Controlled motion of skyrmions in a magnetic antidot lattice

Future spintronic devices based on skyrmions will require precise control of the skyrmion motion. We show that this goal can be achieved through the use of magnetic antidot arrays. We perform micromagnetic simulations and semianalytical calculations based on the Thiele equation, where the skyrmion motion is driven by applied electric current via spin transfer torque (STT) or spin orbit torque (SOT) mechanism. For both torque mechanisms we demonstrate that an antidot array can guide the skyrmions in different directions depending on the parameters of the applied current pulse. Despite the fixed direction of the net driving current, due to the nontrivial interplay between the repulsive potential introduced by the antidots, the skyrmion Hall effect, and the nonuniform current distribution, full control of skyrmion motion in a 2D lattice can be achieved. Moreover, we demonstrate that the direction of skyrmion motion can be controlled by tuning only a single parameter of the current pulse, i.e., current magnitude.

Figure 1

Schematic of the ferromagnet/heavy metal heterostructure that hosts the magnetic skyrmions. Antidots are given by circular holes with diameter $d=250$ nm arranged in the square lattice with lattice constant $a=512$ nm. The antidots repel the skyrmions as indicated by the potential $V(x,y)$ plotted above the heterostructure. The color curves are an example of a skyrmion trajectory induced by applying the current pulse ${\mathbf{j}}_{\text{FM}}$ in the $x$ direction ($j_{\text{FM}}=100{\text{GA/m}}^2$) or ${\mathbf{j}}_{\text{HM}}$ in the $y$ direction (${\theta }_{\text{SH}}{j}_{\text{HM}}=4.13{\text{GA/m}}^2$). The skyrmion is driven from the relaxed position at the potential minimum (at $t=0$) to the antidot wall. When the current pulse is switched off at $t=\mathrm{\Delta }t=62\text{ns}$ (black point), the skyrmion orbits around the antidot and simultaneously relaxes to the neighboring potential minimum. The trajectory calculated by Thiele equation (red curve) is in a reasonable agreement with the trajectory calculated micromagnetically (green curve).

Figure 2

Effective potential $V_{\text{eff}}$ combining the effects of antidot potential $V$ and driving current on the skyrmion motion in the undamped sample. (a)–(d) With increasing ${\mathbf{j}}_{\text{FM}}$ or ${\mathbf{j}}_{\text{HM}}$, the tilt of $V_{\text{eff}}$ along the $y$ direction increases. Red curves show the corresponding skyrmion trajectories starting from the potential minimum (valley) of $V$.

Figure 3

Skyrmion transport without damping driven using rectangular current pulses with the width $\mathrm{\Delta }t$. (a)–(e) Trajectories of skyrmions in the antidot array starting at the bottom of the valley at point $(x,y)=(0,0)$. The value of the pulse width is fixed to $\mathrm{\Delta }t=100$ ns and the values of $j_{\text{FM}}$ are 20, 60, 140, 187, and 240 ${\text{GA/m}}^2$ (or ${\theta }_{\text{SH}}{j}_{\text{HM}}=0.82,2.5,5.8,7.7$, and 9.9 ${\text{GA/m}}^2$), respectively. The color of each trajectory corresponds to a particular region in (f). Black dots denote the position of the skyrmion where the current pulse is switched off. (f) Map of the final positions of the skyrmion as a function of the current pulse width and density. Red dots indicate the parameters defining the trajectories in (a)–(e).

Figure 4

Trajectories of a skyrmion starting in the valley at (0,0) for various values of damping. The skyrmion motion is driven by the uniform current density $j_{\text{FM}}=90\mathrm{GA}/{\mathrm{m}}^2$ (or ${\theta }_{\text{SH}}{j}_{\text{HM}}=3.7\mathrm{GA}/{\mathrm{m}}^2$) applied along the $x$ (or $y$) direction. In the undamped case ($\alpha =0$) the trajectory extends only to the valleys along the $x$ axis [e.g., at (−1,0), (−2,0),...] while, for nonzero damping, the trajectories also reach the valleys in the various directions [e.g., at (−1,1), (0,1),...].

Figure 5

Final position of the skyrmion after the application of uniform current pulse with density $j_{\text{FM}}$ (or $j_{\text{HM}}$) and width $\mathrm{\Delta }t$. Color regions correspond to the parameters with the same final valley. The damping is (a) $\alpha =0.03$, (b) $\alpha =0.1$, (c) $\alpha =0.2$, and (d) $\alpha =0.3$. Shaded region calculated by micromagnetic solver shows the parameters of current pulse at which the skyrmion is annihilated.

Figure 6

(a) Distribution of a periodic nonuniform current density ${\mathbf{j}}_{\text{FM}}(x,y)$ flowing in the ferromagnetic layer depicted in a single unit cell of the antidot lattice. ${\mathbf{j}}_{\text{FM}}(x,y)$ was calculated using the Comsol package where the antidot hole (gray circle) is assumed to be nonconductive. (b) Map of the final positions of the skyrmion after application of current pulses with nonuniform current density in the undamped system.

Figure 7

Final position of the skyrmion driven by STT after the application of nonuniform current pulse with density $j_{\text{FM}}$ and width $\mathrm{\Delta }t$. Color regions correspond to the parameters with the same final valley. The coordinate system is shown in Fig. 4. The damping is (a) $\alpha =0.03$, (b) $\alpha =0.1$, (c) $\alpha =0.2$, and (d) $\alpha =0.3$. Shaded region calculated by micromagnetic solver shows the parameters of current pulse at which the skyrmion is annihilated.

Figure 8

Skyrmion trajectories of damped system with $\alpha =0.1$, nonuniform current density generating STT, fixed width of current pulse at $\mathrm{\Delta }t=50$ ns and varying current densities and polarities. By adjusting a single parameter ($j_{\text{FM}}$) of the current pulse, the skyrmion can be transferred to almost all of the neighboring valleys in the longitudinal as well as transversal direction.

Figure 9

(a) Distribution of a periodic nonuniform current density ${\mathbf{j}}_{\text{HM}}(x,y)$ flowing in the heavy metal layer depicted in a single unit cell of the antidot lattice. ${\mathbf{j}}_{\text{HM}}(x,y)$ was calculated using the Comsol package where the antidot hole (gray circle) is assumed to be nonconductive. (b) Map of the final positions of the skyrmion driven by SOT resulting from current pulses with nonuniform current density in the undamped system.

Figure 10

Final position of the skyrmion driven by SOT after the application of a nonuniform current pulse with density $j_{\text{HM}}$ and width $\mathrm{\Delta }t$. Color regions correspond to the parameters with the same final valley. The coordinate system is shown in Fig. 4. The damping is (a) $\alpha =0.03$, (b) $\alpha =0.1$, (c) $\alpha =0.2$, and (d) $\alpha =0.3$. Shaded region calculated by the micromagnetic solver shows the parameters of current pulse at which the skyrmion is annihilated.

Figure 11

Skyrmion trajectories of damped system with $\alpha =0.1$, nonuniform current density providing SOT, fixed width of current pulse at $\mathrm{\Delta }t=50$ ns, and varying current densities and polarities. By adjusting a single parameter ($j_{\text{HM}}$) of the current pulse, the skyrmion can be transferred to almost all of the neighboring valleys in the longitudinal as well as transversal direction.