is small for wave vectors $|k_z|\ll k^{*}$ where $k^{*}=a_{1}Q/a_2$ ($k^{*}/Q=22.02$ for $h=2.16$). (b) Field dependence of the parameters $a_1$ and $a_2$. The dashed lines in (a) as well as the points marked in (b) refer to Figs. 3 and 4.

Figure 3

Nonlinear spin waves of the skyrmion string studied with micromagnetic simulations for $h=2.16$. The energies of low-energy waves, $X+iY=R_0{e}^{i{k}_{z}z-i\omega t}$, with fixed wave vector $k_z=0.87Q$ but for various amplitudes $R_0$ between 0.1 and 10 nm are fitted to $\omega /{\omega }_{c2}=a_1{(k_z/Q)}^2+a_2{(k_z/Q)}^3-{\left(k_z{R}_0\right)}^2\left[b_1{(k_z/Q)}^2+b_2{(k_z/Q)}^3\right]$, and we obtain $a_1\approx 0.903, a_2\approx 0.041, b_1\approx 0.467, b_2\approx 0.071$. The values for $a_1$ and $a_2$ are in good agreement with linear spin-wave theory [see dots in the inset of Fig. 2]. The coefficients $a_2$ and $b_2$ account for the leading nonreciprocal corrections, that are small for $|k_z|\ll k^{*}$ and are omitted in Eq. (4). The amplitude for both simulation snapshots shown in the insets is $R_0=8$ nm where lengths in the $(x,y)$ plane are upscaled by a factor of 5.

Figure 4

Profile parameters of solitary waves extracted from micromagnetic simulations at $h=2.16$ (symbols) for various values of dimensionless velocity $v$ (in units of 1310 m/s). The profiles generally depend on the sign of $v$, but this nonreciprocity decreases with decreasing $\left|v\right|$ and the profiles approach the analytical predictions of Eq. (7) (lines) valid for $v\to 0$. The insets show a full numerical profile (red lines) for $\alpha \approx 0.04$ (marked by dashed circles in the main panels) with a comparison to the solution of Eq. (6) (dashed lines).