Nonlinear transport, dynamic ordering, and clustering for driven skyrmions on random pinning

Using numerical simulations, we examine the nonlinear dynamics of skyrmions driven over random pinning. For weak pinning, the skyrmions depin elastically, retaining sixfold ordering; however, at the onset of motion there is a dip in the magnitude of the structure factor peaks due to a decrease in positional ordering, indicating that the depinning transition can be detected using the structure factor even within the elastic depinning regime. At higher drives, the moving skyrmion lattice regains full ordering. For increasing pinning strength, we find a transition from elastic to plastic depinning that is accompanied by a sharp increase in the depinning threshold due to the proliferation of topological defects, similar to the peak effect found at the elastic to plastic depinning transition in superconducting vortex systems. For strong pinning and strong Magnus force, the skyrmions in the moving phase can form a strongly clustered or phase-separated state with highly modulated skyrmion density, similar to that recently observed in continuum-based simulations for strong disorder. As the Magnus force is decreased, the density phase-separated state crosses over to a dynamically phase-separated state with uniform density but with flow localized in bands of motion, while in the strongly damped limit, both types of phase-separated states are lost. In the strong pinning limit, we find highly nonlinear velocity-force curves in the transverse and longitudinal directions, along with distinct regions of negative differential conductivity in the plastic flow regime. The negative differential conductivity is absent in the overdamped limit. The Magnus force is responsible for both the negative differential conductivity and the clustering effects since it causes faster moving skyrmions to partially rotate around slower moving or pinned skyrmions.

Figure 1

(a) $S\left({\mathbf{k}}_0\right)$, the magnitude of the structure factor peak at one of the reciprocal lattice vectors of the skyrmion lattice, vs $F_D$ for the weak pinning case $F_p=0.03$ in a sample with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$, where the skyrmions form a pinned crystal that depins elastically. (b) The corresponding $\langle V_{\parallel }\rangle$, the average skyrmion velocity parallel to the driving direction, vs $F_D$. The dip in $S\left({\mathbf{k}}_0\right)$ occurs at the drive for which $\langle V_{\parallel }\rangle$ becomes finite.

Figure 2

$S\left(\mathbf{k}\right)$ from the system in Fig. 1 with $F_p=0.03, n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$, where the depinning is elastic and $F_c=0.0025$. (a) At $F_D=0$, there are six sharp peaks indicative of triangular ordering. (b) Just at depinning for $F_D/F_c=1.0$, the peaks are still present but there is some smearing. (c) At $F_D=0.008$ in the sliding phase, the system is more ordered.

Figure 3

$P_6$, the fraction of sixfold-coordinated skyrmions, vs $F_p$, measured at the depinning transition for the system in Fig. 1 with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$. (b) The corresponding value of the critical depinning force $F_c$ vs $F_p$. A transition occurs from elastic depinning, where the skyrmions maintain sixfold ordering, to plastic depinning, marked by a drop in $P_6$ and a sharp increase in $F_c$. This behavior is similar to the so-called peak effect in type-II superconductors that appears across the transition from elastic to plastic vortex depinning.

Figure 4

(a) $F_c$ vs $F_p$ plotted on a log-log scale for the system in Fig. 1 with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$. The dashed lines are power-law fits to $F_c\propto F_p^{\alpha }$. Blue: $\alpha =2.0$ in the elastic depinning regime; red: $\alpha =1.0$ in the plastic depinning regime. (b) The corresponding $d{F}_c/d{F}_p$ versus $F_p$ curve has a peak near $F_p=0.325$ at the transition from elastic to plastic depinning. (c) The corresponding $P_6$ versus $F_p$ curve shows that the elastic to plastic depinning transition coincides with a sharp drop in $P_6$.

Figure 5

$P_6$ vs $F_D$ for the system in Fig. 1 with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$ at $F_p=0.03$ (red), 0.2 (orange), 0.3 (light blue), and 0.5 (dark blue). For large drives, $P_6$ increases toward $P_6=1.0$ as the system dynamically reorders. The plateau in $P_6$ for $F_p=0.5$ corresponds to a moving liquid phase in which all the skyrmions are moving but the system is disordered.

Figure 6

(a), (c), (e) Real-space images of the skyrmion positions and (b), (d), (f) the corresponding structure factors $S\left(\mathbf{k}\right)$ for the system in Fig. 5 with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$ at $F_p=0.5$. (a), (b) At $F_D=0.3$, the plastic flow phase contains both pinned and moving skyrmions, and $S\left(\mathbf{k}\right)$ shows that the structure is random. (c), (d) At $F_D=1.1$, the skyrmion density is more uniform but the system is still disordered and forms a moving liquid (ML) phase, in which $S\left(\mathbf{k}\right)$ develops a ringlike feature. (d), (e) In the dynamically ordered moving crystal (MC) phase at $F_D=1.75$, the skyrmions form a triangular lattice.

Figure 7

(a) $\langle V_{\parallel }\rangle$ and (b) $\langle V_{\perp }\rangle$, the average skyrmion velocity parallel and perpendicular to the drive, respectively, vs $F_D$ for the system in Fig. 5 with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$ at $F_p=0.5$. Each curve has different nonlinear behavior near the depinning threshold. (c) The corresponding velocity deviations $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$, showing the strong fluctuations in the plastic flow regime. The letters a, c, and e in panel (c) indicate the values of $F_D$ at which the images in Fig. 6 were obtained. Vertical dashed lines indicate the separations between the pinned phase, the plastic flow or phase-segregated (PL-PS) state, the moving liquid (ML), and the moving crystal (MC).

Figure 8

Dynamic phase diagram as a function of $F_D$ vs $F_p$ for the system in Figs. 1, 2, 3, 4, 5, 6, 7 with $n_{sk}=0.16, n_p=0.2$, and ${\alpha }_m/{\alpha }_d=9.95$, highlighting the pinned crystal phase which depins elastically into a moving crystal phase, the pinned skyrmion glass phase, the plastic flow phase, and the moving liquid phase.

Figure 9

(a) $F_c$ vs $n_{sk}$ for samples with $n_p=0.2, F_p=0.02$, and ${\alpha }_m/{\alpha }_d=9.95$. (b) The corresponding $P_6$ vs $n_{sk}$. The transition from elastic to plastic depinning occurs near $n_{sk}=0.125$. For $n_{sk}<0.02$, the behavior of the system is in the single skyrmion limit. (c), (d) Zoomed-in plots of (a) and (b), respectively, for the region near $n_{sk}=0.125$ where the transition from elastic to plastic depinning occurs. A drop in $P_6$ is correlated with an increase in $F_c$, which is similar to the peak effect phenomenon.

Figure 10

The system from Fig. 9 with $n_p=0.2, F_p=0.02$, and ${\alpha }_m/{\alpha }_d=9.95$ in the elastic regime at $n_{sk}=0.207$. (a) $\langle V_{\parallel }\rangle$ vs $F_D$. (b) $R=\langle V_{\perp }\rangle /\langle V_{\parallel }\rangle$ vs $F_D$. (c) ${\theta }_{sk}={tan}^{-1}\left(R\right)$ vs $F_D$. There is a series of steps on which the skyrmion lattice motion locks to specific skyrmion Hall angles of ${\theta }_{sk}={60}^{\circ }, {74}^{\circ }$, and $79.2^{\circ }$, corresponding to the alignment of the direction of motion with a symmetry direction of the skyrmion lattice.

Figure 11

A blowup of the ${\theta }_{sk}=79.2^{\circ }$ locking step in Fig. 10. (a) $\langle V_{\parallel }\rangle$ vs $F_D$ has a linear increase along the locking step. (b) $R$ vs $F_D$ and (c) ${\theta }_{sk}$ vs $F_D$ both have shapes on either end of the locking step that are consistent with what is expected in phase-locking systems.

Figure 12

$\langle V_{\parallel }\rangle$ vs $F_D$ for the system in Fig. 9 with $n_p=0.2, F_p=0.02$, and ${\alpha }_m/{\alpha }_d=9.95$. At $n_{sk}=0.01$ (dark blue), the depinning threshold $F_c$ is close to the single-particle limit and the system does not dynamically order. At $n_{sk}=0.15$ (light blue), the depinning is plastic, $F_c$ is higher than in the elastic limit, and a plateau appears in the velocity response. At $n_{sk}=0.2$ (red), the depinning is elastic. The dashed line indicates the velocity response in the pin-free limit, showing that the introduction of pinning actually increases $\langle V_{\parallel }\rangle$ due to a speedup effect.

Figure 13

Dynamic phase diagram as a function of $F_D$ vs $n_{sk}$ for the system in Figs. 9–12 with $n_p=0.2, F_p=0.02$, and ${\alpha }_m/{\alpha }_d=9.95$ highlighting the pinned glass, pinned crystal, plastic flow, moving liquid, and moving crystal phases.

Figure 14

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ and (b) $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$ for a sample with $F_p=0.2, n_{sk}=0.16$, and $n_p=0.2$ in the overdamped limit of ${\alpha }_m/{\alpha }_d=0$. (c) $\langle V_{\parallel }\rangle$ vs $F_D$ and (d) $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$ for a sample with the same parameters except with ${\alpha }_m/{\alpha }_d=1.0$.

Figure 15

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ and (b) $\delta V_{\parallel }$ (blue) and $\delta V_{\parallel }$ (red) vs $F_D$ for the system in Fig. 14 with $F_p=0.2, n_{sk}=0.16$, and $n_p=0.2$ at ${\alpha }_m/{\alpha }_d=4.0$. (c) $\langle V_{\parallel }\rangle$ vs $F_D$ and (d) $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$ for the same system at ${\alpha }_m/{\alpha }_d=15.79$, where we find a region in which $\langle V_{\parallel }\rangle$ decreases with increasing $F_D$, indicative of negative differential conductivity.

Figure 16

Dynamic phase diagram as a function of $F_D$ vs ${\alpha }_m/{\alpha }_d$ for the system in Figs. 14 and 15 with $F_p=0.2, n_{sk}=0.16$, and $n_p=0.2$, highlighting the pinned, plastic, moving liquid, and moving crystal phases. For ${\alpha }_m/{\alpha }_d=0$, the moving crystal phase is replaced by a moving smectic state. The depinning threshold $F_c$ decreases with increasing ${\alpha }_m/{\alpha }_d$.

Figure 17

Skyrmion positions for a system with $F_p=1.5, n_{sk}=0.44, n_p=0.6$, and ${\alpha }_m/{\alpha }_d=10$. (a) The pinned phase at $F_D=0.2$ has a uniform skyrmion density. (b) The density phase-separated state at $F_D=0.6$. (c) The transition from the density phase-separated state to the moving liquid state at $F_D=1.25$. (d) The moving liquid phase at $F_D=2.0$ where the skyrmion density is uniform.

Figure 18

Skyrmion positions in the density segregated state for the system in Fig. 17 with $n_{sk}=0.44, n_p=0.6$, and ${\alpha }_m/{\alpha }_d=10$ at $F_D/F_c=0.6$ for (a) $F_p=1.0$, (b) $F_p=3.5$, and (c) $F_p=5.0$, showing that as the disorder strength increases, the skyrmion bands become narrower. (d) The $F_p=3.5$ system at $n_{sk}=0.67$, showing the persistence of the bands at higher skyrmion densities.

Figure 19

(a) Skyrmion positions (dots) and trajectories (lines) for the system in Fig. 17 with $n_{sk}=0.44, n_p=0.6$, and ${\alpha }_m/{\alpha }_d=10$ at $F_p=0.3$ and $F_D=0.1$ where the density phase segregation is lost but a dynamical segregation emerges in which the motion is confined to a band. (b) An image of only the skyrmion positions from (a) showing that the skyrmion density is uniform.

Figure 20

Dynamic phase diagram as a function of $F_D$ vs $F_p$ for the system in Figs. 17, 18, 19 with $n_{sk}=0.44, n_p=0.6$, and ${\alpha }_m/{\alpha }_d=10$ showing the pinned crystal (PC), the pinned glass (PG), the dynamically phase-separated state (DPS) illustrated in Fig. 19, the density phase-separated state (PS) illustrated in Figs. 17 and 18, the moving liquid (ML), and the moving crystal (MC).

Figure 21

Dynamic phase digram as a function of $F_D$ vs ${\alpha }_m/{\alpha }_d$ for the system in Figs. 17, 18, 19, 20 with $F_p=1.5, n_{sk}=0.44$, and $n_p=0.6$ showing the pinned state, dynamically phase-separated state (DPS), density phase-separated state (PS), moving liquid (ML), and moving crystal (MC). The PS state appears only for ${\alpha }_m/{\alpha }_d>5.0$.

Figure 22

Skyrmion positions (dots) and trajectories (lines) for the system in Fig. 21 with $F_p=1.5, n_{sk}=0.44$, and $n_p=0.6$ at ${\alpha }_m/{\alpha }_b=15$. (a) At $F_D=0.3$, multiple bands appear consisting of both moving and pinned skyrmions. (b) At $F_D=1.2$, the bands make a steeper angle with the applied drive, and kinks appear as the bands rotate to match the skyrmion Hall angle, which increases with increasing $F_D$.

Figure 23

Dynamic phase diagram as a function of $F_D$ vs $n_p$ for a system with $n_{sk}=0.44, F_p=10$, and ${\alpha }_m/{\alpha }_d=10$ showing the pinned, plastic flow (PF), dynamically phase-separated (DPS), phase-separated (PS), moving liquid (ML), and moving crystal (MC) states. In the PF phase, there are a combination of pinned and flowing skyrmions but there is no dynamical or density phase separation. The DPS and PS states appear only when the disorder is sufficiently dense and strong.