Introduction

Drag, a many-body effect between moving particles or quasiparticles, has been demonstrated in various transport systems, such as (spin) Coulomb drag in bilayer electronic systems1,2,3, vortex drag in superconductors4,5, molecular drag in nanotubes6, and Yukawa particle drag in one-dimensional channels7. As a conventional manifestation, the passive objects gain momentum from active objects via attractive or repulsive forces, making them moving in the same direction as the active objects. For example, in magnetic materials, the momentum transfer from phonons to magnons causes magnon spin current following against the temperature gradient, which results in spin Seebeck effect8,9.

Magnetic skyrmions, swirling magnetic textures in magnetic materials proposed to be of great potential for future applications10,11,12,13,14,15,16,17,18,19,20, usually behave as particles under a driving force, which suggests the possibilities of skyrmion drag effects21. It was proposed recently that an individual skyrmion can be trapped by a local magnetic field22 or focused laser beam23,24, which can provide a dragging force to the skyrmion so that the skyrmion could move together with the potential trap. Similar feature was also reported in ref. 25, where a moving superconducting vortex is found to couple and drag the magnetic skyrmions through the generated stray field. While these reports reveal the skyrmion drag by different active objects, the skyrmion-skyrmion drag remains unexplored.

In this work, we investigate the skyrmion dynamics in magnetic multilayers under a driving current applied to part of the multilayers. We demonstrate that the skyrmions in the active layers with a driving current can cause either a conventional drag in the passive layer with skyrmions following the active layers, or an anomalous drag with distinct skyrmion motions in the passive and active layers, depending on the intensity of the driving current. The origins of both behaviors are well explained by the particular dipolar-field-induced interlayer potential and the characteristic skyrmion dynamics associated with Magnus forces. The ring-shaped interlayer trapping potential can provide either repulsive or attractive forces depending on the skyrmion location, which thus brings in richer skyrmion dynamics beyond those due to the standard repulsive potential between skyrmions in a single layer26,27,28.

Results and discussion

Model and simulations of skyrmion drag

Figure 1a illustrates our theoretical model, which consists of three ferromagnetic layers (labeled by L1–L3) sandwiched separately between heavy metals A and B, whose distinct properties introduce not only a net Dzyaloshinskii-Moriya interaction (DMI)29,30,31 to stabilize the skyrmions in each magnetic layer32 but also spin-orbit torques to trigger the skyrmion dynamics in the presence of a driving current33. The weak interlayer coupling, mainly due to dipolar interaction, allows different skyrmion distributions in the three magnetic layers, as shown in Fig. 1b, c (see Supplementary Note 1 for the calculation details of this state). The skyrmion dynamics in our multilayers are then simulated from MUMAX334 (see Methods for details). By introducing an insulating layer shown in Fig. 1a, the driving current and therefore the spin-orbit torques appear only in two active layers, L2 and L3, leaving L1 as a passive layer.

Fig. 1: Skyrmion drag in magnetic triple-layers.
figure 1

a Sketch of the magnetic multilayers composed of stacked heavy metal/ferromagnet/heavy metal layers. The insulating layer is introduced to ensure the electric current applied only in selected layers. b Initial state used in our micromagnetic simulation, where the black (white) areas denote their magnetization being along the − z (+ z) direction and the colorful regions reflect the orientation of local magnetization in the x-y plane. c Enlarged plot of magnetic textures in different layers. The trajectories of skyrmion centers with the current densities d j = 0.5 MA cm−2 and e j = 5 MA cm−2. The color bars represent the simulation time.

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Two typical trajectories of the skyrmion centers are presented in Fig. 1d, e, where the driving current is applied in y direction. As expected, the skyrmions in the active layers move mainly along x direction in both cases, thanks to the skyrmion Hall effect that refers to, analogous to the Hall effect in electricity, the skyrmion motion transverse to the applied current driven by the Magnus force35,36,37. It is surprising to find that the skyrmions in the passive layer can also move and even exhibit distinct behaviors under different current intensities. For the smaller current at j = 0.5 MA cm−2, the skyrmions in the passive layer directly follow the same dynamics as the active layers. In contrast, the larger current at j = 5 MA cm−2, which is experimentally achievable 36, causes a significant discrepancy of the skyrmion motions in passive and active layers. Specifically, while the active layer shows a simple increase in the speed of skyrmion motion, the skyrmion trajectories in the passive layer become more complicated.

Since there are no direct exchange interactions between the magnetic layers, the skyrmion motion in the passive layer reflects the skyrmion drag effect due to the long-range interlayer dipolar interaction. As the conventional drag dynamics commonly results in either parallel or antiparallel motion between the active and passive objects, the origin of the unconventional noncollinear drag with transverse component in Fig. 1e is to be explored.

Conventional and anomalous drags

One may consider that the unconventional drag might originate from the skyrmion collisions in the passive layer. To examine this mechanism, we load a single skyrmion in L1 to exclude all possible interactions between the skyrmions within the passive layer, as shown in Fig. 2a, where a regular skyrmion lattice is constructed in the active layers to reduce the influence of the fluctuation in the skyrmion distribution. The skyrmion trajectories in such a simplified model are depicted in Fig. 2b, c, which recover all key features in Fig. 1, indicating that both conventional and unconventional characteristics are pure drag effects. Moreover, the velocity of the passive skyrmion in the anomalous case shows a large transverse component compared to the velocity of the active skyrmions.

Fig. 2: Conventional and anomalous drags of a single passive skyrmion.
figure 2

a Constructed magnetic configuration with a single passive skyrmion and a skyrmion lattice in the active layers. b Conventional and c anomalous drags driven by current densities j = 0.5 and 5 MA cm−2, respectively. d Spatiotemporal evolution of the passive skyrmions, where the dashed circles label the locations of two skyrmions in the active layers, indicated by the inset. The colors have the same meaning as those in Fig. 1. e Dragging potential generated by skyrmion lattices. The red curve is the trajectory of the passive skyrmion in the moving potential. f Illustrations of the conventional and anomalous drags with the dashed circle representing the equilibrium configuration.

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To disclose the physics behind the two distinct drag manifestations, we plot a sequence of snapshots to analyze the spatiotemporal evolution of the passive skyrmion in Fig. 2d, where the colored ring-shaped background textures map the dipolar field of the skyrmions in the active layer, with two of them labeled by yellow and blue circles separately (see Supplementary Fig. 2 for the dipolar fields of skyrmions). As seen, for the case of j = 0.5 MA cm−2, the passive skyrmion is always trapped within the yellow circle, which suggests an attractive interaction with the active skyrmion locating beneath. On the other hand, for j = 5 MA cm−2, the passive skyrmion shows a hopping from the yellow circle to the light blue one, which can be qualitatively explained by the high velocity of the attractive potential beyond the threshold to trap the passive skyrmion. The details of these dynamics are provided as Supplementary Movies 1 and 2.

To verify the appearance of the interlayer attraction and evaluate its strength, we calculate the total energy of the entire system with the passive skyrmion sitting at different locations. While the details of this calculation are provided in Supplementary Note 2, we plot the results in Fig. 2e. As the skyrmion lattice in the active layer experiences only negligible modification in its magnetic texture, the variation of the total energy reflects the potential of the passive skyrmion in the dipolar field of the skyrmion lattice. We can see from Fig. 2e that, while the energy map shows maxima for the passive skyrmion sharing the same lateral coordinates with skyrmion centers in the active layer, there is however a ring-shaped blue area around each maximum, indicating that each skyrmion in the active layer can actually generate a circular potential well in the passive layer. As a consequence, the passive skyrmion is trapped in these potential wells and experiences a dragging force in the presence of a lateral motion, which is consistent with the fact that the passive skyrmion in Fig. 2d locates around the edge of the yellow and blue circles, instead of their centers. The red solid curve in Fig. 2e represents the relative trajectory of the passive skyrmion with respect to the skyrmion lattice, which shows clearly its motion along the route defined by the connecting potential wells, just like the water flow in canyon. The transition between the two distinct scenarios in Fig. 2d is illustrated as Fig. 2f.

It is worthy noting that, in contrast to the lateral shift between the skyrmion centers in L1 and L2 layers, the skyrmions in L2 and L3 almost share the same lateral coordinates in the simulation of Fig. 2, which is considered to be related to the asymmetric characteristic of dipolar field above and below the textured magnetic layer L2 (see Supplementary Fig. 2). Similar feature is actually also observed in Fig. 1b, c.

Theoretical analysis of skyrmion drag

In order to understand the formation of potential map in Fig. 2e and interpret the transverse relative motion of passive skyrmion, we then calculate the potential energy from a separate skyrmion unit in the lattice via simulation based on MUMAX3. The obtained potential energy \({U}_{\,{\rm{ss}}}^{{{\rm{inter}}}}({r}_{{{\rm{ss}}}})\) as a function of lateral distance rss, defined in Fig. 3a, is plotted in Fig. 3b, where the local minimum unambiguously indicates the appearance of ring-shaped attractive potential in two dimension (see the inset). The dark red curve represents the fitting by the analytical expression, \({U}_{\,{\rm{ss}}}^{{{\rm{inter}}}}({r}_{{{\rm{ss}}}})=[{u}_{1}\,{\mbox{cos}}(k{r}_{{{\rm{ss}}}}-\delta )+{u}_{2}+{u}_{3}k{r}_{{{\rm{ss}}}}+{u}_{4}{(k{r}_{{{\rm{ss}}}})}^{2}]{{\mbox{e}}}^{-{r}_{{{\rm{ss}}}}/{R}_{1}}\) with extracted parameters listed in Methods. While the choice of other functions may also work and will not affect the results, we here construct this expression by introducing cosine and exponential functions to describe the oscillation and long-distance decay, respectively, together with some additional polynomials for a better fitting of the complicated potential. By using this analytical expression, the summation of all skyrmion units in the entire lattice of L2 and L3 in Fig. 2a then leads to the complete potential map presented in Fig. 3c, which shows very good agreement with that from MUMAX3 in Fig. 2e. Their quantitative comparison along three pathways C1-C3, indicated by dashed lines in Fig. 3c, is presented in Fig. 3d. The fluctuation among the data from MUMAX3 is due to the small variation in the magnetic textures. With similar techniques, we also calculate the interaction energy between the skyrmions in the active layers, which reveals simple repulsion following \({U}_{\,{\rm{ss}}}^{{{\rm{intra}}}}({r}_{{{\rm{ss}}}})={u}_{5}{{\mbox{e}}}^{-{r}_{{{\rm{ss}}}}/{R}_{2}}\)26,27,28, as presented in Supplementary Note 3.

Fig. 3: Trajectory analysis from Thiele equation.
figure 3

a Schematic and b results of dragging potential calculation from a single active skyrmion pair. The inset in (b) shows a circular potential well in the x-y plane. c Dragging potential of the entire skyrmion lattice constructed from (b). The red curve is the trajectory of the passive skyrmion with respect to the potential lattice, calculated from Thiele equation. d Comparison of constructed skyrmion potential and that obtained from MUMAX3. The symbols and curves represent the potentials extracted from Figs. 2e and 3c, respectively. e Analysis of the dragging force (orange arrow) and Magnus force (blue arrow) at different locations of the passive skyrmion in the laboratory coordinates. The black curve corresponds to the trajectory of the passive skyrmion and the dashed circles indicate the circular potential wells.

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The knowledge of the interacting potentials also allows an alternative simulation from Thiele equation38,39,40,41 to examine the skyrmion dynamics. The Thiele equation of ith skyrmion reads42,43,44,45,46

$${{{\bf{G}}}}_{i}\times {\dot{{{\bf{r}}}}}_{{\rm{c}},i}-\Gamma {\dot{{{\bf{r}}}}}_{{\rm{c}},i}+{{{\bf{F}}}}_{j,i}+{{{\bf{F}}}}_{\,{\rm{ss}},i}^{{\rm{inter}}\,}+{{{\bf{F}}}}_{\,{\rm{ss}},i}^{{\rm{intra}}\,}={{\bf{0}}},$$
(1)

where rc,i = (xc,i, yc,i) denotes the position coordinate of the skyrmion center in the x-y plane. The first term in Eq. (1) stands for the Magnus force with the gyrovector G. Γ in the second term corresponds to the dissipation coefficient. The third term, i.e., the driving force due to the current, is provided in Methods. The last two terms represent the forces from the inter- and intra-layer skyrmion interactions, \({{{\bf{F}}}}_{\,{\rm{ss}},i}^{{\rm{inter/intra}}\,}=-{\nabla }_{{{{\bf{r}}}}_{{\rm{c}},i}}[{\sum }_{j\ne i}{U}_{\,{\rm{ss}}}^{{\rm{inter/intra}}\,}(| {{{\bf{r}}}}_{{\rm{c}},i}-{{{\bf{r}}}}_{{\rm{c}},j}| )]\).

By numerically solving the Thiele equation in the configuration of Fig. 2a, we extract the skyrmion trajectories in the presence of driving currents j = 0.5 and 5 MA cm−2, which nicely reproduce both conventional and anomalous dragging dynamics, respectively (see Supplementary Fig. 5). The relative motion of the passive skyrmion with respect to the moving potential in the anomalous case is plotted as the red curve in Fig. 3c, which is also consistent with the simulated curve from MUMAX3 in Fig. 2e. These results suggest that the Thiele approach, although assuming a rigid skyrmion with a stable velocity in its original derivation, remains applicable in the present cases with small variations both in shape and velocity of the skyrmions.

As an advantage of the Thiele equation, we are now allowed to look into the skyrmion motion from the effective forces more explicitly. Specified to the single passive skyrmion, Eq. (1) is reduced to \({{\bf{G}}}\times {\dot{{{\bf{r}}}}}_{{{\rm{c}}}}-\Gamma {\dot{{{\bf{r}}}}}_{{{\rm{c}}}}+{{{\bf{F}}}}_{\,{\rm{ss}}}^{{\rm{inter}}\,}={{\bf{0}}}\). For the conventional drag, its velocity \({\dot{{{\bf{r}}}}}_{{{\rm{c}}}}\) follows that of the active layer, i.e., approximately in x direction according to Fig. 2b. The Magnus force and the dissipation force are therefore along − y and − x directions, respectively. From Fig. 2d, we can see that the trapped passive skyrmion mainly attaches to the lower edge of the yellow circle, indicating the stronger Magnus force compared to the dissipation one, which is also consistent with the ratio ∣G∣/Γ = 4π∣Q∣/(αdxx) ≈ 13.5 derived based on the analytical expression in Methods. Following this mechanism, once the attractive force \({{{\bf{F}}}}_{\,{\rm{ss}}}^{{\rm{inter}}\,}\) is not sufficient to maintain the fully trapped motion, the Magnus force will push the passive skyrmion out of the circular trapping potential and make it move in − y direction. As the passive skyrmion approaches the ring-shaped potential well generated by another active skyrmion nearby and beneath its new location, the increased attractive force from this potential will induce a hopping with a large transverse displacement. These processes together result in the anomalous drag under a large driving current. Figure 2c also reveals that, in addition to the finite motion in the transverse direction, the escaped passive skyrmion also gains a finite longitudinal velocity in − x direction, which is of opposite sign with that of the active skyrmions and thus leads to a surprising counter flow between them. To understand this counter-intuitive behavior, we plot the two dominant forces, i.e., the dragging force and the Magnus force, acting on the passive skyrmion at different moments in Fig. 3e, where the two dashed circles stand for the potential wells induced by two specific skyrmions in the active layer. As seen from the plot at 30 ns, the dragging force is remarkably enlarged when the passive skyrmion falls into the circular well. The increased velocity then leads to stronger accompanying Magnus force, resulting in a quick clockwise round turn around t = 31 ns. Such a process causes a large spatial shift of the passive skyrmion in − x direction.

Dependence on dragging strength

In this section, we discuss the dependence of two distinct drags on dragging strength. In practice, we perform micromagnetic simulations in multilayers by modulating the dipolar-field-induced drag via the number of active layers, as more layers lead to stronger dipolar fields. The comparison of the average skyrmion velocities in three cases is presented in Fig. 4 with varying driving current. The skyrmion configurations in this part are obtained with the same techniques as those in Fig. 2.

Fig. 4: Analysis on skyrmion velocities in different multilayer configurations.
figure 4

a–c Average velocities of passive and active skyrmions as functions of current densities in the configuration illustrated in the insets. The error bars represent the standard derivation. d–f The angle between the average velocities of passive and active skyrmions. In these simulations, there are 5 (81) skyrmions in the passive (active) layer. Other parameters are the same as those used in Fig. 1.

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As seen, the velocities of the skyrmions in the active magnetic layer (hollow symbols in Fig. 4a–c) increase almost linearly with the current strength, as they are directly driven by the current. For the skyrmions in the passive layer, their velocities (solid symbols) vary non-monotonically with increasing current strength, reflecting the transition of the conventional drag and the anomalous scenario. As expected, the current intensity at the transition increases with the number of stacking layers due to the increasing dipolar field. In the anomalous drag of each configuration, the average velocity of passive skyrmions keeps almost a constant, indicating a weak dependence on the velocity of the moving potential. This is due to the fact that the driving source of passive skyrmions, \({{{\bf{F}}}}_{\,{\rm{ss}}}^{{{\rm{inter}}}}=-\nabla {U}_{{{\rm{ss}}}}^{{\rm{inter}}\,}\), involves the spatial derivative of the potential rather than its velocity.

The angle φ12 between the velocities of skyrmions is plotted in Fig. 4d–f as functions of current density, which also show an increase with the stacking layers in the anomalous region. In particular, for the double-layer configuration shown in Fig. 4d, φ12 is less than 90° even in the anomalous region, indicating the absence of the counter flow discussed in Fig. 3e. This can be attributed to the modification of the potential wells, as shown in Supplementary Fig. 3b.

In conclusion, we studied the skyrmion drag effects in the magnetic multilayers based on micromagnetic simulation associated with Thiele equation of skyrmion motion. The skyrmions in the passive layers are found to either follow the skyrmion motion in the active layers as conventional drag, or present an unconventional behavior with both a transverse motion and a counter flow with respect to skyrmion motion in the active layers. The transition between the conventional and anomalous behaviors is attributed to the specific ring-shaped attractive potential generated by the interlayer skyrmion interactions via the dipolar fields. Our results elaborate the physics behind the skyrmion drag in magnetic multilayers, which should be helpful for designing skyrmion-based devices, including logic gates and nano-oscillators.

Methods

Micromagnetic simulations

The simulations of magnetization dynamics in the multilayer are carried out by using micromagnetic software MUMAX3 34 to numerically solve the Landau–Lifshitz–Gilbert (LLG) equation 47

$$\dot{{{\bf{m}}}}=-\gamma {{\bf{m}}}\times {{{\bf{H}}}}_{{{\rm{eff}}}}+\alpha {{\bf{m}}}\times \dot{{{\bf{m}}}}+\gamma {H}_{j}\left({{\bf{m}}}\times {{\bf{p}}}\right)\times {{\bf{m}}},$$
(2)

where \(\dot{{{\bf{m}}}}\) represents the derivative of the reduced magnetization m with respect to time. γ and α stand for the gyromagnetic ratio and the damping constant, respectively. Heff is the effective field including the exchange field, the DMI field, the anisotropy field, the dipolar field and the external magnetic field. p denotes the polarization vector and Hj = jℏθSH/(2μ0etzMs) is the strength of dampinglike spin torque. Here j is the current density, ℏ the reduced Planck constant, θSH the spin Hall angle, μ0 the vacuum permeability constant, e the elementary charge, tz the thickness of ferromagnetic layer, and Ms the saturation magnetization. Note that our simulations adopt the general form of the LLG equation without the high-order terms (such as nutation48), which has been previously demonstrated to be sufficient in describing the current-induced dynamics of the skyrmions very well49,50.

We take [Pt/CoFe/Ir] as the [heavy metal A/ferromagnet/heavy metal B] unit in the multilayer and set the size of each ferromagnet to be 900 × 900 × 1 nm3. In the simulations, the ferromagnets are discretized by 3 × 3 × 1 nm3 grids and the spacing between two adjacent ferromagnets is adopted to be 1 nm. The material parameters are adopted as follows 51: the exchange coefficient A = 9.25 pJ m−1, the DMI constant D = 1.4 mJ m−2, the perpendicular magnetic anisotropy constant K = 0.65 MJ m−3, and the saturation magnetization Ms = 1020 kA m−1. In addition, we take α = 0.05, θSH = 0.1, and p = ex. The magnetic field μ0H = 100 mT is applied along the − z direction. Periodic boundary conditions are used in the x and y directions, while free boundary condition is applied in the z direction. Note that even in realistic samples with defects and natural edges that may cause skyrmion formation or annihilation and affect the skyrmion motion52,53, the conventional and anomalous drags can still be distinguished from the accumulation position of the passive skyrmions at the boundaries, as they drive the passive skyrmions to move in distinct directions, according to Fig. 2. Additionally, the influences of the material parameters on skyrmion dragging dynamics are discussed in Supplementary Note 5.

Expression of forces in Thiele equation

According to ref. 46, the Magnus force in Thiele equation is expressed as

$${{\bf{G}}}\times {\dot{{{\bf{r}}}}}_{{{\rm{c}}}}=(4\pi Q{\mu }_{0}{{M}}_{{{\rm{s}}}}{{t}}_{{z}}/\gamma ){{{\bf{e}}}}_{{z}}\times {\dot{{{\bf{r}}}}}_{{{\rm{c}}}},$$
(3)

with the topological charge \(Q=-1/(4\pi )\int{{\bf{m}}}\cdot \left({\partial }_{x}{{\bf{m}}}\times {\partial }_{y}{{\bf{m}}}\right)dxdy\)54,55. The dissipation force is written as

$$-\Gamma {\dot{{{\bf{r}}}}}_{{{\rm{c}}}}=-(\alpha {\mu }_{0}{M}_{{{\rm{s}}}}{t}_{z}{d}_{xx}/\gamma ){\dot{{{\bf{r}}}}}_{{{\rm{c}}}},$$
(4)

where the structure factor dxx = ∫∂xm ⋅ ∂xmdxdy. The current-induced driving force is described as

$${{{\bf{F}}}}_{j}=-{\mu }_{0}{H}_{j}{M}_{{{\rm{s}}}}{t}_{z}{{\bf{u}}},$$
(5)

with ui = ∫(m × p) ⋅ ∂imdxdy.

According to the magnetic structure of skyrmion, we have Q = − 1, dxx ≈ 18.55, ux ≈ 0, and uy ≈ − 2.09 × 10−7 m. With the values of Q and dxx, we obtain the skyrmion Hall angle θSky = arctan(vcx/vcy) = arctan[− 4πQ/(αdxx)] ≈ 86°, suggesting a transverse dominant motion of skyrmions. The fitting parameters involved in the interaction forces between the skyrmions are as follows u1 = 24.615 × 10−21 J, u2 = − 1.65 × 10−21 J, u3 = − 14.13 × 10−21 J, u4 = 3.5 × 10−21 J, u5 = 1.4 × 10−18 J, k = 0.1 nm−1, δ = 1.1, R1 = 22 nm and R2 = 21 nm.