Chiral skyrmion auto-oscillations in a ferromagnet under spin-transfer torque

A skyrmion can be stabilized in a nanodisk geometry in a ferromagnetic material with Dzyaloshinskii-Moriya (DM) interaction. We apply spin torque uniform in space and time and observe numerically that the skyrmion is set in steady rotational motion around a point off the nanodisk center. We give a theoretical description of the emerging auto-oscillation dynamics based on the coupling of the rotational motion to the breathing mode of the skyrmion and to the associated oscillations of the in-plane magnetization. The analysis shows that the achievement of auto-oscillations in this simple setup is due to the chiral symmetry breaking. Thus, we argue that the system is turned into a spin-torque oscillator due to the chiral DM interaction. We also show injection locking of this skyrmion oscillator demonstrating feasibility of application by using synchronization of multiple skyrmion auto-oscillators.

Figure 1

(a) Vector plot for a skyrmion in the center of a nanodisk with diameter $d=150\mathrm{nm}$ and thickness $d_f=5\mathrm{nm}$. The projection of the magnetization on the plane of the disk is shown. This Néel-type skyrmion is a static magnetization configuration, for the parameter values in Table 1 under a $H_{\mathrm{ext}}=0.1\mathrm{T}$ external field. (b) Contour plot for the out-of-plane component of the magnetization ($m_3$) for the same skyrmion.

Figure 2

(a) Skyrmion position coordinates $(X,Y)$ as functions of time under the influence of spin torque. (b) Trajectory of the skyrmion in the nanodisk. The skyrmion is assumed to be at the point where $m_3=-1$. It is initially located at the nanodisk center, and the current is switched on at time $t=0\mathrm{ns}$. Steady-state periodic motion is established after $t\sim 40\mathrm{ns}$. The red dot in (b) shows the center of the sustained rotational motion, located at $(-4.2\mathrm{nm},11.0\mathrm{nm})$ and the outer blue circle shows the nanodisk boundary. The color code shows time.

Figure 3

Four snapshots of the skyrmion configuration when this is in steady-state rotation (clockwise). We give a vector plot of $(m_1,m_2)$ and, on top of that, a contour plot for the magnetization component $m_3$ with a color code. The time instants $t$ are shown on the corresponding entries.

Figure 4

(a) Frequency and (b) amplitude of oscillations for ${\mathcal{M}}_1$ as functions of external field $H_{\mathrm{ext}}$ for the parameter values in Table 1 and current density $J=113{\mathrm{GA}/\mathrm{m}}^2$. (c) Frequency and (d) amplitude of oscillations for ${\mathcal{M}}_1$ as functions of current density for external field $H_{\mathrm{ext}}=0.1\mathrm{T}$. Every symbol on the graphs correspond to a separate numerical simulation.

Figure 5

(a) The velocity components $({\mathit{v}}_1,{\mathit{v}}_2)$ of the skyrmion as functions of time, found in the simulation, for a few cycles after the steady state has been established (red and blue symbols, respectively). We also plot the corresponding right-hand sides of Eq. (10), in red-solid (for ${\mathit{v}}_1$) and blue-dashed (for ${\mathit{v}}_2$) lines, showing that the formulas are indeed capturing the skyrmion dynamics. The period of rotation is approximately $3.1\mathrm{ns}$ (or $574{\tau }_0$). (b) The quantities $T_1,T_2$ of Eq. (11) as functions of time. They are oscillating in phase with the corresponding velocity components $-{\mathit{v}}_2,{\mathit{v}}_1$.

Figure 6

Total magnetization $\mathcal{M}$, in Eq. (12), as function of time. The period of oscillations is the same as for the velocity components in Fig. 5, that is, the skyrmion is breathing at the same frequency as it is rotating.

Figure 7

Plot of excitation frequency ($f_{\text{exc}}$) vs frequency of injected ac current ($f_{\text{inj}}$) at $H_{\text{ext}}=0.1\mathrm{T}$, $I_{\text{dc}}=2\mathrm{mA}$, and $I_{\text{ac}}=0.2\mathrm{mA}$. The inset shows the plot of locking bandwidth ($f_{\text{bw}}$) vs amplitude of injected ac current ($I_{\text{ac}}$).

Figure 8

(a) The skyrmion is initially perturbed and the breathing mode is excited. We show the total magnetization $\mathcal{M}$ as a function of time. The period of the breathing mode is approximately 0.68 ns. (b) The skyrmion is initially displaced from the center of the nanodisk. We show the coordinates $X$ and $Y$ of the skyrmion position as functions of time, as the skyrmion relaxes back in a spiral trajectory to the center of the nanodisk. The period of the rotation is approximately 4 ns.