Quantized transport for a skyrmion moving on a two-dimensional periodic substrate

We examine the dynamics of a skyrmion moving over a two-dimensional periodic substrate utilizing simulations of a particle-based skyrmion model. We specifically examine the role of the nondissipative Magnus term on the driven motion and the resulting skyrmion velocity-force curves. In the overdamped limit, there is a depinning transition into a sliding state in which the skyrmion moves in the same direction as the external drive. When there is a finite Magnus component in the equation of motion, a skyrmion in the absence of a substrate moves at an angle with respect to the direction of the external driving force. When a periodic substrate is added, the direction of motion or Hall angle of the skyrmion is dependent on the amplitude of the external drive, only approaching the substrate-free limit for higher drives. Due to the underlying symmetry of the substrate the direction of skyrmion motion does not change continuously as a function of drive, but rather forms a series of discrete steps corresponding to integer or rational ratios of the velocity components perpendicular $\left(\langle V_{\perp }\rangle \right)$ and parallel $\left(\langle V_{\left|\right|}\rangle \right)$ to the external drive direction: $\langle V_{\perp }\rangle /\langle V_{\left|\right|}\rangle =n/m$, where $n$ and $m$ are integers. The skyrmion passes through a series of directional locking phases in which the motion is locked to certain symmetry directions of the substrate for fixed intervals of the drive amplitude. Within a given directionally locked phase, the Hall angle remains constant and the skyrmion moves in an orderly fashion through the sample. Signatures of the transitions into and out of these locked phases take the form of pronounced cusps in the skyrmion velocity versus force curves, as well as regions of negative differential mobility in which the net skyrmion velocity decreases with increasing external driving force. The number of steps in the transport curve increases when the relative strength of the Magnus term is increased. We also observe an overshoot phenomena in the directional locking, where the skyrmion motion can lock to a Hall angle greater than the clean limit value and then jump back to the lower value at higher drives. The skyrmion-substrate interactions can also produce a skyrmion acceleration effect in which, due to the nondissipative dynamics, the skyrmion velocity exceeds the value expected to be produced by the external drive. We find that these effects are robust for different types of periodic substrates. Using a simple model for a skyrmion interacting with a single pinning site, we can capture the behavior of the change in the Hall angle with increasing external drive. When the skyrmion moves through the pinning site, its trajectory exhibits a side step phenomenon since the Magnus term induces a curvature in the skyrmion orbit. As the drive increases, this curvature is reduced and the side step effect is also reduced. Increasing the strength of the Magnus term reduces the range of impact parameters over which the skyrmion can be captured by a pinning site, which is one of the reasons that strong Magnus force effects reduce the pinning in skyrmion systems.

Figure 1

(a) The velocity for a skyrmion under a dc driving force $F_D$ in the overdamped limit with ${\alpha }_m/{\alpha }_d=0$ moving over a periodic substrate. The drive is applied in the $x$ direction; the system geometry is illustrated in Fig. 2. $\langle V_{\left|\right|}\rangle$ is the velocity component parallel to the applied drive and $\langle V_{\perp }\rangle$ is the velocity component perpendicular to the drive. Here there is a depinning transition into a sliding state where the skyrmion moves strictly in the direction of the applied drive. The Hall term $R=\langle V_{\perp }\rangle /\langle V_{\left|\right|}\rangle =0.0$ in this case. Dashed line: $\langle V_{\left|\right|}\rangle$ response in the absence of a substrate.

Figure 2

The geometry for the system in Fig. 1 with ${\alpha }_m/{\alpha }_d=0$ where the skyrmion (large red dot) is driven in the $x$ direction under an applied drive $F_D$. The black line is the skyrmion trajectory and the smaller blue dots indicate the locations of the potential maxima in the periodic square substrate lattice.

Figure 3

(a) $\langle V_{\perp }\rangle$ and $\langle V_{\left|\right|}\rangle$ vs $F_D$ for the same system as in Fig. 1 but with ${\alpha }_m/{\alpha }_d=0.45$. (b) $R$ vs $F_D$ in the same sample. The dashed line indicates that in the clean limit $R=-0.45$. There are a series of dips in the velocity-force curves correlated with jumps in $R$, indicating that the skyrmion undergoes a series of transitions between locking to different symmetry directions of the substrate. The largest steps in $R$, corresponding to $n/m$ ratios of $0/1,1/4,1/3,2/5,3/7$, and $1/2$, are marked; there are also additional smaller higher order steps at other integer values of $n$ and $m$.

Figure 4

The skyrmion trajectories and substrate maxima for the system in Fig. 3 with ${\alpha }_m/{\alpha }_d=0.45$. (a) $\left|R\right|=0/1$ state at $F_D=1.0$. (b) $\left|R\right|=1/4$ state at $F_D=1.3$. (c) $\left|R\right|=2/5$ state at $F_D=2.2$. (d) $\left|R\right|=1/2$ state at $F_D=2.75$.

Figure 5

A detailed view of the $R$ vs $F_D$ curve for the same system in Fig. 3 at ${\alpha }_m/{\alpha }_d=0.45$, where we examine the locking for drives up to $F_D=12.0$. The system is locked to the $\left|R\right|=1/2$ value but then jumps to $\left|R\right|=6/13$ and $5/11$ before reaching $\left|R\right|=4/9$. As $F_D$ is further increased, $R$ gradually approaches the clean limit value, indicated by the dashed line.

Figure 6

(a), (b) $\langle V_{\perp }\rangle$ and $\langle V_{\left|\right|}\rangle$ vs $F_D$ for (a) ${\alpha }_m/{\alpha }_d=1.28$ and (b) $1.91$. (c), (d) The corresponding $R$ vs $F_D$ curves. Dashed line in (d): the clean limit value for $R$. In (c) we highlight the $\left|R\right|=0/1$ and $\left|R\right|=1/1$ steps, and in (d) we highlight the $\left|R\right|=1/1$, 4/3, 3/2, 5/3, 7/4, $5/9$, and $33/18$ steps.

Figure 7

The skyrmion trajectories and substrate maxima. (a) The $\left|R\right|=1/1$ step for ${\alpha }_m/{\alpha }_d=1.28$ for the system in Figs. 6 and 6. (b) The $\left|R\right|=5/3$ step for ${\alpha }_m/{\alpha }_d=1.91$ for the system in Figs. 6 and 6. (c) The $\left|R\right|=2/1$ state for ${\alpha }_m/{\alpha }_d=4.925$ for the system in Figs. 8 and 8. (d) The $\left|R\right|=3/1$ state for ${\alpha }_m/{\alpha }_d=4.925$ in Figs. 8 and 8.

Figure 8

(a) $\langle V_{\left|\right|}\rangle$ and $\langle V_{\perp }\rangle$ vs $F_D$ at ${\alpha }_m/{\alpha }_d=4.925$. (b) $\langle V_{\left|\right|}\rangle$ vs $F_D$ for ${\alpha }_m/{\alpha }_d=9.962$. (c) $R$ vs $F_D$ for the sample in panel (a) with ${\alpha }_m/{\alpha }_d=4.925$. The steps at $\left|R\right|=2/1$, 3/1, 7/2, 4/1, 17/4, and 9/2 are highlighted. (d) $R$ vs $F_D$ for the sample in panel (b) with ${\alpha }_m/{\alpha }_d=9.962$, where the steps at $\left|R\right|=5/1$, 6/1, 7/1, 8/1, 17/2, and 9/1 are highlighted.

Figure 9

Skyrmion trajectories and substrate maxima for (a) the $\left|R\right|=5.0$ state for the system in Figs. 8 and 8 at ${\alpha }_m/{\alpha }_d=9.962$, and (b) the $\left|R\right|=8.0$ state for the same system.

Figure 10

The regions of $F_D$ vs ${\alpha }_m/{\alpha }_d$ where the pinned phase and the $\left|R\right|=1/0$, 1/1, 2/1, and $3/1$ locking states occur.

Figure 11

(a) $F_D$ vs ${\alpha }_m/{\alpha }_d$ for the ${\alpha }_m/{\alpha }_d<1.0$ side of the $\left|R\right|=1/1$ locking step from Fig. 10. (b) $F_D$ vs ${\alpha }_m/{\alpha }_d$ showing only the $\left|R\right|=1/1$ locking phase from Fig. 10. The width of the step decreases for large $F_D$.

Figure 12

(a) $R$ vs $F_D$ for the ${\alpha }_m/{\alpha }_d<1.0$ regime of the $\left|R\right|=1/1$ locking phase from Fig. 11. From top right to bottom right, ${\alpha }_m/{\alpha }_d=0.8166$, 0.8418, 0.8668, 0.98041, 0.81475, and $0.9274$. The $\left|R\right|=1/1$ step is marked, and the dashed line indicates the value of $R$ for ${\alpha }_m/{\alpha }_d=0.8166$ in the clean limit. (b) $R$ vs $F_D$ for the ${\alpha }_m/{\alpha }_d>1.0$ regime of the $\left|R\right|=1/1$ locking phase from Fig. 11. From top right to bottom right, ${\alpha }_m/{\alpha }_d=1.084$, 1.1, 1.134, 1.17, 1.207, 1.246, and $1.2885$. The $\left|R\right|=1/1$ step is marked and the dashed line indicates the value of $R$ for ${\alpha }_m/{\alpha }_d=1.2885$ in the clean limit.

Figure 13

(a) The locking phases for $F_D$ vs ${\alpha }_m/{\alpha }_d$ highlighting the fractional locking steps with $\left|R\right|=1/4$, 1/3, 1/2, 2/3, 3/4, 5/4, 4/3, 3/2, 5/3, and $7/4$ steps along with the integer matching steps with $\left|R\right|=0/1$, 1/1, and $2/1$. (b) Integer locking phases only for $F_D$ vs ${\alpha }_m/{\alpha }_d$, for $\left|R\right|=0/1$ through $\left|R\right|=12/1$ from left to right.

Figure 14

(a) The net skyrmion velocity $\langle V\rangle ={({\langle V_{\left|\right|}\rangle }^2+{\langle V_{\perp }\rangle }^2)}^{1/2}$ vs $F_D$ for ${\alpha }_m/{\alpha }_d=9.962$ (upper curve), $0.0$ (lower curve), and the clean limit value (dashed line). Here there are regions of the ${\alpha }_m/{\alpha }_d=9.962$ curve where $\langle V\rangle$ exceeds the clean limit value, indicating a speedup or acceleration effect. (b) $\Delta V=\langle V\rangle -{\langle V\rangle }_{\mathrm{clean}}$ for ${\alpha }_m/{\alpha }_d=9.962$ (upper line), $0.0$ (lower line), and the clean limit value (dashed line). The speedup effect is indicated by regions in which $\Delta V>0.$

Figure 15

$\langle V\rangle$ vs ${\alpha }_m/{\alpha }_d$ for fixed $F_D=0.2$. The dashed line indicates the clean limit value of $\langle V\rangle =0.2$. The speedup effect occurs when $\langle V\rangle >0.2$ and increases in magnitude with increasing ${\alpha }_m/{\alpha }_d$.

Figure 16

A gray-scale image of a portion of the 2D analytic substrate. The external drive is applied along the $x$ direction.

Figure 17

The transport for a skyrmion moving over the 2D sinusoidal substrate illustrated in Fig. 16 for $F_p=1.5$ and ${\alpha }_m/{\alpha }_d=4.925$. (a) The net skyrmion velocity $\langle V\rangle$ vs $F_D$ shows a series of cusps. Dashed line: the clean value limit. (b) $\langle V_{\perp }\rangle$ and $\langle V_{\left|\right|}\rangle$ vs $F_D$ for the same system. (c) $R$ vs $F_D$ for the same system with the $\left|R\right|=1/1$, 2/1, 3/1, and $4/1$ steps highlighted. There are also numerous smaller scale fractional locking steps.

Figure 18

Skyrmion trajectories (lines) for a single skyrmion at different impact parameters $b$ interacting with a single pinning site of radius $R_p=0.35$ and maximum pinning force of $F_p$ for an external drive $F_D$ applied in the positive $y$ direction. (a) ${\alpha }_m/{\alpha }_d=0.0,F_p=0.1$, and $F_D=0.05$. The skyrmions that interact with the pinning site are captured. (b) ${\alpha }_m/{\alpha }_d=0,F_p=0.1$, and $F_D=0.12$. The skyrmions can escape from the pinning site. (c) ${\alpha }_m/{\alpha }_d=10.0,F_p=0.1$, and $F_D=0.05$. The dashed line is a trajectory that a skyrmion would follow in the absence of the pinning site. For a certain range of impact parameters, the skyrmion is captured. For other impact parameters, skyrmions that escape from the pinning site have their trajectories shifted as highlighted by the thick line. (d) ${\alpha }_m/{\alpha }_d=10.0,F_p=0.1$, and $F_D=0.12$. Here the shift in the trajectories of the skyrmions that pass through the pinning site is reduced and there is less curvature in the trajectories.

Figure 19

$\delta r$, the magnitude of the trajectory shift, vs $F_D$ for a skyrmion entering a single pinning site with an impact factor of zero. (a) $F_p=0.1,R_p=0.35$, and ${\alpha }_m/{\alpha }_d=10.0$. $\delta r$ approaches zero at higher drives. (b) The same for ${\alpha }_m/{\alpha }_d=1.0$.

Figure 20

The trajectory shift $\delta r$ vs impact parameter $b$ for a skyrmion moving through a pinning site with $F_p=0.1$ and $R_p=0.35$. (a) ${\alpha }_m/{\alpha }_d=0$ for $F_D=0.12$, 0.16, and 0.20, from upper left to lower left. $\delta r$ is symmetric across $b=0$. (b) ${\alpha }_m/{\alpha }_d=1.0$ for $F_D=0.085$, 0.09, 0.10, 0.11, 0.12, 0.14, 0.16, 0.18, and 0.20, from upper left to lower left. Here the shift is asymmetric across $b=0$. For values of $b$ where there are no points, the skyrmion is captured by the pinning site. (b) ${\alpha }_m/{\alpha }_d=10$ for $F_D=0.03$, 0.05, 0.08, 0.12, and 0.20, from upper left to lower left. The shifts become larger with increasing ${\alpha }_m/{\alpha }_d$.