Wiggling skyrmion propagation under parametric pumping

Magnetic skyrmion motion under parametric pumping is numerically investigated. Parametric pumping with a perpendicularly oscillating electric field can excite the breathing modes of a skyrmion because an electric field can modify the magnetic anisotropy. Interestingly, using an in-plane static magnetic field to break the rotational symmetry, a perpendicularly oscillating electric field can drive a skyrmion undergoing a wiggling motion along a well-defined trajectory. The skyrmion velocity can reach several meters per second when the frequency of the electric field is close to the frequency of the skyrmion breathing motion. The physics is revealed in a generalized Thiele equation where a net spin current excited by the parametric pumping can drive the skyrmion propagation through angular momentum transfer. Our results open new possibilities for manipulating skyrmions in both metals and insulators with low-power consumption. Moreover, the oscillating skyrmion can act as a microwave generator for future spintronic applications such as a nanotool on a diamond nitrogen-vacancy center.

Figure 1

(a) Schematic illustration of a magnetic layered structures. A magnetic field along the $y$-direction breaks the rotational symmetry of a skyrmion about the $z$-axis. An oscillating electric field is applied normally ($z$-direction) to the film. The colors encode $m_z$ values, with the color bar shown in the top-left panel. The skyrmion moves wiggly in the $xy$ plane as illustrated by the yellow arrow. (b) Skyrmion radius as a function of electric field in the absence of magnetic fields. The blue shadowed region below $-3.6\times {10}^6$ V/m is for the unstable skyrmions. The red line is Eq. (2). The insets show the symmetric and asymmetric skyrmion structures for $H=0$ (left) and $-0.3$ T (right), respectively. The dashed lines mark the field variation range within which $R_s$ varies from 20.8 to 15.7 nm.

Figure 2

(a) Evolution of skyrmion center ${\mathbf{r}}_c=(x_c,y_c)$ for $\omega =12.8$ GHz and $\omega =0$. The slopes of the $x_c-t$ and $y_c-t$ curves are the skyrmion velocity $v_x$ and $v_y$, respectively. The right inset is the zoom-in of $x_c\left(t\right)$ (black) and $y_c\left(t\right)$ (red), which show a wiggling motion. The wiggling trajectory is plotted in the left inset. (b) Average skyrmion velocity as a function of electric-field frequency for $\alpha =0.04,0.08,0.12,0.15$, and 0.20. The pink shadow is the absorption spectrum. The inset is the snapshots of distribution of $m_y$ along $x=0$ at different times. The extremes correspond to the skyrmion wall centers. The oscillations of wall centers with time demonstrate a breathing motion under parametric pumping.

Figure 3

(a) Average skyrmion velocity as a function of damping at $\omega =12.8$ GHz. The symbols are simulation data, and the dashed lines are Eq. (4). The inset shows the Hall angle defined as $arctan(v_x/v_y)$ as a function of damping parameter. The solid line is Eq. (4). (b) and (c) Electric-field strength and field frequency dependence of Hall angle under different dampings. The symbols are simulation data, and the horizontal dashed lines are used to guide eyes.

Figure 4

(a) Skyrmion size as a function of time at $\omega =12.8$ GHz for $\alpha =0.02$ (red line), 0.04 (blue line), and 0.20 (purple line). The cyan rectangle is the region of size fluctuation by taking static field $E\in [E_0,-E_0]$. The dashed line is the oscillation electric field. (b) Time dependence of skyrmion size after an electric field is suddenly switched on at $t=0$ ns for $\alpha =0.02$ (black line) and 0.2 (red line). The solid and dashed lines are for $E_0<0$ (dashed horizontal line) and $E_0>0$ (solid horizontal line), respectively.

Figure 5

Skyrmion velocity as a function of the relative strength of the dampinglike torque ${\zeta }_D/{\zeta }_F$. Part (a) is the enlarged version of (b) for ${\zeta }_D/{\zeta }_F<0.012$. $\alpha =0.04$, and $\omega =12.8$ GHz corresponds to the frequency with maximum speed. Phases A (cyan region), B (yellow region), and C (orange region) represent, respectively, regions dominated by fieldlike torque, dominated by dampinglike torque, and with stripe-domains. Dashed lines in (a) are to guide the eye, while the dashed line in (b) marks the position of maximum $v_y$.

Figure 6

(a) Schematic illustration of a FM/NM bilayer with a nanodiamond placed on top of the FM layer. The spin qubit in the NV center of the diamond interacts with the skyrmion via the dipolar field ${\mathbf{B}}^{\mathrm{skx}}$ generated by the oscillating skyrmion (pink lines). (b) Schematic illustration of the energy levels of the ground state of a NV center under a Zeeman field $B_{z^{\prime }}=298$ Oe. (c) The oscillation of the dipolar fields at $d=16$ nm with a phase shift $\pi /2$. (d) Population rate of the excited state ($m_s=-1$) as a function of time for detuning $\delta =0$ (blue line), $0.02{\omega }_0$ (yellow line), and $0.06{\omega }_0$ (red line), respectively.

Figure 7

(a) Skyrmion structures at various in-plane field strengths between 0 and $-1.0$ T. (b) $m_y$ vs $y$ for spins along $x=64$ nm under $H=0$ (red line) and $H=-0.3$ T (black line). The insets are the distributions of $m_y$ in the skyrmion wall regions with and without applied fields denoted by cyan and pink, respectively. For a better view, the peaks (bottom-right inset where spins are parallel to the field) and valleys (top-left inset where spins are antiparallel to the field) are moved together. (c) Skyrmion size as a function of in-plane field strength. The skyrmion becomes unstable for $\left|H\right|>0.9$ T (cyan region). Here the skyrmion size ($R$) is defined by first counting the number of spins with $m_z>0$ ($N$), and then solving $R$ from the algebraic equation $\pi R^2=N{d}^2$, where $d$ is the mesh size.

Figure 8

(a) Skyrmion velocity as a function of electric-field frequency for $\alpha =0.002$ (blue dots) and 0.006 (red squares). (b) Absorption spectrum of the skyrmion for $\alpha =0.04$ (blue line), 0.02 (pink color), 0.01 (black line), and 0.002 (red line). The dashed lines mark the peak positions.

Figure 9

Skyrmion position as a function of time under parametric pumping from simulations (solid dots and squares) and Eq. (C15) (black and red lines). All parameters are the same as those for Fig. 2. The oscillating magnonic current used in the theory is ${\mathbf{j}}^m={\mathbf{j}}_0^m+\mathrm{\Lambda }·{\mathbf{j}}_0^{m}sin(\omega t+\phi )$, where ${\mathbf{j}}_0^m$ is the same as that for the solid line in Fig. 3 and $\mathrm{\Lambda }=\text{diag}(32,4)$.

Figure 10

(a) Time evolution of the position of the Bloch skyrmion under a periodic cycling of the anisotropy constant $K=K_u+\mathrm{\Delta }Ksin\left(\omega t\right)$. The parameters are $A={10}^{-11}$ J/m, $D=3\mathrm{mJ}/{\mathrm{m}}^2$, $K_u=11.57\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, $\mathrm{\Delta }K=0.1\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, $\alpha =0.02$, $\omega =12.8$ GHz, and $H=-0.1$ T. The sampling rate is 1 frame/ns. The inset plots the skyrmion size as a function of time. (b) A static skyrmion profile for $H=-0.1$ T and $D=3\mathrm{mJ}/{\mathrm{m}}^2$.

Figure 11

Skyrmion position as a function of time under a periodic cycling of the exchange constant $A=A_0+0.01A_{0}sin\left(\omega t\right)$ (a) and DMI coefficient $D=D_0+0.01D_{0}sin\left(\omega t\right)$ (b). The inset plots the skyrmion size as a function of time. The parameters are $A_0={10}^{-11}$ J/m, $D_0=3\mathrm{mJ}/{\mathrm{m}}^2$, $K_u=11.57\times {10}^5\mathrm{J}/{\mathrm{m}}^3$, and $H=0.1$ T. The sampling rate is 1 frame/ns.