Controlled motion of skyrmions in a magnetic antidot lattice
J. Feilhauer, S. Saha, J. Tobik, M. Zelent, L. J. Heyderman, M. Mruczkiewicz
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Controlled motion of skyrmions in a magnetic antidot lattice
J. Feilhauer , ∗ S. Saha , , † J. Tobik , M. Zelent , L. J. Heyderman , , and M. Mruczkiewicz ‡ Institute of Electrical Engineering, Slovak Academy of Sciences,Dubravska Cesta 9, SK-841-04 Bratislava, Slovakia Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland Laboratory for Multiscale Materials Experiments,Paul Scherrer Institute, 5232 Villigen PSI, Switzerland and Faculty of Physics, Adam Mickiewicz University in Poznan, Umultowska 85, Poznan, PL-61-614 Poland (Dated: October 17, 2019)Future spintronic devices based on skyrmions will require precise control of the skyrmion motion.We show that this goal can be achieved through the use of magnetic antidot arrays. With micro-magnetic simulations and semi-analytical calculations based on Thiele equation, we demonstratethat an antidot array can guide the skyrmions in different directions depending on the parametersof the applied current pulse. Despite the fixed direction of the net driving current, due to thenon-trivial interplay between the repulsive potential introduced by the antidots, the skyrmion Halleffect and the non-uniform current distribution, full control of skyrmion motion in a 2D lattice canbe achieved. Moreover, we demonstrate that the direction of skyrmion motion can be controlledby tuning only a single parameter of the current pulse, i.e. current magnitude. For lower currentmagnitudes the skyrmion can be moved perpendicularly to the current direction, and can overcomethe skyrmion Hall effect. For larger current magnitudes, the skyrmion Hall effect can be effectivelysuppressed and skyrmions can move parallel to the applied current.
I. INTRODUCTION
In the approximation of continuous magnetization,magnetic skyrmions are topologically non-trivial sta-ble spin textures that can be characterized by a non-zerowinding number. The winding number for skyrmions in2D films is w = ±
1, which signifies that the magneticstate of the skyrmion cannot be unwound into a homoge-neous state by a continuous transformation. Skyrmionsin ferromagnetic materials are stabilized by the inter-play of several magnetic energy contributions, i.e. ex-change, dipole, anisotropy and Zeeman. In particular,the asymmetric Dzyaloshinskii-Moriya exchange interac-tion (DMI) introduces chiral canting between neighbour-ing spins and favours skyrmion stability . DMI canarise due to the spin-orbit coupling and the lack of struc-tural inversion symmetry, which can be realized in asym-metric heavy metal/ferromagnet heterostructures . Dueto the interfacial anisotropy, the ferromagnetic layer ismagnetized in out-of-plane and the skyrmion is a defectin this homogeneous magnetization with diameter from afew to several hundreds of nanometers. The magnetiza-tion of the skyrmion center is oriented in the opposite di-rection with respect to the rest of the ferromagnetic layerand the magnetization of skyrmion continuously rotatesfrom the center to its edges, so reversing in a radiallysymmetric fashion.Another appealing feature of magnetic skyrmions is thepossibilty to manipulate them by electric current viaspin transfer and spin orbit torques . In general,the current density required to drive the skyrmion motionis much smaller than the current density required to movea domain wall . These unique features of skyrmionsmake them a promising candidates for future spintronicapplications such as low dissipation magnetic information storage devices , skyrmion racetrack memories , andlogic devices .Unfortunately, skyrmions have a tendency to mi-grate towards the edges of a magnetic strip, due tothe skyrmion Hall effect, which leads to an unstabletransverse position and annihilation. Furthermore, theskyrmion motion is randomized by the thermal diffu-sion and the presence of pining centers in real samples .Therefore, key challenges regarding skyrmion motion arerelated to stabilization, confinement and control of themovement of skyrmions at room temperature . Theobservation that the edges of the sample and defects re-pel skyrmions led to the idea to use periodic latticesas a medium with well defined and robust motion ofskyrmions .In order to achieve the control over the skyrmion mo-tion, we consider an antidot lattice realized as a thin fer-romagnet/heavy metal bilayer (e.g., Co/Pt, CoFeB/Pt)with circular holes arranged on periodic square lattice.A hole in the magnetic texture of the ferromagnetic layer(i.e. region with zero magnetization) can be viewed asa homogeneous layer superposed with a disk with theopposite magnetization orientation. Since the magneti-zation of this disk is oriented in the same direction asthe magnetization of the skyrmion center, the skyrmionis magnetostatically repelled by the antidot. The repul-sion of skyrmion by the antidot lattice can therefore beexpressed by a periodic effective potential with energyminima (valleys) located between each four neighbour-ing antidots. Due to the dissipation, in the absence ofdriving torques acting on the skyrmion, the skyrmionsare stabilized near the bottom of the valleys. In otherwords, in the presence of the antidot lattice, the relaxedpositions of skyrmions are confined to the discrete squarelattice of valleys. Sufficient spin transfer torque is re-quired to move the skyrmions over the energy barrierfrom one valley to another. The torque acting on theskyrmions can be tuned with current pulses of differentdensity and pulse width. Due to the discrete nature ofthe valleys, the current pulse parameters required for askyrmion to arrive at a particular valley create compactregions in the parameter plane encompassing the pulsewidth, ∆ t and the current density, j . Employing a mi-cromagnetic solver Mumax and a semi-analytical modelbased on the Thiele equation, we calculate a map of theseregions in the (∆ t , j ) plane and study its changes uponvariation of system properties such as damping.The skyrmion trajectories in general are not parallelto the current density due to the skyrmion Hall effect.However, due to the non-trivial interplay between theskyrmion Hall effect and the potential landscape of theantidots, for shorter current pulses with larger magni-tudes, the skyrmion can be moved to the valleys in the di-rection strictly parallel to the net applied current. Thus,for a particular range of current pulse parameters, theskyrmion Hall effect is effectively suppressed. By apply-ing a longer current pulses with smaller magnitudes, theskyrmion can be moved into specific valleys in the di-rection perpendicular to the current. For this range ofcurrent parameters the skyrmion Hall effect is enhanced.We demonstrate that, for an adequate combinationof material, current and antidot lattice parameters, theskyrmion can be transported to almost any nearest-neighbouring valley by applying a specific current pulse.Moreover, transport to the nearest-neighbouring valleyscan be achieved by tuning a single parameter of the cur-rent pulse, namely the current magnitude.We therefore propose a method for unprecedented con-trol of the skyrmion placement, utilizing the induced pe-riodic potential of the antidot lattice. This is an im-portant step towards exploiting antidot arrays in futureskyrmion based spintronic devices.The paper is organized as follows. In Section II, weintroduce the numerical and semi-analytical models forskyrmion transport in the antidot lattice driven by anapplied current. In Section III, we discuss the resultsin three parts. To identify some characteristic featuresof the skyrmion transport in the physical system, wefirst study a much simpler model with negligible dampingand spatially uniform current density (Subsection III A).Then we introduce damping into the model and studythe interplay between the potential of the antidot latticeand the skyrmion Hall effect (Subsection III B). Finally,we utilize the gained knowledge to describe the skyrmiontransport in the physical system with damping and non-uniform current density resulting from the presence ofthe antidots (Subsection III C). II. METHODS
To study the current driven dynamics of skyrmions inthe antidot lattice, we have used the following two meth- ods. First, we employed the Mumax3 solver to performmicromagnetic simulations, which is a proven method forthe investigation of magnetization processes. Second, thesoliton nature of the skyrmion facilitates use of a simplerformalism than that used by the micromagnetic simu-lations to sufficiently describe the skyrmion dynamics,i.e. the Thiele equation. The Thiele equation is solvedby numerical time integration. In this section we pro-vide a detailed description of the calculations for bothapproaches.
Micromagnetic simulations.
The governing equa-tion of magnetization dynamics is the Landau-Lifshitz-Gilbert (LLG) equation, which is a partial differentialequation numerically solved by Mumax3 via a finite dif-ference method. To simulate the skyrmion transport inthe antidot lattice we have used periodic boundary condi-tions with a unit cell of square shape with lattice constantof a = 512 nm and thickness h = 1 . d = 250 nm. The rectangular regular discretizationmesh has dimensions 2 × × . . The simulatedmaterial was CoFeB with material parameters: satura-tion magnetization M S = 1 . × A/m, DMI constant D = 1 mJ/m , exchange constant A ex = 1 . × − J/m,and damping constant α in the interval from 0.03 to 0.3.To stabilize the skyrmions an out-of-plane magnetic fieldof 30 mT was applied.In order to describe the motion of the skyrmions in-duced by the current, we consider the in-plane cur-rent flowing directly in the ferromagnetic material. Forsuch cases the Zhang-Li form of the current-induced spintransfer torque is an adequate model . We neglect thenon-adiabatic contribution to the torque by setting thenon-adiabaticity parameter to ξ = 0. The current den-sities of rectangular current pulses used in the simulationswere varied up to 400 GA/m , which are the values con-sidered in the experiments on skyrmion motion inducedby charge current . The net current was applied in formof a rectangular pulse in the direction parallel to the rowof antidots, along the x-axis (see Fig.1). Thiele equation.
The soliton character of theskyrmion means that it can be described as a quasipar-ticle with a fixed magnetization profile. This approachconsiderably reduces the multidimensional spin degreesof freedom, leaving only two parameters - the coordinatesof skyrmion center r = ( x, y ). The quasiparticle approxi-mation of the skyrmion transforms the LLG equation intothe Thiele equation , which is an ordinary differentialequation describing the dynamics of the skyrmion centerlocated at r . The translational motion of a skyrmion isaffected by the interaction with antidots, which can berepresented by a repulsive potential V ( r ). The motion isdriven by the current density j ( r , t ) which, for the caseof a rectangular current pulse with width ∆ t , is givenby j ( r , t ) = j ( r )Θ(∆ t − t ), where Θ is the Heaviside stepfunction. All these effects are included in the modifiedThiele equation , which takes the form FIG. 1. (Color online) Schematic of the ferromagnet/heavymetal heterostructure that hosts the magnetic skyrmions. An-tidots are given by circular holes with diameter d = 250 nmarranged in the square lattice with lattice constant a =512 nm. The antidots repel the skyrmions as indicated by thepotential V ( x, y ) plotted above the heterostructure. The colorcurves are an example of a skyrmion trajectory induced bythe applied current pulse j in the x -direction. The skyrmionis driven from the relaxed position at the potential minimum(at t = 0) to the antidot wall. When the current pulse isswitched off (black point at t = ∆ t ), the skyrmion orbitsaround the antidot and simultaneously relaxes to the neigh-bouring potential minimum. The trajectory calculated byThiele equation (red curve) is in a reasonable agreement withthe trajectory calculated micromagnetically (green curve). G × ( b j j ( r , t ) − ˙r ) − α D˙r + ∇ V ( r ) = 0 , (1)where G is the gyromagnetic coupling vector, b j = − µ B /eM S ( µ B is the Bohr magneton and e > α is the magnetic damping and D is thedissipative force tensor.The first term in (1) is the topological Magnus force,perpendicular to the velocity of the skyrmion, which isresponsible for the gyrotropic (transverse) motion of theskyrmion . In ultrathin films, the magnetization isconstant in the z direction and the gyromagnetic couplingvector G is perpendicular to the film plane, i.e. G = G e z with G = − πM S hwγ , (2)where γ = gµ B / ~ is a gyromagnetic ratio and g is an elec-tron g-factor. We assume that the magnetization of theskyrmion center is oriented in the negative z -direction.Therefore the winding number of skyrmion used in Eq.(2)is w = −
1. The second term in (1) is the dissipative force, whichis linked to the damping in the material and responsiblefor the friction acting on the skyrmion. In an ultrathinfilm, the dissipative force tensor D is diagonal and hasthe form D = Dδ ij with D = − M S hκγ , (3)where κ is determined by the out-of-plane magnetizationprofile of the skyrmion. Typically, κ is equal to severalunits of π and we use κ = 5 . π , which provides the bestfit of our Thiele equation results with the data obtainedby micromagnetic simulations .The third term in (1) represents the force acting on theskyrmion due to the repulsive interaction with antidotlattice. The corresponding periodic potential V ( r ) wasconstructed numerically with the use of the micromag-netic solver in the following manner: i) a continuous cur-rent was applied to move the skyrmion out of equilibriumto higher energy and the magnetic state was recorded. ii)The saved states were used as the initial states for thetime evolution with reduced damping constant α = 0 . V ( x, y ) which has same periodicity as the cor-responding antidot lattice.The resulting periodic repulsive potential V ( x, y ) isplotted in Fig. 1 with the minimas (valleys) centred be-tween four neighbouring antidots and saddles located be-tween neighbouring antidots.After algebraic manipulation, Eq.1 can be written inthe form ˙ r ( t ) = A j + B F (4)where F = −∇ V and A = Gb j α D + G (cid:20) G αD − αD G (cid:21) , (5) B = 1 α D + G (cid:20) αD − GG αD (cid:21) . (6)We solved the set of Eqs.(4) numerically with the initialcondition r ( t = 0) set at the lowest point of the potentiallandscape V ( r ), which corresponds to the relaxed posi-tion of the skyrmion without an external driving force.The applicability of the Thiele equation approach wastested by comparing the skyrmion trajectories with cal-culations performed using the micromagnetic solver forvarious values of current density and the pulse width,and we found that the trajectories are in agreement. Ex-amples of the two trajectories determined with the dif-ferent methods are shown in the Fig. 1. With the useof Thiele equation, we determined the final position ofthe skyrmion after the application of a current pulse andconstructed a map of the final skyrmion positions as thefunction of the current density and the pulse width. Ap-plying strong current density might lead to skyrmion ani-hilation at the disk boundaries. However, the process ofannihilation of the skyrmion by pushing it to an antidotis beyond the validity of the Thiele equation approach.Therefore, we have used micromagnetic simulations todetermine the critical values of the current pulse param-eters, where the annihilation starts to occur.The antidots realized as holes in the layers do not con-duct electric current. The spatially non-uniform currentdensity distribution for a geometry with holes was cal-culated using Comsol. The non-uniform current densitywas implemented in the micromagnetic as well as in theThiele equation calculations. However, the approxima-tion of the uniform current density simplifies the inter-pretation of results and is valuable for a qualitative un-derstanding of the magnetic processes. Therefore, in thefollowing we present results for both uniform and non-uniform current distributions. III. RESULTS
To understand the origin of several characteristic fea-tures of the skyrmion transport in a real system withdamping and non-uniform current distribution, we firstlystudy much simpler model with negligible damping anduniform current density (Subsection III A). Then we in-troduce damping into the model and study its effect onthe skyrmion trajectory and final position after the cur-rent pulse is applied (Subsection III B). Finally, to deter-mine the behaviour of skyrmions in a physical system, weincorporate a non-uniform current distribution into themodel and discuss the similarities and differences withthe previous simpler cases (Subsection III C).
A. Damping free system with uniform current
The main goal of this paper is to study the transportof skyrmions driven by the applied current pulse in thepresence of a repulsive antidot lattice. For simplicity, westart with the damping free material, i.e. we set α =0. In this case, the matrix A in (4) is diagonal, whichmeans that the torque acting on the skyrmion is parallelto the current density j and there is no Hall effect. Incontrast, the matrix B is off-diagonal, which means thatthe skyrmion is forced to move perpendicularly to thegradient of the antidot potential. Moreover, if we assumethat the current density is spatially uniform, the effect ofcurrent can be easily incorporated into the potential viaan extra term V c = Gb j ( j × r ) z , simplifying Eq. 4 intothe form ˙ r ( t ) = ( e z × F eff ) /G, (7)where F eff = −∇ V eff = −∇ ( V + V c ). The dynamics ofthe skyrmion described by Eq. 7 can be readily under-stood, since the skyrmion moves perpendicularly to the FIG. 2. (Color online) Effective potential V eff combiningthe effects of antidot potential V and driving current j onthe skyrmion motion in the undamped sample. a)-d) Withincreasing j applied in the x direction, the tilt of V eff alongthe y direction increases. Red curves show the correspondingskyrmion trajectories starting from the potential minimum(valley) of V . effective force F eff , i.e. it simply follows the isoenergycontour of the effective potential V eff .When the current is switched off, V eff = V , withclosed isoenergy contours centred around the bottom ofthe valley or around an antidot, depending on whetherthe corresponding energy is smaller or larger than the en-ergy of the saddle point between the valleys (see Fig. 2a).When the uniform current is switched on, the resultingeffective potential V eff is just the antidot potential V tilted in the direction perpendicular to the direction ofthe current. The amount of tilt is proportional to thevalue of the current density, i.e. the larger the current,the larger the tilt of the antidot potential. In particular,when we apply the current in the x direction, the cur-rent density has the form j ( r ) = ( j,
0) and the effectivepotential is simplified to V eff ( x, y ) = V ( x, y ) + Gb j jy, (8)i.e. it is equal to the antidot potential V tilted in the y -direction. The isoenergy contours of this tilted V eff are no longer just closed but open runaway contours alsoexist and extend between the valleys in the x direction(see Fig. 2b-d).We now discuss the three types of skyrmion trajec-tories resulting from the applied current, assuming theskyrmion starts from the bottom of the valley, i.e. thecenter between four neighbouring antidots (red point inFig. 2a). For smaller current density, the correspondingisoenergy contour of V eff is closed and skyrmion oscil-lates inside the valley (Fig. 2b). As the current densityis increased, the tilt of the antidot potential can be largeenough to cause the contour of V eff crossing the start-ing point to open. In this case, the skyrmion escapes FIG. 3. (Color online) Skyrmion transport without damping driven using rectangular current pulses with the density j andwidth ∆ t . a)-e) Trajectories of skyrmions in the antidot array starting at the bottom of the valley at point ( x, y ) = (0 , t = 100 ns and the the values of j are 20, 60, 140, 187 and 240 GA/m respectively. Thecolor of each trajectory corresponds to a particular region in (f). Black dots denote the position of the skyrmion where thecurrent pulse is switched off. f) Map of the final positions of the skyrmion as a function of the current pulse width and density.Red dots indicate the parameters defining the trajectories in (a)-(e). the starting valley and passes along an antidot to theneighbouring valley in the x -direction (Fig. 2c). For suf-ficiently large current density, the tilt of the antidot po-tential is so large that the isoenergy contour along whichthe skyrmion passes to the neighbouring valley can passdirectly through the saddle between the valleys avoid-ing the passage of the skyrmion around the antidot (seeFig. 2d).Our aim is to determine, and therefore control, thefinal positions of the skyrmion after applying a rectangu-lar current pulse with density j and width ∆ t . As shownabove, when the current is switched on, the skyrmion fol-lows the isoenergy contour of the tilted antidot potential V eff . Subsequently, when the current pulse is switchedoff, the tilted antidot potential returns back to the origi-nal antidot potential (without current) and the skyrmionfollows a closed isoenergy contour, i.e. it stays trappedinside a valley or orbits around an antidot depending onthe parameters j and ∆ t . Five examples of such trajec-tories are shown in Fig. 3a-e for a skyrmion starting atthe valley at ( x, y ) = (0 ,
0) driven by a current pulse withfixed width ∆ t = 100 ns and various current densities j .The final positions of skyrmion resulting from the ap-plied current pulse are summarized in a pulse width-current density map in Fig. 3f. This map was calculatedusing the Thiele equation (4). The regions of parame-ters j and ∆ t which correspond to the same final posi-tion of the skyrmion are given by the same color. Thegray regions correspond to the case where the skyrmionorbits around the antidots marked I, II, or III (see e.g.Fig. 3b). The other colored regions correspond to thesituation where the skyrmion ends up trapped inside avalley in the horizontal direction (e.g. at (x,y) = (0,0), (-1,0), ...). For simplicity, we plot only the data for thevalleys with distance smaller than 3 a from the origin.The data for valleys further than 3 a are all located inthe white region in Fig. 3b. When the current densityis small or the pulse is very short, the skyrmion stayspinned inside the starting valley at (0,0) as depicted bythe dark green region. When increasing the pulse param-eters, the current pulse is sufficient to move the skyrmionover the saddle point and the skyrmion orbits around theantidot I after the current pulse is turned off (light grayregion, AD I). Increasing the current pulse parameterseven more, the skyrmion is able to move to the neighbour-ing valley (region (-1,0) in light green), orbits around theantidot II (region AD II in dark gray), moves to the next-neigbouring valley (region (-2,0) in light blue) and so on.There is a discontinuity in Fig. 3b at j u = 187 GA/m ,which separates two types of trajectories. For j < j u theskyrmion trajectories crossing the valleys pass around theantidots I, II, ... (e.g. the trajectory for j = 140 GA/m in Fig. 3a) while for j > j u the skyrmions reach thesevalleys directly through the saddles of the antidot poten-tial (e.g. the trajectory for j = 240 GA/m in Fig. 3a).At j = j u the skyrmion passes to the saddle point ofthe effective potential V eff located between the antidotsI and A where F eff = ∇ V eff = 0. Here, the torque act-ing on the skyrmion due to the applied current is com-pletely compensated by the repulsion of the antidot lat-tice. Then, since the right side of the Thiele equation(7) vanishes, the skyrmion velocity becomes zero and itstops, i.e. the saddle point of V eff is an unstable positionfor the skyrmion. After the current pulse is switched offthe skyrmion starts to orbit around the antidot I. Thecorresponding skyrmion trajectory is shown in Fig. 3d FIG. 4. (Color online) Trajectories of a skyrmion startingin the valley at (0,0) for various values of damping. Theskyrmion motion is driven by the uniform current density j =90 GA/m applied along the x direction. In the undampedcase ( α = 0) the trajectory extends only to the valleys in thedirection of the current (e.g. at (-1,0), (-2,0), ...) while, fornon-zero damping, the trajectories also reach the valleys inthe transverse direction (e.g. at (-1,1), (0,1), ...). where the black point indicates the unstable position ofthe skyrmion where the skyrmion is fixed until the cur-rent pulse is switched off.As shown above, without damping, it would be possibleto transfer the skyrmions only to neighbouring valleys inthe direction parallel with the current. As we show in thefollowing section, to move the skyrmion in the transversedirection with respect to the current, nonzero dampingand the resulting skyrmion Hall effect is essential. B. Damped system with uniform current
In order to determine how the motion of the skyrmionis affected by damping, we include a non-zero α in theoff-diagonal elements of matrix A in Eq. 4, which is re-sponsible for the skyrmion Hall effect. In the unpatternedthin film, the skyrmion would move at an angle with re-spect to the current direction given by a Hall angle | Θ H | = arctan κα π . (9)From Eq. 9, we found that the value of Θ H varies from2 . ° to 23 . ° when changing α from 0 .
03 to 0 .
3. Anothereffect resulting from the damping is the relaxation of theskyrmion to a lower energy configuration when the cur-rent pulse is switched off. Due to the damping, the diag-onal elements of B in Eq. 4 are nonzero and the skyrmion motion converges to the local energy minima, i.e. it endsup at the bottom of a valley.Due to the nontrivial manifestation of damping in theThiele equation (8), the simple concept of a tilted anti-dot potential described by (7) is not valid for non-zerodamping. Therefore the shapes of the skyrmion trajec-tories are more complex, which is illustrated in Fig. 4 forfixed uniform current density j = 90 GA/m and variousvalues of damping α . In the undamped case ( α = 0),the trajectory extends in the x direction with a periodicform with a period of a single lattice constant a . This is aresult of the fact that the direction of the applied currentand the corresponding torque acting on the skyrmion areparallel to the symmetry axis of the antidot lattice (i.e. x axis). If the torque applied on the skyrmion had anonzero component in the y direction, the skyrmion tra-jectory would on average follow the torque direction butit would be formed from discrete steps with the period-icity larger than a single a . For non-zero damping, sincethe current flows along the x direction, the effective y component of the torque is generated via the skyrmionHall effect. Therefore, as was also shown in Ref. [14], de-pending on the amount of damping and density of the ap-plied current, the skyrmion trajectories in the presence ofan antidot lattice form a series of periodic discrete stepsextending in both x and y directions with periods thatcan be multiples of a , e.g. the trajectory for α = 0 . x axis also increases. Due to the presence of aperiodic antidot potential, the trajectory angle is muchlarger than the corresponding Hall angle for the unpat-terned film. For example, the trajectory for α = 0 . . ° . Therefore, thepresence of the antidot lattice enhances the skyrmionHall effect. Similar enhancement of skyrmion Hall ef-fect was observed in Ref.14 for a skyrmion moving on atwo-dimensional periodic substrate.As we have seen, the skyrmion trajectories in a dampedsystem extend not only to the valleys in the direction ofthe current but also to the valleys in the transverse direc-tion. Therefore, the number of valleys that are reachableby the skyrmion is significantly increased. When consid-ering a finite current pulse, we can distinguish two scenar-ios for the skyrmion transfer between the neighbouringvalleys. First, the skyrmion is directly transferred to thedesired valley by an applied current and then, when thecurrent is switched off, the skyrmion relaxes to the valleybottom. The second case is illustrated in Fig. 1 where, af-ter the current pulse is switched on, the skyrmion climbsfrom the bottom of the valley to the antidot wall untilthe pulse is switched off (black point). Without damp-ing the skyrmion would orbit around the antidot foreverbut, since damping is present ( α = 0 . FIG. 5. (Color online) Final position of the skyrmion afterthe application of uniform current pulse with density j andwidth ∆ t . Color regions correspond to the parameters withthe same final valley. The damping is a) α = 0 .
03, b) α = 0 . α = 0 . α = 0 .
3. Shaded region calculated bymicromagnetic solver shows the parameters of current pulseat which the skyrmion is annihilated. the final positions of the skyrmion that result after ap-plying a uniform current pulse with density j and width∆ t . The maps in Fig. 5 were calculated using the Thieleequation (4) for various values of damping constant α .For small damping, α = 0 .
03 (Fig. 5a), we can iden-tify several features that also appear in the undampedcase. When the current density or width of the currentpulse is too small, the skyrmion just stays pinned in thestarting valley (dark green region) after the pulse is ap-plied. The current j u defining the trajectories passingto the unstable position of the skyrmion located betweenthe antidots I and A is shifted to the larger values, i.e. j u = 234 GA/m . For j > j u , the region correspondingto the transport of skyrmion to the neighbouring valley(-1,0) directly through the saddle between the antidotsI and A has almost the same area and shape as in theundamped case. However, due to the enhanced skyrmionHall effect, the same valley is almost unreachable via thetrajectories passing around the antidot I and the region(-1,0) is much smaller for j < j u than for the undampedcase. The former light gray region ascribed to the tra-jectories orbiting around the antidot I (region AD I inFig. 3f) is now decomposed into the set of thin regionscorresponding to the four valleys at ( x, y ) = (0 , j and ∆ t .For larger damping α = 0 . j u = 430 GA/m exceeds the displayed interval of cur-rent densities. Therefore, in Fig. 5b all skyrmions trav-eling along the trajectories which leave the starting val-ley pass around the antidot I. Then the only way thatthe skyrmion can reach the valley (-1,0) is by descendingfrom the wall of antidot I after the current pulse is off(light green region in Fig. 5b). Since the potential of theantidots V ( x, y ) and the applied electric current j haveperiodicity of the antidot lattice, the points related tothe unstable position of the skyrmion are periodically lo-calized throughout the whole antidot lattice. As we havealready discussed, the periodicity of skyrmion trajecto-ries can exceed a single lattice constant, which meansthat a skyrmion can pass to an unstable point severallattice constants away from the starting point. As anexample, the discontinuity appearing in the blue regionat (-2,1) for j = 240 GA/m corresponds to the trajec-tory that passes to the unstable point located betweenthe antidots III and B defined in Fig. 4.For even larger values of damping, the valley (-1,0) isunreachable within the given range of current densities asis demonstrated in Fig. 5c,d for α = 0 . α = 0 .
3, re-spectively. To summarize, for α = 0 .
1, the regions in theparameter plane (∆ t, j ) corresponding to the different fi-nal positions of the skyrmion are more compact than forthe case of small damping α = 0 .
03 where the situationis quite chaotic. In contrast, for larger values of damp-ing α = 0 . , .
3, the region corresponding to the valley(-1,0) is shifted to the large values of current outside ofthe displayed interval. Therefore, the number of neigh-bouring valleys the skyrmion can be moved to is reducedfor α = 0 . , .
3. Therefore, to maximize the number ofreachable valleys by the skyrmion, the value of damping α = 0 . j or ∆ t , the skyrmion inelasticallyscatters off the antidots or even annihilates at the antidotedges. The skyrmion annihilation is not captured by theThiele equation since it is based on the assumption thatthe magnetization profile of skyrmion is fixed. The trans-parent gray regions in Fig. 5 highlight the parameters ofcurrent pulses which lead to the skyrmion annihilationcalculated using micromagnetic simulations. For smallervalues of damping, the regions of annihilation cover asignificant part of the displayed parameter maps. FIG. 6. (Color online) a) Distribution of a periodic non-uniform current density j ( x, y ) in a single unit cell of the antidot lattice. j ( x, y ) was calculated using the Comsol package where the antidot hole (gray circle) is assumed to be nonconductive. b) Mapof the final positions of the skyrmion after application of current pulses with non-uniform current density in the undampedsystem.FIG. 7. (Color online) Final position of the skyrmion afterthe application of non-uniform current pulse with density j and width ∆ t . Color regions correspond to the parameterswith the same final valley. The coordinate system is shownin the Fig. 4. The damping is a) α = 0 .
03, b) α = 0 . α = 0 . α = 0 .
3. Shaded region calculated bymicromagnetic solver shows the parameters of current pulseat which the skyrmion is annihilated.
C. System with non-uniform current
Here we employ a more realistic model of electric cur-rent density. Up till now we have assumed that the elec-tric current density is distributed uniformly and flowsalong the x -direction in the whole antidot lattice, i.e. j ( x, y ) = ( j, j ( x, y ) assuming anet voltage drop applied in the x direction. This pe-riodic non-uniform current distribution j ( x, y ) is shownin Fig. 6a. Due to the symmetry of the square antidotlattice, the current density averaged along the transver-sal ( y ) direction is the same as in the uniform case, i.e. a R a j ( x, y ) dy = ( j,
0) and, as a result of current conser-vation, it is independent of x . We have incorporated thisnon-uniform current distribution into the micromagneticsimulation and Thiele equation (4) and recalculated themaps of final skyrmion positions in the current pulse pa-rameter plane.The map for non-uniform current density and zerodamping is shown in Fig. 6b. This map is very simi-lar to the case of uniform current in Fig. 3b, but thediscontinuity corresponding to the unstable position ofskyrmion is shifted to smaller values of current.The skyrmion transport driven by short current pulseswith non-uniform current density is summarized in Fig. 7for various values of damping. Again, these maps are sim-ilar to the maps for uniform current in Fig. 5, but thereare some significant differences. In general, as a resultof the current flowing around the antidot, the currentsneeded to move the skyrmion directly through the saddleto the neighbouring valley (avoiding the orbiting aroundan antidot) are smaller. For small damping, α = 0 . x direction from the starting valley. Thelight green region corresponding to the valley (-1,0) isbroader, which makes this valley accessible for a largerrange of current pulse parameters. Most importantly, themicromagnetic simulations reveal that the annihilationregions are shifted to the larger values of current pulseparameters than for uniform current in Fig. 5. This is FIG. 8. (Color online) Skyrmion trajectories of damped sys-tem with α = 0 .
1, non-uniform current density, fixed widthof current pulse at ∆ t = 50 ns and varying current densitiesand polarities. By adjusting a single parameter (j) of the cur-rent pulse, the skyrmion can be transfered to almost all of theneighbouring valleys in the longitudinal as well as transversaldirection. expected since, for uniform current, the current flow atone side of the antidot pushes the skyrmion directly to-wards the antidot edge, while for non-uniform current,the current flowing around the antidot reduces the con-tact of the skyrmion with the antidot edge. For α = 0 . x direc-tion with fixed width ∆ t = 100 ns and varying densityand polarity. The skyrmion trajectories correspondingto j = ± , ± , ±
220 GA/m are shown in Fig. 8.By tuning a single parameter, current density j , theskyrmion starting at the bottom of a valley can be trans-ported to six of the eight neighbouring valleys. IV. SUMMARY
In this work, we have developed a semi-analyticalmodel to determine the skyrmion motion driven by short in-plane current pulses in the presence of a magnetic an-tidot array. Due to the repulsion of the skyrmion bythe antidot edges, the antidot lattice results in an effec-tive potential of attractive valleys located between eachfour neighboring antidots. We have demonstrated thatskyrmion transport between individual valleys can becontrolled by applying a rectangular current pulse withadequate density and width. As a result of the inter-play between the antidot potential, skyrmion Hall effectand non-uniformity of the current, skyrmions can be ma-nipulated in the longitudinal and even in the transversedirection with respect to the current flow. We have iden-tified two mechanisms determining the final position ofthe skyrmion: i) the skyrmion is directly driven by theapplied current to the desired valley, ii) after the cur-rent pulse is switched off, the skyrmion relaxes down theantidot wall to the desired valley.We have calculated maps showing the regions of thecurrent pulse parameters that give a particular final po-sition of the skyrmion after the pulse is switched off.Starting from the bottom of a valley, our calculationsshow that, by applying an adequate unidirectional cur-rent pulse, it is possible to move the skyrmion to al-most all of the neighboring valleys horizontally and ver-tically. Therefore, by using a sequence of electrical cur-rent pulses, the magnetic antidot arrays can be used as amedium for well controlled skyrmion motion. Our resultsare therefore an important step towards skyrmion baseddevices.
V. ACKNOWLEDGMENTS
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