Nonequilibrium phases and segregation for skyrmions on periodic pinning arrays

Using particle-based simulations, we examine the collective dynamics of skyrmions interacting with periodic pinning arrays, focusing on the impact of the Magnus force on the sliding phases. As a function of increasing pinning strength, we find a series of distinct dynamical phases, including an interstitial flow phase, a moving disordered state, a moving crystal, and a segregated cluster state. The transitions between these states produce signatures in the skyrmion lattice structure, the skyrmion Hall angle, the velocity fluctuation distributions, and the velocity-force curves. The moving clustered state is similar to the segregated state recently observed in continuum-based simulations with strong quenched disorder. The segregation arises from the drive dependence of the skyrmion Hall angle, and appears in the strong pinning limit when the skyrmions have nonuniform velocities, causing different portions of the sample to have different effective skyrmion Hall angles. We map the evolution of the dynamic phases as a function of the system density, the ratio of the Magnus force to the dissipative term, and the ratio of the number of skyrmions to the number of pinning sites.

Figure 1

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for skyrmions moving in a square periodic pinning array with a filling factor of $f=N_{sk}/N_p=1.0117$, pinning strength $F_p=2.0$ and pinning density $n_p=0.1975$ at ${\alpha }_m/{\alpha }_d=9.96$. (b) The corresponding velocity deviations $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding fraction of sixfold-coordinated skyrmions $P_6$ vs $F_D$. Dashed lines indicate the boundaries between the pinned state (phase I), the flow of interstitial skyrmions (phase II), the disordered flow state (phase III), the segregated or phase-separated state (phase PS), and the moving crystal (phase MC). The letters a to d in panel (c) indicate the values of $F_D$ at which the skyrmion images in Fig. 2 were obtained.

Figure 2

Skyrmion (blue dots) and pinning site (red circles) locations at one instant in time for the system in Fig. 1 with $f=1.0117,F_p=2.0,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$. (a) The pinned phase I at $F_D=0.$ (b) The disordered flow phase III at $F_D=1.0$. (c) The segregated or phase-separated phase PS at $F_D=3.0$. (d) The moving crystal phase MC at $F_D=5.5$.

Figure 3

(a) $d\langle V_{\parallel }\rangle /d{F}_D$ vs $F_D$ for the system in Fig. 1 with $f=1.0117,F_p=2.0$, $n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$ showing peaks at the transitions between different phases. I: pinned; II: interstitial flow; III: disordered flow; PS: phase separated; MC: moving crystal. (b) $R=\langle V_{\perp }\rangle /\langle V_{\parallel }\rangle$ vs $F_D$ for the same system.

Figure 4

${\theta }_{sk}$ vs $F_D$ for the system in Fig. 3 with $f=1.0117,F_p=2.0,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$, highlighting the directional locking steps in phase II (interstitial flow). (b) ${\theta }_{sk}$ vs $F_D$ for the same system in phases III (disordered flow), PS (phase separated), and MC (moving crystal), showing a jump down in ${\theta }_{sk}$ across the III-PS transition and a jump up across the PS-MC transition. The horizontal dashed lines in both panels indicate the disorder-free skyrmion Hall angle ${\theta }_{sk}^{\mathrm{int}}$.

Figure 5

The structure factor $S\left(\mathbf{k}\right)$ for the system in Fig. 1 with $f=1.0117,F_p=2.0,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$. (a) Pinned phase I. $S\left(\mathbf{k}\right)$ for the interstitial flow phase II (not shown) is similar in appearance. (b) Disordered flow phase III. (c) The PS phase. (d) The MC phase.

Figure 6

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for a system with $f=1.0117,n_p=0.1975$, ${\alpha }_m/{\alpha }_d=9.96$, and larger $F_p=3.0$. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$. Dashed lines indicate the boundaries between the phases. I: pinned; II: interstitial flow; ${\mathrm{I}}_{re}$: reentrant pinned; III: disordered flow; PS: phase separated; MC: moving crystal.

Figure 7

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for a system with $f=1.0117,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$ at $F_p=0.75$ where a diagonal phase-separated state, phase DPS, appears as illustrated in Fig. 8. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$. Dashed lines indicate the phase boundaries. I: pinned; II: interstitial flow; DPS: diagonal phase separated; III: disordered flow; PS: phase separated; MC: moving crystal.

Figure 8

(a), (b) Skyrmion (blue dots) and pinning site (red circles) locations at one instant in time for the system in Fig. 7 with $f=1.0117,n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=0.75$. (a) The diagonal phase-separated state DPS at $F_D=0.25$, where multiple diagonal phase-separated bands appear. (b) The phase-separated PS state at $F_D=1.5$, where the skyrmion density is more uniform. The pinning sites are not shown for clarity. (c), (d) Skyrmion and pinning site locations for a system with $f=1.0117,n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=0.2$. (c) The DPS phase at $F_D=0.075$, where only one band of skyrmions forms. (d) Phase III flow at $F_D=0.55$. The pinning sites are not shown for clarity.

Figure 9

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for a system with $f=1.0117,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$ at $F_p=0.2$. Here, phase II is lost and the system depins directly from phase I to phase DPS. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$. Dashed lines indicate the phase boundaries. I: pinned; DPS: diagonal phase separated; III: disordered flow; MC: moving crystal.

Figure 10

(a) A blowup of $\langle V_{\parallel }\rangle$ vs $F_D$ for the system in Fig. 9 with $f=1.0117$, $n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=0.2$ in the region containing the I-DPS and DPS-III transitions. A dip in $\langle V_{\parallel }\rangle$ appears at the DPS-III transition, producing negative differential conductivity $d\langle V_{\parallel }\rangle /d{F}_D<0$. (b) $d\langle V_{\parallel }\rangle /d{F}_D$ vs $F_D$ for the same system. For clarity, these data have been subjected to a running average in order to reduce the noise.

Figure 11

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for a system with $f=1.0117,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$ at $F_p=0.015$, where the skyrmions depin elastically from a floating triangular solid to a MC state. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$.

Figure 12

Dynamic phase diagram as a function of $F_D$ vs $F_p$ for the system in Fig. 1 with $f=1.0117,n_p=0.1975$, and ${\alpha }_m/{\alpha }_d=9.96$, showing the pinned phase I, the interstitial flow phase II, the reentrant pinned phase ${\mathrm{I}}_{re}$, the disordered phase III, the phase-separated state PS, the diagonal phase-separated state DPS, and the moving crystal state MC.

Figure 13

(a) A blowup of the weak pinning region in the dynamic phase diagram from Fig. 12 as a function of $F_D$ vs $F_p$ showing the transition in the pinned phase I from a pinned commensurate solid (PC) to an incommensurate floating solid (IC) near $F_p=0.02$. This transition also produces a jump down in $F_c$. (b) $P_6$ vs $F_p$ at $F_D=0$ in the same system showing a drop in $P_6$ across the IC-PC line.

Figure 14

$\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$ for a system with $f=1.0117,n_p=0.1975$, and $F_p=1.5$. (a) ${\alpha }_m/{\alpha }_d=6.591$, where there is no PS phase. (b) ${\alpha }_m/{\alpha }_d=13.3$. (c) ${\alpha }_m/{\alpha }_d=26.7$, where there is a reentrant phase III above the PS phase. (d) ${\alpha }_m/{\alpha }_d=39.98$. I: pinned; II: interstitial flow; III: disordered flow; PS: phase separated; MC: moving crystal.

Figure 15

Dynamic phase diagram as a function of $F_D$ vs ${\alpha }_m/{\alpha }_d$ for the system in Fig. 12 with $f=1.0117$ and $n_p=0.1975$ at $F_p=1.5$. Here, the PS phase only occurs for ${\alpha }_m/{\alpha }_d>7.0$. There is also a reentrant phase III that appears when ${\alpha }_m/{\alpha }_d>15$. I: pinned; II: interstitial flow; III: disordered flow; PS: phase separated; MC: moving crystal.

Figure 16

Skyrmion (blue dots) and pinning site (red circles) locations for the system in Fig. 1 with $n_p=0.1975$ and ${\alpha }_m/{\alpha }_d=9.96$ at $F_p=2.0$ and $F_D=0$. (a) $f=0.5$, where an ordered checkerboard state appears. (b) $f=1.65$. (c) $f=2.0$, where there is another ordered state. (d) $f=3.0$, where a partially ordered dimer state occurs.

Figure 17

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for the system in Fig. 16 with $n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96,F_p=2.0$, and $f=2.5$. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. III: disordered flow; PS: phase separated; MC: moving crystal. (c) The corresponding $P_6$ vs $F_D$. The letters a–d indicate the values of $F_D$ at which the images in Fig. 18 were obtained.

Figure 18

Skyrmion (blue dots) and pinning site (red circles) locations at one instant in time for the system in Fig. 17 with $n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=2.0$ at $f=2.5$. (a) A uniform distribution of skyrmions and pinning sites at $F_D=0.0$. (b) The PS phase at $F_D=2.75$. (c) The latter part of the PS phase at $F_D=6.0$. (d) The MC phase at $F_D=7.25$. Note that due to the periodic boundary conditions, the upper portion of the stripe region in panel (c) appears at the bottom of the figure.

Figure 19

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for the system in Fig. 16 with $n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=2.0$ at a filling factor of $f=0.56$, where only phases I (pinned), III (disordered flow), and MC (moving crystal) appear. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$.

Figure 20

Dynamic phase diagram as a function of $F_D$ vs $f$ for the system in Fig. 16 with $n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=2.0$. For clarity, the reentrant phase I region falling between $2.04\le f\le 2.4$ is marked only by lines and not with data points. I: pinned; II: interstitial flow; III: disordered flow; PS: phase separated; MC: moving crystal.

Figure 21

(a) $\langle V_{\parallel }\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) at $f=2.33$ for the system in Fig. 20 with $n_p=0.1975,{\alpha }_m/{\alpha }_d=9.96$, and $F_p=2.0$ showing a reentrant pinning effect ${\mathrm{I}}_{re}$. (b) A blowup of the dynamic phase diagram from Fig. 20 as a function of $F_D$ vs $f$ showing the reentrant pinning which occurs for $2.04\le f<2.4$.

Figure 22

(a) $\langle V_{\parallel }\rangle$ vs $F_D$ for a system with $f=1.0117,F_p=2.0,{\alpha }_m/{\alpha }_d=9.96$, and pinning density $n_p=0.436$. (b) The corresponding $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. The letters a–f indicate the values of $F_D$ at which the images in Fig. 24 were obtained. (c) The corresponding $P_6$ vs $F_D$. DPS: diagonal phase separated; III: disordered flow; PS: phase separated; MC: moving crystal.

Figure 23

A blowup of the $\delta V_{\parallel }$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$ curves from Fig. 22 highlighting the subphases ${\mathrm{DPS}}_1$ and ${\mathrm{DPS}}_2$ within the DPS phase. The subphases have different stripe orientations as illustrated in Figs. 24 and 24. The letters a, b, and c indicate the values of $F_D$ at which the images in Fig. 24 were obtained. The vertical dashed lines are guides to the eye to show the different phases. II: interstitial flow; ${\mathrm{DPS}}_1$ and ${\mathrm{DPS}}_2$: the two subphases of the diagonal phase-separated state; III: disordered flow; PS: phase separated.

Figure 24

Skyrmion (blue dots) and pinning site (red circles) locations for the system in Figs. 22 and 23 with $f=1.0117,F_p=2.0,{\alpha }_m/{\alpha }_d=9.96$, and $n_p=0.436$ obtained at values of $F_D$ marked by the letters a–f in those figures. (a) At $F_D=0.7$ in the ${\mathrm{DPS}}_1$ subphase, the stripes are aligned at an angle of nearly ${45}^{\circ }$ with respect to the driving direction. (b) At $F_D=1.0$ in the ${\mathrm{DPS}}_2$ subphase, the stripes are aligned at an angle of nearly ${67}^{\circ }$ with respect to the driving direction. (c) Disordered phase III flow at $F_D=1.8$. (d) At $F_D=4.47$ in the PS phase, there are two moving stripes aligned with the driving direction. (e) At $F_D=7.0$, close to the PS-MC transition, the two stripes begin to merge. (f) The MC phase at $F_D=7.5$.

Figure 25

Dynamic phase diagram as a function of $F_D$ vs $n_p$ for the system in Fig. 22 with $f=1.0117,F_p=2.0$, and ${\alpha }_m/{\alpha }_d=9.96$. I: pinned; II: interstitial flow; III: disordered flow; DPS: diagonal phase separated; PS: phase separated; MC: moving crystal.