Magnus-Induced Ratchet Effects for Skyrmions Interacting with Asymmetric Substrates
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Magnus-Induced Ratchet Effects For Skyrmions Interacting with AsymmetricSubstrates
C. Reichhardt, D. Ray, and C. J. Olson Reichhardt ∗ Theoretical Division and Center for Nonlinear Studies,Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA (Dated: July 16, 2018)When a particle is driven with an ac force over an asymmetric potential, it can undergo a ratcheteffect that produces a net dc motion of the particle. Ratchet effects have been observed in numeroussystems such as superconducting vortices on asymmetric pinning substrates. Skyrmions, stabletopological spin textures with particle-like properties, have many similarities to vortices but theirbehavior is strongly influenced by non-disspative effects arising from a Magnus term in their equationof motion. We show using numerical simulations that pronounced ratchet effects can occur for acdriven skyrmions moving over asymmetric quasi-one-dimensional substrates. We find a new typeof ratchet effect called a Magnus-induced transverse ratchet that arises when the ac driving forceis applied perpendicular rather than parallel to the asymmetry direction of the substrate. Thistransverse ratchet effect only occurs when the Magnus term is finite, and the threshold ac amplitudeneeded to induce it decreases as the Magnus term becomes more prominent. Ratcheting skyrmionsfollow ordered orbits in which the net displacement parallel to the substrate asymmetry directionis quantized. Skyrmion ratchets represent a new ac current-based method for controlling skyrmionpositions and motion for spintronic applications.
Ratchet effects can arise when particles placed inan asymmetric environment are subjected to suitablenonequilbrium conditions such as an externally appliedac drive, which produces a net dc motion of the particles .Ratchet effects have been realized for colloidal systems ,cold atoms in optical traps , granular media on sawtoothsubstrates , and the motion of cells crawling on patternedasymmetric substrates , and they can be exploited tocreate devices such as shift registers for domain wallsmoving in an asymmetric substrate . There has been in-tense study of ratchet effects for magnetic flux vorticesin type-II superconductors interacting with quasi-one-dimensional (q1D) or two-dimensional (2D) asymmetricpinning substrates. The vortices can be effectively de-scribed as overdamped particles moving over an asym-metric substrate, and under ratcheting conditions theapplication of an ac driving current produces a net dcflux flow . The simplest vortex ratchet geometry wasfirst proposed by Lee et al. , where an effectively 2D as-sembly of vortices was driven over a q1D asymmetricallymodulated substrate using an ac driving force orientedalong the direction of the substrate asymmetry.Skyrmions in chiral magnets, initially discoveredin MnSi and subsequently identified in numerousother materials at low and room temperatures ,have many similarities to superconducting vortices.Skyrmions are spin textures forming topologically sta-ble particle-like objects that can be set into motionby the application of a spin-polarized current . Asa function of the external driving, skyrmions can ex-hibit a depinning transition similar to that found forvortices, and from transport measurements it is possi-ble to construct skyrmion velocity versus applied forcecurves . It has been shown that the dynamics ofskyrmions interacting with pinning can be captured byan effective particle equation of motion, which produces pinning-depinning and transport properties that agreewith those obtained using a continuum-based model ,just as a particle-based description for vortices can ef-fectively capture the vortex dynamics in the presence ofpinning sites . Due to their size scale and the low cur-rents required to move them, skyrmions show tremen-dous promise for applications in spintronics such as race-track type memory devices originally proposed fordomain walls , and it may be possible to create skyrmionlogic devices similar to those that harness magnetic do-main walls . In order to realize such skyrmion applica-tions, new methods must be developed to precisely con-trol skyrmion positions and motion.The most straightforward approach to a skyrmionratchet is to utilize the same types of asymmetricsubstrates known to produce ratchet effects in vor-tex systems. There are, however, important differ-ences between skyrmion and vortex dynamics due to thestrong non-dissipative Magnus component of skyrmionmotion that is absent in the vortex sys-tem. The Magnus term produces a skyrmion velocitycontribution that is perpendicular to the direction of anapplied external force. In the presence of pinning, theMagnus term reduces the effectiveness of the pinning bycausing the skyrmions to deflect around the edges of at-tractive pinning sites rather than being capturedby the pinning sites as overdamped vortices are. TheMagnus term also produces complex winding orbits forskyrmions moving in confined regions or through pin-ning sites . The ratio of the strength of theMagnus term α m to the dissipative term α d can be 10or higher for skyrmion systems . In contrast, althoughit is possible for a Magnus effect to appear for vorticesin superconductors, it is generally very weak so that thevortex dynamics is dominated by dissipation . A keyquestion is how the Magnus force could impact possible F ac F ac x y FIG. 1:
Asymmetric substrate geometry : A schematicof the sample geometry where a skyrmion is placed on aq1D asymmetric periodic substrate. An ac driving force F ac cos( ωt ) is applied parallel ( F ac k ) or perpendicular ( F ac ⊥ )to the substrate asymmetry direction. ratchet effects for skyrmions, and whether new types ofratchet phenomena can be realized for skyrmion systemsthat are not accessible in overdamped systems .In this work we numerically examine a 2D assembly ofskyrmions interacting with a q1D asymmetric periodicsubstrate shown schematically in Fig. 1. An ac drive isapplied either parallel ( F ac k ) or perpendicular ( F ac ⊥ ) tothe substrate asymmetry direction, which runs along the x axis. In the overdamped limit, no ratchet effect ap-pears for F ac ⊥ ; however, we find that the Magnus effectproduces a novel ratchet effect for F ac ⊥ when it curvesthe skyrmion trajectories into the asymmetry or x di-rection. We model the skyrmion dynamics in a samplewith periodic boundary conditions using a particle baseddescription and vary the ratio of the Magnus to dis-sipative dynamic terms, the amplitude and frequency ofthe ac drive, and the strength of the substrate. Simulation
The equation of motion for a single skyrmion with ve-locity v moving in the x − y plane is α d v + α m ˆ z × v = F SP + F ac k , ⊥ (1)The damping term α d keeps the skyrmion velocityaligned with the direction of the net external force, whilethe Magnus term α m rotates the velocity toward the di-rection perpendicular to the net external forces. To ex-amine the role of the Magnus term we vary the relativestrength α m /α d of the Magnus term to the dissipativeterm under the constraint α d + α m = 1, which maintainsa constant magnitude of the skyrmion velocity. The sub-strate force F SP = −∇ U ( x ) ˆx arises from a ratchet po-tential U ( x ) = U [sin(2 πx/a ) + 0 .
25 sin(4 πx/a )] , (2)where a is the periodicity of the substrate and we de-fine the strength of the substrate to be A p = 2 πU /a . -202468 < V > || , ⊥ -6-4-202 < V > || , ⊥ -6-4-20 < V > || , ⊥ F ac|| -8-6-4-20 < V > || , ⊥ α m / α d = 0.0 α m / α d = 4.58 α m / α d = 7.858 (a) α m / α d = 1.6 (b)(c)(d) B C FIG. 2:
Parallel and perpendicular ratchet veloci-ties for parallel drive : The skyrmion velocity parallel h V i k (red) and perpendicular h V i ⊥ (blue) to the substrate asym-metry vs F ac k for a substrate strength of A p = 1 .
5. (a) At α m /α d = 0 .
0, the overdamped limit, only h V i k is nonzeroand shows quantized steps with h V i k = n , where n is an inte-ger. (b) At α m /α d = 1 .
6, there is quantized motion in boththe parallel and perpendicular directions. (c) α m /α d = 4 . α m /α d = 7 .
85. The points labeled B and C indicate wherethe skyrmion orbits in Fig. 3(b,c) were obtained.
The ac driving term is either F ac k = F ac k cos( ωt ) ˆx or F ac ⊥ = F ac ⊥ cos( ωt ) ˆy . We measure the time-averagedskyrmion velocities parallel h V i k ≡ π h v · ˆx i /ωa or per-pendicular h V i ⊥ ≡ π h v · ˆy i /ωa to the substrate asym-metry direction as we vary the ac amplitude, substratestrength, or ω . Here we focus primarily on sampleswith A p = 1 . ω = 2 . × − inverse simulationtime steps, and we consider the sparse limit of a sin-gle skyrmion which can also apply to regimes in whichskyrmion-skyrmion interactions are negligible. Results and DiscussionA. ac Driving Parallel to Substrate Asymmetry
We first apply the ac driving force parallel to the asym-metry direction, a configuration previously studied forthe same type of substrate in vortex systems . InFig. 2 we plot h V i k and h V i ⊥ versus F ac k for sampleswith A p = 1 .
5, so that the maximum magnitude of thesubstrate force is F s max = 2 . x directionand F s max = 1 . x direction. The over-damped limit α m /α d = 0 . h V i ⊥ = 0 . x -direction for F ac k > .
25. The skyrmionvelocity is quantized and forms a series of steps with h V i k = n where n is an integer. On the highest stepin Fig. 2(a) near F ac k = 2 . n = 7. The quantization in-dicates that during one ac driving period, the skyrmionmoves a net distance of na in the parallel direction.For very large values of F ac k , the velocity quantizationis lost and the parallel ratchet effect gradually dimin-ishes. In Fig. 2(b) we plot h V i k and h V i ⊥ for a samplewith α m /α d = 1 .
6, showing a nonzero skyrmion veloc-ity component for both the parallel and perpendiculardirections. There is again a quantization of the parallelvelocity h V i k , and the Magnus term transfers this quanti-zation into the perpendicular direction even though thereis no periodicity of the substrate along the y direction.Thus, we find h V i ⊥ = nα m /α d . For F ac k > .
0, wefind windows in which the ratcheting effect is lost and h V i k = h V i ⊥ = 0. As α m /α d increases, the maximumvalue of n decreases, so that at α m /α d = 4 .
58 in Fig. 2(c),only the n = 1 step appears. At the same time, thewidth of the non-ratcheting windows increases, as shownin Fig. 2(d) for α m /α d = 7 . ω or increasing the sub-strate strength A p , it is possible to increase the range andmagnitude of the skyrmion ratchet effect. In Fig. 3(b,c)we show representative skyrmion orbits for the system inFig. 2(b,c). Figure 3(b) shows the orbits at α m /α d = 1 . F ac k = 2 .
2, corresponding to the n = 3 step inFig. 2(b). Here the skyrmion moves along straight linesoriented at an angle to the direction of the applied acdrive. During each ac cycle, the skyrmion first translatesa distance 3 a in the positive x direction and then travelsin the reverse direction for a distance of a/ x direction for this magnitude of acdrive. It repeats this motion in each cycle. In Fig. 3(c) weshow the skyrmion orbit at α m /α d = 4 .
58 for F ac k = 2 . n = 1 orbit in Fig. 2(c). The anglethe skyrmion motion makes with the ac driving direc-tion is much steeper, and the skyrmion is displaced a netdistance of a in the x direction during each ac drivingcycle.In order to clarify the evolution of the ratchet phasesas a function of α m /α d and F ac k , in Fig. 3(a) we plot theratchet envelopes for the n = 1 to n = 6 steps. For the n = 1 case we show, in blue, all of the regions in the F ac k − α m /α d plane where n = 1 steps appear. For n = 2to n = 6, to prevent overcrowding of the graph, insteadof plotting all ratchet regions we show only the inner stepenvelope for each n = n i , obtained from simulations inwhich F ac k is swept up from zero, and defined to extendfrom the drive at which n first reaches n i to the drive atwhich n first drops below n i . We find that the minimum F ac k required to induce a ratchet effect increases linearlywith increasing α m /α d , and that the ratcheting regionsform a series of tongue features (shown for n = 1 only).The ratchet effects persist up to and beyond α m /α d = 10. α m / α d F a c || A CB D (a) x(b)y x(c)y
FIG. 3:
Ratchet region envelopes and example ratchetorbits for parallel ac driving : ( a ) A plot of F ac k vs α m /α d indicating all n = 1 (blue) ratcheting regions, and the innerenvelopes of the n = 2 (orange), 3 (purple), 4 (green), 5(red) and 6 (yellow) ratcheting regions. The boundaries of theinner envelopes are defined as the point at which the ratchetvelocity first reaches the value n followed by the point atwhich it first drops below n . The y-axis marked A and thedashed lines marked B − D are the values of α m /α d for whichthe velocity curves in Fig. 2 were obtained. The minimum acamplitude required to produce a ratchet effect increases withincreasing α m /α d . ( b,c ) Skyrmion trajectory images. White(green) regions are high (low) areas of the substrate potential.Thin black lines indicate forward motion of the skyrmion andthick purple lines indicate backward motion. The red circleis the skyrmion. ( b ) The n = 3 orbit at α m /α d = 1 . F ac k = 2 . a in the x direction during one ac drivecycle. ( c ) The n = 1 orbit at α m /α d = 4 .
58 for F ac k = 2 . a inthe x direction during every ac drive cycle. For α m /α d = 0, the strongest ratchet effects with thelargest values of n occur near F ac k = 2 .
3, correspondingto a driving force that is slightly higher than the max-imum force exerted on the skyrmion by the substratewhen it is moving with a negative x velocity component.As F ac k increases above this value, the skyrmion can slipbackward over more than one substrate plaquette duringeach ac cycle, limiting its net forward progress. -4-202468 < V > || , ⊥ -8-6-4-202 < V > || , ⊥ F ac ⊥ -8-6-4-20 < V > || , ⊥ F ac ⊥ < V > || ac ⊥ -1-0.500.5 < V > || α m / α d = 0.0 α m / α d = 0.855 α m / α d = 7.018 (a)(b)(c) α m / α d = 4.0 FIG. 4:
Parallel and perpendicular ratchet veloci-ties for perpendicular drive : The skyrmion velocity par-allel h V i k (red) and perpendicular h V i ⊥ (blue) to the sub-strate asymmetry vs F ac ⊥ for samples with A p = 1 .
5. ( a ) At α m /α d = 0 . F ac ⊥ > .
5, and h V i k = n withinteger n . Inset: At α m /α d = 0, there is no ratchet effect.( b ) α m /α d = 4 .
0. ( c ) α m /α d = 7 . h V i k from the main panel at the third n = 1 step showingthat additional fractional steps appear at velocity values suchas n/m = 1 / B. ac Driving Perpendicular to SubstrateAsymmetry
In Fig. 4 we plot h V i k and h V i ⊥ versus F ac ⊥ for A p =1 .
5. The inset in Fig. 4(a) shows that for α m /α d = 0, h V i k = h V i ⊥ = 0, indicating that there is no ratchet ef-fect in either direction. When the Magnus term is finite,as in Fig. 4(a) where α m /α d = 0 . h V i k is quantized at integer values,with the largest step in Fig. 4(a) falling at n = 5, and h V i ⊥ / h V i k = α m /α d . As α m /α d increases, Fig. 4(b,c)shows that the maximum value of F ac ⊥ at which ratchet-ing occurs decreases, as does the maximum value of n andthe widths of the ratcheting windows. In addition to theinteger steps in h V i k , we also observe fractional steps, asshown in the inset of Fig. 4(c) for α m /α d = 7 . n = 1 step. On the upper n = 1step edge there is a plateau at h V i k = 1 / h V i k = 3 /
4, 2/3, 1/3, and 1 /
4, which are accom-panied by even smaller steps that form a devil’s staircasestructure . For higher values of F ac ⊥ , smaller plateausappear at rational fractional velocity values h V i k = n/m with integer n and m , while full locking to the integer n = 1 step no longer occurs. The n/m rational lock-ing steps also occur for parallel ac driving but are muchweaker than for perpendicular ac driving. x(a)y x(b)yx(c)y x(d)y FIG. 5:
Example ratchet orbits for perpendicularac driving : Skyrmion trajectory images from the systemin Fig. 4. White (green) regions are high (low) areas of thesubstrate potential. Black lines indicate the motion of theskyrmion; the red circle is the skyrmion. ( a ) The n = 2 stepat α m /α d = 0 .
855 for F ac ⊥ = 2 .
0. ( b ) The n = 1 step at α m /α d = 4 . F ac ⊥ = 0 .
7. ( c ) A non-ratcheting orbit at α m /α d = 4 . F ac ⊥ = 0 .
76. ( d ) For higher ac drives the or-bits become more complicated, as shown here for α m /α d = 4 . F ac ⊥ = 1 . The skyrmion orbits for perpendicular ac driving differmarkedly from those that appear under parallel ac driv-ing, since the Magnus term induces a velocity compo-nent perpendicular to the direction of the external driv-ing force. As a result, perpendicular ac drives induceparallel skyrmion motion that interacts with the sub-strate asymmetry. Figure 5(a) shows a skyrmion orbiton the h V i k = 2 step for F ac ⊥ = 2 . α m /α d = 0 . a in the x direction during every ac drivecycle. In Fig. 5(b) we plot an n = 1 orbit at F ac ⊥ = 0 . α m /α d = 4 . a in the x direction in each ac drive cycle.Figure 5(c) shows a non-ratcheting orbit for α m /α d = 4 . F ac ⊥ = 0 .
75, where the skyrmion traces out a compli-cated periodic closed cycle. For higher ac amplitudes, theratcheting orbits become increasingly intricate, as shownin Fig. 5(d) for α m /α d = 4 . F ac ⊥ = 1 . n = 1 ratchet regions andthe inner envelopes of the n = 2 to 5 ratchet regions asa function of F ac ⊥ and α m /α d . For low values of α m /α d , α m / α d F a c ⊥ A p F a c ⊥ A B C (a) (b) α m / α d =4.92 FIG. 6:
Ratchet region envelopes for perpendicularac driving : ( a ) A plot of F ac ⊥ vs α m /α d indicating all n = 1(blue) ratcheting regions, and the inner envelopes of the n = 2(orange), 3 (purple), 4 (green), and 5 (red) ratcheting regions.The dashed lines marked A − C are the values of α m /α d forwhich the velocity curves in Fig. 4 were obtained. There is noratchet effect at the overdamped limit of α m /α d = 0, and theminimum ac amplitude required to produce a ratchet effectdecreases with increasing α m /α d . ( b ) A plot of F ac ⊥ vs A p for α m /α d = 4 .
92 indicating the ratcheting regions with n = 1(blue), 2 (orange), 3 (purple), and 4 (green). As the substratestrength increases, stronger ratchet effects appear. there is no ratcheting since the Magnus term is too weakto bend the skyrmion trajectories far enough to run alongthe direction parallel to the asymmetry. The minimum acamplitude required to generate a ratchet effect decreasesas the strength of the Magnus term increases, opposite towhat we found for parallel ac driving in Fig. 3(a). Here,the skyrmion trajectories are more strongly curved intothe parallel direction as α m /α d increases, so that a lowerac amplitude is needed to push the skyrmions over thesubstrate potential barriers. In contrast, for a parallel acdrive the Magnus term curves the skyrmion trajectoriesout of the parallel direction, so that a larger ac ampli-tude must be applied for the skyrmions to hop over thesubstrate maxima. We also find that as α m /α d increases,the minimum ac force needed to produce a ratchet effectdrops below the pinning force exerted by the substrate.This cannot occur in overdamped systems, and is an in-dication that the Magnus term induces some inertia-likebehavior. The threshold ac force value F th for ratchetingto occur can be fit to F th ⊥ ∝ ( α m /α d ) − for perpendicu-lar driving, while for parallel driving, F th k ∝ α m /α d . As α m /α d increases, the extent of the regions where ratch-eting occurs decreases; however, by increasing the sub-strate strength, the regions of ratcheting broaden in ex-tent and the magnitude of the ratchet effect increases.In Fig. 6(b) we show the evolution of the n = 1, 2, 3and 4 ratchet regions as a function of F ac ⊥ and A p forfixed α m /α d = 4 .
92. Here, as A p increases, the mag-nitude of the ratchet effect increases, forming a seriesof tongues. The minimum ac force needed to induce aratchet effect increases linearly with increasing A p . Theseresults indicate that skyrmions can exhibit a new type FIG. 7:
Parallel ratchet velocity heat maps for par-allel and perpendicular ac driving : The magnitude of h V i k in samples with A p = 1 . α m /α d = 3 . a ) As afunction of F ac ⊥ and ω in units of 10 − inverse simulation timesteps, h V i k has clear tongue structures, and its magnitude isnon-monotonic for varied frequency. ( b ) h V i k as a functionof F ac k and ω has similar features. of Magnus-induced ratchet effect that appears when thedriving force is applied perpendicular to the asymmetrydirection of the substrate. C. Frequency Dependence
In Fig. 7(a) we plot a heat map of the parallel ratchetvelocity h V i k for perpendicular ac driving as a function of F ac ⊥ and ω in a sample with A p = 1 . α m /α d = 3 . F ac ⊥ atlow ac driving frequencies, while for higher ω the ratchet-ing regions form a series of tongues. The ratchet magni-tude is non-monotonic as a function of ω , and in severalregions it increases in magnitude with increasing ω , suchas on the first tongue where the maximum ratcheting ef-fect effect occurs near ω = 10 − inverse simulation timesteps before decreasing again as ω increases further. Wefind very similar ratchet behaviors for parallel ac driving,as shown in Fig. 7(b) where we plot h V i k as a functionof F ac k and ω and observe a series of tongues.Our results show that skyrmions are an ideal sys-tem in which to realize ratchet effects in the presenceof asymmetric substrates. As expected, they undergoratcheting when an ac drive is applied parallel to thesubstrate asymmetry direction; however, the skyrmionsalso exhibit a unique ratchet feature not found in over-damped systems, which is the transverse ratchet effect.Here, when the ac drive is applied perpendicular to thesubstrate asymmetry direction, the Magnus term curvesthe skyrmion orbits and drives them partially paral-lel to the asymmetry, generating ratchet motion. Theasymmetric substrates we consider could be created us-ing methods similar to those employed to create quasi-one-dimensional asymmetric potentials in superconduc-tors and related systems, such as nanofabricated asym-metric thickness modulation, asymmetric regions of ra-diation damage, asymmetric doping, or blind holes ar-ranged in patterns containing density gradients. Since the skyrmion ratchet effects persist down to low frequen-cies, it should be possible to directly image the ratchet-ing motion, while for higher frequencies the existence ofratchet transport can be deduced from transport studies.Such ratchet effects offer a new method for controllingskyrmion motion that could be harnessed in skyrmion ap-plications. We also expect that skyrmions could exhibita rich variety of other ratchet behaviors under differentconditions, such as more complicated substrate geome-tries, use of asymmetric ac driving, or collective effectsdue to skyrmion-skyrmion interactions. Acknowledgments
This work was carried out under the auspices of theNNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396. DR acknowledges support providedby the Center for Nonlinear Studies. Reimann, P., Brownian motors: noisy transport far fromequilibrium.
Phys. Rep. , 57–265 (2002). Rousselet, J., Salome, L., Ajdari, A. & Prost, J., Direc-tional motion of Brownian particles induced by a peri-odic asymmetric potential.
Nature (London) , 446–448(1994). Salger, T. et al. , Directed transport of atoms in a Hamil-tonian quantum ratchet.
Science , 1241–1243 (2009). Farkas, Z., Tegzes, P., Vukics, A. & Vicsek, T., Transitionsin the horizontal transport of vertically vibrated granularlayers.
Phys. Rev. E , 7022–7031 (1999). Mahmud, G. et al. , Directing cell motions on micropat-terned ratchets.
Nature Phys. , 606-612 (2009). Franken, J. H., Swagten, H. J. M. & Koopmans, B., Shiftregisters based on magnetic domain wall ratchets with per-pendicular ansisotropy.
Nature Nanotechnol. , 499–503(2012). Lee, C. S., Jank´o, B., Der´enyi, I. & Barab´asi, A. L., Re-ducing vortex density in superconductors using the ’ratcheteffect.’
Nature (London) , 337–340 (1999). Wambaugh, J. F., Reichhardt, C., Olson, C. J., March-esoni, F., & Nori, F., Superconducting fluxon pumps andlenses.
Phys. Rev. Lett. , 5106–5109 (1999). Olson, C. J., Reichhardt, C., Jank´o, B. & Nori, F.,Collective interaction-driven ratchet for transporting fluxquanta.
Phys. Rev. Lett. , 177002 (2001). Villegas, J. E. et al. , A superconducting reversible rectifierthat controls the motion of magnetic flux quanta.
Science , 1188-1191 (2003) de Souza Silva, C. C., Van de Vondel, J., Morelle, M.& Moshchalkov, V. V., Controlled multiple reversals of aratchet effect. Nature (London) , 651–654 (2006). Lu, Q., Reichhardt, C. J. O. & Reichhardt, C., Reversiblevortex ratchet effects and ordering in superconductorswith simple asymmetric potential arrays.
Phys. Rev. B ,054502 (2007). M¨ulbauer, S. et al. , Skyrmion lattice in a chiral magnet.
Science , 915-919 (2009). Yu, X. Z. et al. , Real-space observation of a two-dimensional skyrmion crystal.
Nature , 901-904 (2010). Heinze, S. et al. , Spontaneous atomic-scale magneticskyrmion lattice in two dimensions.
Nature Phys. , 713-718 (2011). Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y., Observationof skyrmions in a multiferroic material.
Science , 198–201 (2012). Nagaosa, N. & Tokura, Y., Topological properties anddynamics of magnetic skyrmions.
Nature Nanotechnol. ,899-911 (2013). Woo, S. et al. , Observation of room temperature magneticskyrmions and their current-driven dynamics in ultrathinCo films. arXiv:1502.07376 (unpublished). Moreau-Luchaire, C. et al. , Skyrmions at room tempera-ture : From magnetic thin films to magnetic multilayers.arXiv:1502.07853 (unpublished). Tokunaga, Y. et al. , A new class of chiral materialshosting magnetic skyrmions beyond room temperature.arXiv:1503.05651 (unpublished). Jonietz, F. et al. , Spin transfer torques in MnSi at ultralowcurrent densities.
Science , 1648-1651 (2010). Schulz, T. et al. , Emergent electrodynamics of skyrmionsin a chiral magnet.
Nature Phys. , 301–304 (2012) Yu, X.Z. et al. , Skyrmion flow near room temperature in anultralow current density.
Nature Commun. , 988 (2012). Iwasaki, J., Mochizuki, M. & Nagaosa, N., Universalcurrent-velocity relation of skyrmion motion in chiral mag-nets.
Nature Commun. , 1463 (2013). Lin, S.-Z., Reichhardt, C., Batista, C. D. & Saxena, A.,Driven skyrmions and dynamical transitions in chiral mag-nets.
Phys. Rev. Lett. , 207202 (2013). Lin, S.-Z., Reichhardt, C., Batista, C. D. & Saxena, A.,Particle model for skyrmions in metallic chiral magnets:Dynamics, pinning, and creep.
Phys. Rev. B , 214419(2013). Reichhardt, C., Ray, D. & Reichhardt, C. J. O., Collectivetransport properties of driven skyrmions with random dis-order.
Phys. Rev. Lett. , in press (2015). arXiv:1409.5457. Liang, D., DeGrave, J. P., Stolt, M. J., Tokura, Y. &Jin, S., Current-driven dynamics of skyrmions stabilizedin MnSi nanowires revealed by topological Hall effect.arXiv:1503.03523 (unpublished). Brandt, E. H., The flux-line lattice in superconductors.
Rep. Prog. Phys. , 1465–1594 (1995). Fert, A., Cros, V. & Sampaio, J., Skyrmions on the track.
Nature Nanotechnol. , 152-156 (2013). Sampaio, J., Cros, V., Rohart, S., Thiaville, A. & Fert, A.,Nucleation, stability and current-induced motion of iso-lated magnetic skyrmions in nanostructures.
Nature Nan-otechnol. , 839-844 (2013). Parkin, S. S., Hayashi, M. & Thomas, L. Magnetic domain-wall racetrack memory.
Science , 190–194 (2008). Allwood, D. A. et al. , Magnetic domain-wall logic.
Science , 1688–1692 (2005). B¨uttner, F. et al. , Dynamics and inertia of skyrmionic spin structures.
Nature Phys. , 225–228 (2015). Liu, Y.-H. & Li, Y.-Q., A mechanism to pin skyrmionsin chiral magnets.
J. Phys.: Condens. Matter , 076005(2013). M¨uller, J. & Rosch, A., Capturing of a magnetic skyrmionwith a hole.
Phys. Rev. B , 054410 (2015). Reichhardt, C., Ray, D. & Reichhardt, C. J. O., Quantizedtransport for a skyrmion moving on a two-dimensional pe-riodic substrate.
Phys. Rev. B , 104426 (2015). A ratcheting motion of skyrmions was discused inMochizuki, M. et al. , Thermally driven ratchet motion ofa skyrmion microcrystal and topological magnon Hall ef-fect.
Nature Mater. , 241-246 (2014); however, this isfor a system with no imposed asymmetric substrate and isdifferent from the ratchet effects we study in this work. Bak, P. & Bruinsma, R., One-dimensional Ising model andthe complete devil’s staircase.
Phys. Rev. Lett.49