Nonlinear Transport, Dynamic Ordering, and Clustering for Driven Skyrmions on Random Pinning

Nonlinear Transport, Dynamic Ordering, and Clustering for Driven Skyrmions on Random Pinning

C. Reichhardt and C. J. O. Reichhardt Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
July 27, 2019
Abstract

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.

I Introduction

There are a wide variety of systems that can be effectively described as a collection of particles that couple to a randomly disordered substrate 1 (); 2 (). Specific examples include vortices in type-II superconductors with naturally occurring defects 3 (); 4 (); 5 (); 6 (), colloidal particles on disordered landscapes 16 (); 17 (); 18 (); 19 (); 20 (), pattern forming systems on rough surfaces 21 (); 22 (); 23 (); 24 (), Wigner crystals in the presence of charged impurities 25 (); 26 (); 27 (); 28 (), active matter or self-propelled particles in complex environments 29 (); 30 (), fluid flow over disordered surfaces 31 (); 32 (); 33 (), granular matter flowing over disordered backgrounds 34 (), various models of sliding friction 35 (), dislocation dynamics 36 (); 37 (), and geological systems such as plate tectonics 38 (). Under an applied drive, these systems typically exhibit a pinned phase at low drive that transitions to a sliding state at higher drives, and additional transitions can occur within the sliding state between different types of flowing phases 1 (); 2 (). In an elastic depinning transition, the particles keep their same neighbors, while during plastic depinning, the particles exchange neighbors, leading to a proliferation of topological defects in the form of dislocations 1 (); 2 (); 3 (); 39 (); 40 (). Some systems can be described in terms of the depinning of polycrystalline states composed of large ordered regions separated by mobile grain boundaries 41 (), a process that produces distinct transport features compared to strongly disordered systems in which local ordering extends only a few lattice spacings or less 39 (); 41 (). These different types of depinning phenomena produce different features in the velocity-force curves, differential conductivity, velocity fluctuation spectra, and global structure of the particles 1 (); 2 (); 3 (). Plastic depinning can be followed by a dynamical transition at higher drives from the disordered moving plastic state into a moving crystal 1 (); 3 (); 5 () or moving smectic state 42 (); 43 (); 44 (); 45 (); 46 (), where the system regains considerable ordering when the high velocity motion of the particles reduces the effectiveness of the pinning.

In certain systems such as vortices in type-II superconductors, when the pinning strength is increased or the elastic constant of the vortex lattice is reduced, there can be a transition from elastic to plastic depinning accompanied by a sharp increase in the critical depinning force, called the “peak effect,” which is also associated with changes in the shape of the velocity-force curves 2 (); 3 (); 4 (); 47 (); 48 (). The velocity typically increases monotonically with driving force when the substrate is disordered; however, the velocity-force curve can be nonlinear and exhibit scaling of the form , where is the depinning force and the value of depends on whether the depinning transition is elastic or plastic 1 (); 2 (); 39 (); 40 (). If the substrate is periodic, the velocity-force curves can show a decrease in the particle velocity with increasing , giving rise to negative differential conductivity 1 (); however, for random substrates, negative differential conductivity is generally not observed.

In 2009, the discovery of skyrmions in chiral magnets opened the study of a new type of particle-like objects that can form an elastic lattice. The initial neutron scattering measurements 49 () revealed sixfold ordering, indicating that the skyrmions form a triangular lattice similar to the vortex lattices found in type II superconductors 49 (). A short time later, hexagonal skyrmion lattices were directly observed with Lorentz microscopy 50 (). Since then, there has been an enormous increase in skyrmion studies, and a variety of new materials have been identified that are capable of supporting skyrmions, including some in which the skyrmions are stable at room temperature 51 (); 52 (); 53 (); 54 (); 55 (); 56 (). It was shown that skyrmions can be set into motion by the application of a current, leading to depinning transitions that can be detected via changes in the transport properties 57 (); 58 (); 59 (); 60 (); 61 () or through direct imaging 51 (); 53 (); 54 (); 56 (); 62 (); 63 (); 64 (). Skyrmions show great promise for numerous applications due to their size and mobility, so understanding their collective dynamics is of key importance for manipulating dense skyrmion arrays 65 ().

Among systems that undergo nonequilibrium dynamical transitions when driven over quenched disorder, skyrmions exhibit a new class of behavior, since unlike previously studied systems, the topological nature of the skyrmions causes their dynamics to be dominated by a non-dissipative Magnus force 58 (). The Magnus force generates a velocity component perpendicular to the net force experienced by the skyrmion, and as a result, the skyrmions move at an angle, called the skyrmion Hall angle , to an applied driving force 54 (); 57 (); 58 (); 62 (); 63 (); 64 (); 66 (); 67 (); 68 (); 69 (). The Magnus force also strongly modifies the interactions of the skyrmions with pinning sites or barriers. For localized pinning, the Magnus force causes the skyrmion to rotate around the edge of the pinning site, greatly reducing the depinning threshold compared to the overdamped limit. This effect was argued to be one of the reasons that low critical depinning forces have been observed in certain skyrmion systems 54 (); 57 (); 58 (); 58 (); 59 (); 66 (); 67 (); 68 (); 69 (). For line defects or spatially extended pinning sites, the skyrmions cannot avoid the pinning by moving around it, so the effective pinning strength remains large even when the Magnus force is finite 70 (); 71 (); 72 ().

Although the intrinsic skyrmion Hall angle is a constant determined by the material parameters 41 (), the measured has a strong drive dependence in the presence of pinning, as initially observed in particle-based simulations of skyrmions moving in random 67 (); 73 (); 74 () and periodic pinning arrays 75 (). At the depinning threshold, , and as the skyrmion velocity increases with increasing drive, increases until it saturates at . Imaging experiments of driven skyrmions reveal a pinned regime, a creep regime where flow occurs in jumps giving , and a viscous flow regime, where increases with increasing and saturates at higher drives 62 (). Other experiments have also shown an increase in with increasing drive amplitude 63 (); 64 (). The drive dependence of arises when the Magnus force induces a shift in the motion of the skyrmions as they pass over the pinning sites 67 (); 69 (); 73 (); 74 (); 75 (), similar to the side jump effect found for an electron scattering off of magnetic impurities. The faster the skyrmion moves, the smaller the shift, as illustrated in particle-based simulations. Continuum-based simulations of driven skyrmions show that in the absence of pinning, is constant, while in the presence of pinning, the velocity-force curves are nonlinear and increases with increasing drive from zero up to the intrinsic value 54 (); 68 (), indicating that the particle-based skyrmion simulations successfully capture many features of skyrmion dynamics even though they do not include the internal degrees of freedom of the skyrmions.

Simulations reveal that a transition occurs from a skyrmion crystal to a disordered skyrmion glass state when the pinning strength increases 67 (). The crystal depins elastically while the glass depins plastically, but at higher drives the glassy state can dynamically order into a moving skyrmion crystal in a process that is similar to the dynamical ordering transitions observed in the depinning and sliding of vortices 1 (); 3 (); 5 (); 44 (); 45 (); 46 (); 47 (), colloids 16 (); 18 (); 19 (), and Wigner crystals 27 (). The Magnus force causes the dynamical fluctuations of the moving skyrmions to be much more isotropic than those found in driven overdamped systems, favoring the emergence of an isotropic moving crystal rather than the moving smectic state observed in overdamped systems 74 (). Continuum-based simulations of skyrmions on random substrates show similar dynamical ordering under increasing drive when the substrate is only moderately strong 76 (), and recent experimental neutron scattering data gives evidence for the dynamic ordering of driven skyrmions at high drives 77 ().

In this work we expand upon our previous particle-based studies of skyrmion dynamics in random pinning, and perform a detailed study of the skyrmion transport, structure, and dynamics as we vary the pinning strength, the ratio of skyrmions to pinning sites, and the ratio of the Magnus force to the damping term. For weak pinning, the skyrmions form a triangular lattice that depins elastically into a moving triangular lattice. Even though there is no structural transition between the pinned and moving states, we find a dip in the weight of the structure factor peaks at the depinning transition due to the deviations of the skyrmions from the lattice positions, followed by a sharpening of the peaks with increasing drive as the effectiveness of the pinning decreases. For increasing pinning strength, a transition from a pinned crystal state to a disordered skyrmion glass occurs that is accompanied by a sharp increase in the critical depinning force , similar to the peak effect phenomenon found at the transition from elastic to plastic depinning in superconducting vortex systems 3 (); 6 (); 47 (); 48 (). We also find that the scaling of with the pinning strength crosses over from a quadratic form in the elastic depinning regime to linear scaling in the plastic depinning regime 1 (); 2 (); 4 (). These results suggest that a peak effect phenomenon could be a general feature in skyrmion systems that occurs as a function of magnetic field, temperature, or lattice shearing. We observe a change in the scaling of the velocity-force curves across the transition from elastic to plastic depinning, including a region of negative differential mobility, and we explain these effects in terms of the drive dependence of . When the pinning strength is high or the Magnus force is large, we observe clustered or density segregated states, where an effective attraction between the skyrmions leads to the formation of dense moving bands separated by low density regions. In recent continuum-based simulations 76 (), a similar clustered or segregated state appeared at strong pinning and was attributed to the generation of spin waves by the moving skyrmions. In our work, the effective attraction is a result of the Magnus force and occurs when the pinning causes the skyrmions to move at different relative velocities, giving them a tendency to rotate around each other rather than moving parallel to each other. The phase separated states only occur when both the Magnus force and the pinning are sufficiently strong. For weaker pinning, we find a dynamical phase separated state in which the skyrmion density is uniform but there is a coexistence of localized bands of motion and pinned bands.

The paper is organized as follows. In Section II we describe the details of our simulation technique. In Section III we show a transition from elastic to plastic depinning as a function of pinning strength that is accompanied by a sudden change in the critical depinning force, called a peak effect, and we demonstrate that the nonlinearity in the velocity-force curves is distinct from that found for overdamped systems. In Section IV we study the impact of the skyrmion density on the peak effect in the critical depinning force, describe the appearance of transverse locking in the elastic depinning regime, and show crossing of the velocity-force curves in samples on either side of the transition from elastic to plastic depinning. Section V shows how the dynamic phases change when the ratio of the Magnus force to the damping term is varied. The density phase separation and dynamic phase separation that appear for strong pinning and strong Magnus force are described in Section VI. We summarize our results in Section VII.

Ii Simulation

We consider a two-dimensional system of size with periodic boundaries in the and directions containing rigid skyrmions modeled as point particles. The dynamics of the skyrmions is obtained by integrating a modified version of the Thiele equation that takes into account skyrmion-skyrmion interactions, skyrmion-pin interactions, and an external driving force 19 (); 20 (); 21 (); 31 (). The equation of motion of a skyrmion is

(1)

where is the skyrmion velocity, is the damping constant which tends to align the skyrmion velocity in the direction of the external forces, and is the strength of the Magnus term which tends to align the skyrmion velocity in the direction perpendicular to the external forces. When both and are finite, the skyrmions move at an angle called the intrinsic skyrmion Hall angle, , with respect to an externally applied driving force. The skyrmion-skyrmion interaction is a short range repulsive force of the form , where is the modified Bessel function, is the distance between skyrmion and skyrmion , and 19 (); 30 (); 31 (). We place pinning sites in random but non-overlapping positions and model each pinning site as a parabolic trap of range that can exert a maximum pinning force of on a skyrmion of the form , where is the distance between skyrmion and pin , , and is the Heaviside step function. The skyrmion density is , the pinning density is , and . The driving force is , and we measure the average velocity per skyrmion parallel, , and perpendicular, , to the applied drive. We compute the drive-dependent skyrmion Hall angle , as well as quantities related to the standard deviation of the skyrmion velocities in the parallel and perpendicular directions, and . We also characterize the dynamics as a function of driving force by measuring the structure factor and the average fraction of sixfold-coordinated particles , where is the coordination number of skyrmion obtained from a Voronoi tessellation.

Iii Elastic to Plastic Depinning

Figure 1: (a) , the magnitude of the structure factor peak at one of the reciprocal lattice vectors of the skyrmion lattice, vs for the weak pinning case in a sample with , , and , where the skyrmions form a pinned crystal that depins elastically. (b) The corresponding , the average skyrmion velocity parallel to the driving direction, vs . The dip in occurs at the drive for which becomes finite.
Figure 2: from the system in Fig. 1 with , , , and , where the depinning is elastic and . (a) At , there are six sharp peaks indicative of triangular ordering. (b) Just at depinning for , the peaks are still present but there is some smearing. (c) At in the sliding phase, the system is more ordered.

Previous work with the particle-based skyrmion model showed that sufficiently strong pinning causes the system to form a glassy state that depins plastically and then dynamically orders at higher velocities into a moving crystal that can be detected using the structure factor or 67 (); 74 (). Here we show that even when the pinning is weak, so that the skyrmions form a pinned lattice rather than a glass and retain their sixfold ordering across the depinning transition, there are still detectable changes in the positional ordering of the skyrmions at the depinning transition. In Fig. 1(a) we plot , the magnitude of the structure factor at one of the reciprocal lattice vectors of the skyrmion lattice, versus in a sample with , , , and , while in Fig. 1(b) we show versus for the same sample. Here the skyrmions depin elastically, so there is no generation of topological defects at the depinning transition, which occurs at . Nevertheless, there is a dip in at the depinning transition, and for , gradually increases with increasing . In Fig. 2(a) we show for the same sample at in the pinned state where there is clear sixfold ordering. Figure 2(b) indicates that at the depinning transition , maintains its sixfold ordering but the peaks become slightly smeared, while in Fig. 2(c) at in the moving phase, the peaks sharpen again. We find a similar trend for other values of within the elastic depinning regime.

Figure 3: , the fraction of sixfold coordinated skyrmions, vs , measured at the depinning transition for the system in Fig. 1 with , , and . (b) The corresponding value of the critical depinning force vs . A transition occurs from elastic depinning, where the skyrmions maintain sixfold ordering, to plastic depinning, marked by a drop in and a sharp increase in . 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.

By conducting a series of simulations and examining the velocity-force curves and skyrmion positions for the system in Fig. 1, we construct a plot of the fraction of particles with sixfold ordering at the depinning transition versus , shown in Fig. 3(a). For , , since the skyrmions retain six neighbors at the elastic depinning transition, while for , drops since there is a proliferation of topological defects in the form of 5-7 paired dislocations during plastic depinning. In Fig. 3(b) we plot the critical depinning force versus . For the value of at which the drop in occurs, we find a sharp increase in , indicating that the skyrmions have become much more strongly pinned. The sudden increase in at the transition from elastic depinning to plastic depinning is very similar to the phenomenon known as the peak effect in type-II superconductors, where the weakly pinned elastic vortex lattice undergoes a sharp increase in the depinning force when the vortex lattice disorders and becomes well coupled to the pinning 1 (); 3 (); 6 (). In superconducting systems, the strength of the pinning sites is fixed, but the vortex-vortex interaction strength can be modified by changing the temperature or magnetic field, producing a transition to plastic depinning once the vortex-vortex interaction strength drops below the vortex-pin interaction strength. Our results indicate that a similar peak effect phenomenon should be observable in skyrmion systems.

Figure 4: (a) vs plotted on a log-log scale for the system in Fig. 1 with , , and . The dashed lines are power law fits to . Blue: in the elastic depinning regime; red: in the plastic depinning regime. (b) The corresponding versus curve has a peak near at the transition from elastic to plastic depinning. (c) The corresponding versus curve shows that the elastic to plastic depinning transition coincides with a sharp drop in .

In Fig. 4(a) we plot versus on a log-log scale for the system in Fig. 3, while in Fig. 4(b) we show the corresponding versus curve. In two-dimensional systems that exhibit elastic depinning, the depinning force is expected to scale as with , while in the plastic depinning regime, there is a similar scaling with 1 (); 4 (). In Fig. 4(a), the dashed lines indicate power law fits with at smaller in the elastic depinning regime and at larger in the plastic depinning regime, showing good agreement with the expected scalings. At the transition from elastic to plastic depinning, a peak appears in separating a linear increase of with in the elastic depinning regime from a constant in the plastic depinning regime, consistent with the power law scaling in the vs curves. The peak in coincides with a sharp drop in , as shown in Fig. 4(c), and continues to decrease with increasing throughout the plastic flow regime.

Figure 5: vs for the system in Fig. 1 with , , and at (red), (orange), , (light blue) and (dark blue). For large drives, increases toward as the system dynamically reorders. The plateau in for corresponds to a moving liquid phase in which all the skyrmions are moving but the system is disordered.

In Fig. 5 we plot versus for the system in Fig. 4 at , , , and . For , at all values of , indicating that the skyrmions retain their sixfold ordering. At where the depinning is plastic, dips down to in the plastic flow phase above depinning but increases to near when the skyrmions dynamically regain their triangular ordering, similar to what was observed previously 67 (); 74 (). For and , we find an additional shoulder feature above the plastic depinning transition, and for curve this shoulder is followed by a plateau with over the range , above which the system dynamically orders. In the plastic flow phase, there is a coexistence of moving and pinned skyrmions, while in the moving liquid phase on the plateau, all of 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 for the system in Fig. 5 with , , and at . (a,b) At , the plastic flow phase contains both pinned and moving skyrmions, and shows that the structure is random. (c,d) At , the skyrmion density is more uniform but the system is still disordered and forms a moving liquid (ML) phase, in which develops a ringlike feature. (d,e) In the dynamically ordered moving crystal (MC) phase at , the skyrmions form a triangular lattice.

In Fig. 6(a,b) we show the skyrmion positions and for the system in Fig. 5 at in the plastic flow phase at , where the skyrmion structure is disordered and there is a tendency for a density phase separation to occur, as discussed in more detail in Section VI. The structure factor is nearly featureless, as expected for a random spatial distribution. Figure 6(c,d) illustrates the skyrmion positions and for the same system in the moving liquid phase at , where the skyrmion density is more uniform but the skyrmion arrangement is still disordered, and where the structure factor contains a ring feature consistent with a liquid state. We note that it is possible to distinguish between different types of random structures. For example, a random arrangement of particles created using a Poisson process gives a structure factor that has finite weight for , while in an arrangement of particles with what is called a disordered hyperuniform structure, approaches zero as 78 (). It has been argued that in the presence of random pinning, an assembly of superconducting vortices exhibits a disordered hyperuniform structure when the vortex-vortex interactions are sufficiently strong, while when the quenched disorder dominates all other energy scales, a Poisson random arrangement of vortices appears 79 (). One possibility is that the plastic flow phase of the skyrmions is Poisson random while the moving liquid phase has a disordered hyperuniform structure. In Fig. 6(e,f) we plot the skyrmion positions and for the moving crystal phase at , where the skyrmions have strong triangular ordering and shows sharp sixfold peaks.

iii.1 Nonlinear Velocity-Force Curves

Figure 7: (a) and (b) , the average skyrmion velocity parallel and perpendicular to the drive, respectively, vs for the system in Fig. 5 with , , and at . Each curve has different nonlinear behavior near the depinning threshold. (c) The corresponding velocity deviations (blue) and (red) vs , showing the strong fluctuations in the plastic flow regime. The letters a, c, and e in panel (c) indicate the values of 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).

In Fig. 7(a,b) we plot and versus for the system in Fig. 5 at . Here we find a pinned phase for , a plastic flow or phase segregated (PL-PS) phase for , a moving liquid (ML) phase for , and a moving crystal (MC) phase for . In the pinned phase, , and only small shifts in the particle positions occur as is increased. In the plastic flow or partially phase separated state, remains nearly constant while the magnitude of increases linearly with increasing . A cusp in appears at the transition from plastic flow to the moving liquid phase, above which begins to increase linearly with . There is little change in the behavior of through the PL-PS, ML, and MC phases, nor does any signature of the transition from the ML to the MC phase appear in . In Fig. 7(c) we plot the velocity deviations and versus . In the PL phase, these velocity fluctuations are largest in the perpendicular direction, while in the ML and MC phases, the fluctuations are reduced and become mostly isotropic. These results show that the strong nonlinearity of the velocity-force curves at lower drives has a very different character than that found for overdamped particles such as superconducting vortices that exhibit plastic depinning, where increases monotonically with according to with 1 (); 2 (). No such scaling can be applied to the vs curve in the skyrmion system, while the curve has . Clearly the Magnus force strongly modifies the scaling of the velocity-force curves. It has been argued that non-dissipative effects can change the nature of the depinning transition from continuous to discontinuous when inertia or stress overshoots are included 1 (); 80 (). In our system, we find that the versus curves begin to develop a discontinuous jump at depinning when the Magnus term is finite.

Figure 8: Dynamic phase diagram as a function of vs for the system in Figs. 1 to 7 with , , and , 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.

By conducting a series of simulations for varied and analyzing features in the velocity-force and curves, we construct the dynamic phase diagram as a function of versus shown in Fig. 8. The pinned crystal phase depins elastically into a moving crystal phase. In contrast, the pinned skyrmion glass phase depins into a plastic flow phase, which transitions into a moving crystal phase for lower or a moving liquid for higher . The moving liquid phase transitions into a moving crystal at higher drives. For , we find an increasing amount of clustering occurring in the plastic flow phase. Overall the phase diagram is similar to that found in overdamped systems, except that in the latter, the moving crystal phase is replaced by a moving smectic phase since the dynamically generated fluctuations experienced by the particles are highly anisotropic. In the skyrmion system, the Magnus force mixes the fluctuations in the driving direction into the transverse direction, giving more isotropic fluctuations that cause the particles to reorder into a moving crystal rather than a moving smectic state. The isotropy of the fluctuations increases as the Magnus force increases, as studied in detail in previous work 74 ().

Iv Dynamics as a Function of Skyrmion Density

Figure 9: (a) vs for samples with , , and . (b) The corresponding vs . The transition from elastic to plastic depinning occurs near . For , the behavior of the system is in the single skyrmion limit. (c,d) Zoomed in plots of panels (a) and (b), respectively, for the region near where the transition from elastic to plastic depinning occurs. A drop in is correlated with an increase in , which is similar to the peak effect phenomenon.

We next consider the effect of varying the skyrmion density while holding the pinning density and pinning strength fixed. Such a situation could be achieved experimentally by varying the magnetic field. In Fig. 9(a,b) we plot versus for samples with , , and . An elastic solid with triangular ordering forms when , where we find and low . For , we observe a pinned skyrmion glass that depins plastically, as indicated by the increase in and the drop in . For , the distance between neighboring skyrmions becomes so great that the system enters the limit of non-interacting skyrmions, in which and the skyrmions are better described as a pinned gas. Figure 9(c,d) shows a blowup of Fig. 9(a,b) near where the transition from elastic to plastic depinning occurs. There is a clear increase in at the transition produced when topological defects appear in the system and drops below 1. The transition to plastic depinning with decreasing skyrmion density occurs due to the corresponding decrease in the shear modulus of the skyrmion lattice. Once the shear modulus becomes small enough, topological defects percolate within the lattice, producing a field-induced peak effect phenomenon.

iv.1 Transverse Locking in the Elastic Depinning Regime

In the elastic depinning regime, the skyrmions depin as a unit and maintain their triangular lattice ordering. The moving lattice can become oriented in the direction of drive or may remain oriented in some particular direction due to the geometry of the sample. In some cases, it may be necessary for topological defects to nucleate inside the lattice in order to permit it to rotate and orient with the driving direction, and since these defects are energetically costly, the lattice orientation may remain fixed in the moving state even when the drive, and the effective skyrmion Hall angle, increase.

Le Doussal and Giamarchi argued that a moving superconducting vortex lattice interacting with pinning and subjected to a longitudinal drive can exhibit a finite threshold for transverse depinning when an additional drive is applied perpendicular to the longitudinal drive, and that the lattice can begin sliding along the transverse direction as well as the longitudinal direction when is large enough 43 (). This transverse critical force has been observed in simulations of moving superconducting vortices 45 (); 81 (); 82 (); 83 (); 84 () and Wigner crystals 27 (); 83 () as well as in superconducting vortex experiments 85 (); 86 (). Le Doussal and Giamarchi also predicted that if the system forms a moving triangular lattice which remains oriented in a particular direction, then under an additional transverse drive, the initial transverse depinning threshold is followed by a series of higher-order transverse depinning thresholds that appear whenever the vector of net applied force aligns with a symmetry direction of the moving triangular lattice. Such directional locking effects were observed in simulations of triangular vortex lattices moving over random substrates under a fixed longitudinal drive and an increasing transverse drive 83 (), and similar effects were demonstrated for particles moving over periodic 87 (); 88 (); 89 () and quasiperiodic substrates 91 (); 92 () .

Figure 10: The system from Fig. 9 with , , and in the elastic regime at . (a) vs . (b) vs . (c) vs . There is a series of steps on which the skyrmion lattice motion locks to specific skyrmion Hall angles of , and , corresponding to the alignment of the direction of motion with a symmetry direction of the skyrmion lattice.

In the skyrmion case, we find that transverse directional locking arises in the elastic flow regime in the absence of any additional transverse applied force, which does not occur in the overdamped limit. The directional locking of the skyrmion lattice is a result of the velocity dependence of the skyrmion Hall angle, which causes the net direction of motion of the skyrmion lattice to change as the velocity increases. When the direction of motion aligns with a symmetry direction of the skyrmion lattice, we find a locking effect, as shown in Fig. 10 for the system in Fig. 9 in the elastic regime with . In Fig. 10(a) we plot versus where we observe three step features. The versus curve in Fig. 10(b) shows more clearly that along these steps, is constant, while in the plot of versus in Fig. 10(c), the steps correspond to skyrmion Hall angles of , and . In each case the skyrmion lattice remains locked to a specific skyrmion Hall angle over a fixed interval of . For a triangular lattice, the locking occurs when , where and are integers. At and , ; at and , ; and at and , . Some smaller steps appear for higher values of .

Figure 11: A blowup of the locking step in Fig. 10. (a) vs has a linear increase along the locking step. (b) vs and (c) vs both have shapes on either end of the locking step that are consistent with what is expected in phase locking systems.

In Fig. 11 we show a blowup of the curves in Fig. 10 on the , step at . Here increases along the step while remains flat and on either side of the step has the shape expected for a phase locked system such as a Shapiro step. Figure 11(c) shows that is locked to a fixed angle as well. We find that the dynamical fluctuations are generally reduced on each step and that the peaks in are sharper, similar to the enhanced ordering in step regions observed in other systems that exhibit phase locking. These results show that an elastic skyrmion lattice exhibits a self-induced phase locking that has the same features as phase locking in an overdamped system but that occurs under the application of a single fixed direction dc drive rather than two superimposed dc drives or one rotating dc drive. In the plastic depinning regime, the dynamically reordered skyrmion lattices that appear at high drives generally still contain a small number of topological defects, which smear out the directional locking features.

Figure 12: vs for the system in Fig. 9 with , , and . At (dark blue), the depinning threshold is close to the single particle limit and the system does not dynamically order. At (light blue), the depinning is plastic, is higher than in the elastic limit, and a plateau appears in the velocity response. At (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 due to a speed up effect.

In Fig. 12 we plot versus for the system in Fig. 9 for varied skyrmion density. At a low density of , the system is in the single skyrmion limit where the critical depinning force is high and the skyrmions do not dynamically order. Here we find a sharp depinning threshold followed by a monotonic increase in with . At , the skyrmions depin plastically and dynamically reorder at higher drives. In this case, just above the depinning threshold we find a window in which remains roughly constant with increasing in the plastic flow regime before switching to a monotonic increase in the dynamically reordered regime. For a higher density of , is small, the skyrmions depin elastically, and increases monotonically with . In the plastic flow window of the sample just above depinning, where only some of the skyrmions have depinned and are moving while the rest remain pinned, is larger than the value found over the same range of drives for the elastically flowing sample, even though in the denser sample, all of the skyrmions are moving. The two velocity-force curves cross near , above which is higher for the denser elastically flowing system. At higher values of , converges to the pin-free value (shown as a dashed line) for all values of . Over a large region of drive, extending from the depinning transition to and higher, we find a pinning-induced speed up effect in which the skyrmions move faster in the direction of the applied drive than a freely moving overdamped particle, as indicated by the fact that the velocity-force curves fall above the dashed line. The pinning-induced speed up effect results when the interaction of the skyrmion with the pinning site is partially transformed into motion along the driving direction by the Magnus force. Such effects have been observed previously in simulations with periodic pinning or linelike defects. For higher drives, the skyrmions move faster, the effectiveness of the pinning is reduced, and gradually approaches its pin-free value. If the relative strength of the Magnus term is lowered, the size of the speed up effect diminishes, and in the damping-dominated limit, is always equal to or less than the pin-free value. These results show that speed up effects for skyrmions persist even when the disorder is random rather than periodic.

Figure 13: Dynamic phase diagram as a function of vs for the system in Figs. 9 to 12 with , and highlighting the pinned glass, pinned crystal, plastic flow, moving liquid, and moving crystal phases.

By conducting a seres of simulations, we construct a dynamic phase diagram as a function of versus skyrmion density , as shown in Fig. 13. For , the system forms a pinned glass, and it may be possible to further distinguish the formation of a pinned gas phase for where the system enters the single skyrmion limit. For , the pinned crystal depins into a moving crystal, while for , the pinned glass undergoes plastic depinning into a plastic flow phase which transitions into a moving liquid and finally into a moving crystal at high drives. For , above depinning the system always remains in a moving liquid phase and dynamical reordering never occurs.

V Dynamics as a function of Magnus Force

Figure 14: (a) vs and (b) (blue) and (red) vs for a sample with , , and in the overdamped limit of . (c) vs and (d) (blue) and (red) vs for a sample with the same parameters except with .

In Fig. 14(a,b) we plot , , and versus for a system with , , and in the overdamped limit of . Under these conditions, , and the initial increase in with just above depinning has the nonlinear form with to , as previously observed for plastic depinning in vortex matter 1 (); 39 (). At higher drives where all the skyrmions are moving, the velocity-force curve becomes linear. We find that in this overdamped system, for all drives since the fluctuating force generated by the pinning sites is aligned with the driving direction. The strongly anisotropic pinning-induced fluctuation forces cause the system to form a moving smectic, as predicted from theory 42 (); 43 () and observed in previous simulations 1 (); 45 (); 46 (); 74 () and experiments 44 (). In Fig. 14(c,d), we plot , , and versus in a system with the same parameters except with . Here, the onset of linear behavior in the velocity-force curve shifts closer to the depinning transition, while only in the strongly plastic flow region . Within the plastic flow regime, is zero at depinning and slowly increases toward the intrinsic value with increasing . For in the moving crystal phase, we find . The crossover in the magnitude of the velocity fluctuations occurs because the Magnus force induces fluctuations that are perpendicular to the forces exerted by the pinning sites. At higher drives, as the effectiveness of the pinning is reduced.

Figure 15: (a) vs and (b) (blue) and (red) vs for the system in Fig. 14 with , , and at . (c) vs and (d) (blue) and (red) vs for the same system at , where we find a region in which decreases with increasing , indicative of negative differential conductivity.

In Fig. 15(a,b) we plot , , and for the system in Fig. 14 at . Here the velocity-force curve develops a plateau above depinning, destroying the scaling associated with plastic flow in the overdamped limit. For , moving and pinned skyrmions coexist, and we find strongly pronounced velocity fluctuations with . The velocity-force curves become linear for ; however, dynamical reordering does not occur until a much higher , and the system is in a moving liquid phase. In Fig. 15(c,d) we show , , and versus for a sample with the same parameters but at . Here, there is a region in which decreases with increasing , indicative of negative differential conductivity. For , increases linearly with , while and are largest for . In general, we find that increasing either the Magnus force or increases the range over which is flat or decreasing with increasing , with the maximum range corresponding to . Negative differential conductivity, , appears for sufficiently large Magnus force in the regime containing a combination of moving and pinned skyrmions, and occurs when the increase in combined with the Magnus force generates an increasing velocity component in the direction perpendicular to the drive due to collisions between moving and pinned skyrmions, lowering the velocity response parallel to the drive. For , all the skyrmions depin and the negative differential conductivity is lost.

Figure 16: Dynamic phase diagram as a function of vs for the system in Figs. 14 and 15 with , and , highlighting the pinned, plastic, moving liquid, and moving crystal phases. For , the moving crystal phase is replaced by a moving smectic state. The depinning threshold decreases with increasing .

By conducting a series of simulations and examining the nonlinear features in the velocity-force and versus curves, we can construct a dynamical phase diagram as a function of versus , as shown in Fig. 16. At , the pinned and plastic flow phases are followed at higher by dynamical reordering into a moving smectic state, as described in detail in Ref. 74 (). For , the system dynamically orders into a moving crystal phase when . For , we find a growing window of a moving liquid state in which all of the skyrmions are moving but there is still considerable topological disorder. The moving liquid regime increases in extent with larger due to the rotational character of the fluctuations generated by the Magnus force. If is sufficiently large, the Magnus force enhances the net force experienced by the skyrmions when , and the resulting fluctuations melt the skyrmion lattice. The magnitude of the fluctuations grows roughly linearly with , while the magnitude of the forces exerted by the pinning sites decrease as 1 (), so the value of at which dynamical reordering into the moving crystal phase occurs increases roughly linearly with . For the illustrated value of , the velocity-force curves develop a window in the plastic flow regime in which remains constant or decreases with increasing . We note that it is possible for the smectic phase observed at to persist for finite but small values of , as will be discussed elsewhere. The value of marking the depinning transition into the plastic flow state shifts to lower with increasing as also found in previous simulations, and it has been argued that the Magnus force is one of the reasons that the critical depinning force is small in skyrmion systems 58 ().

Vi Density Phase Separation and Dynamic Phase Separation

When both the pinning and the Magnus force are strong, we observe a dynamically induced density segregation or skyrmion clustering effect. This effect was first found in continuum-based simulations of moving skyrmions in the strong substrate limit, where it was attributed to an attraction between the skyrmions arising from spin wave excitations generated by the fluctuating internal modes of the skyrmions 76 (). Recent simulations with periodic pinning arrays show that a strong clustering effect occurs for moving skyrmions when both the Magnus force and the pinning strength are sufficiently large 93 (). In these simulations, which contain no spin waves, the effect was attributed to the drive dependence of the skyrmion Hall angle, which causes different regions of the skyrmions to move toward each other due to their differing values of if a sufficiently strong velocity gradient can be induced by the pinning. In the case of periodic pinning, this situation arises both in the plastic flow regime and at higher drives. Here we show that a similar dynamical density phase separation can occur for strong random pinning, but that it is restricted to the regime . We also find two possible types of phase separation: a density modulated state for strong pinning, and bands of moving skyrmions coexisting with bands of pinned skyrmions for weaker pinning.

Figure 17: Skyrmion positions for a system with , , , and . (a) The pinned phase at has a uniform skyrmion density. (b) The density phase separated state at . (c) The transition from the density phase separated state to the moving liquid state at . (d) The moving liquid phase at where the skyrmion density is uniform.
Figure 18: Skyrmion positions in the density segregated state for the system in Fig. 17 with , , and at for (a) , (b) , and (c) , showing that as the disorder strength increases, the skyrmion bands become narrower. (d) The system at , showing the persistence of the bands at higher skyrmion densities.

The density phase separation is most pronounced in denser systems, as illustrated in Fig. 17 where we show the skyrmion positions at different drives for a system with , , , and . In Fig. 17(a) at , we find a uniform disordered pinned glass state, while at in the plastic flow state, Fig. 17(b) shows that a dense band of skyrmions emerges that is surrounded by a region of low skyrmion density. At in Fig. 17(c), near the transition from the density phase separated state to the moving liquid, the density banding is reduced, while in the moving liquid phase at in Fig. 17(d), the density phase separation is lost and the skyrmion density becomes uniform. In general, the density phase separation occurs when , , , and , in a regime where there is a combination of moving and pinned skyrmions. The density phase separation takes the form of bands of skyrmions roughly aligned with the direction of motion , similar to what was observed in the continuum-based simulation studies. In Fig. 18(a) we show the skyrmion positions for in the system from Fig. 17 at , where we find that the bands of skyrmions are wider but still present. At the same value of for and 5.0, as shown in Figs. 18(b, c), respectively, we observe a compression of the skyrmion bands as increases. Figure 18(d) illustrates the system at a higher skyrmion density of , showing that the banding persists for higher skyrmion densities.

Figure 19: (a) Skyrmion positions (dots) and trajectories (lines) for the system in Fig. 17 with , , and at and 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 panel (a) showing that the skyrmion density is uniform.

For the parameters illustrated in Figs. 17 and 18, we find that when , the density phase separation disappears and is replaced by a dynamical phase segregation in which the density is uniform but the motion occurs only in localized bands. This is shown in Fig. 19(a) where we plot the skyrmion trajectories over a period of time in a sample with at constant . Here a wide band of moving skyrmions coexists with a pinned band. Figure 19(b) shows only the particle positions from Fig. 19(a) where it is clear that the skyrmion density is uniform. For the same parameter set at , we find a uniform plastic flow phase, and for , the depinning is elastic.

Figure 20: Dynamic phase diagram as a function of vs for the system in Figs. 17 to 19 with , , and 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 vs for the system in Figs. 17 to 20 with , , and 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 .
Figure 22: Skyrmion positions (dots) and trajectories (lines) for the system in Fig. 21 with , , and at . (a) At , multiple bands appear consisting of both moving and pinned skyrmions. (b) At , 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 .

In Fig. 20 we plot a dynamical phase diagram as a function of versus for the system in Figs. 17 to 19. For , a pinned crystal (PC) phase appears which depins directly into the moving crystal (MC) phase. In the moving liquid (ML) phase, the system is disordered but there is no dynamical or density phase separation. The dynamically phase separated state (DPS), illustrated in Fig. 19, has uniform skyrmion density but phase separated moving regions, while the density phase separated state (PS), illustrated in Figs. 17 and 18, has nonuniform skyrmion density. We find a pinned glass region (PG) at low when is sufficiently large. In Fig. 21 we show the dynamical phase diagram as a function of versus for the same system at fixed . The PS phase appears only when , while the DPS phase extends from . At , the MC phase is replaced by a moving smectic phase (not shown). When , we find that the bands in the PS phase sharpen and split into multiple lanes, as illustrated in Fig. 22(a) at and . Here, multiple bands appear that are each composed of both moving and pinned skyrmions. As is increased, the bands simultaneously widen and move at a steeper angle with respect to the applied drive, since the skyrmion Hall angle increases with increasing drive. The rotation of the bands as they follow the skyrmion Hall angle generally occurs via the formation of kinks, in which one portion of the band is moving at the lower angle while another portion is moving at the new, higher angle, as shown in Fig. 22(b) at a higher .

Figure 23: Dynamic phase diagram as a function of vs for a system with , , and 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 is 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.

In Fig. 23 we plot a dynamic phase diagram as a function of versus for samples with , , and . We observe the PS phase when and the DPS phase in the range . The skyrmion density remains uniform when and/or , but in this regime we can distinguish between a plastic flow (PF) phase in which moving and pinned skyrmions coexist, and a ML phase in which all the skyrmions are moving but the system is disordered. The transition to the MC phase shifts to higher values of with increasing , while the transition into the ML phase remains roughly constant with increasing since it is controlled by the value of .

The dynamic phase diagrams in Figs. 20, 21, and 22 indicate that a moving segregated state can arise in particle-based skyrmion models when both the pinning and the Magnus force are sufficiently strong. We note that our particle-based model does not include the generation of spin waves by the moving skyrmions, which was the mechanism proposed to be responsible for the clustering transition observed in continuum simulations. The clustering transition we observe arises due to the velocity differential exhibited by the skyrmions as they move over the strong random pinning. The pinned or slowly moving skyrmions have a smaller skyrmion Hall angle than the more rapidly moving skyrmions, and as a result, these two types of skyrmions tend to move toward each other. It is not clear whether the clustering effect we observe is actually the same as that found in the continuum simulations 76 () or whether skyrmion cluster states can be generated via more than one mechanism. Recent experiments involving nonequilibrium quenches of a skyrmion system also show the formation of clustered or phase separated states 94 (), so it is possible that density segregated states are a general phenomenon in nonequilibrium skyrmion systems.

Vii Summary

Using a particle-based model, we examine the various types of collective dynamic phases that can occur for skyrmions moving over random pinning under a dc drive as we vary the Magnus force, pinning strength, skyrmion density, and pinning density. For weak pinning, the skyrmions form a triangular lattice that depins elastically, and although there is no proliferation of topological defects at the elastic depinning transition, we find that the sixfold peaks in the structure factor diminish in weight due to smearing of the skyrmion positions near depinning, while the peaks sharpen again at higher drives when the skyrmion lattice becomes more ordered. As a function of increasing pinning strength or decreasing skyrmion density, we find a transition into a pinned skyrmion glass phase that depins plastically into a disordered moving state. This transition is accompanied by a sharp increase in the critical depinning force which resembles the peak effect phenomenon found in superconducting vortex systems at the transition from elastic to plastic depinning.

We find that the skyrmion Hall angle is zero at very low drives and increases with increasing driving force before saturating to the intrinsic value at high drives. In the elastic depinning regime for weak pinning, a directional locking effect occurs in which the direction of skyrmion motion locks to a symmetry direction of the triangular pinning substrate. As the drive increases, the skyrmion Hall angle also increases, producing a series of directional locking phases. These directional locking effects are similar to those predicted by Le Doussal and Giamarchi for elastic vortex lattices moving under a longitudinal drive that are subjected to an additional transverse applied driving force; however, in the skyrmion case, the direction of the net driving force remains fixed and the skyrmion Hall angle changes due to interactions with the pinning sites. We show that the skyrmion velocity-force curves differ substantially from those found in overdamped systems with random pinning, and that when the Magnus force is strong, there can be a plateau or even a decrease in the skyrmion velocity with increasing driving force, leading to negative differential conductivity. The scaling of the velocity-force curves changes under strong Magnus force from the typical form observed in overdamped systems to a more discontinuous behavior similar to that predicted by Schwartz and Fisher for depinning in systems that include non-dissipative effects such as stress overshoots or inertia. As the Magnus force increases, we find a region of what we call a moving liquid phase in which the skyrmions are all in motion but the non-dissipative fluctuations induced by the skyrmion-pin interactions are large enough to disorder the skyrmion lattice. Under the same conditions in the overdamped limit, a moving smectic state appears instead.

One of the most striking features we observe is a density segregation or skyrmion clustering effect that occurs for strong pinning and strong Magnus force. Here the skyrmions form dense bands aligned with the driving direction that are separated by low density regions, a behavior similar to that recently found in continuum-based simulations in the strong random substrate limit as well as in particle-based simulations of skyrmions driven over periodic pinning arrays. As the pinning becomes weaker, we find that the density phase segregated state transitions into a dynamically phase separated state in which the skyrmion density is uniform but localized bands of moving skyrmions coexist with bands of pinned skyrmions, while for very weak pinning or small Magnus force, both the skyrmion density and the skyrmion motion are uniform. The density segregation arises due to the velocity-dependent skyrmion Hall angle induced by the Magnus force. When the pinning is strong enough, the skyrmion velocity can develop a large local differential, causing different groups of skyrmions to move at different Hall angles until the groups collide to form a region of enhanced skyrmion density at the cost of a density depleted region. The clustering effects occur in the plastic flow regime where there is a combination of pinned and moving skyrmions. Our results show that a particle-based model that neglects spin waves is sufficient to capture such clustering effects; however, it is not clear whether the skyrmion clustering observed in continuum based simulations is produced by the same mechanism as the clustering found in the particle-based model.

Acknowledgements.
We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD program for this work. This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396 and through the LANL/LDRD program.

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