Magnetic bilayer-skyrmions without skyrmion Hall effect
MMagnetic bilayer-skyrmions without skyrmion Hall effect
Xichao Zhang , Yan Zhou , , ∗ and Motohiko Ezawa † Department of Physics, University of Hong Kong, Hong Kong, China Center of Theoretical and Computational Physics, University of Hong Kong, Hong Kong, China and Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
Arising from emergent electromagnetic field of magnetic skyrmions due to their nontrivial topology, theskyrmion Hall effect might be a roadblock for practical applications since any longitudinal motions ofskyrmions in nanotrack is accompanied by a transverse motion. A direct consequence of such an effectis easy destruction of skyrmions at the nanotrack edges during their fast motions along the nanotrack,despite their topological protection. Here we propose an entirely novel solution of completely inhibit-ing such skyrmion Hall effect without affecting its topological properties based on a antiferromagnetic-coupling bilayer system. We show that a pair of magnetic skyrmions can be nucleated in such a bilayersystem through vertical current injection or converted from a current-driven domain-wall pair. Oncenucleated, the skyrmion pair can be displaced through current-induced spin torque either from a verticalinjected current or in-plane current. The skyrmion Hall effect is completely suppressed due to the cancel-lation of back-action forces acting on each individual skyrmion, resulting in a straight and fast motion ofskyrmions along the current direction. This proposal will be of fundamental interests by introducing thebilayer degree of freedom into the system. Moreover, it provides an easy way to engineer the transportproperties of the skyrmionic devices to achieve desired performance, making it highly promising for prac-tical applications such as ultradense memory and information-processing devices based on skyrmions.
Since the first experimental observations of magneticskyrmion lattices in bulk non-centrosymmetric magnets andfilms , there has been tremendous interest in these topolog-ically protected spin configurations with a quantized topo-logical number . However, the creation and transmissionof isolated magnetic skyrmion in magnetic thin films are re-quired for any practical applications such as encoding in-formation in individual skyrmion to allow entirely novel de-vices and circuitry . Significant efforts and progress havebeen made towards realizing such ultrathin film based on per-pendicularly magnetized magnetic layer | heavy metal struc-ture to host a rich variety of chiral spin textures includingskyrmions . The strong spin orbit coupling at the inter-faces between the magnetic layer with perpendicular magneticanisotropy (PMA) and the underlying heavy metal layer pro-vides a sizeable Dzyaloshinsky-Moriya interaction (DMI) tostabilize skyrmions .To move a skyrmion in nanotrack for information-processing applications, a convenient and efficient way is bymeans of spin current which can transfer the angular momen-tum from itinerant conduction electrons to the magnetic mo-ments of the skyrmion . However, one major roadblock tothe manipulation and transmission of skyrmions in nanotrackis the skyrmion Hall effect, i.e. , skyrmions exhibit the Hall ef-fect driven by spin currents due to the presence of the Magnusforce which in turn originates from its nontrivial topology .Thus a skyrmion will not move parallel with the direction ofthe current. Instead it will gain a transverse velocity with amagnitude proportional to the spin current density which dis-places the skyrmion towards the edge of the nanotrack. There-fore a skyrmion will be easily destroyed for a distance ofmuch less than 1 µ m for a nanotrack made of typical magneticlayer | heavy metal system .Since both the skyrmion Hall effect and its topological pro-tection arise from the same mechanism, i.e ., its nontrivialtopological number of ± , it seems to be impossible to in- hibit the skyrmion Hall effect without breaking its topolog-ical protection. In this work, we propose a novel solutionof two perpendicularly magnetized sublayers strongly cou-pled via the antiferromagnetic (AFM) exchange interactionwith a heavy metal layer beneath the bottom magnetic layer (Fig.1a, b, c). When a skyrmion is created in one of thesublayers, another skyrmion is simultaneously created in theother sublayer under certain conditions. We refer to such apair of magnetic skyrmions as a magnetic bilayer-skyrmion(Fig.1d, e). Moreover, we show a bilayer-skyrmion can bedisplaced over arbitrarily long distances driven by spin cur-rents without touching the edges due to the absence of theskyrmion Hall effect, which is distinct from a skyrmion inthe monolayer thin-film structure. In addition to address theabove-mentioned dilemma, bilayer-skyrmions have many dis-tinct characteristics from conventional skyrmions in mono-layer thin film, which allow for versatile and multifunctionalultradense and ultrafast information processing and logic ap-plications.These features are true for both the current-in-plane (CIP)and current-perpendicular-to-plane (CPP) cases. The CIP im-plementation has been extensively studied in the past ten yearsfor displacing domain walls or switching magnetization. Re-cently, the spin Hall effect has been demonstrated to be amore efficient means of manipulating magnetization. The cur-rent for the CIP case is along the nanowire axial direction.By contrast, the spin current is perpendicular to the heavymetal/ferromagnetic interface for the CPP case. In our nomen-clature for CPP, the spin current is perpendicular-to-film planewhereas the charge current can be either in the film plane(such as the spin Hall effect scenario) or perpendicular to thefilm plane (such as the perpendicular MRAM where a per-pendicularly magnetized polarizer is incorporated). The CPPscheme has been proven to be much more efficient to movethe domain wall or skyrmion than the CIP method .In this work, we first describe the nucleation process of an a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r nanodisknanotrack (CIP)nanotrack (CPP) Spin-polarized currentSpin-polarized currentTop FM layerInsulating spacerBottom FM layerHeavy metalTop FM layerInsulating spacerBottom FM layer
Top FM layerInsulating spacerBottom FM layer
Heavy metal Charge current Spin current
Spin currentAFM-coupled ab dec m z +1-10 FIG. 1:
Schematics of the bilayer nanodisk, nanotrack and the bilayer-skyrmion. a , The bilayer nanodisk for the creation of skyrmions, ofwhich the diameter is 100 nm. The spin current (polarized along − z ) is injected into the top layer in the central circle region with a diameterof 40 nm. b , The bilayer nanotrack (500 nm ×
50 nm × x -direction, which leads to the generation of spin current (polarized along + y ) perpendicularly injected to the bottom layer due to the spin Hall effect. The skyrmion in the bottom layer is driven by the spin current,while the skyrmion in the top layer moves remotely due to the interlayer exchange coupling. c , The bilayer nanotrack (500 nm ×
50 nm × i.e. , the charge currents flow along the - x -direction. The skyrmions in both the top and bottom layers are driven by the spin current. In allmodels, the thickness of both the top ferromagnetic (FM) layer, the bottom FM layer and the insulating spacer are equal to 1 nm. The toplayer and bottom layer is antiferromagnetically exchange-coupled, where the initial state of the top layer is almost spin-up (pointing along+ z ) and that of bottom layer is almost spin-down (pointing along - z ). d , Illustration of the bilayer-skyrmion in a nanodisk, which is a set ofantiferromagnetically exchange-coupled skyrmions. e , Sideview of the bilayer-skyrmion along the diameter of d . The color scale representsthe out-of-plane component of the magnetization, which is used throughout this paper. isolated skyrmion within the top layer by utilizing the CPP in-jection into a disk composed of two AFM-coupled magneticsublayers. Owing to the interlayer AFM exchange coupling,another skyrmion will automatically emerge in the bottomlayer. In so doing we explore various properties of a bilayer-skyrmion. Furthermore, we move a bilayer-skyrmion in thenanotracks either by the CPP or CIP. We also study the gener-ation of a domain-wall (DW) pair and the conversion processfrom a DW pair to a skyrmion in the bilayer nanotracks. Bilayer System coupled with AFM interactionHamiltonian.
We investigate the bilayer system where thetop and bottom ferromagnetic (FM) layers are coupled anti-ferromagnetically by the exchange interaction, as illustratedin Fig.1a. The Hamiltonian for each layer reads H τ = − A intra (cid:88) (cid:104) i,j (cid:105) m τi · m τj + (cid:88) (cid:104) i,j (cid:105) D · ( m τi × m τj )+ K (cid:88) i [1 − ( m τ,zi ) ] + H DDI , (1)where τ is the layer index ( τ = T, B), m τi represents the lo-cal magnetic moment orientation normalized as | m τi | = 1 at the site i , and (cid:104) i, j (cid:105) runs over all the nearest neighbor sites ineach layer. The first term represents the FM exchange inter-action with the FM exchange stiffness A intra . The second termrepresents the DMI with the DMI vector D . The third termrepresents the PMA with the anisotropic constant K . H DDI represents the dipole-dipole interaction. There exists an AFMcoupling between the top and bottom layers, H inter = − A inter (cid:88) i m T i · m B i . (2)The sign of A inter is negative reflecting that the interlayer in-teraction is antiferromagnetic. We assume that the spins inthe top layer are pointing upward. Then the spins in the bot-tom layer are pointing downward due to the interlayer AFMcouplings. Topological Number.
The classical field m τ ( x ) is intro-duced for the spin texture in the FM system by consideringthe zero limit of the the lattice constant, a → . The ground-state spin textures are m T = (0 , , and m B = (0 , , − .A magnetic skyrmion is a spin texture which has a quan-tized topological number. Spins swirl continuously aroundthe core and approach the ground-state value asymptotically. Top
Bottom Top Bottom
Bottom c u rr en t o ff c u rr en t on I n t e r l a y e r e xc hange ( p J m - ) Current density (x 10 Am -2 )A inter =-2 pJ m -1 , j =20x10 Am -2 D =4 mJ m -2 D =4 mJ m -2 A inter =0 pJ m -1 , j =10x10 Am -2 Current density (x 10 Am -2 ) I n t e r l a y e r e xc hange ( p J m - ) bilayer skyrmionbilayer AFM state D (mJ m -2 ) E t o t a l ( x - J ) a b TopBottomA inter =-2 pJ m -1 , j =20x10 Am -2 A inter =0 pJ m -1 , j =10x10 Am -2 TopBottom
Pulse on Pulse off t (ns) S ky r m i on nu m be r d ec FIG. 2:
Creation of skyrmions in the bilayer nanodisk and time evolution of the skyrmion number. a , Injection of skyrmions in the bilayernanodisk ( D = 4 mJ m − ) with/without interlayer AFM exchange coupling. A 0.5-ns-long spin current ( P = 0.4) is injected into the top layer(denoted by solid yellow circles), followed by a 1.5-ns-long relaxation. The interlayer exchange constant A inter is set as 0 or -2 pJ m − ,whereas the corresponding interface exchange constant σ equals to 0 or -2 mJ m − (see Supplementary Movies 1-2). b , The time evolutionof the skyrmion number of the top and bottom layers in the nucleation process of skyrmions corresponding to a . c , Total micromagneticenergy E total (including the intralayer exchange, interlayer exchange, dipolar, anisotropy and DMI energy) for a bilayer-skyrmion and theAFM-coupled ground state as a function of the DMI constant D . Relaxed state of ( d ) the top layer and ( e ) the bottom layer after the injectionof a 0.5-ns-long spin current for various current density j and interlayer exchange constant A inter . The skyrmion is characterized by the topological number Q τ in each layer, Q τ = − π (cid:90) d x m τ ( x ) · ( ∂ x m τ ( x ) × ∂ y m τ ( x )) . (3)We obtain Q τ = ± for a skyrmion in a sufficiently largearea. We also call Q τ the skyrmion number. Even if theskyrmion spin texture is deformed, its skyrmion number doesnot change, as far as the boundary condition is not modified.It can be neither destroyed nor separated into pieces, i.e. , askyrmion is topologically protected.The spins in the top and bottom layers are tightly boundeddue to the AFM coupling. Accordingly, if one skyrmion iscreated in the top layer, a second skyrmion is also createdin the bottom layer (Fig.2a) simultaneously. The topologi-cal number of the bottom layer is opposite to that of the toplayer since all the spins are inverted, Q B = − Q T . See Fig.2bhow these topological numbers evolve after the creation of askyrmion in the top layer. Excitation Energy.
We may determine numerically thespin profile of each skyrmion and estimate the excitation en-ergy based on the Hamiltonian H total = H T + H B + H inter . We compared the energy of one bilayer-skyrmion state and the en-ergy of the homogeneous state in Fig.2c as a function of theDMI. Similar result is obtained for any value of the interlayerAFM coupling strength. This is because the spin directionsbetween the top and bottom layers are always opposite.The energy of the bilayer-skyrmion becomes lower thanthat of the uniform ground state for D (cid:38) mJ m − , which isthe threshold value. Consequently, the bilayer-skyrmion ex-citation is energetically favorable when the DMI strength islarger than the threshold value. However, spontaneous gen-eration of skyrmions does not occur due to the topologicalprotection. Nevertheless, the topological protection can be vi-olated in condensed matter physics because of the existenceof the lattice structure and the boundary of the sample. Wemay create skyrmions by leveraging these properties. Skyrmion Hall effect.
It is well understood that thecenter-of-mass motion of a skyrmion is determined by the La-grangian, L = L B − U, (4) a b Top Bottom A inter =-6 pJ m -1 , j =50x10 Am -2 A inter =0 pJ m -1 , j =50x10 Am -2 A inter =-6 pJ m -1 , D =3.5 mJ m -2, j =50x10 Am -2 FIG. 3:
Creation of a bilayer DW pair, its conversion into a bilayer-skyrmion, and their motions driven by vertical current in a bilayernanotrack.
The length, wide width and narrow width of the nanotrack ( D = 3.5 mJ m − ) equal 400 nm, 100 nm, and 20 nm, respectively. a ,A local vertical spin current ( j = 50 × A m − , P = 0.4, polarized along − z ) is perpendicularly applied to the top layer of the narrow side(85 nm < x < 115 nm) before t = 50 ps. When the top and bottom layers are decoupled ( A inter = 0 pJ m − ), only one DW pair is generated inthe top layer. However, when the top and bottom layers are coupled ( A inter = -6 pJ m − ), a bilayer DW pair is created at t = 50 ps. b , A globalvertical spin current is perpendicularly applied to the bottom layer (towards + z , polarized along + y ) when the bilayer DW pair is created at t =50 ps. The current density j in wide part equals 5 × A m − , which is proportional to that inside the narrow part with respect to the ratioof narrow proportion (200 nm ×
20 nm) to wide proportion (200 nm ×
100 nm). The 45-dgree notches are employed to reduce the requiredcurrent density for the DW-skyrmion conversion. When the global driving current is turned on at t = 50 ps, the local DW injection current isturned off at the same time. See Supplementary Movie 3. where L B is the Berry phase term of the spin texture, L B = G XY − X ˙ Y ) = 12 G · ( ˙ R × R ) , (5)and U is the potential. Here, G = (0 , , G ) is the gyromag-netic coupling constant representing the Magnus force with G = 4 πQ , and R = ( X, Y ) is the center-of-mass coordinateof the skyrmion. The Euler-Lagrange equation yields ˙ R = 1 G e z × F , (6)where e z = (0 , , , and F = −∇ U is the force acting on theskyrmion. Consequently, a moving skyrmion feels the Mag-nus force and bends toward the direction perpendicular to theforce. It is called the skyrmion Hall effect.The direction of the Magnus force is opposite when thesign of the skyrmion number Q is opposite. For instance, if askyrmion is driven by the current along the + x direction it willbe bended toward the + y ( − y ) direction in the top (bottom)layer. Landau-Lifshitz-Gilbert-Slonczewski equation for CPP
We may apply a CPP spin-polarized current injection frommagnetic tunnel junction (MTJ) or the spin Hall effect inheavy metal layer . We numerically solve the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation, which gov-erns the dynamics of the magnetization m i at the lattice site i . By suppressing the layer index, it reads d m i dt = − | γ | m i × H eff i + α m i × d m i dt + | γ | u ( m i × p × m i ) − | γ | u (cid:48) ( m i × p ) , (7) where H eff i = − ∂H total /∂ m i is the effective magnetic fieldinduced by the Hamiltonian H total = H T + H B + H inter , γ is the Gilbert gyromagnetic ratio, α is the Gilbert-dampingcoefficient originating from spin relaxation, u is the Slon-czewski STT coefficient, u (cid:48) is the out-of-plane STT coef-ficient, and p represents the electron polarization direction.Here, u = | (cid:126) µ e | j | p | dM s with µ the vacuum magnetic permit-tivity, d the film thickness, M s the saturation magnetization,and j the current density. We take − z direction for creatingthe skyrmion, while + y direction for moving the skyrmion.The STT is induced either by injection from a magnetic tun-nel junction polarizer or by the spin Hall effect . LLG equation for CIP
Alternatively we may apply a CIP injection to moveskyrmions. We numerically solve the LLG equation, d m i dt = − | γ | m i × H eff i + α m i × d m i dt + p | a eM s ( j ( r ) · ∇ ) m i − p | a β eM s [ m i × ( j ( r ) · ∇ ) m i ] , (8)where β is the strength of the non-adiabatic torque and a isthe lattice constant. Creation of a bilayer-skyrmion by vertical spin current.
We employ a CPP injection with a circular geometry in a nan-odisk. The spin-polarized current (polarized along − z ) is in-jected into the top layer in the central circle region, as illus-trated in Fig.1a.We demonstrate how the spin textures develop in Fig.2a.The spins start to flip in both layers following the spin cur- a CPP bc d
CIP
Interlayer exchange coupled Interlayer exchange decoupled de c oup l ed c oup l ed ( A i n t e r = - . p J m - )( A i n t e r = - p J m - ) de c oup l ed c oup l ed ( A i n t e r = - . p J m - )( A i n t e r = - p J m - ) FIG. 4:
Motion of skyrmions in the top and bottom layers of a bilayer nanotrack.
Top-views of motion of skyrmions at selected interlayerexchange coupling and times driven by (a) the CPP injection and (b) the CIP injection (see Supplementary Movies 4-7). The parameters ofthe bilayer nanotrack are 500 nm ×
50 nm × D = 3.5 mJ m − . The skyrmions are initially created by MTJ skyrmion injector placedon the top layer at x = 100 nm. For the CPP case, the spin current ( P = 0.4) in the bottom layer is applied along + z but polarized along + y .The skyrmion in the bottom layer moves towards right driven by the spin current, while the skyrmion in the top layer remotely moves due tothe interlayer coupling. For the CIP case, the skyrmions in both the top and bottom layers are driven by in-plane spin currents ( P = 0.4). Thevelocities of skyrmions in the top and bottom layers as functions of current density j with (c) large interlayer exchange A inter = -6 pJ m − and (d) small interlayer exchange A inter = -0.06 pJ m − . The cross symbol denotes the decoupling and destruction of skyrmions in the top andbottom layers due to large current density and small interlayer exchange coupling, where the velocities are calculated before the destruction ofskyrmion. rent injection only in the top layer. When there is no inter-layer AFM coupling, a skyrmion is formed only in the toplayer (see Supplementary Movie 1). By contrast, a skyrmionis formed also in the bottom layer upon the current injectionin the presence of the interlayer AFM coupling (see Supple-mentary Movie 2).We show the evolution of the skyrmion number in Fig.2b.It oscillates at the initial stage for t < . ns, and rapidlyincreases to . The skyrmion remains stable even when thecurrent is switched off, demonstrating that it is topologicallyprotected. During this process, the spins in the top and bot-tom layers are always anti-parallel. A comment is in order.The saturated skyrmion number is not exactly Q = 1 but Q = 0 . . This is due to the fact that there is a backgroundskyrmion number which originates from the tilting the edgespins. It is Q = − . . Accordingly, the calibrated skyrmionnumber is Q = 0 . , which is almost unity.We present a nucleation phase diagram of a bilayer-skyrmion pair as a function of the current density and the in-terlayer AFM coupling in Fig.2d and Fig.2e. When the mag-nitude of the injected current is strong enough, the bilayer-skyrmion is created. This is due to the fact that spin flip costsa certain energy. On the other hand, if the interlayer AFMcoupling is too strong, the bilayer-skyrmion is suppressed dueto the fact that the nucleation field and the coercivity increaseswith the interlayer AFM exchange, leading to a larger currentdensity for nucleation. Creation of a bilayer-skyrmion from a bilayer DW pair.
A magnetic skyrmion can be created from a DW pair by us-ing a junction geometry . In this scenario we first make a DWpair into a nanotrack of the top layer through the local CPP in-jection with − z direction. We show how the spins start to flipin the top layer and subsequently in the bottom layer driven bythe AFM exchange force in Fig.3a. Then, the bilayer DW pairis shifted by applying CPP current, as shown in the processfrom t = 50 ps to t = 120 ps in Fig.3b. Here we considerthe vertical injection of a spin current towards + z and polar-ized along + y in the bottom layer. The CPP injection movesthe bilayer DW in the rightward direction. When the bilayerDW arrives at the junction interface ( t = 170 ps), both the endspins of the DW are pinned at the junction, whereas the centralpart of the DW continues to move due to STT in the wide partof the nanotrack. Therefore, the structure is deformed into acurved shape and a bilayer-skyrmion texture forms at t = 190 ps (see Supplementary Movie 3). Current-driven motion of a bilayer-skyrmion in a nan-otrack.
The magnetic bilayer-skyrmion can be displaced bythe vertical spin-polarized current as in the case of the mag-netic skyrmion. We may employ the CPP injection or the CIPinjection to drive a bilayer-skyrmion. In general, a movingskyrmion is easily destroyed by the sample edges due to theskyrmion Hall effect. Therefore, the maximum velocity ofskyrmion in FM nanotrack is typically much less than 10 m/s,limited by the edge confining force of ∼ D /J .The skyrmion in the top layer follows the motion of theskyrmion in the bottom layer even when the current is not in-jected into the top layer. This is because that two skyrmionsare bounded by the interlayer AFM coupling. There is noskyrmion Hall effect for a magnetic bilayer-skyrmion. Thiscan be explained as follows. If there is no interlayer AFMcoupling, a skyrmion in the top layer moves left-handedand the skyrmion in the bottom layer moves right-handed.However, when the AFM coupling is strong enough, twoskyrmions are tightly bounded and the Magnus forces actingon the skyrmions between the top and bottom layers are ex-actly cancelled. Accordingly the bilayer-skyrmion will movestraightly. This mechanism works both for the CPP and CIPcases (see Supplementary Movies 4-5).We show the relation between the magnitude of the injectedcurrent and the velocity in Fig.4. The velocity is proportionalto the injected current density. For strong enough current, thebilayer-skyrmion is destroyed and split into two independentskyrmions. This is because that the skyrmion Hall effect in-creases as the current increases, which acts as the repulsiveforce between two skyrmions (see Supplementary Movies 6-7).With strong interlayer AFM exchange coupling, the cou-pled skyrmions move along the central line of the nanotrackat a high speed of a few hundred meters per second, withoutany transverse motion. However, with small interlayer AFMexchange coupling, the skyrmions in the top and bottom lay-ers will be decoupled due to the fast motion of the skyrmion inthe bottom layer driven by large current. Once the skyrmionsin the top and bottom layers are decoupled, the skyrmion Hall effect becomes active, leading to the destruction of skyrmionsin the top and/or bottom layer by touching edge. At the sametime, in the CPP case, the skyrmion in the top layer will stopmotion. It can be seen that the coupled skyrmions driven byCIP current doesn’t require a fine tuning of damping and non-adiabatic torque coefficients.It is also worth noting that the interlayer AFM couplingdoes not produce a mass of the bilayer-skyrmion. When thedriving current is suddenly turned off, the bilayer-skyrmionstops high-speed motion immediately (see SupplementaryMovie 8). On the other hand, we also investigated the casewhere the DMI constant D is different between the top andbottom layers. It is found that the results of current-drivenmotion of the bilayer-skyrmion do not change much since theDMI only changes the radius of the skyrmion (see Supplemen-tary Movies 9-10). The massless property and its robustnessmake the bilayer-skyrmion an ideal candidate for practical ap-plications. Perspectives
We have presented a novel solution of inhibiting the Halleffect of skyrmions without affecting their topological proper-ties, by exploring a new device made of antiferromagneticallyexchange-coupled bilayer nanodisks and nanotracks. Com-pared with the mostly investigated skyrmion in the FM mono-layer system, the bilayer-skyrmion exhibits entirely distinctcharacteristics with regard to the current-transport behaviourand robustness. First, it can move strictly along the direc-tion of the spin current flow, which makes it more appeal-ing for motions in nanowires for ultradense memory appli-cations. This is in high contrast with the case of monolayerskyrmion, where the skyrmion information carrier can be eas-ily destroyed by the edges of the nanotracks. Second, it willbe immune to magnetic field perturbations which might begenerated externally or internally within the device circuitrysince the net magnetic moment is zero. Third, by introducingthe bilayer degree of freedom, the transport properties of thedevice can be engineered to achieve desired performance. Forexample, the skyrmion Hall effect can be intentionally sup-pressed or enhanced by tuning the magnetic properties of in-dividual layers. This newly proposed solution of transportingskyrmion information carrier for arbitrarily long distances atmuch enhanced velocity may be very appealing for versatileapplications such as ultradense memory and information pro-cessing. Similar ideas can be extended to multilayer or su-perlattice where the skyrmions are strongly coupled to realizea better manipulation of skyrmions in nanotrack or extendedthin films.
MethodsModeling and simulation.
The micromagnetic simula-tions are performed using the Object Oriented MicroMag-netic Framework (OOMMF) including the Dzyaloshinskii-Moriya interaction (DMI) extended module . Thetime-dependent magnetization dynamics is governed bythe Landau-Lifshitz-Gilbert (LLG) equation including spintorque . The average energy density E is a function of M , which contains the intralayer exchange, the interlayer ex-change, the anisotropy, the applied field (Zeeman), the demag-netization and the DMI energy terms. For micromagnetic sim-ulations, the intrinsic magnetic parameters are adopted fromRefs. : Gilbert damping coefficient α = 0 . and the valuefor Gilbert gyromagnetic ratio is -2.211 × m A − s − .Saturation magnetization M S = 580 kA m − , intralayer ex-change stiffness A = 15 pJ m − , DMI constant D = 0 ∼ − and perpendicular magnetic anisotropy (PMA) K = 0.8MJ m − unless otherwise specified. The interlayer exchangecoefficient A inter is set from 0 to -10 pJ m − , whereas the cor-responding interface exchange coefficient σ equals from 0 to-10 mJ m − ( σ = A inter / 1 nm), where "-" denotes that theinterface is antiferromagnetically coupled. The field-like out-of-plane STT coefficient u (cid:48) is set to zero. All samples arediscretized into cells of 2 nm × × ∼ x = 100 nm) by alocal spin current perpendicular to the plane of the top layer.Then the system is relaxed to an energy minimum state with-out applying any current. Next, we start the timer and the spincurrent ( P = 0.4) is injected into the nanotrack with the geom-etry of current-in-plane (CIP) or current-out-of-plane (CPP)as shown in Fig.1. In the configuration of CIP, the electronsflow toward the right in both the top and bottom layers, i.e. ,the currents flow toward the left, while in the configurationof CPP, the electrons flow toward the top only in the bottomlayer. ∗ Corresponding author: [email protected] † Corresponding author: [email protected] S. Mhlbauer et. al. Science , 915 (2009). X. Z. Yu et. al. Nature , 901 (2010). S. Heinze et. al. Nature Phys. , 713 (2011). N. Nagaosa and Y. Tokura, Nat. Nanotech. , 899 (2013). A. Fert, V. Cros, and J. Sampaio, Nat. Nanotech. , 152 (2013). J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat.Nanotech. , 839 (2013). Y. Tchoe and J. H. Han, Phys. Rev. B , 174416 (2012). Y. Zhou, E. Iacocca, A. Awad, R. K. Dumas, F. C. Zhang, H. B.Braun, and J. Akerman, cond-mat/arXiv:1404.3281. M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A. Itoh,L. Duo, A. Kirilyuk, Th. Rasing, and M. Ezawa, Phys. Rev. Lett. , 177205 (2013). X. C. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. , 9400 (2015). X. C. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia,and F. J. Morvan, Sci. Rep. , 7643 (2015). C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, N. VanHorne, C.A.F. Vaz, K. Bouzehouane, K. Garcia, C. Deranlot, P.Warnicke, P. Wohlhüter, J.M. George, J. Raabe, V. Cros, and A.Fert, arXiv:1502.07853 (2015). S. Woo, K. Litzius, B. Krüger, M. Y. Im, L. Caretta, K. Richter, M.Mann, A. Krone, R. Reeve, M. Weigand, P. Agrawal, P. Fischer,M. Kläui, G. S. D. Beach, arXiv:1502.07376 (2015). S. Parkin, and S. Yang, Nat. Nanotech. , 195 (2015). N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP , 101(1989). N. Bogdanov and A. J. Hubert, Magn. Magn. Mater. , 255(1994). U. K. Roessler, N. Bogdanov, and C. Pfleiderer, Nature , 797(2006). J. Iwasaki, W. Koshibae, and N. Nagaosa, Nano Lett., , 4432(2014). J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotech. , 742(2013). J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Commun. ,1463 (2012). S. H. Yang, K. S. Ryu, and S. Parkin, Nat. Nanotech. , 221(2015). M. Stone, Phys. Rev. B , 16573 (1996). A.V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi,K.A. Zvezdin, A. Anane, J. Grollier, and A. Fert, Phys. Rev. B ,020402(R) (2013). R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri,and G. Finocchio, Sci. Rep. , 6784 (2014). Y. Zhou and M. Ezawa, Nat. Com. , 4652 (2014). M. J. Donahue and D. G. Porter, National Institute of Stan-dards and Technology, Interagency Report NISTIR, (1999OOMMF user’s guide, version 1.0.). S. Rohart and A. Thiaville, Phys. Rev. B , 184422 (2013). W. F. Brown J., Micromagnetics (Krieger, New York, 1978). T. L. Gilbert, Phys. Rev. , 1243 (1955). L. Landau and E. Lifshitz, Physik. Z. Sowjetunion , 153 (1935). A. Thiaville, S. Rohart, E. Jue, V. Cros, and A. Fert, Europhys.Lett. , 57002 (2012). A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. , 990 (2005). Acknowledgements
Y.Z. thanks the support by the Seed Funding Programfor Basic Research and Seed Funding Program for AppliedResearch from the HKU, ITF Tier 3 funding (ITS/171/13),the RGC-GRF under Grant HKU 17210014, and UniversityGrants Committee of Hong Kong (Contract No. AoE/P-04/08). M.E. thanks the support by the Grants-in-Aid forScientific Research from the Ministry of Education, Science,Sports and Culture, No. 25400317. M.E. is very much grate-ful to N. Nagaosa for many helpful discussions on the subject.
Author contributions
M.E. conceived the idea and designed the project. M.E.and Y.Z. coordinated the project. X.Z. performed the numer-ical simulations supervised by Y.Z. All authors discussed theresults and wrote the manuscript.
Additional information
Supplementary information is available.