Shear banding, intermittency, jamming, and dynamic phases for skyrmions in inhomogeneous pinning arrays

We examine driven skyrmion dynamics in systems with inhomogeneous pinning where a strip of strong pinning coexists with a region containing no pinning. For driving parallel to the strip, we find that the initial skyrmion motion is confined to the unpinned region and the skyrmion Hall angle is zero. At larger drives, a transition occurs to a phase in which motion also appears within the pinned region, creating a shear band in the skyrmion velocity, while the skyrmion Hall angle is still zero. As the drive increases further, the flow becomes disordered and the skyrmion Hall angle increases with drive until saturating at the highest drives when the system transitions into a moving crystal phase. The different dynamic phases are associated with velocity and density gradients across the pinning boundaries. We map out the dynamic phases as a function of pinning strength, skyrmion density, and Magnus force strength, and correlate the phase boundaries with features in the velocity-force curves and changes in the local and global ordering of the skyrmion structure. For large Magnus forces, the shear banding instability is replaced by large-scale intermittent flow in the pinned region accompanied by simultaneous motion perpendicular to the direction of the drive, which appears as oscillations in the transport curves. We also examine the case of a drive applied perpendicular to the strip, where we find a jamming effect in which the skyrmion flow is blocked by skyrmion-skyrmion interactions until the drive is large enough to induce plastic flow.

Figure 1

Image of the system showing skyrmion positions (blue dots) and pinning site locations (open red circles). There are half as many skyrmions as pinning sites and only the lower half of the sample contains pinning. The drive can be applied along either the $x$ or the $y$ direction.

Figure 2

(a) $\langle V_{\left|\right|}\rangle$ (blue solid line) and $\langle V_{\perp }\rangle$ (red line) vs $F_D$ for a sample with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=1.0$. Vertical dashed lines indicate the four dynamical phases: I, longitudinal flow in the $x$ direction in only the pin-free channel; ${\mathrm{II}}_{sb}$, the shear banding phase; ${\mathrm{III}}_{pl}$, plastic flow; ML, a moving liquid; and MC, a moving crystal. (b) The corresponding skyrmion Hall angle ${\theta }_{sk}$ vs $F_D$ showing that ${\theta }_{sk}$ increases from zero in phase ${\mathrm{III}}_{pl}$ and reaches a saturation value in the ML and MC phases. (c) The corresponding fraction of sixfold coordinated skyrmions $P_6$ vs $F_D$, showing changes across each of the four dynamic phase transitions.

Figure 3

Skyrmion locations (blue dots), pinning site locations (open circles), and skyrmion trajectories (green lines) during a fixed time interval for the system in Fig. 2 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=1.0$. (a) Phase I at $F_D=0.1$, where the flow is only in the unpinned region. (b) Phase ${\mathrm{II}}_{sb}$ at $F_D=0.2$, where there is flow in both the unpinned and pinned regions but the skyrmion Hall angle is zero. (c) Phase ${\mathrm{III}}_{pl}$ at $F_D=0.5$ where the skyrmion Hall angle is finite. (d) The pinning sites and skyrmion locations without trajectories in the moving crystal phase MC at $F_D=1.5$.

Figure 4

A blowup of the $\langle V_{\left|\right|}\rangle$ (blue solid line) and $\langle V_{\perp }\rangle$ (red line) vs $F_D$ curves from Fig. 2 highlighting the I-${\mathrm{II}}_{sb}$ transition at $F_D=0.16$. The blue dashed line is the velocity $\langle V_{\left|\right|}\rangle$ in the pin-free limit.

Figure 5

The structure factor $S\left(\mathbf{q}\right)$ for the system in Fig. 2 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=1.0$. (a) Phase I; (b) Phase ${\mathrm{II}}_{sb}$; (c) Phase ${\mathrm{III}}_{pl}$; (d) ML phase; (e) MC phase. (f) The same system at $F_p=1.0$ has a moving square lattice phase described in Sec. 4.

Figure 6

The velocities spatially averaged over the $x$ direction as a function of $y$ for the system in Fig. 2 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=1.0$. (a) $\langle V_{\left|\right|}\rangle$ and (b) $\langle V_{\perp }\rangle$ in phase I at $F_D=0.04$. (c) $\langle V_{\left|\right|}\rangle$ and (d) $\langle V_{\perp }\rangle$ in phase ${\mathrm{II}}_{sb}$ at $F_D=0.25$. The vertical dashed lines indicate the separation between the pinned region $\left(y\le 18\right)$ and the pin-free region $\left(y>18\right)$.

Figure 7

The velocities spatially averaged over the $x$ direction as a function of $y$ for the system in Fig. 2 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=1.0$. (a) $\langle V_{\left|\right|}\rangle$ and (b) $\langle V_{\perp }\rangle$ in phase ${\mathrm{III}}_{pl}$ at $F_D=0.5$. (c) $\langle V_{\left|\right|}\rangle$ and (d) $\langle V_{\perp }\rangle$ in the MC phase at $F_D=1.5$. The vertical dashed lines indicate the separation between the pinned region $\left(y\le 18\right)$ and the pin-free region $\left(y>18\right)$.

Figure 8

(a) $\langle V_{\left|\right|}\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) vs $F_D$ for the system in Fig. 2 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=0.$ We find phase I and a moving smectic (MS) phase as illustrated in Fig. 9. (b) The corresponding fraction of sixfold coordinated skyrmions $P_6$ vs $F_D$ showing that in the MS phase there is a jump up to $P_6=0.95$.

Figure 9

The Voronoi construction for the skyrmion locations for the system in Fig. 8 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=0$ at $F_D=1.0$ in the MS phase. The coordination number of individual skyrmions is 5 (red), 6 (black), 7 (blue), or outside the range of 5–7 (green). Fivefold and sevenfold coordinated defects form pairs that have their Burgers vector aligned with the driving direction. (b) The corresponding structure factor $S\left(\mathbf{k}\right)$ indicates the presence of smectic order.

Figure 10

(a) $\langle V_{\left|\right|}\rangle$, (b) $\langle V_{\perp }\rangle$, and (c) $P_6$ vs $F_D$ for the sample in Fig. 2 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=0.0125$. The vertical lines distinguish the different dynamic phases. I, longitudinal flow in only the pin-free channel; MS, moving smectic; ML, moving liquid; MC, moving crystal. A drop in $P_6$ marks the window of disordered moving liquid between the moving smectic (MS) and moving crystal (MC) phases.

Figure 11

Skyrmion locations (blue dots) and pinning site locations (open circles) for the system in Fig. 10 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=0.0125$. (a) The MS phase at $F_D=2.5$, with skyrmion trajectories during a fixed time interval drawn as green lines. (b) The same as (a) with only the skyrmion locations shown, indicating that the skyrmion lattice is aligned in the $x$ direction, parallel to the drive. (c) The ML phase at $F_D=7.0$. (d) The MC phase at $F_D=7.5$.

Figure 12

(a) $\langle V_{\left|\right|}\rangle$ (blue) and $-\langle V_{\perp }\rangle$ (red) vs $F_D$ for the system in Fig. 10 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=0.25$. Here we have multiplied $\langle V_{\perp }\rangle$ by $-1$ for clarity. (b) The corresponding $P_6$ vs $F_D$ showing the absence of the MS phase.

Figure 13

The dynamic phase diagram as a function of $F_D$ vs ${\alpha }_m/{\alpha }_d$ for the low Magnus force regime. Phase I, olive green; moving smectic (MS) phase, teal; moving liquid (ML) phase, brown; plastic flow phase ${\mathrm{III}}_{pl}$, pink; shear band phase ${\mathrm{II}}_{sb}$, red; and the moving crystal (MC) phase, light blue. The MS phase occurs only when ${\alpha }_m/{\alpha }_d<0.1$. (b) The same data on a log-log scale. The dashed orange line is a fit to $F_D\propto 1/({\alpha }_m/{\alpha }_d)$, showing that the drives at which the MS-ML and ML-MC phase transitions occur diverge as ${\alpha }_m/{\alpha }_d$ decreases.

Figure 14

(a) $\langle V_{\left|\right|}\rangle$ (blue) and $-\langle V_{\perp }\rangle /6$ (red) vs $F_D$ for a system with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=6.0$. For clarity, we have multiplied $\langle V_{\perp }\rangle$ by $-1$ and divided it by the value of ${\alpha }_m/{\alpha }_d$ in order to normalize $\langle V_{\perp }\rangle$ to the value of $\langle V_{\left|\right|}\rangle$ at high drives. (b) The corresponding velocity deviations $\delta V_{\left|\right|}$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$, where we find an extended window of moving liquid (ML) phase. ${\mathrm{III}}_{pl}$, plastic flow phase; MC, moving crystal phase.

Figure 15

Skyrmion locations (blue dots), pinning site locations (open circles), and skyrmion trajectories (green lines) during a fixed time interval for a system with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/\alpha =2.5$, where the shear banding phase is replaced with the avalanche phase $I{I}_a$. Moving skyrmions from the unpinned region enter the pinned region in the form of avalanches at (a) $F_D=0.1$, (b) $F_D=0.2$, and (c) $F_D=0.3$. (d) The plastic flow phase ${\mathrm{III}}_{pl}$ at $F_D=0.5$.

Figure 16

$\langle V_{\left|\right|}\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) vs $F_D$ in a sample with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=10$. Above phase ${\mathrm{II}}_a$, there is a region of large oscillations in the intermittent phase ${\mathrm{I}}_{in}$, where large-scale phase-separated flow occurs as shown in Fig. 17.

Figure 17

Skyrmion locations (blue dots), pinning site locations (open circles), and skyrmion trajectories (green lines) during a fixed time interval for the system in Fig. 16 with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=10$ in phase ${\mathrm{I}}_{in}$. (a) $F_D=0.12$, (b) $F_D=0.25$, (c) $F_D=0.3185$, and (d) $F_D=0.325$.

Figure 18

The dynamic phase diagram as a function of $F_D$ vs ${\alpha }_m/{\alpha }_d$ in samples with $x$ direction driving at $F_p=0.75$, showing phases I (olive green), ${\mathrm{II}}_{sb}$ (red), ${\mathrm{III}}_{pl}$ (pink), ${\mathrm{II}}_a$ (light purple), ${\mathrm{I}}_{in}$ (violet), ML (brown), and MC (blue). Here phase ${\mathrm{II}}_a$ appears when ${\alpha }_m/{\alpha }_d>1.9$ and phase ${\mathrm{I}}_{in}$ appears when ${\alpha }_m/{\alpha }_d\ge 7.0$.

Figure 19

The evolution of ${\theta }_{sk}$ vs $F_D$ for samples with $x$ direction driving at $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=10$, 7, 5, 2.5, 1.75, 1.25, 1.0, 0.75, 0.7, 0.5, 0.375, 0.25, 0.175, and 0.125, from top to bottom.

Figure 20

A system with $x$ direction driving at ${\alpha }_m/{\alpha }_d=1.0$ and $F_p=0.01$, where an elastic depinning transition in which the skyrmions keep their same neighbors separates the pinned phase P from the MC phase. (a) $\langle V_{\left|\right|}\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) vs $F_D$. (b) The corresponding $P_6$ vs $F_D$.

Figure 21

Dynamic phase diagram as a function of $F_D$ vs $F_p$ in the weak pinning regime for samples with $x$ direction driving and ${\alpha }_m/{\alpha }_d=1.0$, showing the pinned phase (yellow), moving crystal MC (light blue), plastic flow ${\mathrm{III}}_{pl}$ (pink), and phase I (olive green).

Figure 22

Velocity deviations $\delta V_{\left|\right|}$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$ for samples with $x$ direction driving and ${\alpha }_m/{\alpha }_d=1.0$ showing a transition from a triangular moving crystal MC to a square moving crystal ${\mathrm{MC}}_{sq}$. (a) $F_p=0.8$. (b) $F_p=1.0$.

Figure 23

${\theta }_{sk}$ vs $F_D$ showing the evolution of the skyrmion Hall angle across the ML-MC-${\mathrm{MC}}_{sq}$ transitions for the system in Fig. 21 with $x$ direction driving and ${\alpha }_m/{\alpha }_d=1.0$. (a) $F_p=0.8$. (b) $F_p=1.0$.

Figure 24

Dynamic phase diagram as a function of $F_D$ vs $F_p$ for the intermediate pinning regime in samples with $x$ direction driving and ${\alpha }_m/{\alpha }_d=1.0$ showing phases I (olive green), ${\mathrm{II}}_{sb}$ (red), ${\mathrm{III}}_{pl}$ (pink), ML (brown), MC (light blue), and ${\mathrm{MC}}_{sq}$ (dark blue).

Figure 25

(a) $\langle V_{\left|\right|}\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) vs $F_D$ for a sample with $x$ direction driving, ${\alpha }_m/{\alpha }_d=1.0$, and $F_p=4.5$ showing regions in which $d\langle V_{\left|\right|}\rangle /d{F}_D<0.0$, also known as negative differential mobility, due to the trapping of multiple skyrmions by individual pinning sites. (b) The corresponding ${\theta }_{sk}$ vs $F_D$ where multiple regimes appear. (c) $\langle V_{\left|\right|}\rangle$ and $\langle V_{\perp }\rangle$ vs $F_D$ for a sample with $x$ direction driving, ${\alpha }_m/{\alpha }_d=4.0$, and $F_p=4.5$. (d) The corresponding ${\theta }_{sk}$ vs $F_D$.

Figure 26

Skyrmion locations (blue dots) and pinning site locations (open circles) for the system in Fig. 25 with $x$ direction driving at ${\alpha }_m/{\alpha }_d=1.0$ and $F_p=4.5$ for $F_D=1.0$, just after the drop down in $\langle V_{\left|\right|}\rangle$, showing multiple skyrmion trapping by individual pinning sites.

Figure 27

Driving in the $y$ direction, perpendicular to the pinning strip, for the system in Fig. 2 with $F_p=0.75$ and ${\alpha }_m/{\alpha }_d=1.0$. The vertical dashed lines denote transitions among a jammed phase J, a moving plastic flow phase ${\mathrm{III}}_{pl}$, a ML, and a ${\mathrm{MC}}_{sq}$ phase. (a) $\langle V_{\left|\right|}\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) vs $F_D$. (b) $\delta V_{\left|\right|}$ (blue) and $\delta V_{\perp }$ (red) versus $F_D$. (c) $P_6$ vs $F_D$. Velocities are always measured parallel or perpendicular to the direction of the pinning strip.

Figure 28

Skyrmion locations (blue dots), pinning site locations (open circles), and skyrmion trajectories (green lines) during a fixed time interval for the system in Fig. 27 with $y$ direction driving, $F_p=0.75$, and ${\alpha }_m/{\alpha }_d=1.0$. (a) Transient rearrangements in the jammed phase J at $F_D=0.06$. (b) Phase ${\mathrm{III}}_{pl}$ at $F_D=0.25$, where the skyrmions can cross the entire system. (c) The ML phase at $F_D=1.0$, showing the skyrmion locations only. (d) The moving square lattice ${\mathrm{MC}}_{sq}$ phase at $F_D=1.5$ showing the skyrmion locations only.

Figure 29

(a) $80\langle V_{\left|\right|}\rangle$ (blue) and $\langle V_{\perp }\rangle$ (red) vs $F_D$ for the system with $y$ direction driving at ${\alpha }_m/{\alpha }_d=0.0125$ and $F_p=0.75$. For clarity, we have divided $\langle V_{\left|\right|}\rangle$ by the value of ${\alpha }_m/{\alpha }_d$ in order to normalize $\langle V_{\left|\right|}\rangle$ to the value of $\langle V_{\perp }\rangle$ at high drives. (b) The corresponding $\delta V_{\left|\right|}$ (blue) and $\delta V_{\perp }$ (red) vs $F_D$. (c) The corresponding $P_6$ vs $F_D$. We observe a jammed phase (J), plastic flow phase $\left({\mathrm{III}}_{pl}\right)$, partially locked phase (PL), and a moving crystal phase (MC).

Figure 30

The Voronoi construction for the skyrmion locations for the system in Fig. 29 with $y$ direction driving, $F_p=0.75$, and ${\alpha }_m/{\alpha }_d=0.0125$. The red dashed line indicates the edge of the pinned region. (a) The PL phase at $F_D=1.0$, where there is a change in the orientation of the crystal across the boundaries separating the unpinned and pinned regions. (b) The MC phase at $F_D=2.5$. The dislocation pair in the pinned region is in the process of gliding and temporarily has more than seven neighbors on one end.

Figure 31

Dynamic phase diagram as a function of $F_D$ vs ${\alpha }_m/{\alpha }_d$ for driving in the $y$ direction for a system with $F_p=0.75$. Partially locked phase (PL), dark green; jammed phase (J), tan; plastic flow $\left({\mathrm{III}}_{pl}\right)$, pink; intermittent $\left({\mathrm{I}}_{in}\right)$, violet; moving liquid (ML), brown; moving crystal (MC), light blue. At ${\alpha }_m/{\alpha }_d=1.0$ there is a moving square lattice $\left({\mathrm{MC}}_{sq}\right)$, dark blue dashed line.