Current-induced dynamics of skyrmion tubes in synthetic antiferromagnetic multilayers
Jing Xia, Xichao Zhang, Kai-Yu Mak, Motohiko Ezawa, Oleg A. Tretiakov, Yan Zhou, Guoping Zhao, Xiaoxi Liu
CCurrent-induced dynamics of skyrmion tubes in synthetic antiferromagnetic multilayers
Jing Xia,
1, 2, ∗ Xichao Zhang, ∗ Kai-Yu Mak, Motohiko Ezawa, Oleg A. Tretiakov, Yan Zhou, † Guoping Zhao, ‡ and Xiaoxi Liu § College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan School of Physics, The University of New South Wales, Sydney 2052, Australia (Dated: February 18, 2021)The topological spin textures can be found in both two-dimensional and three-dimensional nanostructures,which are of great importance to advanced spintronic applications. Here we report the current-induced skyrmiontube dynamics in three-dimensional synthetic antiferromagnetic (SyAF) bilayer and multilayer nanostructures.It is found that the SyAF skyrmion tube made of thinner sublayer skyrmions is more stable during its motion,which ensures that a higher speed of the skyrmion tube can be reached effectively at larger driving current. Inthe SyAF multilayer with a given total thickness, the current-induced deformation of the SyAF skyrmion tubedecreases with increasing number of interfaces, namely, the rigidity of the SyAF skyrmion tube with a giventhickness increases with the number of consisting ferromagnetic (FM) layers. For the SyAF multilayer with aneven number of consisting FM layers, the skyrmion Hall effect can be eliminated when the thicknesses of allconsisting FM layers are identical. Larger damping parameter leads to smaller deformation and slower speedof the SyAF skyrmion tube. Larger field-like torque leads to larger deformation and higher speed of the SyAFskyrmion tube. Our results are useful for understanding the dynamic behaviors of three-dimensional topologicalspin textures, and may provide guidelines for building SyAF spintronic devices.
I. INTRODUCTION
Nanoscale spin textures in magnetic materials may exhibitunique static and dynamic properties due to their topologi-cal structures [1–17]. An exemplary topological spin textureis the skyrmion texture, which was theoretically predicted in1989 [1] and experimentally observed in 2009 [2]. The mag-netic skyrmion has been extensively studied in the past decadedue to its intriguing physical properties and broad potentialapplications in functional spintronic devices [7–15]. In partic-ular, the magnetic skyrmion can be used as a nonvolatile infor-mation carrier in magnetic memory [18–23] and logic com-puting [24–27] applications that meet future commercial re-quirements, such as the ultrahigh storage density and ultralowenergy consumption.Towards the applications of skyrmions in magnetic andspintronic devices, several different skyrmion-hosting sys-tems, ranging from quasi-two dimensional to three dimen-sional structures, have been developed and investigated usinga variety of theoretical and experimental methods [2, 5, 7–15, 28–39]. For example, the existence of magnetic skyrmionswas first realized in magnetic ultrathin films and bulk materi-als, where skyrmions are stabilized by Dzyaloshinskii-Moriya(DM) interactions [2, 5]. Recently, the community has fur-ther focused on the skyrmions in ferromagnetic (FM) mul-tilayers with interface-induced DM interactions, where boththe magnitude of DM interaction and the thermal stability of ∗ These authors contributed equally to this work. † [email protected] ‡ [email protected] § [email protected] skyrmions can be enhanced due to the multilayer nanostruc-ture [39–48].However, FM skyrmions, either in single or multilayerfilms, may show the skyrmion Hall effect when they are drivenby spin currents [49–51], which is a dynamic phenomenonassociated with the topological nature of skyrmions and usu-ally leads to the accumulation or destruction of skyrmionsat sample edges [50–53]. Hence, many strategies have beenproposed to eliminate the skyrmion Hall effect for spintronicapplications based on in-line motion of skyrmions [16, 52–61]. A most important strategy is to create and manipu-late skyrmions in synthetic antiferromagnetic (SyAF) bilayerand multilayer nanostructures [16, 52–54, 60–63], where theSyAF skyrmions carry a net topological charge of zero andthus are free from the skyrmion Hall effect. For example, abilayer SyAF skyrmion consists of two skyrmions with oppo-site topological charges, where the topological Magnus forcesacted on the two skyrmions are identical in magnitude but op-posite in directions [52, 53]. Therefore, the Magnus forcesare adequately canceled out and the bilayer SyAF skyrmioncan move straightly along the driving force direction. Re-cent state-of-the-art experiments have demonstrated the stabi-lization [61] and current-driven motion [60] of bilayer SyAFskyrmions at room temperature.In thick SyAF multilayer structures, the SyAF skyrmionis more like a three-dimensional tube instead of a two-dimensional object. If the multilayer SyAF skyrmion consistsof even number of antiferromagnetically exchange-coupledskyrmions, the total skyrmion number of the SyAF skyrmiontube is equal to zero and the skyrmion Hall effect can be elim-inated in principle [53, 54]. However, a large driving forcemay result in the distortion of the skyrmion tube in the thick-ness dimension and may further lead to more complex dy-namic behaviors of the skyrmion tube [64, 65]. Although a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b N = 6 N = 2 N = 2 N = 4 N = 6 N = 12 (a)(b) O R x R y (d)(c) x y FIG. 1. (a) Schematics of the simulation models. The total samplethickness is fixed at nm. N denotes the number of FM layers ina sample. For N = 2 , the thickness of each FM layer equals nm.For N = 12 , the thickness of each FM layer equals nm. In eachsample, the adjacent FM layers are antiferromagnetically exchange-coupled, forming a SyAF structure. (b) Illustration of a SyAF -layerskyrmion tube (i.e. N = 2 ). Black arrows indicate the Magnus forceacted on each FM layer. (c) Illustration of a SyAF -layer skyrmiontube (i.e. N = 6 ). (d) Definitions of R x and R y , which are usedto describe the size and shape of the skyrmion in the x − y plane ofeach FM layer. the dynamics of FM skyrmion tube have been studied in re-cent years [32, 33, 36–39, 64–67], the complex dynamics of aSyAF skyrmion tube still remain elusive. In this work, we sys-tematically study the current-induced dynamics of skyrmiontubes in SyAF multilayers using both theoretical and compu-tational approaches. II. METHODS
Figure 1(a) illustrates the SyAF multilayer nanotracks. TheSyAF N -layer nanotrack ( N ≥ ) includes N FM lay-ers, which are strictly exchange-coupled in an antiferromag-netic (AFM) manner by interlayer AFM exchange interac-tions. In all SyAF multilayer nanotracks, the length along the x -direction, the width along the y -direction, and the thicknessalong the z -direction are equal to nm, nm, and nm, respectively. The periodic boundary conditions (PBCs)are applied in the x and y directions. It should be mentionedthat two adjacent FM layers should be separated by a nonmag-netic metal spacer in real experimental samples, however, weignore the thickness of nonmagnetic spacer but preserve theeffect of nonmagnetic spacer in the simulation for the sake ofsimplicity, which saves the computational power.In this work, we explicitly consider the SyAF N -layer nan-otracks with N = 2 , , , . For the SyAF multilayer nan-otrack with N = 2 , two -nm-FM layers are antiferromag-netically exchange-coupled. For the SyAF multilayer nan-otrack with N = 4 , four -nm-FM layers are antiferromag-netically exchange-coupled. For the SyAF multilayer nan-otrack with N = 6 , six -nm-FM layers are antiferromagnet-ically exchange-coupled. For the SyAF multilayer nanotrack with N = 12 , 12 -nm-FM layers are antiferromagneticallyexchange-coupled. At the initial state, the skyrmion tube isrelaxed at the position of x = 50 nm, y = 50 nm. The to-tal skyrmion number Q tot of the SyAF N -layer skyrmion tubeis equal to zero due to the nature of SyAF nanotrack [53].We consider a current-perpendicular-to-plane (CPP) geome-try, where the driving spin current is injected into all FM lay-ers vertically.The total Hamiltonian H is decomposed into the Hamilto-nian for each FM layer H n and the interlayer AFM exchangecoupling H inter between neighboring FM layers, H = N (cid:88) n =1 H n + H inter . (1)The Hamiltonian for each FM layer reads H n = − A intra (cid:88) (cid:104) i,j (cid:105) m ni · m nj + K (cid:88) i (cid:104) − ( m n,zi ) (cid:105) + D ij (cid:88) (cid:104) i,j (cid:105) ( ν ij × ˆ z ) · (cid:0) m ni × m nj (cid:1) + H DDI , (2)where n is the FM layer index ( n = 1 , , · · · , N ), m ni rep-resents the local magnetic moment orientation normalized as | m ni | = 1 at the site i , and (cid:104) i, j (cid:105) runs over all the nearest-neighbor sites in each FM layer. The first term represents theintralayer FM exchange interaction with the intralayer FM ex-change stiffness A intra . The second term represents the DMI,where D ij is the DMI coupling energy and ν ij is the unitvector between sites i and j . The third term represents theperpendicular magnetic anisotropy (PMA) with the anisotropyconstant K . H DDI represents the dipole-dipole interaction.The Hamiltonian for the interlayer AFM exchange interac-tions reads H inter = − N − (cid:88) n =1 A inter (cid:88) i m ni · m n +1 i . (3)Here the interlayer exchange stiffness A inter is negative due tothe interlayer AFM exchange interaction.For the current-induced dynamics, we numerically solvethe Landau-Lifshitz-Gilbert (LLG) equation including thedamping-like and field-like spin-orbit torques (SOTs), givenas [53, 54, 68] d m dt = − γ m × h eff + α (cid:18) m × d m dt (cid:19) − u m × ( m × p ) − ξu ( m × p ) . (4)Here, h eff = − µ M S · ∂H∂ m is the effective field. µ is the vac-uum permeability constant, and M S is the saturation magne-tization. γ is the gyromagnetic ratio with its absolute value,and α is the Gilbert damping coefficient. u = | γ (cid:126) µ e | jθ SH aM S is thedamping-like SOT coefficient, and ξ is the relative strength ofthe field-like torque. p = − y represents the unit spin polariza-tion vector, (cid:126) is the reduced Planck constant, e is the electroncharge, j is the applied driving current density, θ SH = 0 . isthe spin Hall angle, and a = 1 nm is the thickness of cell size.The simulation is performed by using the 1.2a5 release ofthe Object Oriented MicroMagnetic Framework (OOMMF)developed at the National Institute of Standards and Technol-ogy (NIST) [68]. The simulation uses the OOMMF extensi-ble solver (OXS) objects of the standard OOMMF distribu-tion along with the OXS extension modules for the interface-induced DMI [69, 70]. The cell size used in the simulation is nm × nm × nm, which guarantees both numerical accu-racy and computational efficiency. The magnetic parametersused in the simulation are [19, 21, 22, 52, 53]: α = 0 . ∼ . with a default value of . ; γ = − . × m/(As); M S = 1000 kA/m; A intra = 10 pJ/m; A inter = − pJ/m (i.e. σ = − mJ/m ); D = 1 . mJ/m (for N = 2 ); D = 1 . mJ/m (for N > ); K = 0 . MJ/m . III. RESULTS AND DISCUSSIONS
We start with a computational investigation of the current-velocity relation of the skyrmion tubes in SyAF N -layer nan-otracks with N = 2 , , , , where we initially consider onlythe damping-like torque (i.e. ξ = 0 ). It is found that the veloc-ity of the skyrmion tube is proportional to the driving currentdensity, as shown in Fig. 2(a).For the steady motion of the rigid skyrmion tubes in SyAF N -layer nanotracks, we use the Thiele equation [22, 71] tointerpret the simulation results. The Thiele equation for theskyrmion in each FM layer reads as G n × v n − α D n · v n + p · B n + F n = , (5)with n being the layer index. D n , v n , B n , and F n repre-sent the dissipative tensor, the skyrmion velocity, the tensorrelated to the driving current, and the effective force due tothe AFM interlayer exchange coupling, respectively. G n = T n M S γ (0 , , Q n ) is the gyromagnetic coupling constant repre-senting the Magnus force with Q n being the skyrmion num-ber, where T n is the thickness of the FM sublayer. It shouldbe noted that the Thiele equation (i.e. Eq. 5) essentially doesnot include the thickness since the contributions of the thick-ness are same in all terms, especially for the two-dimensionalmodel. The skyrmion number in each FM layer is defined as Q n = − π (cid:90) m n · ( ∂ x m n × ∂ y m n ) dxdy. (6)We have taken the same damping coefficient α for all FM lay-ers. D n is the dissipative tensor with D nµν = T n M S γ (cid:82) ∂ µ m n · ∂ ν m n dxdy/ π . B n is the tensor related to the driving forcewith B nµν = − T n M S γ u (cid:82) ( ∂ µ m n × m n ) ν dxdy/ π .First, we assume that all sublayer skyrmions of a skyrmiontube move together with the same velocity v since they aretightly bound in an AFM configuration. Summing all n ThieleEqs. (5), we can phenomenologically obtain − α D · v + p · B = , (7)where the interlayer AFM forces are canceled out, i.e., (cid:80) F n = 0 . The Magnus forces are also canceled out, i.e., N = 2 N = 4 N = 6 N = 1 2 v (m/s) j ( 1 0 A / m )( a ) ( b ) N = 2 N = 4 N = 6 N = 1 2 D y (nm) j ( 1 0 A / m )( c ) N = 2 N = 4 N = 6 N = 1 2 Rx (nm) j ( 1 0 A / m ) N = 2 N = 4 ( d ) N = 6 N = 1 2 Ry (nm) j ( 1 0 A / m )( e ) N = 2 N = 4 N = 6 N = 1 2 Ry-Rx (nm) j ( 1 0 A / m ) ( f ) D Rx (nm) N N D Ry (nm) FIG. 2. (a) Skyrmion tube velocity v as a function of driving cur-rent density j for a SyAF N -layer skyrmion. (b) Horizontal distancebetween the top-layer and bottom-layer skyrmion centers in the y di-rection ∆ y as a function of driving current density j . Note that whenthe SyAF N -layer skyrmion is driven into the motion, the skyrmioncenter position in the x direction are the same in all FM layers, i.e., ∆ x = 0 . (c) R x as a function of driving current density j for theskyrmion in the bottom FM layer. (d) R y as a function of driving cur-rent density j for the skyrmion in the bottom FM layer. (e) R y − R x as a function of driving current density j for the skyrmion in the bot-tom FM layer. (f) ∆ R x (i.e. R bottom x − R top x ) as a function of N when j = 20 × A/m. The inset shows the corresponding ∆ R y (i.e. R bottom y − R top y ). (cid:80) G n = 0 . Solving Eq. 7, the velocity of the SyAF skyrmiontube can be obtained v x = uIα D , v y = 0 , (8)where I = πr sk / and D = π / . The theoretical solutionsshow that the skyrmions in each FM layer steadily move alongthe x direction given that they are strictly exchange-coupledantiferromagnetically. The skyrmion velocity is proportionalto the driving force, which is in line with the simulation re-sults.As shown in Fig. 2(a), the dynamic stability of the SyAFskyrmion tube is enhanced when the number of FM layersincreases. For example, the SyAF -layer skyrmion tube isdestroyed when the driving current density j > × A/m. The SyAF -layer skyrmion tube is destroyed when j > × A/m. The SyAF -layer skyrmion tube is destroyed y (nm) L a y e r I n d e x N = 2 N = 4 N = 6 N = 1 2 ( a ) N = 2 N = 1 2 ( b ) Skyrmion Area (nm2)
L a y e r s I n d e x( c )
FIG. 3. (a) Schematic illustrations of deformed moving SyAFskyrmion tubes. The total thickness is nm, i.e., spins in thethickness direction. N denotes the number of FM layers in a sam-ple. For N = 2 , the thickness of each FM layer equals nm. For N = 12 , the thickness of each FM layer equals nm. At the samedriving current density, the Magnus-force-induced deformation ofthe SyAF -layer skyrmion tube is larger than that of the SyAF -layer skyrmion. (b) Sublayer skyrmion center locations (in the y direction) of deformed SyAF N -layer skyrmion tubes driven by acurrent density of j = 40 × A/m. The layer index indicatesthe single-spin-thick sublayer position, for example, and denotethe most bottom and top layers of the SyAF structure. (c) Sublayerskyrmion areas of a deformed SyAF -layer skyrmion tube drivenby a current density of j = 40 × A/m. when j > × A/m. The SyAF -layer skyrmion tubeis destroyed when j > × A/m.The destruction of the moving skyrmion tube is caused bythe fact that the Magnus forces acted on sublayer skyrmionswith opposite skyrmion number Q n are pointing in oppositedirections, which may deform and pull apart the skyrmiontube when the Magnus forces are larger than a certain thresh-old. The magnitude of the Magnus force [i.e. G n × v n (seeEq. 5)] is proportional to the skyrmion speed as well as themagnetization and sublayer thickness [72], which can be seenfrom the definition G n = T n M S γ Q n = − T n M S γ π (cid:90) m n · ( ∂ x m n × ∂ y m n ) dxdy, (9)where T n is the thickness of the FM sublayer. Hence, it canbe seen that in the SyAF multilayers with identical total thick-ness, the skyrmion tube with fewer layers (i.e. smaller N )could be easier to be deformed by the Magnus force. To bemore specific, the Magnus force will lead to the shift of sub-layer skyrmions in the ± y directions. Due to the Magnus-force induced deformation, the SyAF skyrmion tube velocitiesare slightly different for the SyAF nanotracks with different N , especially when the driving current density is large.Figure 2(b) shows the distance (i.e. ∆ y ) in the y directionbetween the top sublayer and bottom sublayer skyrmions asa function of the driving current density. ∆ y increases withincreasing driving current density. When the driving currentdensity increases, the Magnus force acting on skyrmions ineach FM layer increases, leading to larger shift of sublayerskyrmion centers. However, ∆ y decreases when the numberof FM layers (i.e. N ) increases at a given driving current den-sity. For example, when j = 100 × A/m, ∆ y = 7 nm for the SyAF -layer skyrmion, and ∆ y decreases to nm for theSyAF -layer skyrmion. Note that the total thickness of theSyAF nanotracks is fixed at nm.We further investigate the deformation of SyAF skyrmiontubes. The geometries of bottom sublayer skyrmions are de-scribed by R x , R y , and R y − R x in Fig. 2(c)-(e). The sublayerskyrmions of a moving SyAF skyrmion tube is elongated inthe y direction. The deformation is significant when the driv-ing current density is large as the Magnus force [i.e. G n × v n (see Eq. 5)] acting on each FM sublayer increases with thecurrent-induced velocity. However, it can be seen that thedeformation of the SyAF -layer skyrmion tube is smallercompared to that of the SyAF -layer and -layer skyrmiontubes when j > × A/m. The reason is that the Mag-nus force also decreases with decreasing thickness of the FMsublayer (see Eq. 9). For the SyAF -layer skyrmion tube, thethickness of each FM sublayer equals nm, while it is equalto nm for the SyAF -layer skyrmion tube.We also study the geometries of sublayer skyrmions in themost top and bottom FM layers. Fig. 2(f) shows ∆ R x (i.e. R bottom x − R top x ) and ∆ R y (i.e. R bottom y − R top y ) as functionsof N . For the SyAF -layer skyrmion tube, ∆ R x and ∆ R y areabout nm. For the SyAF -layer skyrmion tube, ∆ R x and ∆ R y are almost zero. The reason behind this phenomenoncould be the effect of the dipole-dipole interaction. Namely,when the thickness of FM layers is thick, the dipole-dipoleinteraction may result in certain nonuniformity and tilt of theskyrmion tube in the thickness direction.In Fig. 3(a), we illustrate two deformed SyAF skyrmiontubes driven by a current density of j = 40 × A/m.The slanted deformation of the SyAF -layer skyrmion tubeis obviously larger than that of the SyAF -layer skyrmiontube. For the SyAF -layer skyrmion tube, the Magnus forcesacted on the top FM and bottom FM layers are large (due tothe thick thickness of FM sublayers) and are pointing in oppo-site directions, which lead to the deformation of the skyrmiontube along the direction of Magnus forces (i.e., the ± y di-rection). In contrast, for the SyAF -layer skyrmion tube,the magnitude of Magnus forces is much smaller due to thereduced thickness of each FM sublayer. At the same time,the Magnus forces acted on FM sublayers are oppositeto each other in a staggered manner, which leads to a bettercancellation of Magnus forces and smaller deformation of theSyAF skyrmion tube. As shown in Fig. 3(b), for the SyAFmultilayer with a given total thickness of nm, the current-induced deformation of the SyAF N -layer skyrmion tube inthe Magnus force direction (i.e., the ± y directions) driven by j = 40 × A/m decreases with increasing number of FMsublayers. Namely, the deformation decreases with decreas-ing thickness of the FM sublayers. For the case of N = 2 , thehorizontal spacing between the most top and bottom sublayerskyrmions equals ∼ nm, while it equals ∼ nm for the caseof N = 12 .On the other hand, it is worth mentioning that the large leapof the N = 2 case in Fig. 3(b) indicates that the slanted de-formation of the SyAF -layer skyrmion tube is most signifi-cant at the antiferromagnetically exchange-coupled interface,where the shear strain is maximum from a phenomenologi-
02 0 04 0 06 0 08 0 01 0 0 0 v (m/s) (cid:1) ( a ) ( b ) ( c ) D y (nm) (cid:1) B o t t o m l a y e r R x R y T o p l a y e r R x R y R (nm) (cid:1) FIG. 4. (a) Damping dependence of the SyAF -layer skyrmiontube velocity v at j = 100 × A/m. (b) Damping dependence of ∆ y of the SyAF -layer skyrmion tube at j = 100 × A/m. (c)Damping dependences of R x and R y of the SyAF -layer skyrmiontube at j = 100 × A/m. v (m/s) (cid:1) ( a ) ( b ) D y (nm) (cid:1) ( c ) B o t t o m l a y e r R x R y T o p l a y e r R x R y R (nm) (cid:1) FIG. 5. Effect of the field-like torque strength ξ on the current-induced motion of a SyAF -layer skyrmion at j = 100 × A/m.(a) Velocity, (b) ∆ y , (c) R x , and (c) R y as functions of ξ . cal point of view. However, for other cases with N > , thereduced Magnus forces as well as increased number of anti-ferromagnetically exchange-coupled interfaces cannot lead toobvious shear strain (i.e. leaps) at interfaces.Note that, as mentioned above [see Fig. 2(f)], the sublayerskyrmion size is not uniform in the thickness direction, asshown in Fig. 3(c), which may be caused by complex dipole-dipole interactions in the SyAF multilayer structure. For ex-ample, the size of the sublayer skyrmion is larger near thetop and bottom multilayer surfaces for the SyAF -layerskyrmion tube, while it is smaller in the mid interior of themultilayer. In particular, the sublayer skyrmion size in themost bottom layer is larger than that in the most top layer. Asthe magnitude of Magnus force acting on each sublayer FMskyrmion is also proportional to the sublayer skyrmion size(i.e., in addition to the sublayer thickness), the nonuniformityand asymmetry of the SyAF skyrmion tube in the thicknessdirection may result in the fact that the Magnus forces cannotbe canceled perfectly, especially during the acceleration of theSyAF skyrmion tube upon the application of driving current.Consequently, the uncompensated Magnus forces may leadto complex dynamic deformation and transverse shift of theSyAF skyrmion tube. Namely, when the SyAF skyrmion tubereaches the steady motion, it may show certain deformationin three dimensions as well as a certain transverse shift of itsaverage center in the ± y direction, which are most significantfor the case of N = 2 [see Fig. 3(b)]. v (m/s) T t o p ( n m )( a ) B o t t o m l a y e r R x R y T o p l a y e r R x R y R (nm) T t o p ( n m )( c ) (cid:1) SkHE (rad) T t o p ( n m )( b ) FIG. 6. The current-induced motion of a SyAF bilayer skyrmion (i.e. N = 2 ). Here the total thickness of the sample is fixed at nm. Thethicknesses of the top and bottom FM layers are defined as T top and T bottom , respectively. Namely, T top + T bottom = 6 nm. (a) Velocity,(b) skyrmion Hall angle θ SkHE , (c) R x , and (c) R y as functions of T top at j = 20 × A/m.
The effect of damping parameter α on the current-inducedmotion of SyAF skyrmion tube is also investigated. Fig-ure 4 shows the results for the current-induced motion of aSyAF -layer skyrmion tube, which is the most stable SyAFskyrmion tube studied in this work. The skyrmion tube ve-locity decreases with increasing α [see Fig. 4(a)], which fol-lows the theoretical solution given in Eq. 8. The shift of thesublayer skyrmion centers in the y direction also decreaseswhen α increases, as shown in Fig. 4(b). Figure 4(c) shows R x and R y of sublayer skyrmions in the most top and bot-tom FM layers. When α = 0 . , the deformation of sublayerskyrmions both in top and bottom FM layers are significant,where R y − R x reaches nm. When α = 0 . , R x and R y arealmost identical, indicating insignificant distortion.We also study the effect of the field-like torque on thecurrent-induced motion of a SyAF -layer skyrmion. Fig-ure 5(a) shows the velocity of the skyrmion tube as a functionof the field-like torque strength ξ . The field-like torque canincrease the size of sublayer skyrmions, which results in therise of the skyrmion tube velocity as the skyrmion velocityis proportional to the skyrmion size at a given current den-sity [59]. The shift of the sublayer skyrmion centers in the y direction slightly increases with increasing ξ , as shown inFig. 5(b). The field-like torque can also lead to the expansionof sublayer skyrmions as well as the deformation of skyrmiontube [see Fig. 5(c)].In the above simulations we assume a fixed thickness ofeach FM layer. Here we proceed to investigate the effectof sublayer thickness T on the skyrmion tube dynamics, asshown in Fig. 6. In this part, we consider a SyAF bilayer nan-otrack (i.e. N = 2 ) with a fixed total thickness of nm (i.e. T top + T bottom = 6 nm). We simulate three cases, i.e., T top = 2 , , and nm. Figure 6(a) shows the current-driven motion ofthe SyAF bilayer skyrmion tube. Due to the AFM exchangecoupling, the sublayer skyrmions in top and bottom FM lay-ers are exchange-coupled tightly and move together. When T top = T bottom = 3 nm , the velocity reaches m/s and theskyrmion Hall angle is equal to zero [see Fig. 6(b)]. When T top (cid:54) = T bottom , the skyrmion tube velocity is reduced andthe skyrmion tube shows the skyrmion Hall effect. As shownin Fig. 6(c), the skyrmion tube deformation increases when T top (cid:54) = T bottom . IV. CONCLUSION
In conclusion, we have studied the current-induced motionof skyrmion tubes in SyAF multilayer nanotracks. The SyAF N -layer skyrmion tubes consist of N sublayer FM skyrmions,which are strictly exchange-coupled antiferromagnetically. Itis found that for SyAF N -layer multilayers with identical to-tal thickness, the current-driven dynamic stability of the SyAFskyrmion tube increases with increasing N . As a result, theSyAF N -layer skyrmion with a higher N can be driven by alarger current density and thus, can reach a higher speed. Fur-thermore, we have studied the effects of damping parameterand field-like torque on the moving SyAF N -layer skyrmiontube. When the damping parameter is large, the motion of aSyAF N -layer skyrmion will be more stable while its speedwill be reduced. The field-like torque can deform the SyAFskyrmion tube but it can also lead to a speed increase of theSyAF skyrmion tube. In addition, we computationally demon-strated the effect of sublayer thickness on the skyrmion Halleffect of a SyAF bilayer skyrmion tube. For the SyAF bilayerskyrmion, when the thicknesses of the top and bottom FM lay-ers are identical, the SyAF skyrmion shows no skyrmion Halleffect due to the cancellation of the Magnus forces. However,when the thicknesses of the top and bottom FM layers are dif-ferent, the skyrmion Hall effect cannot be eliminated. We be-lieve our results are useful for understanding the the dynamicstability and mobility of the skyrmion tubes in SyAF struc- tures. We also believe our results can provide guidelines forbuilding SyAF spintronic devices based on topological spintextures. ACKNOWLEDGMENTS
X.Z. was an International Research Fellow of Japan Societyfor the Promotion of Science (JSPS). X.Z. was supported byJSPS KAKENHI (Grant No. JP20F20363). M.E. acknowl-edges the support by the Grants-in-Aid for Scientific Re-search from JSPS KAKENHI (Grant Nos. JP18H03676 andJP17K05490) and the support by CREST, JST (Grant Nos.JPMJCR20T2 and JPMJCR16F1). O.A.T. acknowledgesthe support by the Australian Research Council (Grant No.DP200101027), the Cooperative Research Project Program atthe Research Institute of Electrical Communication, TohokuUniversity (Japan), and by the NCMAS grant. X.L. acknowl-edges the support by the Grants-in-Aid for Scientific Researchfrom JSPS KAKENHI (Grant Nos. JP20F20363, 17K19074,26600041, and 22360122). G.Z. acknowledges the supportby the National Natural Science Foundation of China (GrantNos. 51771127, 51571126, and 51772004), the ScientificResearch Fund of Sichuan Provincial Education Department(Grant Nos. 18TD0010 and 16CZ0006). 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