Reversible Vector Ratchets for Skyrmion Systems
We show that ac driven skyrmions interacting with an asymmetric substrate provide a realization of a new class of ratchet system which we call a vector ratchet that arises due to the effect of the Magnus term on the skyrmion dynamics. In a vector ratchet, the dc motion induced by the ac drive can be described as a vector that can be rotated clockwise or counterclockwise relative to the substrate asymmetry direction. Up to a full rotation is possible for varied ac amplitudes or skyrmion densities. In contrast to overdamped systems, in which ratchet motion is always parallel to the substrate asymmetry direction, vector ratchets allow the ratchet motion to be in any direction relative to the substrate asymmetry. It is also possible to obtain a reversal in the direction of rotation of the vector ratchet, permitting the creation of a reversible vector ratchet. We examine vector ratchets for ac drives applied parallel or perpendicular to the substrate asymmetry direction, and show that reverse ratchet motion can be produced by collective effects. No reversals occur for an isolated skyrmion on an asymmetric substrate. Since a vector ratchet can produce motion in any direction, it could represent a new method for controlling skyrmion motion for spintronic applications.
In a rocking ratchet, a particle or collection of particles interacting with an asymmetric substrate undergoes a net dc drift when subjected to an ac drive 1 (); 2 (), as observed for vortices in type-II superconductors interacting with one-dimensional (1D) 3 (); 4 (); 5 () or two-dimensional (2D) asymmetric substrates 4w (); 5w (); 6 (); 7 (). In the single particle limit, the ratchet motion is typically in the easy flow direction of the substrate asymmetry; however, when collective effects come into play, it is possible for a reverse ratchet effect to occur in which the particles move along the opposite or hard flow direction of the substrate asymmetry. Reversals of the ratchet direction can occur when parameters such as the ac amplitude, particle density, or substrate strength are varied 1 (); 2 (); 8 (); 9 (); 10 (); 11 (); 12 (); 13 (). It is also possible to observe a transverse ratchet effect in which the net dc drift of the particles is perpendicular to applied ac drive. For such transverse ratchets, when the ac drive is applied transverse to the substrate asymmetry direction, the resulting dc drift is parallel to the substrate asymmetry in either the easy or hard flow direction 9 (); 15 (); 16 (); 17 ().
In many of the experimentally studied systems where ratchet effects occur, such as vortices in type-II superconductors 8 (); 10 (); 12 (); 13 (); 14 () or colloids 20 (); 21 (), the motion of the particles is effectively overdamped. Recently a new particlelike excitation called skyrmions was discovered in chiral magnets 22 (); 23 (); 24 (). These skyrmions have many similarities to vortices in type-II superconductors in that they exhibit particlelike properties and have a mutually repulsive interaction that leads to the formation of a triangular skyrmion lattice 22 (); 23 (). Skyrmions can be driven with an applied current 24 (); New (); 25 (); 26 (); 27 (); 28 (); 29 () and exhibit pinning-depinning phenomena 25 (); 27 (); 29 (). A key difference between superconducting vortex and skyrmion systems is that in addition to the damping, skyrmion motion involves a strong non-dissipative Magnus effect which rotates the skyrmion velocity into the direction perpendicular to the net applied external forces. This Magnus term can be ten or more times larger than the damping term 24 (); 25 (); 27 (); 30 (). In the absence of pinning, under a dc drive the Magnus effect causes the skyrmions to move at an angle, the skyrmion Hall angle , with respect to the driving direction, where and is the ratio of the Magnus term to the damping term. In the presence of pinning, the skyrmion Hall angle has a strong drive dependence 31 (); 32 (); 33 (); 34 (); litzius (). Skyrmions have now been stabilized at room temperature 29 (); 35 (); 36 () making them promising candidates for a variety of spintronic applications 37 (), any of which would require the ability to precisely control the skyrmion motion. One method for achieving such control would be to exploit ratchet effects.
In previous numerical work, it was shown that an individual skyrmion in a 2D system interacting with a quasi-1D asymmetric substrate exhibits a rocking ratchet effect when the ac drive is applied along the substrate asymmetry direction 38 (). In this case, the resulting dc skyrmion velocity has components both parallel and perpendicular to the substrate asymmetry direction due to the Magnus term. A new type of ratchet effect, called a Magnus ratchet, was shown to occur when the ac drive is applied perpendicular to the substrate asymmetry direction 38 (). Here, the Magnus term induces skyrmion velocity components both parallel and perpendicular to the ac drive. As a result, the skyrmions translate partially along the substrate asymmetry direction, permitting ratcheting motion to occur. In the overdamped limit, this Magnus ratchet effect is lost. In the single skyrmion limit for both longitudinal and transverse ac driving, the ratchet flux is always aligned with the easy flow direction of the substrate asymmetry, so an open question is whether it is possible to realize a reversible skyrmion ratchet effect.
In this work we consider skyrmions driven by ac forces over gradient pinning arrays. Previous studies of such arrays in the overdamped limit for superconducting vortices demonstrated that both longitudinal and transverse ratchet effects as well as ratchet reversals occur as a function of ac amplitude and vortex density 17 (); 39 (). Here we show that for ac drives applied either parallel or perpendicular to the substrate asymmetry direction, when a finite Magnus term is present, ratchet effects occur even in regimes where there is no ratchet motion in the overdamped limit, while multiple reversals of the ratchet effect can appear when the ac amplitude, the skyrmion density, or the ratio of the Magnus term to the damping term is varied. The net dc drift of the skyrmions can be described as a vector which contains information about the magnitude of the drift and the angle between the drift direction and the substrate easy flow direction. With changing , ac amplitude, or skyrmion density, the ratchet vector undergoes either a clockwise or counterclockwise rotation of up to 360, indicating that ratcheting motion can occur in any direction for a 2D system. It is even possible to have a reversal in the direction of rotation of the ratchet vector. This system thus represents a new class of ratchet which we call a vector ratchet, and we predict that vector ratchets should be a general feature of any system in which Magnus effects are important, including skyrmions in chiral magnets 24 (), skyrmion phases in p-wave superconductors 40 (); 41 (); 42 (), rotating colloids 43 (), and charged particles in magnetic fields such as dusty plasmas 44 (); 45 (). Additionally, since vector ratchets allow for motion in any direction, they also could serve as a new method to control skyrmion motion for spintronic applications.
We model a 2D system of size with periodic boundary conditions in the - and -directions containing skyrmions at a density of . We place pinning sites in one of the periodic gradient array configurations illustrated in Fig. 1. We focus primarily on the conformal array shown in Fig. 1(a), which is produced by performing a conformal transformation on a uniform triangular array of pinning sites, as described in detail in previous work on pinning 46 (); 47 () and ratchet effects 17 (); 39 () for superconducting vortices in conformal pinning arrays. Successful experimental realizations of conformal pinning arrays for superconducting vortex systems 48 (); 49 () suggest that similar nanofabrication techniques could be used to create such arrays for skyrmion systems. Figure 1(b) illustrates the square gradient array, produced by subjecting a square pinning lattice to a gradient along the direction, while Fig. 1(c) shows the random gradient array, generated by introducing the same direction pinning density gradient to a random pinning array. We apply an ac driving force to the skyrmions of either , in the longitudinal or direction, or , in the transverse or direction, and measure the average net displacement of the skyrmions as a function of ac cycle.
To simulate the skyrmion motion we use a modified Theile equation 50 () described in Refs. 30 (); 32 (); 33 () that takes into account skyrmion-skyrmion interactions and skyrmion-pinning interactions. The equation of motion of a single skyrmion is
Here is the location of skyrmion and is the skyrmion velocity. The damping term with prefactor generates a skyrmion velocity component in the direction of the net external forces, while the Magnus term with prefactor generates a skyrmion velocity component perpendicular to the net external force direction. The repulsive skyrmion-skyrmion interactions are given by , where is the distance between skyrmions and , and is the modified Bessel function which falls off exponentially for large . The pinning force is modeled as arising from attractive nonoverlapping harmonic traps of radius which can exert a maximum pinning force of . The ac driving force is , where for longitudinal driving and for transverse driving, as shown schematically in Fig. 1. To characterize the ratchet effect, we measure the average net displacement of the skyrmions over time in both the and directions to obtain and , where is the position of skyrmion at time and is the initial reference time. We use a measurement interval of ac drive cycles, and the initial reference time is taken to be no less than 50 ac drive cycles after the system is initialized. The system size and the spacing between repeated tilings of our gradient pinning arrays is . The average spacing between individual pinning sites is . In this work we focus on samples with skyrmion density , filling fraction of , pinning radius of , and pinning force of .
Iii dc Depinning
We first apply a dc drive to the conformal pinning array sample in order to determine the depinning threshold. In Fig. 2(a) we plot and versus the dc drive amplitude for driving in the positive -direction in a sample with . The inset shows the skyrmion Hall angle versus , where and . The depinning threshold is close to . The Hall angle at low drives, and gradually increases with increasing until it reaches the expected pin-free value of . This strong dependence of the skyrmion Hall angle on the external drive in the presence of pinning was observed in previous studies of particle-based 32 (); 33 () and continuum-based 31 () simulations as well as in experiments 34 (). For dc driving in the positive direction, Fig. 2(b) shows that the depinning threshold has a lower value of . Near depinning, there is a stronger guiding effect in the -direction as the skyrmions move through the low pinning density region of the conformal array. As a result, the motion just above depinning is almost completely locked in the direction, giving a Hall angle close to zero, as shown in the inset of Fig. 2(b).
Iv Ratchet Effects with Longitudinal and Transverse AC Drives
To analyze the ratchet effect, we apply an ac drive to the conformal pinning array sample in Fig. 1(a) along the longitudinal () or transverse () direction, as indicated by the arrows in Fig. 1. In the overdamped case, only two types of ratchet effects occur: a net dc motion along the positive or negative direction, parallel to the drive, for longitudinal driving, and a net dc motion along the positive or negative direction, perpendicular to the drive, for transverse driving. In contrast, there can be up to eight types of motion for a Magnus induced ratchet. As shown in Fig. 3, these are type I, with net motion in the positive direction only; type II, with net motion in the positive and positive directions; type III, with net motion in the positive direction only; type IV, with net motion in the negative and positive directions; type V, with net motion in the negative direction only; type VI, with net motion in the negative and negative directions; type VII, with net motion in the negative direction only; and type VIII, with net motion in the positive and negative directions. We also refer to type IX, where there is no net motion in either direction, indicating the lack of a ratchet effect. Overdamped systems exhibit ratchet types I and V.
We now consider a case where there is no ratchet effect in the overdamped limit for either longitudinal or transverse ac driving, and we vary the ratio of the Magnus term to the damping term. In Fig. 4(a,b) we plot the average cumulative displacement per skyrmion in the direction and in the direction versus time in ac cycles for a system with in the longitudinal or direction. At , and , indicating the absence of a ratchet effect. For , the skyrmions move in the negative direction and the positive direction, which in the notation of Fig. 3 is a type IV ratchet. The negative direction is the easy flow direction of the substrate asymmetry. As increases from 4 to 20, a reversal of the ratchet effect occurs in which becomes positive so that the skyrmions are moving in the hard flow direction of the substrate asymmetry. The corresponding remains in the positive direction for , resulting in a type II ratchet, while for , and , giving a type I ratchet. For and , there is a direction reversal with and , producing a type VIII ratchet. The sequence of ratchet types that appear as a function of increasing , including the lack of a ratchet effect at , is IX-IV-III-II-I-VIII, so that the ratchet direction is moving clockwise around the diagram in Fig. 3.
In Fig. 4(c,d) we show and versus time in ac cycles for transverse or direction ac driving with , where there is again no ratchet effect for We find that is always negative but that there is a reversal in , which is negative for , giving a type VI ratchet, and positive for , producing a type IV ratchet. The ratchet sequence in this case is IX-VI-V-IV. The maximum ratchet flow magnitude is times larger for transverse ac driving than for longitudinal ac driving.
We also observe ratchet reversals at fixed as we vary , as shown in Fig. 5(a,b) where we plot and versus time in ac cycles. At there is no ratchet effect, while at , there is a weak ratchet effect in the negative direction that crosses over to a positive ratchet for and . The ratchet effect in the -direction is always negative. At , the motion is predominately in the negative direction with almost no direction movement, so the resulting sequence of ratchet types is IX-V-VIII-VI.
In Fig. 5(c,d) we show and versus time for the sample under transverse ac driving. The ratchet effect is always in the negative direction, with the largest ratchet flow occurring at , a drive at which a single skyrmion translates a distance larger than the entire system length during half of an ac drive cycle. The ratchet motion transitions from weak to strong negative direction flow with increasing before switching to positive direction flow for , giving a ratchet sequence of IX-VI-V-IV.
In Fig. 6(a) we plot the values of and after 400 ac cycles as a function of for the system in Fig. 4(a,b). At there is no ratchet effect, which we term a type IX ratchet, while for , the ratchet motion is in the negative and positive directions, which is a type IV ratchet. The ratchet motion passes through zero in the direction at while continuing to flow in the positive direction, giving a type III ratchet. This is also an example of a transverse ratchet effect in which a longitudinal dc drive produces drift motion strictly in the transverse direction. In the interval we find a type II ratchet with positive and positive motion, followed by a type I or strictly positive direction ratchet at . Finally, for , we observe type VIII flow with positive and negative motion. The sequence of ratchet types as a function of is indicated in the inset of Fig. 6(a), where the flow begins in region IV and gradually rotates clockwise by nearly . For driving in the direction, Fig. 6(b) shows that initially the system exhibits a type VI ratchet effect with negative and motion, passes through a type V ratchet in which motion occurs only in the negative direction despite the fact that the driving is applied along the direction, and then finally enters a broad type IV ratchet region in which the flow is in the negative and positive directions. The flow sequence is thus IX-VI-V-IV, and the flow rotates clockwise in the inset of Fig. 6(b) by about 90 as a function of .
In Fig. 7(a) we plot and versus for the system in Fig. 4 under direction driving with . A series of ratchet types appear, and there is a double reversal in from negative to positive and then back to negative, as well as in , which transitions from negative to positive and back to negative. The resulting ratchet sequence is IX-VI-V-IV-III-II-I-VIII-VII-VI, showing that the flow undergoes clockwise rotation through all the possible ratchet types or a rotation of in the inset of Fig. 7(a). For direction driving, , Fig. 7(b) shows that the ratchet sequence is IX-VI-V-IV, giving a clockwise rotation of .
From the ratchet behavior shown in Figs. 6 and 7, we can describe the direction of ratchet motion in terms of a vector with an amplitude of and an orientation of . This ratchet vector rotates as the parameters of the system are changed, and it can in principle point along any direction in the plane even though the asymmetry of the substrate exists only along the -direction. This represents a new type of ratchet that arises due to the skyrmion Hall angle, which depends on both and drive amplitude as shown in Fig. 2. For the parameters we consider, increasing the ac drive or the ratio increase the skyrmion Hall angle in the clockwise direction.
In order to get a better understanding of the evolution of the ratchet flow in Figs. 6 and 7, in Fig. 8 we show a heat map of the direction and magnitude of the net flux for direction ac driving based on the value of (Fig.8(a)) and (Fig.8(b)) after 400 ac cycles as a function of vs . Here for and , the ratchet effect is weak or absent. It is also clear that a reversal occurs in both and as functions of and . Figure 8(d,e) shows similar heat maps for direction ac driving. In this case the maximum intensity of the ratchet effect is stronger and is always negative, while there is a reversal in . Since is always negative, the ratchet sequence is limited to types III-IV-V-VI-VII.
V Skyrmion Density Dependence and Commensuration Effects
We next consider the effect of varying the skyrmion density for a fixed pinning site density of at . In Fig. 9(a) we plot and after 400 ac cycles for direction driving of with over the range . There is a strong type VIII ratchet flux , and the ratchet sequence IV-III-II-I-VIII progresses clockwise around the diagram in the inset of Fig. 9(a). In general we do not observe any ratchet motion in the single skyrmion limit of , indicating that the skyrmion ratchet motion on the conformal array is a collective effect, unlike the ratchet effect observed for a single skyrmion on a quasi-one-dimensional asymmetric substrate 38 (). There is a weak dip in the ratchet flux at , and the maximum ratchet flux occurs near , above which the flux decreases again. In general, the ratchet flux diminishes for large where the skyrmions form a stiff lattice that only weakly couples to the substrate. Similar effects appear in a superconducting vortex system for the ratchet flux at high vortex densities in the presence of a conformal pinning array 39 (). In Fig. 9(b) we show the same system with direction driving of . For , the data is fairly noisy, but for , ratchet flow occurs in both and with a ratchet sequence of IX-IV-V-VI-VII-VIII, indicating a counter-clockwise rotation of the flow by as indicated in the inset.
In Fig. 10(a) we show the same system as in Fig. 9(a) driven in the -direction with a pinning strength of and an ac amplitude of that have both been increased by a factor of five. In this case, the net ratchet flux is up to times larger than that produced when and . Here, is generally larger than , and there are multiple reversals in the direction motion as well as one reversal in the direction motion. The ratchet sequence is IX-IV-V-VI-VII-VIII-I-II for , giving a counter-clockwise rotation of the flow direction by as shown in the leftmost inset of Fig. 9(a), while for , the ratchet sequence is II-I-VIII, giving a clockwise rotation of as shown in the rightmost inset. This indicates that it is also possible to have reversals in the direction of the ratchet flow rotation, leading to what we term a reversible vector ratchet. Near , the ratchet flux is strongly reduced due to enhanced pinning from a commensuration effect with the underlying substrate. In Fig. 10(b), we show the ratchet flux in the same system for driving in the -direction with . There is a strong type IV ratchet effect with a maximum flux near . These results show that the skyrmion ratchet effect is robust over a wide range of skyrmion densities, ac drive amplitudes, and ratios.
We have also examined the effect of varying the ac driving frequency. In Fig. 11(a,b) we plot and after 400 ac cycles versus ac frequency in samples with and . The ratchet flux drops with increasing , in agreement with observations made in overdamped systems 39 (). In the insets of Fig. 11(a,b), we show the normalized quantities and , where is the number of simulation time steps required to complete 400 ac drive cycles at a driving frequency , is the value of at , and is the value of at . The normalized measures indicate that the net ratchet flux remains roughly constant when adjusted for the amount of time spent ratcheting at the different ac drive frequencies.
Vi Particle Trajectories
We image the skyrmion trajectories on either side of a ratchet reversal in order to understand how the geometry of the pinning array affects the skyrmion motion and how the amplitude of the ac drive can change the direction of the net ratchet flux. In Fig. 12(a,b) we plot the skyrmion positions, pinning site locations, and skyrmion trajectories in a sample with under a direction ac drive of , which produces a type VI ratchet with strong flux in the negative direction and weak flux in the negative direction. During the positive portion of the ac drive cycle, shown in Fig. 12(a), the skyrmions predominantly move in the positive direction. The flow is concentrated in the regions of lower pinning density, and there is a small amount of skyrmion hopping in the positive direction, which is the hard flow direction of the substrate asymmetry. If the pinning sites were not present, during the positive portion of the ac drive cycle the skyrmions would move with a Hall angle of relative to the positive axis. Instead, in Fig. 12(a), the Hall angle is nearly zero since skyrmion motion in the positive direction is blocked by the regions of dense pinning. The Magnus term couples the and motion and causes the positive direction to act like a hard flow direction even though there is no asymmetry in the substrate along the direction. During the negative portion of the ac drive cycle, illustrated in Fig. 12(b), the motion is mostly in the negative direction, with some hopping in the negative direction. Since the negative direction is the easy flow direction of the ratchet asymmetry, the Magnus coupling causes the negative direction to act like an easy flow direction, and the net ratchet flux during the entire cycle is larger in the negative direction than in the positive direction, producing a net negative and negative flow. Figure 12(c) shows the positive portion of the ac cycle for a drive of , while Fig. 12(d) shows the negative portion of the ac cycle at the same drive. For this ac drive amplitude, there is a strong ratchet flux in the negative direction and a weaker ratchet flux in the positive direction, giving a type IV ratchet effect. The ac drive is strong enough that, during the positive portion of the ac cycle in Fig. 12(c), the skyrmions can pass through the densely pinned regions, and the resulting Hall angle is larger than that observed at the lower ac amplitude of . During the negative portion of the ac cycle, shown in Fig. 12(d), the skyrmions continue to pass through the densely pinned regions, but since the negative direction is the easy flow direction of the substrate asymmetry, the net amount of negative motion is increased compared to that which occurs during the positive portion of the ac cycle, and correspondingly the amount of motion in the negative direction is decreased. Thus, for fixed pinning strength and skyrmion density, the ratchet flow rotates with increasing ac amplitude due to the depinning process in the direction and the increasing Hall angle, as shown in Fig. 1.
Vii Square and Random Gradient Arrays
In Fig. 13(a,b) we show and after 400 ac cycles versus in samples with containing either the square gradient array illustrated in Fig. 1(b) or the random gradient array shown in Fig. 1(c). Also shown for comparison is a sample with a conformal array. Here the square gradient array produces a large ratchet flux for low , and in some cases the flow is in the opposite direction to that observed in the conformal array. The random gradient array in general shows a much smaller ratchet flux that is primarily in the negative and positive directions, which is opposite to the flux observed for the conformal array. Figure 13(c) shows vs for the same systems under direction ac driving, . In this case, the conformal array always produces a negative ratchet flux, while the square gradient array shows a weaker ratchet flux as well as a reversal from positive to negative flow near . The random gradient array does not show any appreciable ratchet flux. In Fig. 13(d), the corresponding versus plot indicates that the ratchet flux of the square gradient array is comparable to or even higher than that of the conformal array for , while the random gradient array shows almost no ratchet flux. We observe similar effects for fixed and varied ac amplitude . In general, the conformal array produces the largest ratchet flux, while the ratchet flux for the square gradient array is weaker, and that of the random gradient array is the weakest.
We have shown that ac driven skyrmions interacting with two-dimensional gradient pinning arrays represent a realization of a new type of ratchet system that we call a vector ratchet. In overdamped systems, the ratchet flux is limited to flowing parallel to the substrate asymmetry direction in the forward or reverse direction. In contrast, the strongly non-dissipative Magnus term found in skyrmion systems produces a skyrmion Hall angle that couples the motion parallel and perpendicular to the substrate asymmetry direction. The resulting dc ratchet drift generated by the ac drive can be described as a vector which can rotate counter-clockwise or clockwise in the plane as the ac amplitude or the ratio of the Magnus term to the dissipative term is varied, so that it is possible to realize reversals in the ratchet flux in both the and directions. We show that this vector ratchet appears for ac driving both parallel to and perpendicular to the substrate asymmetry direction. The ratchet reversals we observe are a result of collective skyrmion interactions, as previous work on individual skyrmions interacting with asymmetric substrates showed no ratchet reversals. In addition to reversals in the ratchet flux in the and directions, the angular rotation of the ratchet vector itself can also show a reversal. We find that it is possible to have rotations of the ratchet vector of up to , indicating that vector ratchets can be used to direct skyrmion motion in any in-plane direction. Thus, the vector ratchet could serve as a powerful new method for controlling skyrmion motion. Vector ratchets should be general to systems of collectively interacting particles driven over asymmetric substrates where Magnus type effects are present.
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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