Shapiro Steps for Skyrmion Motion on a Washboard Potential with Longitudinal and Transverse ac Drives
We numerically study the behavior of two-dimensional skyrmions in the presence of a quasi-one-dimensional sinusoidal substrate under the influence of externally applied dc and ac drives. In the overdamped limit, when both dc and ac drives are aligned in the longitudinal direction parallel to the direction of the substrate modulation, the velocity-force curves exhibit classic Shapiro step features when the frequency of the ac drive matches the washboard frequency that is dynamically generated by the motion of the skyrmions over the substrate, similar to previous observations in superconducting vortex systems. In the case of skyrmions, the additional contribution to the skyrmion motion from a non-dissipative Magnus force shifts the location of the locking steps to higher dc drives, and we find that the skyrmions move at an angle with respect to the direction of the dc drive. For a longitudinal dc drive and a perpendicular or transverse ac drive, the overdamped system exhibits no Shapiro steps; however, when a finite Magnus force is present we find pronounced transverse Shapiro steps along with complex two-dimensional periodic orbits of the skyrmions in the phase-locked regimes. Both the longitudinal and transverse ac drives produce locking steps whose widths oscillate with increasing ac drive amplitude. We examine the role of collective skyrmion interactions and find that additional fractional locking steps occur for both longitudinal and transverse ac drives. At higher skyrmion densities, the system undergoes a series of dynamical order-disorder transitions, with the skyrmions forming a moving solid on the phase locking steps and a fluctuating dynamical liquid in regimes between the steps.
Phase locking or synchronization effects can arise in coupled oscillators when the different frequencies lock together over a certain range of parameter space, an effect that was first reported by Huygens for the synchronization of pendulum clocks 1 (). Phase locking has been extensively studied for numerous dynamical systems ranging from a pair of coupled oscillators to an entire coupled oscillator array 2 (); 3 (). A single particle moving over a tilted one-dimensional washboard potential can also experience phase locking when an additional ac driving force is applied. The substrate periodicity produces intrinsic periodic modulations of the particle velocity in the absence of an ac drive which increase in frequency as the magnitude of the tilt or dc drive increases. Addition of an external fixed-frequency ac drive produces locking regimes in which the average dc velocity remains constant even as the magnitude of the dc drive is increased. The same picture can be applied to Josephson junctions, where the analog of a velocity-force curve is the voltage-current curve, which exhibits a series of phase locking regions called Shapiro steps under an applied ac drive for single junctions 4 (); 5 () and coupled arrays of junctions 6 (). One of the hallmarks of Shapiro steps is that the step width oscillates as a function of the ac drive amplitude 4 (); 5 (); 6 (). Shapiro step phenomena also arise in dc and ac driven charge density waves 7 (); 8 (); 9 (), spin density waves 10 (), and Frenkel-Kontorova models consisting of commensurate or incommensurate arrangements of particles moving over ordered or disordered substrates 11 (); 12 (); 13 (). In the case of vortex motion in type-II superconductors, Martinoli et al. reported the first observation of Shapiro steps for dc and ac driven vortices interacting with a periodic one-dimensional (1D) substrate created by periodic thickness modulations of the sample 14 (); M (), while similar effects were observed for vortices driven over 1D 15 (); 16 () or two-dimensional (2D) 17 (); 18 () periodic substrates. More recently, Shapiro steps have been found for ac and dc driven colloidal particles moving over a quasi-1D periodic substrate 19 ().
Shapiro steps can also occur when a lattice of collectively interacting particles moves over a random substrate under combined dc and ac drives. Here, the effective elastic coupling between the particles comprising the lattice generates an intrinsic washboard frequency that can lock to the applied ac driving frequency. Such steps have been studied for vortices moving over random disorder 20 (); 21 (); 22 (); 23 (); S () or through confined channel geometries 24 (). For particles confined to 2D and moving over a quasi-1D substrate, both the ac and dc drives must be applied in the same direction to produce Shapiro steps; however, for vortices moving over 2D periodic or egg-carton substrates, it is possible to obtain what are called transverse phase locking steps when the ac drive is perpendicular to the direction of the dc drive 25 (); 26 (); 27 (). These phase locking steps are distinct from Shapiro steps, and their widths grow quadratically with increasing ac amplitude rather than showing the oscillatory behavior associated with Shapiro steps. Phase locking effects can also occur for overdamped particle motion in 2D periodic systems under combinations of two perpendicular ac drives, producing localized and delocalized motion as well as rectification effects 28 (); 29 (); 30 (); 31 (); 32 (); 33 ().
In systems such as vortices and colloidal particles, an overdamped description of the equations of motion is appropriate. In contrast, the skyrmions that were recently discovered in chiral magnets have particle-like properties and many similarities to superconducting vortices, but have the important distinction that there is a strong non-dissipative Magnus force in their motion 34 (); 35 (); 36 (); 37 (); 38 (); 39 (); 40 (); 41 (); 42 (); 43 (). The skyrmions can be set into motion by an applied current and are observed to have a very small depinning threshold 36 (); 37 (); 38 (); 39 (); 44 (); 45 (), in part because the effectiveness of the Magnus force can be up to ten times stronger than the dissipative force component. The Magnus force introduces a velocity component of the skyrmion that is perpendicular to the direction of an imposed external force, so a skyrmion deflects from or spirals around an attractive pinning site rather than moving directly toward the potential minimum as would occur for systems governed by overdamped dynamics 38 (); 39 (); 45 (); 46 (); 47 (); 48 (); 49 (). Since skyrmions are particle-like objects, many of their dynamical properties can be captured using a point particle model based on a modified Theile’s equation that takes into account repulsive skyrmion-skyrmion interactions, the Magnus force, damping, and substrate interactions 45 (); A (). Such as approach has been shown to match well with micromagnetic modeling 45 () of the depinning of skyrmions in periodic 46 () and random pinning arrays 47 (). Particle-based skyrmion models were used to describe the motion of skyrmions interacting with single pinning sites 48 (); 49 () as well as skyrmion motion in confined regions 50 (). Since skyrmions can easily be driven with an applied external drive they potentially open a new class of experimentally accessible dynamical systems where the Magnus force has a dominant effect. It should be possible to create various types of potential energy landscapes for skyrmions through techniques such as thickness modulations, periodic applied stain, controlled irradiation, or spatially periodic doping. An open question is how known phase locking phenomena would be affected by the inclusion of a Magnus force, and whether new types of phase locking effects might appear that are absent in overdamped systems. Skyrmions also have potential for various spintronic applications 51 () which would require the skyrmions to move in a controlled manner, so an understanding of skyrmion phase locking dynamics could be useful for producing new methods for precision control of skyrmion motion.
In this work we examine Shapiro steps for skyrmions moving over a quasi-1D periodic washboard substrate similar to the geometry considered by Martinoli et al. for vortices moving over substrates with a periodic thickness modulation 14 (); M (); 15 (). We consider the motion of single skyrmions and collectively interacting skyrmions over a periodic substrate, as illustrated in Fig. 1. Here the longitudinal or -direction is aligned with the direction of the periodicity of the substrate, while the -direction corresponds to the transverse direction. In this geometry in the overdamped limit, Shapiro steps occur when the dc () and ac () driving forces are both applied along the -direction. When a finite Magnus force is present, we find that phase locking continues to occur; however, the net skyrmion motion is rotated at an angle with respect to the substrate periodicity direction. As the magnitude of the Magus force prefactor is increased, the width and number of steps for a fixed dc drive gradually decreases, and the steps shift to higher values of . In the overdamped limit, if the ac drive is applied in the transverse or -direction, no Shapiro steps occur, but for a finite Magnus force a new set of Shapiro steps can arise, with the width and the number of resolvable steps in the velocity-force curve increasing with increasing Magnus force prefactor. On a phase-locked step, the skyrmion motion takes the form of intricate periodic 2D orbits. As a function of the ac amplitude, we show that the widths of the phase locking steps for both longitudinal and transverse ac driving exhibit the oscillatory behavior associated with Shapiro steps.
We note that the Shapiro steps we observe are different than the previously studied phase locking motion for purely dc-driven skyrmions moving over a 2D periodic substrate 46 (). In the latter case, the phase locking was associated with a directional locking effect in which the skyrmion motion locks to certain symmetry directions of the substrate potential. In the present study there is no phase locking without an ac drive.
Ii Simulation and System
We consider a 2D system of size with periodic boundary conditions in the and directions containing skyrmions at a density of . Single () or multiple skyrmions interact with a quasi-1D periodic sinusoidal potential with a periodicity direction running along the direction, as illustrated in Fig. 1. The equation of motion for a single skyrmion with velocity moving in the plane is
Here is the location of the skyrmion and is the prefactor of the damping force that aligns the skyrmion velocity in the direction of the net external forces. The second term is the Magnus force with prefactor , which rotates the velocity into the direction perpendicular to the net external forces. In order to maintain a constant magnitude of the skyrmion velocity we impose the constraint and vary the relative importance of the Magnus force to the damping force by changing the ratio . In the overdamped limit , while for skyrmions can be ten or larger 39 (); 45 ().
The skyrmion-skyrmion interaction force is where , , and is the modified Bessel function. This interaction is repulsive and falls off exponentially for large . For most of this work we remain in the limit where skyrmion-skyrmion interactions are weak so that we can consider the dynamics of a single skyrmion; however, we show that most of our results are robust under the inclusion of skyrmion-skyrmion interactions. The substrate force arises from a washboard potential
where , is the periodicity of the substrate, and we define the substrate strength to be . Unless otherwise noted, we take . The dc driving term is slowly increased in magnitude to avoid any transient effects. The ac driving term is either for transverse driving or for perpendicular driving.
We measure the time-averaged skyrmion velocities in the direction and -direction . Here, due to the periodicity of the substrate, phase locked steps occur when the skyrmions travel integer multiples of the substrate periodicity during each ac drive cycle, allowing us to label the steps for the pinned phase and for the higher order steps. We focus on the two ac frequencies inverse simulation time steps for the longitudinal ac driving and inverse simulation time steps for the transverse ac driving, and use a substrate lattice constant of .
We use two different driving protocols as illustrated in Fig. 1. For longitudinal driving, we have
corresponding to the conditions under which Shapiro steps arise for an overdamped system. For transverse driving, we have
which would produce no Shapiro steps in the overdamped limit.
In Fig. 1(a) we show the skyrmion trajectories for , , , and a skyrmion density of . In this case the skyrmions are pinned and form a triangular lattice that is commensurate with the substrate. The ac drive causes the skyrmions to oscillate in the potential minima; however, their motion is not strictly in the -direction but is tilted at an angle of with respect to the direction due to the Magnus force, which induces a velocity component perpendicular to the ac driving direction. In the absence of a substrate, a dc or ac drive applied in the -direction causes the skyrmions to move at an angle with respect to the driving direction, so that in the overdamped limit of the skyrmion moves parallel to the direction of the net external driving force. In Fig. 1(b), we rotate the direction of the ac drive to be in the transverse direction with and for a sample with . In this case the skyrmion motion follows strictly 1D paths aligned with the -direction that pass through the potential minima of the substrate. For , as shown in Fig. 1(c), the skyrmions rotate in counterclockwise elliptical patterns, showing that the Magnus force can induce -direction motion even when the drive is applied only in the -direction. In the absence of the substrate the ac drive would produce only 1D trajectories at an angle with respect to the -axis. This highlights the fact that the Magnus force affects how the skyrmions move when interacting with forces induced by the substrate.
Iii Longitudinal ac Driving
We first consider the case illustrated in Fig. 1(a) of ac driving in the longitudinal direction. We conduct a series of simulations for increasing and focus on the single skyrmion limit. In general we find that the Shapiro steps we observe remain robust when finite skyrmion-skyrmion interactions are included; however, additional features can arise for varied fillings when the skyrmion structure is incommensurate with the substrate, as we discuss in Section V. In Fig. 2(a) we plot and versus for the system in Fig. 1(a) at in the overdamped limit of . Here while shows a series of steps indicative of the phase locking. These features are similar to those observed for other overdamped systems moving over quasi-1D periodic substrates such as vortices in type-II superconductors moving over quasi-1D substrate modulations. In Fig. 2(b), when , both and are finite and have a ratio of . Here the phase locking is still occurring, but the intervals of in which the phase locking steps appear are shifted. Figure 2(c) shows that at , both and some of the step widths have increased in size, and there are no clear regions between the steps where no phase locking is occurring. In Fig. 2(d), at , there is only a single phase locking step.
To more clearly demonstrate the behavior of the steps for varied , in Fig. 3(a) we plot versus for ranging from to , with the evolution of the first three locking steps to 3 highlighted. For a given value of , the step in has a fixed value regardless of the choice of , and each step shifts to higher values of with increasing . The corresponding versus plot in Fig. 3(b) shows that the steps in are not quantized in integer multiples of . The quantization of the arises from the periodicity of the substrate in the -direction, and since the -direction has no periodicity, there is no quantization of . In Fig. 4 we highlight the evolution of the widths of the through steps as a function of and at . At , the largest number of phase locking steps can be resolved. We observe two general trends as increases. First, for , the widths of the locking regions decrease and the intervals of over which the locking occurs shift to higher values of , with the magnitude of this shift increasing with increasing . Second, the width of the , 2, and steps initially increases for increasing before reaching a maximum and then decreasing again. The width of the step reaches a maximum with increasing and then saturates. The shift in the locations of the phase locking regions arises because the angle at which the skyrmions move with respect to the -axis increases with increasing , causing the skyrmions to spend larger intervals of time interacting with the repulsive portion of the substrate potential. As a result, higher values of must be applied to cause the skyrmion to translate in the -direction at larger .
We next determine if the phase locking steps at high Magnus force prefactor are of the Shapiro type. In Fig. 5 we plot versus for at , 4.2, and to show the variation in the widths of the , , and steps. In Fig. 6 we plot the widths and of the and steps, respectively, versus . Each step shows the characteristic oscillation expected for Shapiro steps, where the width of step is proportional to , where is the th-order Bessel function 5 (). The solid lines in Fig. 6(a,b) are fits to and , respectively. The higher order steps obey similar fits. This indicates that in the Magnus-dominated limit, Shapiro step phase locking is occurring.
Iv Transverse ac Driving
We next consider the case illustrated in Fig. 1(b,c), where the ac drive is applied transverse to the direction of the substrate potential. In the overdamped limit of , such a drive causes the skyrmion to oscillate in the -direction as shown in Fig. 1(b), and when a finite dc drive is applied in the longitudinal direction, a single washboard oscillation frequency in the -direction is generated by the motion of the skyrmion over the periodic substrate. Since only one frequency is present, there is no coupling between two frequencies, so mode locking does not occur. When the Magnus force is finite, the transverse ac drive induces an oscillating velocity component in the longitudinal or -direction as well as in the -direction, as illustrated in Fig. 1(c), so that it is possible for the dc-induced washboard frequency to couple to the transverse ac frequency and generate a transverse Shapiro step. In Fig. 7 we plot vs for a single skyrmion moving with . At , shown in Fig. 7(a), there are no steps in , indicating the lack of phase locking, while the corresponding Depinning occurs at the threshold value of . Figure 7(b) shows that at , the depinning threshold has dropped substantially to and a series of steps are now visible for , indicating that phase locking is occurring. For , is finite and the versus curve has exactly the same form as versus , but the magnitude of is multiplied by . In Fig. 7(c,d) we plot versus for samples with and , respectively. Here, the widths of the locking steps increase with increasing and the step locations are shifted to higher values of . In samples with and , as shown in Fig. 7(e,d), respectively, the steps extend out to larger values of , and the non-phase locking regions between the steps are also extended. The steps in once again occur at quantized values of due to the periodicity in the -direction, while the steps in do not have quantized values.
In Fig. 8 we plot the location of the upper edge of the step as a function of and for the system shown in Fig. 6 with . This is equivalent to the threshold depinning force . Here at and it decreases to zero at . There is a local maximum in at , followed by another minimum near and a broad plateau for higher values of . This oscillatory behavior in the step width is absent for longitudinal ac driving, as shown in Fig. 5 where exhibits only monotonic behavior. The dips and maxima in for the transverse ac driving are associated with transitions in the shape of the skyrmion orbits during a single ac drive cycle for increasing .
In Fig. 9 we illustrate the skyrmion trajectories in a subsection of the system on the step at the points labeled (a-d) in Fig. 8. Figure 9(a) shows that for and , the skyrmion moves in a 1D path in the -direction along the potential minimum. At and , in Fig. 9(b), the skyrmion forms an elliptical orbit that is confined within a single potential trough. On the local maximum in the step marked point c in Fig. 8, at and , the skyrmion forms a more complicated 2D orbit that has three lobes. In a single ac drive cycle the skyrmion translates back and forth by two substrate lattice constants. The dip in at shown in Fig. 8 corresponds to the point at which the skyrmion orbit transitions from being confined in one potential minimum to traversing two potential minima. Above the second local minimum at in Fig. 8, the skyrmion orbit becomes even more complex, as illustrated in Fig. 9(d) for and . The skyrmion now moves between three substrate potential minima in a single ac drive cycle. The local minimum in the step width at then corresponds to the transition in the skyrmion motion from traversing two substrate minima to traversing three substrate minima. For higher values of , additional minima in could occur that would be correlated with orbits traversing four or more substrate minima. We expect that additional substrate minima would be resolvable in samples with a smaller substrate lattice constant .
In Fig. 10(a) we highlight the regions of phase locking as a function of and for steps through for the system in Fig. 7. When , all the steps with vanish, as illustrated in Fig. 10(b) where we plot the regime . As the Magnus force increases, a larger number of steps can be resolved. In general, the step widths increase with increasing ; however, certain steps such as , 2, and show step width oscillations. In the case of longitudinal ac driving, the skyrmion orbits along the different locking steps are always 1D in nature. In contrast, the orbits are much more complicated for transverse ac driving. In Fig. 11(a) we show the skyrmion orbit from Fig. 10 at and . The skyrmion translates in the positive -direction and negative -direction, making an angle close to with the -axis. During a single orbit the skyrmion passes through a loop and translates by one lattice constant in the -direction. Figure 11(b) illustrates the orbit for at , where the skyrmion translates two lattice constants in the -direction per ac cycle. On the step for , and , shown in Fig. 11(c), the skyrmion moves at a steeper angle of from the -axis. In Fig. 11(d), which shows the orbit at and , during a single ac drive cycle the skyrmion initially moves in the positive -direction before moving in the negative -direction, producing a net translation in the -direction of a distance per ac cycle. Figure 11(e) shows the system in the orbit at , where the skyrmion translates by in a single ac cycle. On the step at and , plotted in Fig. 11(f), the skyrmion moves in the positive -direction during the first portion of the ac drive cycle followed by in the negative -direction during the second portion of the ac drive cycle, producing a net translation of in the direction during a single ac cycle. We observe similar orbits for the other values of , and find that the net angle of the skyrmion motion with respect to the axis increases with increasing .
iv.1 Dependence on Substrate Strength and ac Amplitude
We next consider the effect of the substrate strength on the transverse locking steps at and . In Fig. 12(a) we plot versus for , 2.0, 4.0, and . At the lower values of , the phase locking steps decrease in width, and the steps completely vanish when . This is highlighted in Fig. 12(b) where we plot the widths of the , 1, 2, and steps as a function of and . The width of the locking regions oscillates with increasing , and for all the locking phases shift linearly to higher values of with increasing . The step width oscillations arise due to variations in the number of potential minima through which the skyrmion orbit passes during a single ac drive cycle, similar to what was observed for fixed and varied . This result shows that the transverse phase locking is a generic feature that appears in both the strong and weak substrate regimes, and that it is more pronounced for stronger substrates.
We next examine the dependence of the step widths at a fixed on the ac driving amplitudes, as shown in Fig. 13 where we plot and versus for and . The solid lines are fits to , indicating that the transverse phase locking steps are also of the Shapiro step type, similar to the longitudinal phase locking steps.
V Collective Effects
We next consider assemblies of interacting skyrmions for the system shown in Fig. 1. In general, when the skyrmion density is commensurate with the substrate and the skyrmions can form a triangular lattice, skyrmion-skyrmion interactions cancel and we find the same types of phase locking observed in the single skyrmion systems. For incommensurate fillings where dislocations are present or when the skyrmion structure becomes distorted or anisotropic in the pinned phase, we find that it is possible for additional fractional phase locking to occur between the integer phase locking steps. These fractional locking steps occur when a portion of the skyrmions are locked to step and the remainder of the skyrmions are locked to step . In Fig. 14(a) we plot versus for a system with ac driving in the -direction at and to compare the results for a single skyrmion with a system at a skyrmion density of . There are no fractional steps in the single skyrmion system; however, when interacting skyrmions are present we find fractional steps , where and are integers. Figure 14(b) shows the same system for ac driving in the -direction, where the same types of fractional steps arise. The fractional steps appear at incommensurate fields when it is possible to have two effective particle species in the system. One species is commensurate and the other is associated with interstitials, dislocations, or vacancies. In overdamped systems such as superconducting vortices moving over 2D periodic substrates, similar integer steps for individual or non-interacting vortices appear at commensurate matching fillings while additional fractional locking steps arise at non-matching fields 18 ().
At much higher skyrmion densities and for sufficiently strong substrate strengths, the pinned skyrmion structures become highly anisotropic due to the confinement in the 1D pinning rows. In the moving phase just above depinning, the effectiveness of the pinning is partially reduced and the repulsive skyrmion-skyrmion interactions favor a more uniform structure. The competition between skyrmion-skyrmion and skyrmion-substrate interactions produces a series of order-disorder transitions in the moving state. On the phase-locked steps, the skyrmions form an ordered moving anisotropic lattice and travel in a synchronized fashion, while between the phase locking steps the skyrmions adopt a more isotropic or liquid like configuration.
In Fig. 15(a) we plot versus for a sample with , , and a skyrmion density of , showing the , 3, and 4 phase locking steps. Figure 15(b) illustrates the corresponding fraction of six-fold coordinated skyrmions , where is the coordination number of skyrmion obtained from a Voronoi construction. On the phase locking steps, increases to , while between the steps on average and shows strong fluctuations. In Fig. 16(a,b) we show the real space locations of the skyrmions and the corresponding structure factor on the step at from Fig. 15. The skyrmions are all moving together and form a partially ordered but anisotropic lattice. Even though the system is anisotropic, most of the skyrmions have six neighbors, so that . Figure 16(c,d) shows the same sample at , corresponding to the non-phase locking region labeled b in Fig. 15. Here the skyrmions form a disordered structure that is less anisotropic than the phase locked state. We observe similar sets of dynamical order-disorder transitions between step and non-step regions for increasing and find similar effects for ac driving in the -direction. Studies in overdamped systems of collections of interacting vortices also show that the vortices are more ordered and exhibit suppressed noise fluctuations in a phase locked region 22 (); S (). At higher , the effectiveness of the substrate gradually diminishes, the phase locking steps disappear, and the skyrmions can reorder into a more uniform moving crystal state as shown in Fig. 16(e,f) at . Similar dynamical reordering to a triangular lattice for high drives has been observed for skyrmions interacting with random pinning 47 () as well as for vortices driven over random pinning arrays 52 (); 53 (). These results show that Shapiro steps for skyrmions interacting with a periodic substrate are a robust feature that occurs for a variety of skyrmion densities and substrate strengths. The change in the skyrmion lattice structure as the system passes in and out of phase locked states as a driving current is swept could be observed using neutron scattering or noise measurements.
We have analyzed Shapiro steps for skyrmions interacting with periodic quasi-one-dimensional substrates in the presence of combined dc and ac drives, with a specific focus on the role of the Magnus force in the dynamics. When the dc and ac drives are both applied in the longitudinal direction, which is aligned with the substrate periodicity, phase locking occurs, and as the role of the Magnus force increases, the phase locking steps gradually reduce in width and shift to higher values of the driving force. The skyrmions move at an angle to the direction of the external dc drive that increases as the contribution of the Magnus force increases. When the ac drive is applied perpendicular to the dc drive and the substrate periodicity direction, there is no phase locking in the overdamped limit; however, when there is a finite Magnus force, phase locking can occur. On the phase locked steps the skyrmions move in intricate two-dimensional periodic orbits. We map out the evolution of the phase locked regions for the transverse and longitudinal ac driving for varied contribution of the Magnus force, ac driving amplitudes, and substrate strength. When collective interactions between skyrmions are introduced, fractional Shapiro steps can appear. For strong substrate strengths and higher skyrmion densities, both longitudinal and transverse phase locking steps occur that are associated with dynamically induced transitions between an ordered anisotropic solid on the steps to a fluctuating liquid state in the non-phase locked regimes. Such transitions could be observed with neutron scattering.
Acknowledgements.This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.
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