Resonant collective dynamics of the weakly pinned soliton lattice in a monoaxial chiral helimagnet

We study the spin dynamics of a confined chiral soliton lattice whose ends are weakly held. We demonstrate that in this case the system possesses its own resonant frequency. To study features of the resonant dynamics, we analyze the collective motion of the system driven by an oscillating magnetic field directed along the chiral axis. By using the method of collective coordinates we find analytically the resonant frequency and verify the result by numerical simulation of the spin dynamics with the aid of Landau-Lifshitz-Gilbert equations. The numerical simulation shows an appearance of the asymmetric profile of the frequency response function with increasing ac field, which is typical for a nonlinear resonance. To give an explanation of this behavior, we invoke the multiple-time-scale method and predict an emergence of hysteresis phenomena. We also demonstrate that the spin-motive force is strongly amplified by the resonant oscillations.

Figure 1

(a) $\theta$ and $\phi$ angles of the local spin $\mathit{n}$ and (b) the collective coordinates $Z$ and ${\xi }_0$, which describe collective motion of the soliton lattice.

Figure 2

Schematic view of the CSL weakly pinned at both ends $z=\pm L/2$. The solid curve represents the $z$ dependence of the phase at the ground state. Once the center of coordinate $Z$ shifts from zero, the total energy increases because of the energy cost associate the phase mismatch at the boundaries. This energy costs give the confinement potential for the CSL.

Figure 3

Schematic picture of the CSL (top view from the positive side of the $x$ axis). The stripe pattern represents the distribution of the topological charge [${\partial }_z{\phi }_0\left(z\right)$]. SMF generation needs static transverse field $H^x$ and time-dependent longitudinal field $H^z\left(t\right)=H_0^{z}sin\left(\mathrm{\Omega }t\right)$. The black bars at both ends represent the external pinning.

Figure 4

(a) Raw time series of $\phi \left(t\right)$ mode with beats. (b) The spectral decomposition of the raw time series. The boundary pinning field, ${\beta }_0={\tilde{H}}_{\text{p}}/JS=0.002$; the driven field, ${\beta }_z\left(t\right)=0.0001sin\mathrm{\Omega }t$; the driving frequency, $\mathrm{\Omega }=0.00045$ (line 2). Line 1 corresponds to the intrinsic mode, which is a primary resonance. The additional small two peaks may be related with subharmonic and superharmonic resonances generated by nonlinearity of the system or with finite-size effects. The dotted line is the two-mode approximation $0.7cos\left(0.00035t\right)-0.55cos\left(0.00045t\right)$ delivered by the spectral analysis.

Figure 5

(a) Raw time series of $\theta \left(t\right)$ mode with beats. (b) The spectral decomposition of the raw time series. The parameters are the same as in Fig. 1. The dotted line is the two-mode approximation $\pi /2+0.0097sin\left(0.00035t\right)-0.0059cos\left(0.00045t\right)$ delivered by the spectral analysis.

Figure 6

Dependence of the resonant intrinsic frequency ${\mathrm{\Omega }}_0$ on the boundary field ${\beta }_0$ extracted from beats (circles) and a square-root fitting (solid line). The amplitude of the field ${\beta }_z$ is 0.0001.

Figure 7

Dependence of the resonant intrinsic frequency ${\mathrm{\Omega }}_0$ on the chain length $L$ extracted from beats (circles) and the linear fitting (solid line). The amplitude of the field ${\beta }_z$ is 0.0001.

Figure 8

Resonant amplitude of the $\phi$ mode. The boundary field, ${\beta }_0=0.002$; the Gilbert damping, $\alpha =0.001$; the driving field ${\beta }_z,{10}^{-5} (•),5\times {10}^{-5} \left(▵\right),{10}^{-4} (\circ )$.

Figure 9

Resonant amplitude of the $\theta$ mode. The parameters are the same as in Fig. 8.

Figure 10

Frequency dependence of the squared amplitude of the center-of-mass position $Z$. In the shaded region (corresponding to $\mathrm{\Lambda }<0$), hysteresis phenomena occurs. The amplitude of the ac field increases from bottom to top. For $H_{z0}$ exceeding the critical value, the hysteresis phenomena appears. The response profile is categorized into center $\left(d{a}_1^2/d\sigma >0\right)$ and saddle $\left(d{a}_1^2/d\sigma <0\right)$ flows, which are represented in the blue and red curves, respectively.