Realization and Current‐Driven Dynamics of Fractional Hopfions and Their Ensembles in a Helimagnet FeGe

3D topological spin textures—hopfions—are predicted in helimagnetic systems but are not experimentally confirmed thus far. By utilizing an external magnetic field and electric current in the present study, 3D topological spin textures are realized, including fractional hopfions with nonzero topological index, in a skyrmion‐hosting helimagnet FeGe. Microsecond current pulses are employed to control the dynamics of the expansion and contraction of a bundle composed of a skyrmion and a fractional hopfion, as well as its current‐driven Hall motion. This research approach has demonstrated the novel electromagnetic properties of fractional hopfions and their ensembles in helimagnetic systems.


DOI: 10.1002/adma.202210646
inversely proportional to the volume of spin textures, [8,9,16,17] and the novel particle-like behavior of skyrmions under ultralow energy stimulations. [18,19] Several types of topological magnetic textures have been found, including 1D chiral soliton lattice in hexagonal helimagnets with uniaxial magnetic anisotropy, [20] 2D magnetic skyrmions with topological number N = −1 in cubic chiral-lattice magnets with isotropic Dzyaloshinskii-Moriya interaction (DMI), [10][11][12] their antiparticles-antiskyrmions with N = +1 in tetragonal chiral-lattice magnets with anisotropic DMI [21,22] -and 2D skyrmionium comprising a skyrmion and an antiskyrmion with a total topological number of zero. [23,24] Recently, the 3D chiral soliton-hopfion-has been discovered in liquid crystals doped with a ferromagnetic hexaferrite, exhibiting uniaxial anisotropy [25,26] and magnetic multilayered system. [27] The topological charge of the hopfion can be characterized by the Hopf index where A is the vector potential of F = ∇ × A, and the components of vector F can be expressed via the unit vector n pointing along the local magnetization direction, and Levi-Civita permutation symbol ε as F i = ε ijk n· (∂ j n × ∂ k n). [28] The hopfion lattice and isolated hopfion can be controlled by external magnetic and electric fields, [29] and have been detected by nonlinear optical microscopy. [26] Theoretically, the dynamics of hopfions under an external magnetic field [30] and electric current [31][32][33] are expected to show intriguing behaviors that have not been experimentally clarified thus far. The formation of hopfions in the aforementioned material systems generally requires strong surface anisotropy or geometrical confinement. Hence, further exploring the nontrivial current-driven dynamics of hopfions is challenging. In contrast, bulk helimagnets are promising candidates to host nanometer-scale hopfions, where their dynamics can be thoroughly investigated. In skyrmion-hosting helimagnets with the non-centrosymmetric cubic structure, the hopfion has been predicted as a closed loop of a twisting skyrmion string (Figure 1a). [34,35] However, the hopfion has not been observed in helimagnets experimentally due to the technical challenges in 3D magnetic imaging with nanometer-scale spatial resolution. Figure 1b shows the simulated XY-plane projection of the field (B) map for a hopfion with a Hopf index of 1. The corresponding 2D magnetic configuration should 3D topological spin textures-hopfions-are predicted in helimagnetic systems but are not experimentally confirmed thus far. By utilizing an external magnetic field and electric current in the present study, 3D topological spin textures are realized, including fractional hopfions with nonzero topological index, in a skyrmion-hosting helimagnet FeGe. Microsecond current pulses are employed to control the dynamics of the expansion and contraction of a bundle composed of a skyrmion and a fractional hopfion, as well as its current-driven Hall motion. This research approach has demonstrated the novel electromagnetic properties of fractional hopfions and their ensembles in helimagnetic systems.

Introduction
Topology is a branch of mathematics that characterizes objects using topological indices to describe their invariant properties under continuous deformation. [1] The concept of topology has been extended to condensed matter physics to reconsider electronic states [2][3][4] in both momentum and real space, and thus to explain novel quantum phenomena in actual materials. [5][6][7][8] Nanometer-scale spin textures, such as magnetic skyrmions or antiskyrmions possessing a topological index, have recently attracted considerable attention in condensed matter physics and spintronics, [9][10][11][12][13][14][15] particularly after the discovery of vortex-like skyrmions in a helimagnet MnSi. [10] The reason for this interest is the emergent electromagnetic response being be detectable by Lorentz transmission electron microscopy (TEM) (Figure 1c, simulated image). Although such a 2D magnetic structure and the related B-field maps in the XY-plane are the same for a hopfion and a skyrmionium (Figure 1g-i), they are topologically distinct. One can define by the iso-spin curve (ISC) on which the spin direction is the same. The hopfion has a characteristic 3D structure and linked ISCs between any two specific spin directions as a manifestation of its integer Hopf index, while the skyrmionium does not have any linked ISCs and only shows a 2D structure in the XY-plane with a rather uniform magnetization along the z-axis. However, the deformed skyrmionium (Figure 1d,e) with a nonzero and noninteger Hopf index Q H shows linked ISCs for some spin directions (Figure 1f). Thus, the deformation leads to a 3D topological object with partial linking defined as a fractional hopfion. These differences of spin textures between the fractional hopfion and skyrmionium can be directly discerned from the oblique projections at a tilt angle to the z-axis in real space although their projections in the XY-plane are the same. Both phase images (Figure 1j,l) and B-field maps (Figure 1k,m) at a 30° tilt for the hopfion (Figure 1j,k) are distinct from those for a skyrmionium (Figure 1l,m). Meanwhile, the topological distinction of the hopfion from the skyrmionium can be revealed by their dynamics upon electric current stimulations; the Hall motions emerge for the fractional hopfion, while they should be absent for the skyrmionium (see details in Figure S1 in the Supporting Information). Here, we report the realization of fractional hopfions with nonzero topological index, their bound state with skyrmion and skyrmion strings, and their dynamics in the prototype skyrmion-hosting helimagnet FeGe. [36] To generate such 3D spin textures and observe their dynamics, a field cooling (FC) procedure from a room temperature (>T C ≈ 278 K) to 100 K was employed in Lorentz TEM observations (see the "Experimental Section"), except for the current-driven dynamics of spin textures observed at 120 K. The fractional hopfions and their ensembles have been created by flipping the external field or flowing a pulsed current. Stretching and compression of the bundle of hopfions and skyrmions by the electric current, as well as the current-driven Hall motion of the bundles, have been demonstrated. The experimental observations agree well with micromagnetic simulations. . g-i) The 3D spin map of g) a skyrmionium with a zero topological index, h) its cross section, and i) ISCs for S = (−1, 0, 0) (cyan) and S = (+1, 0, 0) (red). j,l) The simulated phase images and k,m) the B-field maps for j,k) a hopfion and a l,m) a skyrmionium, respectively, by a 30° tilt. Dashed circles indicate the symmetric spin textures to the tilt axis, in which helicities are the same.

Figure 2a
shows a schematic of the B20-type cubic crystal structure for the helimagnet FeGe with the broken inversion symmetry. In FeGe, the skyrmion and its lattice state have been directly observed under an external magnetic field in terms of real-space imaging. The thermodynamic phase diagram ( Figure 2b) of a (110) FeGe plate with dimensions of 9 µm × 10 µm × 0.15 µm (see a scanning electron microscopy image in Figure S2 in the Supporting Information) reveals that the skyrmion lattice (SkL) phase can be stabilized in a narrow window of an external magnetic field H and temperature T close to T C . In contrast, the helical state at lower H (<100 mT) and the conical phase at higher H (100 mT < H < 500 mT) are thermodynamically stable at T < 200 K. Here, H was normally applied to the sample plane. To induce 3D topological spin textures such as hopfions, we chose the low-T conical state to avoid the formation of SkL, which appears under higher H after zero-field cooling (ZFC) or even at zero field after FC when the cooling field was appropriate to stabilize SkL. [37] The skyrmion clusters sometimes appear under a high external field and lower T such as 100 K. [19] Meanwhile, flipping an external field normally  applied to the 2D SkL plane or applying an in-plane magnetic field is expected to modify the topological spin textures and their topological charge. The FC from the random spin state is also supposed to induce metastable spin-ordered states and generate exotic spin textures, such as vortex-like 3D chiral bobbers, [38] skyrmion bags, [39] and skyrmion braids [40] in FeGe.
Figure 2e,f presents a phase image (Figure 2e)and the corresponding B-field (Figure 2f, magnetic induction derived from the underfocused (Figure 2c) and overfocused (Figure 2d) Lorentz TEM images via transport of intensity equation (TIE)) for a 3D magnetic object projected on the XY-plane in a FeGe slab. The Lorentz TEM images were observed under 170 mT at 100 K. The B-field map viewed in the XY-plane reveals a double-ring spin textures; the spin helicity is opposite for the double rings; between the rings, the out-of-plane field directions (dark color) should be opposite, although the projected images in the XY-plane at a zero tilt angle cannot clarify the up or down direction. We then tilt the double-ring textures at the zero external field to clarify its 3D structure. A bound state of a skyrmion (bottom-left vortex-like domain in Figure 2i-l) and a hopfion-like object (similar to images shown in Figure 2c-f) have been observed upon flipping the external magnetic field H applied along the normal to the FeGe plate, after the FC process. First the sample was cooled from the room temperature (RT) to 100 K under a normal −100 mT field (H) to generate a metastable SkL (see Figure S2a in the Supporting Information) composed of skyrmions with a counterclockwise (CCW) helicity; then, by flipping the field from −100 to +220 mT at a rate of 2 mT s −1 , a conical domain was created with a wavevector q parallel to the external field direction (black color in Figure S2b in the Supporting Information) accompanied by a bundle of skyrmions and double-ring textures with opposite helicities. The magnified phase image derived from defocused Lorentz TEM images (Figure 2i,j) and the corresponding B-field map are presented in Figure 2k,l, respectively. With the further increase in the magnetic field up to 250 mT, an SkL ( Figure S2c, Supporting Information) appears again, while the skyrmion helicity is opposite to that observed at H = −100 mT. The reversal of the cooling field can also generate bundles of fractional hopfions and skyrmions (see Figure S2d-h, Supporting Information), indicating that such 3D topological spin textures depend less on the sign of the cooling field, but that the flip of the external field is crucial.
Regardless of the deformation, the topological nature of the double-ring spin textures in a bound state should be the same as that for the individual double-ring structure shown in  (Figure 2p). The fractional hopfion with interlinked spin textures is topologically distinct from the skyrmionium, which has a zero topological number, lacks links in its spin texture (Figure 1i), and does not show Hall motion upon the electric current excitation ( Figure S1d, Supporting Information). In contrast to the stable fractional hopfions created under a relatively high external field in a conical state, the skyrmionium, comprising a skyrmion and an antiskyrmion with an opposite polarity for core spins, should be energetically unstable at a high magnetic field and hence could barely be detected by Lorentz TEM with an exposure time in the order of a millisecond. Meanwhile, our detailed calculations for the fractional hopfion with a systematically varying external field reveal that the topological charge of the hopfion cannot approach an integer value in the FeGe slab. The formation of an ideal torus-like hopfion (the closed loop of a skyrmion string shown in Figure 1a) usually requires the strong perpendicular magnetic anisotropy on top and bottom surfaces of the confined nanometric helimagnet, [35] thus, it might be difficult to observe the ideal hopfion in the FeGe slab without surface anisotropy.

In-Plane Field-Induced Motions of Hopfion Ensembles
Besides the procedures of flipping the external magnetic field, the oblique-field cooling process is more effective in manipulating and driving the bundle of fractional hopfions and skyrmions in addition to the topological state transformation accompanied by the flipping of the direction of skyrmion strings via the conical state ( Figure S3, Supporting Information) and exotic 3D spin textures ( Figure S4, Supporting Information), as detailed in the Supporting Information. To avoid the skyrmion formation in FeGe, a relatively high cooling field was employed. We first applied a 200 mT field, tilted at 15° from the normal to the sample plane. Subsequently, we cooled the sample from RT to 100 K. Upon gradually increasing the external field to 250 mT, the hopfion-like textures bound with several vortices and a skyrmion appear (Figure 3a,b). The observed spin textures indicated by orange dashed lines in Figure 3a are similar to the fractional hopfion texture, as shown in Figures 1c and 2e, respectively. Notably, the observed bundles are topologically different from skyrmion clusters, such as skyrmion bags [39] and skyrmion braids: [40] the bundles comprise a fractional hopfion and skyrmion strings, while the skyrmion bags and braids are composed of only several skyrmions. Such 3D spin textures have been evidenced by the external field-and current-driven dynamics as in the following, which differ from 2D spin texture's dynamics, such as current-driven motions of skyrmions and skyrmioniums. In addition to the magnetically created bundle of skyrmions and fractional hopfions, the large extension of the bundle (Figure 3c Figure 3d) can also move almost along the field direction ( Figure 3e). When we reverse the in-plane field while keeping its magnitude constant, the bundle (Figure 3f) tends to move in the opposite way ( Figure 3g).

Asymmetric Hall Motions of a Hopfion Ensemble upon Directional Current Excitations
To elucidate the bundle' texture, we propose a model of a bound state composed of a cylindrical skyrmion, deformed skyrmion strings [41] and a fractional hopfion (Figure 4a) and simulated the Lorentz TEM image in the XY-plane (Figure 4b). Figure 4c-e shows the 3D magnetization map (Figure 4c), and slices at the top (Figure 4d) and bottom surfaces (Figure 4e). Interestingly, such 3D textures have been realized in the conical state (appearing under 100 mT at 120 K) by flowing a microsecond current pulse (pulse width = 0.1 ms, current density = 3.5 × 10 9 A m −2 ) through a FeGe-based microdevice (the left panel of Figure 4f). Both the Lorentz TEM image (the right panel of Figure 4f; the defocus distance is −0.3 mm) and its B-field map (Figure 4g) agree well with the simulations (Figure 4b,e). Then, we examine the dynamics of the bundle of a fractional hopfions, a skyrmion and skyrmion strings with sequential current-pulse stimulations. Figure 4h-l shows Hall motions accompanied by stretching dynamics for the bundle of skyrmions and hopfions (Figure 4h). The initial state is a conical state accompanied by a bundle (see the overfocused Lorentz TEM image in Figure 4h). The Lorentz TEM image is in accordance with a simulated Lorentz TEM configuration for a bound state of skyrmions and hopfions ( Figure 4b). Notably, the underlying black band originates from the bend contour, which did not change during the sequential Lorentz TEM observations. The Lorentz TEM images clearly reveal that the bundle shows Hall motions for the current above 7 mA (current density = 3.5 × 10 9 A m −2 ). This effect can be seen in Figure 4i after applying a single 7.5 mA (current density = 3.75 × 10 9 A m −2 ) pulse (the current direction is indicated by red dashed arrows). Upon further increasing the current amplitude while maintaining its flow direction and pulse duration constant, the hopfion bundle extends and moves away from the current-flow direction (see Movie S1 in the Supporting Information). When we reverse the current direction while keeping the current duration constant, the bundle (Figure 4j) starts to move by extending the in-plane skyrmion strings almost along the current direction at the current higher than −6 mA (Figure 4k; current density = 3 × 10 9 A m −2 ).  sign here means the current flows from the bottom left (skyrmion site) to the top right (hopfion site)) and relatively slow motions of the bundle under negative current stimulations. The current-driven dynamics of the bundle clearly confirmed that the observed 3D textures are topologically distinct from the skyrmionium which should show no Hall motion with the current flow, agreeing well with our micromagnetic simulations, as shown in Figures S1 and S5 (Supporting Information).

Elastic Deformations of Hopfion Ensembles upon Electric Currents
In addition to the Hall motion of the 3D spin textures under electric current flow, elastic deformations (stretching and compression) of the bundles of fractional hopfions and skyrmions emerge under microsecond current-pulse stimulations. Sequential Lorentz TEM observations (Figure 5a-h) reveal that by varying the strength and direction of the current we can extend or compress the skyrmion strings in the bundle composed of the skyrmion and hopfion and that the deformational motions of the hopfion occur via the spin transfer torque. [31][32][33]42] The experimental observations for the currentinduced stretching and compression (see Movies S2 and S3 in Supporting Information) of the hopfion-containing bundle can be reproduced by micromagnetic simulations (Figure S5a-c, Supporting Information). Figure 5i shows the effect of current on the relative bundle length d − d 0 , where d 0 is the initial distance between the skyrmion center and hopfion center; the variation of d − d 0 with the current density is nonlinear both in stretching and compression processes. In contrast to the motion of hopfion bunch accompanying extending and shortening skyrmion strings, the bottom skyrmion in the bundle is pinned. Although the extension and compression of the hopfion bundles are observed almost along the current flow direction (i.e., with a small Hall angle) as the skyrmion is pinned, the bundles exhibit longitudinal and transversal movements in the skyrmion's depinned state. They qualitatively agree with the numerical simulations of hopfion dynamics under electric current. [31][32][33]

Conclusion
Metastable 3D spin textures, such as fractional hopfion and its bound states with skyrmions, were realized in the helimagnet FeGe. Their dynamics, driven by the external magnetic field and pulsed electric current, present novel phenomena of the  bound states of fractional hopfions and skyrmions, such as the current-induced extension and compression of the bundle as well as the asymmetric Hall motion with respect to the current direction. These dynamics differ from the behavior of a skyrmion and its cluster under electric current excitation, [19] indicating much richer phenomena realized in 3D spin textures as compared with 2D ones.

Experimental Section
Sample Preparations, Lorentz TEM Observations, and Analyses: A single crystal of FeGe was grown by chemical vapor transport, [37] and the (110) plate was thinned from bulk FeGe to ≈150 nm by using a focused ion beam (FIB) system (Hitachi NB5000). Various 3D spin textures (Figures 2-5; Figures S2-S4, Supporting Information) were observed using two transmission electron microscopes, JEM-2800 (JEOL) and Talos F200X (Thermo Fisher Scientific), equipped with a liquid nitrogen holder (Gatan 636) and electric feedthrough holder (HC3500, Gatan) to perform cryogenic Lorentz TEM observations under an external magnetic field and electric current. The external magnetic field was induced by the electric current flowing through the magnetic objective lens of the microscope. The in-plane field was generated by tilting the sample. A source measure unit (Keithley 2612A) supplied the current pulses. A current of 1 mA corresponds to a current density of 5 × 10 8 A m −2 . The Joule heating from applied current was weak and did not affect the spin texture dynamics, as verified in the previous study. [19] The B-field (magnetic induction) maps were obtained by solving the transport-ofintensity equation (TIE) for Lorentz TEM images [43] Here, I,ϕ, λ, e, ℏ, t, and n are the electron beam intensity, the phase of electron wave, electron wavelength, charge, Planck's constant, sample thickness, and unit vector perpendicular to the sample surface, respectively.
Micromagnetic Simulations: Micromagnetic simulations were performed using the open-source package MuMax3 [44] Here A is the exchange stiffness, m is the normalized magnetization, D is the isotropic bulk DMI constant, H is the external magnetic field, and B d is the demagnetizing field. The saturation magnetization M s = 384 kA m −1 was obtained from the literature. [37] The A/D ratio was fixed to 5.6 to match the measured period of helical stripes d = 4πA/D = 70 nm. The DMI constant D and the stiffness A were set to 0.5 mJ m −2 and 2.79 mJ m −1 , respectively, to account for the experimentally observed disappearance of skyrmions when the external magnetic field was raised to ≈400 mT. The cell size was set to 4 × 4 × 4 nm 3 , and the sample size was 1 µm × 1 µm × 160 nm. The total energy of all simulated spin textures was minimized using the conjugate gradient method based on the Runge-Kutta solver (RK45), built in MuMax3. [44] The fractional hopfion was simulated under a 100 mT magnetic field without any magnetic anisotropy. Lorentz TEM images and the corresponding magnetic field maps were simulated with the opensource package PyLorentz. [45,46] The fractional hopfion bundles were simulated as follows. First a skyrmionium was generated by minimizing the energy of a random spin state at 100 mT. Then, a uniaxial anisotropy of 8 × 10 5 J m −3 , directed