Introduction

Topological magnetic textures such as skyrmions1,2,3, vortices4,5,6,7 and (bi)merons8,9,10, are exotic and versatile topological objects in magnetic material that may provide applications in spintronic devices. The manifestation of these topological textures strongly relies on the material parameters and their environment. For example, an isolated skyrmion can be either of circular shape or of stripe configuration, while the skyrmions in a condensed phase usually form a lattice with circular shapes11,12,13. Studies on skyrmions in two dimensions (2D) have been recently extended to their variants, such as skyrmion bags14,15,16 and high-order skyrmions17, as well as their intriguing dynamics driven by spin-transfer torques18,19,20 or spin waves21,22,23. A further extension from 2D to three dimensions (3D) brings in additional family members of the topological magnetic textures, including hopfions24,25,26,27,28, torons29,30, chiral bobbers (ChBs)31,32,33,34, etc35,36,37,38,39,40, which has recently attracted tremendous attention because of their fantastic dynamic properties41,42,43,44,45,46. Recently, it was demonstrated both theoretically47 and experimentally31,32,48 that, in a thick magnetic film, skyrmion tubes (SkTs), skyrmion-type 2D spin texture penetrating through the thickness, and ChBs, broken SkTs terminating at Bloch points (BPs)32 within the film, can be stabilized simultaneously by a bulk Dzyaloshinskii-Moriya interaction (DMI) with rather good robustness against disturbances thanks to their topological characteristics. The differentiation between them is however challenging as most of the magnetic imaging approaches reflect only their very similar skyrmion-type 2D magnetic texture near the sample surface. As the X-ray ptychography is applicable to reveal their different 3D textures32,49, the appearance of the BP may provide alternatively easier solutions to distinguish ChBs from SkTs in spintronic devices, according to the distinctive BP-induced phenomena43,50,51,52,53,54.

To date, electrical manipulation of 3D magnetic structures has been realized via the spin-transfer34,55,56,57 or spin-orbit torques46,58 generated by a spin-polarized current. The typical magnitudes of the current density in those studies however are too large to satisfy the requirement of low power consumption applications. More seriously, the enormous current-induced heating is detrimental to the stability of ChB/SkT. In contrast, the magnetic dynamics driven by spin waves can efficiently reduce the energy consumption and heating effects59,60,61,62, which is however still largely unexplored in 3D systems, especially for the ChBs and SkTs and their back-actions to spin waves. This motivates our present work to investigate the interplay between ChBs/SkTs and spin waves.

Based on micromagnetic simulations, we unveil distinct behaviors between the ChB and SkT dynamics driven by spin waves. Specifically, we find that the velocity of the SkT is proportional to the power of the driving field, while the ChB motion presents a threshold in the driving power of spin waves. In addition, we find tornado-like dynamics of SkTs and ChBs, namely, a dynamic modulation of the magnetic textures during their lateral motion. In the meantime, the spin waves diffracted from the BPs of ChBs are found to deviate from the Rutherford-type scattering previously predicted from an analytic derivation50. These results provide insights for understanding and applying the dynamics of 3D topological magnetic structures.

Results and discussion

Theoretical model

The geometry of our system is schematically shown in Fig. 1. With a perpendicular magnetic anisotropy and an external magnetic field Hex applied normal to the film plane, the total micromagnetic energy density thus reads,

$${{{{{{{\mathscr{H}}}}}}}}= A{\left(\nabla {{{{{{{\bf{m}}}}}}}}\right)}^{2}-{K}_{{{{{{{{\rm{u}}}}}}}}}{\left({m}_{z}\right)}^{2}-{\mu }_{0}{H}_{{{{{{{{\rm{ex}}}}}}}}}{M}_{{{{{{{{\rm{s}}}}}}}}}{m}_{z}+D{{{{{{{\bf{m}}}}}}}}\cdot \left(\nabla \times {{{{{{{\bf{m}}}}}}}}\right)\\ -\frac{1}{2}{\mu }_{0}{M}_{{{{{{{{\rm{s}}}}}}}}}{{{{{{{{\bf{H}}}}}}}}}_{{{{{{{{\rm{dm}}}}}}}}}\cdot {{{{{{{\bf{m}}}}}}}},$$
(1)

where A is the stiffness constant, Ku is the anisotropy constant, Ms is the saturation magnetization and D is the bulk DMI coefficient. Hdm in the last term corresponds to the demagnetization field. The magnetic parameters and other modeling details are provided in the Methods. Starting from a bobberlike initial state with a proper magnetic field, the energy relaxation due to the Gilbert term leaves a stable ChB structure shown in Fig. 1. Specifically, we set an initial state as all magnetizations within a bobber-shaped region of a given length and a finite radius, pointing opposite to those of the background. After the relaxation process, the initial state finally goes to a stable SkT, ChB, or uniform ferromagnetic phase. For a ChB final state, its penetration length depends on the strength of the external magnetic field and the length of the initial bobber-shaped region. Unless otherwise specified, the magnetic field in the following is set to be μ0Hex = 0.18 T, which stabilizes ChB with a penetration length LB = 23 nm. Alternatively from an initial SkT configuration, the same magnetic field is found to be able to stabilize a SkT as well, where, as shown in Fig. 1, the lateral spin texture presents a dependence on depth due to the demagnetization field.

Fig. 1: Schematic diagram for the simulation of the chiral bobber (ChB) and skyrmion tube (SkT) dynamics driven by the spin waves.
figure 1

Sketch of the simulated geometry with the spin-wave source represented by the khaki region. The left and right solitons correspond to the isosurfaces of mz = 0 for ChB with a penetration length (LB) and SkT, respectively, with the colors indicating the orientation of the in-plane magnetization. The arrows around the end of ChB show the spin configuration of the magnetic Bloch point (BP). The two gray dashed curves describe the spatial profile of the skyrmion centers in SkT along the thickness direction at the static and dynamic states. The thin dark blue arrows indicate the trajectories of the skyrmion centers in the top, middle, and bottom layers. The rightmost panel shows the magnetic structures of the top, middle, and bottom layers in SkT.

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The spin waves are then introduced to trigger the dynamics of magnetic textures. As the detail of the spin-wave generation may modify the spatial profile of spin waves and hence the response of the magnetic textures, it cannot affect qualitatively the physics. Therefore, in practice, we consider that the spin waves are generated by an AC magnetic field, \({{{{{{{\bf{h}}}}}}}}(t)={h}_{{{{{{{{\rm{rf}}}}}}}}}\sin (2\pi ft)\hat{x}\) with a frequency f = 120 GHz, in a thin plane of two-unit-cell width illustrated as the khaki region in Fig. 1. In a realistic case with a wider excitation region, a comparable spin-wave intensity can be achieved by a weaker AC magnetic field. The excited spin waves then propagate in the x direction. In order to avoid the reflection at the lateral boundaries, absorbing boundary conditions with gradually enhanced damping is used, while the free boundary condition is applied for the upper and lower surfaces.

To describe the dynamics of ChBs and SkTs, we calculate the temporal evolution of the in-plane topological centers at different values of z from

$${{{{{{{\bf{R}}}}}}}}(z,t)=\frac{\int{{{{{{{\bf{r}}}}}}}}\rho ({{{{{{{\bf{r}}}}}}}},z,t)dxdy}{\int\rho ({{{{{{{\bf{r}}}}}}}},z,t)dxdy},$$
(2)

in which the topological density is expressed as \(\rho =\frac{1}{4\pi }{{{{{{{\bf{m}}}}}}}}\cdot ({\partial }_{x}{{{{{{{\bf{m}}}}}}}}\times {\partial }_{y}{{{{{{{\bf{m}}}}}}}})\)63. While the topological centers at the equilibrium configuration form nearly a straight string across the thickness, as shown by the gray dashed curves in Fig. 1 and discussed explicitly below, the presence of spin waves introduces an interesting distortion of linked topological centers and tornado-like dynamics.

Spin-wave-driven dynamics of SkTs and ChBs

The trajectories of the topological centers under the driving field hrf = 0.1 T are plotted in Supplementary Fig. 1 with different values of z. As seen, the topological centers of ChB show only a tiny rotation, while those of the SkT move collectively towards the wave source with a finite Hall angle ΘH ≃ 13. 5°. A more explicit analysis of the ChB dynamics (at z ≃ − LB = − 23 nm) reveals that the BP actually keeps standing still. The relative motions of the skyrmion centers in ChB with respect to the BP are plotted in Fig. 2a, which clearly shows a larger rotation radius for the skyrmion centers approaching the surface. For the SkT, the trajectories of topological centers at z = 0 and − 40 nm present clearly wavy vibrations while that at z = − 20 nm is nearly a simple straight line (see Supplementary Fig. 1). The internal motion of the topological centers within the SkT is thus described by the relative trajectories with respect to the instantaneous location of the topological center at z = − 20 nm. As depicted in Fig. 2b, the topological center in each layer of SkT, similar to the ChB case, shows nearly circular rotations with the corresponding radius reaching maximal values around the surfaces of the film. As shown in Fig. 2c, the relative rotations of the 2D topological centers in the upper and lower half film are of the same direction but with approximately a π-phase difference, very similar to the spin dynamics of the low-frequency mode in an easy-axis antiferromagnet. Another intriguing feature is that the connection of the rotation centers of SkT surprisingly presents a complicated profile in the y−z plane, i.e., the cross-section perpendicular to the spin-wave propagation direction. Such a distortion is found to be almost a quasi-static configuration during the dynamics and its magnitude becomes larger under a stronger excitation field, which suggests that this distortion should be closely related to the propagation of spin waves. In contrast, the rotation centers of ChB only show a small tilting in the thickness direction. Simulations with different spatial discretization schemes result in very similar features (see Supplementary Fig. 2), which excludes the artificial origin of such tornado-like dynamics. The typical dynamic of the skyrmion at the upper surface of a SkT is shown in Supplementary Movie 1 with hrf = 0.4T.

Fig. 2: Spin-wave-driven tornado-like dynamics of chiral bobber (ChB) and skyrmion tube (SkT).
figure 2

The trajectories of the topological centers and their projections in the x−z and y−z planes for a ChB and b SkT (with respect to the one at z = − 20 nm) under the excitation of hrf = 0.1 T. The pink curves stand for the spatial profile of the instantaneous topological centers. c The trajectories of the top and bottom topological centers in one period with the arrows and the pink dots indicating the direction of their movements and their positions at a certain moment, respectively.

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Origin of the deformation of SkT and ChB

In order to understand the above dynamic features, we derive the bulk dispersion relation of the spin waves as

$$2\pi f=\gamma \left({\mu }_{0}{H}_{{{{{{{{\rm{ex}}}}}}}}}+\frac{2{K}_{{{{{{{{\rm{u}}}}}}}}}}{{M}_{{{{{{{{\rm{s}}}}}}}}}}\right)+\gamma \frac{2A}{{M}_{{{{{{{{\rm{s}}}}}}}}}}{k}^{2}+\gamma \frac{2D}{{M}_{{{{{{{{\rm{s}}}}}}}}}}{k}_{z}.$$
(3)

As the derivation details are provided in Supplementary Note 1, the spectrum with ky = 0 in our excitation configuration is plotted in Fig. 3a. The presence of the bulk DMI shifts the center of the circular iso-energy surfaces away from k = 0 by \({k}_{O{O}^{{\prime} }}\hat{z}\) (see Supplementary Fig. 3a). As a consequence, the group velocity of the generated spin-wave with kz = 0 has both x and z components, meaning that the spin waves also travel across the thickness during their propagation in the x direction and undergo reflection at the surfaces. The x and z components of the group velocity of the reflected beam are the same as and opposite to those of the initial spin-wave, respectively, corresponding to the transition of the wave vector from k0 to kr (see Supplementary Fig. 3a). The coherently reflected beam will then cause interference and lead to a considerable modulation of the spin-wave intensity in the thickness direction, giving by \(\sim \exp (\,{{\mbox{i}}}z{k}_{0}^{z})+\exp ({{\mbox{i}}}z{k}_{r}^{z})\), where \({k}_{0}^{z}\) and \({k}_{r}^{z}\) correspond to the z components of k0 and kr, respectively. By substituting \({k}_{0}^{z}=0\) and \({k}_{r}^{z}=2{k}_{O{O}^{{\prime} }}\), the modulation of spin-wave intensity can be expressed as \(1+\cos (2z{k}_{O{O}^{{\prime} }})\), which leads to a period of about \(\pi /{k}_{O{O}^{{\prime} }} \sim 12.6\) nm with \({k}_{O{O}^{{\prime} }}=0.25\) nm−1, consistent with the numerical results in Fig. 3b and Supplementary Fig. 3b for the distribution of mx and my in the x−z cross-section, respectively. It is found that such a modulation coincides nicely with the spatial distortion of the ChB and SkT shown in Fig. 3d. However, the puzzle remains because the spin waves are pulling the ChB and SkT mainly in the x direction, but the distortion of the topological centers along the thickness counter-intuitively occurs in the y direction.

Fig. 3: Deformation of chiral bobber/skyrmion tube caused by the nonuniform distribution of spin-wave intensity in the thickness direction.
figure 3

a Dispersion relations of the spin waves in the ky = 0 plane. The closed curves in the kx − kz plane correspond to the projections of the isofrequency contours at f = 5, 15, 30, 60, and 120 GHz. b An instantaneous magnetization plot of the spin waves in the x−z plane at hrf = 0.1 T. The color bar represents the x component of the magnetization. c The schematic of force analysis for the skyrmions in neighboring layers, S1 and S2, with strong and negligible driving force Fdrive, respectively. FM represents the Magnus force perpendicular to the velocity v and the purple spring stands for the attractive interaction between the skyrmions. d The spatial profile of the skyrmion positions in the y−z plane under the driving field hrf = 0.1 T. The orange areas indicate the regions of high-intensity spin waves.

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The solution to this puzzle lies in the requirement of the force balance for the quasi-stationary state of ChB and the uniform lateral motion of SkT. To deduce the driving force from the spin-wave, we consider a skyrmion located in the region with a rather weak spin-wave intensity, i.e., one of the white regions in Fig. 3d, where the driving force acting on the skyrmion is negligibly small. As analyzed in Supplementary Note 2 and sketched in Fig. 3c, the dissipation force is also very weak due to the small damping constant and the Magnus force FM of the moving SkT, perpendicular to its velocity, then should be balanced by the dragging force from the 2D skyrmions in the neighboring sublayers. The latter is apparently in the same direction as the relative shift between adjacent 2D skyrmions. Because of the small Hall angle, the Magnus force approximately points to − y direction with \(| {F}_{{{{{{{{\rm{M}}}}}}}}}^{y}| /| {F}_{{{{{{{{\rm{M}}}}}}}}}^{x}| \approx 4.2\), making the weakly driven skyrmions fall behind those experiencing strong driving force along the y-axis (see the pink dots in the white windows in Fig. 3d). For the ChB, the Magnus force, which is proportional to the velocity, is significantly suppressed because of the absence of the net uniform motion and the driving force is mainly compensated by the pinning force of the BP. As a result, the topological centers of the 2D skyrmions form a simple profile in the y−z plane as shown by the green stars in Fig. 3d. Note that in the cases of SkT and ChB, varying the frequencies will not affect our qualitative conclusions. The result of SkT excited by the AC magnetic field at 60 GHz is shown in Supplementary Fig. 4.

Origin of the lateral rotation of topological centers

To further explore the origin of the lateral rotation shown in Fig. 2, an important task is to justify whether it is a persistent rotation or a dissipative one. Therefore, we perform the same simulation as that for Fig. 2 but with a larger damping constant α = 0.01. The trajectories of the topological center at z = − 40 nm are plotted in Fig. 4a for different values of driving field and out-of-plane exchange strength. It can be seen that the wavy amplitude decreases for all cases, indicating clearly the dissipative nature of the lateral rotation. The long-time relative motions of the topological center can therefore be schematically described by the inset in Fig. 4a, where the radius of the rotation orbit gradually shrinks to a single point at the rotation center. This behavior looks similar to the skyrmion motion in an attractive central force field64. It is also noteworthy that, the number of the wavy periods in the trajectories is almost independent of the strength of the driving field but strongly relies on the strength of out-of-plane exchange constant A⊥, which suggests that the interlayer exchange coupling may provide an attractive force in the present case65,66,67. Note that the value of A⊥ can be different from that of A in layered materials or artificial magnetic multilayers68,69. For the latter case, A⊥ is determined by an adjustable Ruderman-Kittel-Kasuya-Yosida interaction70. For a qualitative analysis, we illustrate in the inset of Fig. 4b the forces acting on a skyrmion S2 which experiences a relative rotation around the other skyrmion S1 in a neighboring layer. Focusing on the relative motion between S1 and S2, we here discard the driving force from spin waves and consider only the radial components of the interlayer pulling force Fd and the Magnus force FM = Gυr, where G and υr represent the gyrovector and the relative motion velocity of the adjacent layer skyrmion, respectively. For the sake of simplification, the dissipation of the S2 motion is assumed to be sufficiently weak so that the rotation orbit can be regarded approximately as a well-defined circle with the force balance Fd = FM. Assuming further Fd ≈ ηA⊥Δ with η and Δ being a coupling constant and the distance between S1 and S2, respectively65, we derive the frequency of the rotational motion f ∝ A⊥, which is confirmed nicely by our micromagnetic simulations (see Fig. 4b). A detailed analysis with the numerical evaluation of η can be found in Supplementary Note 3 and Supplementary Fig. 5.

Fig. 4: Lateral rotation of the topological centers controlled by out-of-plane exchange strength.
figure 4

a Trajectories of topological centers at z = − 40 nm for the first 50 ns, with different values of driving field (hrf) and out-of-plane exchange strength (A⊥). A relatively large damping constant α = 0.01 is used to magnify the decay rate of the rotation radius. The inset in panel a illustrates the long-period evolution of the dissipative rotation of a topological center. b Rotation frequency of the topological centers as a function of A⊥. The inset in panel b is the force analysis of the skyrmion S2 with a relative velocity υr with respect to the skyrmion S1 in its neighboring layer. Fd stands for the attractive force from interlayer coupling.

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Velocities of SkT/ChB motion vs. driving field

We now explore the relationship between the velocities of the SkT/ChB and the strength of the driving source. As shown in Fig. 5a, the velocity of the SkT is proportional to the driving power (\(\propto {h}_{{{{{{{{\rm{rf}}}}}}}}}^{2}\)). For the ChB, as discussed above, the spin waves generated by a weak driving field are not sufficient to move the ChB, leading to a vanishing velocity. However, when the driving force is enhanced by increasing the driving field beyond a threshold, the pinning force will be fully compensated, resulting in a collective motion of the ChB. Note that the ChB with a certain penetration length LB can be actually stable under different values of magnetic field and, similarly, the same magnetic field can stabilize ChB with different values of LB (see Supplementary Fig. 6). This is due to the pinning effect of BP in the thickness direction. The origin of the pinning effects in both lateral and thickness directions results from the crystal field in real materials, which is mimicked by the spatial discretization in our simulation34. In order to analyze the influences of the penetration length and the external magnetic field on the critical driving field of ChB motion, the results of ChB with different values of LB and Hex are plotted in Fig. 5a. The extracted critical driving fields are plotted as a function of penetration length and described nicely by a linear fitting in Fig. 5b, while the external field plays only a marginal role on the threshold. In general, the motion of a longer ChB requires a smaller driving field, as it contains more skyrmions to generate a stronger driving force. Such a relation may be useful to distinguish ChBs from SkTs and estimate the penetration length of ChBs in the experiment. Above the threshold of the driving field, the skyrmions of each layer in a ChB still display internal rotations relative to the BP shown in Fig. 2a. The missing ChB data in the strong driving regime in Fig. 5a is because the ChBs under such large driving fields become unstable. Note also that SkTs will also be destroyed by a too-strong driving field. The maximal velocity of SkTs is expected to be limited by the same mechanisms as those in the spin-wave-driven skyrmions in two dimensions ~100 m s−1(ref. 61). The quantitative evaluation of the maximal value requires a systematical simulation and is left for future study.

Fig. 5: Velocities of chiral bobber (ChB)/skyrmion tube (SkT) and the threshold of the driving field for the ChB motion.
figure 5

a The velocities of the SkT and ChBs with different penetration lengths LB as functions of the driving field hrf. The solid curve corresponds to a fitting of the SkT velocity with a quadratic function. b Threshold of the driving field \({h}_{{{{{{{{\rm{rf}}}}}}}}}^{{{{{{{{\rm{c}}}}}}}}}\) for the ChB motion as a function of penetration length. The blue solid line is a linear fit of the ChB data and the dashed line shows the extrapolation to the length of the SkT. The error bars in b represent half of the step size (0.025 T) of the magnetic field during the calculation.

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Scattering of spin waves by SkT and ChB

Figure 6 plots the snapshots of real space spin-wave patterns scattered by SkT (Fig. 6a–c) and ChB (Fig. 6d–f), with the green circles and dot indicating the locations of the skyrmion and the BP, respectively. A general feature of the scattering patterns is that, in contrast to the highly asymmetric scattering commonly seen in 2D case (see Supplementary Fig. 7), the interference patterns of the scattered beams in the present 3D case are clearly visible on both sides of skyrmions (see Fig. 6a–d). In the meantime, even for SkT, the scattering patterns present a strong layer dependence, which can be attributed to the layer-dependent spin-wave intensity in the thickness direction shown in Fig. 3b. In addition, the spin-wave scattering near the BP point for the ChB in Fig. 6e is found to exhibit asymmetric pattern as well, indicating its deviation from the Rutherford-like scattering predicted previously50. This reflects the influence of the neighboring skyrmions, which are closely connected with the BP.

Fig. 6: Scattering of spin waves by skyrmion tube and chiral bobber.
figure 6

Scattered spin-wave patterns of a–c skyrmion tube and d–f chiral bobber at different layers. The green circles in a–d and the solid green dot in e are defined as mz = 0, indicating separately the location of skyrmions and the Bloch point in the corresponding layers. The color bar represents the x component of the dynamical magnetization.

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Conclusions

In summary, we have demonstrated that ChB and SkT can be driven by spin waves in 3D systems. Bulk DMI causes nonuniform spin-wave intensity in the thickness direction, resulting in tornado-like dynamics of SkT and ChB. In contrast to the vanishing threshold of driving power for SkT motion, ChB can gain a finite velocity only when the driving power is beyond a critical value, which depends linearly on the penetration length of ChB. These reveal the potential directions to experimentally distinguish SkT and ChB. As a back action, the spin waves are scattered by SkT and ChB with the scattering spectrum distinguished from the 2D skyrmions and a symmetric Rutherford-like pattern from a single individual BP. The rich dynamics of ChB and SkT provide opportunities for exploring spin-wave devices and other spintronics applications.

Methods

Micromagnetic simulations

The stable magnetic textures are simulated by numerically solving the Landau-Lifshitz-Gilbert equation71,72:

$$\frac{d{{{{{{{\bf{m}}}}}}}}}{dt}=-\gamma {{{{{{{\bf{m}}}}}}}}\times {{{{{{{{\bf{B}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}+\alpha {{{{{{{\bf{m}}}}}}}}\times \frac{d{{{{{{{\bf{m}}}}}}}}}{dt},$$
(4)

where γ is the gyromagnetic ratio, α is the Gilbert damping constant and \({{{{{{{{\bf{B}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=-\delta {{{{{{{\mathscr{H}}}}}}}}/\left({M}_{{{{{{{{\rm{s}}}}}}}}}\delta {{{{{{{\bf{m}}}}}}}}\right)\) is the effective magnetic field. In our simulation, we take A = 0.1 pJ m−1, Ku = 5 kJ m−3, Ms = 100 kA m−1, D = 0.05 mJ m−2 (ref. 26), and an experimentally accessible damping coefficient α = 0.00173,74,75. With these parameters, the discrimination parameter for isolated skyrmions is estimated to be π 2D2/(16AKu) > 176, suggesting the necessity of a proper external magnetic field to stabilize an isolated circular skyrmion, which is confirmed by our simulation. The magnetic film with the size of 300 × 300 × 41 nm3 is discretized by 1 × 1 × 1 nm3 units.