Sewing skyrmion and antiskyrmion by quadrupole of Bloch points

We report three-dimensional topological spin configurations including a novel form of skyrmion-antiskyrmion coupling in D2d chiral magnets. The skyrmion-antiskyrmion coupled string, consisting of skyrmions located in the near-surface layers and antiskyrmions in the interior of the nanostructured sample that are sewn together by emergent Bloch points, is a hybrid three-dimensional soliton solution resulting from the application of micromagnetic equations. We further provide experimental evidence of these 3D topological skyrmionic strings in a FeNiPdP chiral magnet with S4 symmetry. Our results demonstrate reversible topological magnetic transformations between skyrmion-antiskyrmion and skyrmion-bubble strings, which are mediated by the creation and annihilation of Bloch points.

Magnetic skyrmions are particle-like spin swirls and are characterized by a topological charge, which is a quantized integer that is conserved under continuous deformation of the spin configuration [1].Skyrmions have been proposed as potential information carriers in next-generation magnetic data storage and processing devices due to their emergent topology-related electron-magnetic properties [2].Magnetic Bloch points are typically considered magnetization singularities, where magnetization vanishes [3][4][5].In recent years, there has been increasing interest in the interaction between magnetic Bloch points and magnetic skyrmions [3][4][5].Theory and simulations have shown the important role of Bloch points in stabilizing Y-shaped skyrmion strings [4], biskyrmions [3], and bobbers [5].Understanding the interplay between magnetic Bloch points and skyrmions is therefore of great importance for the development of new materials and technologies based on these exotic magnetic structures [3][4][5].
Here, we show that the quadrupole of Bloch points can sew skyrmions and antiskyrmions in the depth dimension to form a new style of skyrmion-antiskyrmion coupling.Antiskyrmions are reported in few non-centrosymmetric D2d magnets [6,7].
In D2d magnets, complex multiple magnetic interactions also contribute to the stabilization of elliptic Bloch-skyrmions [8].The skyrmion-antiskyrmion coupling pair is not only a candidate for exploring fascinating particle-antiparticle interactions [9], but also a promising information carrier with zero skyrmion Hall effects for topological spintronic device applications [10].However, in D2d magnets, skyrmion and antiskyrmion reveal repulsive interaction with each other in two dimensions [11].
In this communication, we report the direct connection between Bloch points, skyrmions, and antiskyrmions by extending the topological magnetism in the third dimension of a FeNiPdP alloy with S4 symmetry [7], in a combination of 3D micromagnetic simulations with both Fresnel and electronic holography modes of Lorentz-transmission electronic microscopy (TEM) magnetic imaging [12].Our results not only build complete physical models of topological spin textures in the popular antiskyrmion-hosting magnet FeNiPdP, but also propose a new skyrmionantiskyrmion coupling style that is applicable in topological spintronic devices.
In 3D magnets, skyrmions form with spin twists along the depth orientation because of the magnetic dipole-dipole interactions or conical modulations [13,14].
Using the MuMax3 approach, we simulated the 3D skyrmionic textures in chiral magnets with both D2d interactions and uniaxial magnetic anisotropies Ku.It has been established two types of skyrmionic textures in D2d magnets [8]: skyrmion and antiskyrmion (Fig. S1).In a 136-nm-thick magnet, our simulations show a 3D skyrmion string with a depth-modulated Bloch-to-Néel spin-twisting from the middle layer to the surface layer (Fig. S2a), which is attributed to the magnetic dipole-dipole interaction.Similarly, setting a uniform antiskyrmion string as the initial state, we obtain a stable twisted 3D antiskyrmion string (Figs.S2b and S3).A pair of magnetic singularities (Fig. S2b) supported by D2d interaction maintains the surface layers and contributes to the positive integer topological charge  = 1/(4π) , which essentially counts how many times the magnetization vector field m at the position r = (x, y) winds around the unit sphere [15].However, the crosssectional configurations along the locations of singularities are not energetically supported from the view of magnetic dipole-dipole interaction (Fig. S3c).Setting the skyrmion and antiskyrmion strings as two end states in the nudged elastic band (NEB) simulation [16], we obtain that their mutual transformation must be through two stable 3D hybrid strings mediated by the emergence and annihilation of Bloch points (Fig. 1c).First, through the emergence of a dipole of Bloch points, the skyrmion string transforms to a skyrmion-bubble string (Fig. S4) [17], whose surface magnetizations maintain the skyrmion with Q = −1 while interior magnetizations turn to a bubble with Q = 0. Second, the skyrmion-bubble string transforms to the skyrmion-antiskyrmion string, whose topological reversals are mediated by the quadrupole of Bloch points (Fig. 1a).Finally, the topological skyrmion-antiskyrmion string to antiskyrmion string transformation is achieved through the annihilation of the quadrupole of Bloch points.It should be noted that Q is the product of vorticity and polarity (Fig. S1).Specifically, at negative magnetic fields, the Q values for skyrmions and antiskyrmions are 1 and −1, respectively, due to the polarity reversal compared to the values at positive magnetic fields.
The simulations reveal that the skyrmion-antiskyrmion string is more stable than the antiskyrmion string from the view of total energy (Fig. 1c) and can be stabilized in a broad field region (Fig. S5).By fixing the thickness t, the exchange interaction stiffness A, and the saturated magnetization Ms, we obtain the stable phase diagram of the skyrmion-antiskyrmion string as a function of Ku and D2d, as shown in Fig. 1d.
The skyrmion-antiskyrmion string can be stabilized in a broad region of magnetic parameters, and typically in chiral magnets with uniaxial anisotropies and relatively weak D2d interactions.The measured magnetic parameters (marked by a star symbol "★" in Fig. 1d) of (Fe0.67Ni0.3Pd0.07)3Palloy with S4 symmetry satisfy the stabilization region of the skyrmion-antiskyrmion string [7].We synthesized the single crystal of (Fe0.67Ni0.3Pd0.07)3Pusing the self-flux method.Macroscopic magnetization measurements (Fig. S6 and S7) indicate that the magnetic parameters of our samples closely resemble those of the targeted sample [7].Magnetic domains in 136-nm thick (Fe0.67Ni0.3Pd0.07)3Plamella are then experimentally observed using Lorenz-TEM, which images the average in-plane magnetic induction fields [12].For consistency, we further obtain the simulated magnetic induction fields of 3D spin configurations ).Similar phenomena have been observed and discussed in 3D dipolar skyrmions [14].Further analysis shows that the outer twisted antiskyrmion-like configurations are contributed by antiskyrmions Q = 1 in the interior layers (Fig. S11).Thus, these distinct characteristics of the hybrid skyrmion-antiskyrmion configurations can be applied for experimental verifications of the 3D topological spin configurations with depth-modulated spin twists in (Fe0.67Ni0.3Pd0.07)3Palloys.After carefully examining our Lorentz-TEM studies, we identify the theory-predicted weak Bloch-twisted skyrmion-like cores of the 3D skyrmion-antiskyrmion configurations in experiments, as shown in Fig. 1g.In experiments, we also identify the over-defocused Fresnel images of the skyrmion-antiskyrmion strings with black and white dots (Fig. S10), corresponding to the counterclockwise and clockwise circulations of Bloch-twisted skyrmion-like cores, respectively.These experimental signatures (Fig. 1g) are highly reproducible and fully consistent with micromagnetic simulations (Fig. 1e), providing unambiguous experimental proof of these 3D hybrid spin configurations.
Notably, the in-plane magnetic induction field mappings of skyrmionantiskyrmion strings are quite like that of antiskyrmion strings (Figs.S8 and S12) [7].
After carefully examining the detailed in-plane spin configurations, here we provide a protocol for distinguishing skyrmion-antiskyrmion strings from antiskyrmion strings using Lorentz-TEM.When integrating the in-plane magnetic induction fields in nearsurface layers of the antiskyrmion string, the magnetic induction field mappings in the top and bottom surface layers with Q = 1 both show two magnetic singularities, around which the in-plane magnetic induction fields in all layers through the depth all point to a same orientation (Fig. S12).Thus, the overall amplitude of the average inplane magnetic induction field near the magnetic singularities in the four corners of the rectangle shows maximum values for antiskyrmion strings (Fig. S13).In contrast, for the skyrmion-antiskyrmion string, the surface magnetic induction field in the corner of the rectangle shows a reversed orientation as that in the interior layers (Fig. S11), leading to the greatly reduced average in-plane magnetic induction field in the corners.Here, we define |Bxy-max| as the maximum amplitude of the average in-plane magnetic induction field in a line along the radial axis from the center (Fig. S13),  is the angle between the radial axis and +x axis.Thus, |Bxy-max| as a function of  could be taken as a protocol for distinguishing between the skyrmion-antiskyrmion and antiskyrmion strings.For the antiskyrmion string, the peak values of |Bxy-max| are in the four corners of the rectangle (Fig. S13), i.e.  45, 135, 225, 315 °.In contrast, the peak values of |Bxy-max| are in the central edges of the rectangle for the skyrmionantiskyrmion string (Fig. 1f), i.e.  0, 90, 180, 270 °.Such a protocol works for all shapes of strings (Figs.1f and S13c).We further extract the  vs. |Bxy-max| of representative spin configurations in experiments, as shown in Fig. 1h.We confirm the stabilization of skyrmion-antiskyrmion strings from the characteristics of the proposed protocol.Note that the magnetic field configurations retrieved from the TIE analysis of Fresnel images keep consistency for a defocused distance smaller than 300 μm (Fig. S14).Excessive defocusing in Fresnel imaging can introduce noticeable artificial distortions.Additionally, the average magnetic induction fields of skyrmionantiskyrmion strings are also determined using electronic holography magnetic imaging (Fig. S15).Both the Fresnel and electronic holography modes provide the stabilization proof of skyrmion-antiskyrmion strings.Moreover, the antiskyrmion string is hardly visible in our samples (Fig. S16), which is understood by the instability of high-energy antiskyrmion strings from the NEB simulation (Fig. 1c).
Fig. 2a shows the stabilized phase diagram as a function of temperature T and magnetic field B. In the (Fe0.67Ni0.3Pd0.07)3Plamella, we observe only stripe-toskyrmion transformations in the B-increasing process.Magnetic configurations of skyrmion strings (Fig. S17) are consistent with previous studies [7].Magnetic configurations in the B-decreasing process show different trends.At T 300 K, the ferromagnet (FM) prefers to transform to skyrmion-bubble strings (Fig. 2b) in the Bdecreasing process.Skyrmion-antiskyrmion strings are hardly observed at T 300 K.
In the high-temperature region near the Curie temperature Tc 380 K, the skyrmionantiskyrmion strings are the most stable phases in low-field regions from skyrmionbubble strings in the B-decreasing process.Importantly, the topological transformations between skyrmion-antiskyrmion and skyrmion-bubble strings are reversible by varying magnetic fields (Fig. 2c), suggesting the controlled injection and annihilation of Bloch points.Using micromagnetic simulations, such transformations are understood from the energy landscape.The energy difference shows that skyrmion-antiskyrmion and skyrmion-bubble strings are the more stable phases at low field and high field, respectively (Fig. S18).In our zero-temperature simulations, we cannot directly achieve such transformations, which agrees with the absence of skyrmion-antiskyrmion strings at low temperatures in experiments.At high temperatures, the significant thermal fluctuation Etherm can assist a high-energy state in overcoming an energy barrier to achieve reversible topological transformations between skyrmion-antiskyrmion and skyrmion-bubble strings (Fig. 2c).
Despite that skyrmion-antiskyrmion strings are spontaneous low-energy states at high temperatures, we can also obtain metastable skyrmion-antiskyrmion strings at low temperatures through a field-cooling process as shown in Fig. S19.Increasing the field, the skyrmion-antiskyrmion string can transform to the skyrmion-bubble string (Figs.S19b and S20).But the reversed transformation in the B-decreasing process is not supported for T < 320 K.We finally obtain the stabilization diagram as a function of temperature and field in the B-increasing process from initial skyrmionantiskyrmion strings at zero fields (Fig. S19c), which shows that the skyrmionantiskyrmion strings can stabilize at a broad temperature and field region as a metastable phase.Skyrmion-antiskyrmion strings have similar overall in-plane magnetizations as that of antiskyrmion strings [7].Thus, without consideration of 3D spin twists, skyrmion-antiskyrmion strings could be easily mistaken as antiskyrmion strings observed by Lorentz-TEM.Despite the similarity in overall average in-plane magnetic induction field mappings and Fresnel contrasts for antiskyrmion and skyrmion-antiskyrmion strings (Figs.1e and S13a), they have entirely different topologies and related topological magnetism (Fig. 1), such as skyrmion Hall effects [18].The skyrmion-antiskyrmion string with a greatly decreased averaged Q could provide a new strategy to realize Hall balance (see detail in Supplemental Section II) [10].
Thus, the observation of the skyrmionic textures and physical clarifications of magnetic nature in these popular antiskyrmion-hosting materials should redefine their potential applications in topological spintronic devices.Simulations based on measured magnetic parameters of (Fe0.67Ni0.3Pd0.07)3Pshow the lower energy of the skyrmion-antiskyrmion string and negligible energy barrier from the antiskyrmion string to the skyrmion-antiskyrmion string (Fig. 1c), which suggests that the skyrmion-antiskyrmion string could be achieved much easier than the antiskyrmion string in (Fe0.67Ni0.3Pd0.07)3Pand experimentally identified in our experiment.
Nevertheless, despite our ability to relax equilibrium antiskyrmion strings from initial configurations of antiskyrmions in zero-temperature simulations (Fig. S2b), the energy barrier for the transformation from antiskyrmion strings to skyrmion-bubble strings is too small to be reached in our NEB simulation.As a result, thermal fluctuations in realistic experimental conditions can readily lead to the annihilation of antiskyrmion strings.This explains why we consistently observed skyrmionantiskyrmion strings instead of antiskyrmion strings (Fig. S16).
In summary, in combination with 3D micromagnetic simulations, we have demonstrated the stabilization and observation of 3D topological spin textures including skyrmion-antiskyrmion strings in chiral materials with both D2d and Ku.We propose a protocol to distinguish between skyrmion-antiskyrmion and antiskyrmion strings using Lorentz-TEM.Our results suggest a strategy to design skyrmion-antiskyrmion coupled states through the emergent quadrupole of Bloch points, clarify 3D spin configurations of topological spin textures in the popular antiskyrmionhosting magnets, and provide more chances to develop potential topological spintronic devices.

𝛾𝐦 𝐇 𝐮 𝐯 • 𝛁 𝐦 𝐦 𝛽 𝐮 𝛼 𝐯 • 𝛁 𝐦 (S2)
Here,  is the Gilbert damping and  is the non-adiabatic parameter. is the total effective field.The strength of spin transfer torque is characterized by   with parameters  , μ ,  , and P are the Landé factor, Bohr magneton, electron charge, and the polarization rate, respectively.v is the drift velocity.

Fabrication of FeNiPdP microdevices:
The 136-nm thick FeNiPdP films were fabricated from a bulk single crystal using a standard lift-out method, with a focused ion beam and scanning electron microscopy dual beam system (Helios Nanolab 600i, FEI).

Fig. 1 .
Fig. 1. (Color online) Theoretical prediction and experimental observation of the skyrmionantiskyrmion string in D 2d magnets with perpendicular magnetic anisotropies.(a) 3D configuration of skyrmion-antiskyrmion string.White arrows indicate the locations of the quadrupole of Bloch points.(b) Depth z dependence of topological charge of the skyrmion-antiskyrmion string.(c) Topological transformation between the skyrmion string and the antiskyrmion string obtained using the NEB method.The energy is expressed by the differenced energy between each state and

(
Figs.S8 and S9).Figs.1e and 1gshow the representative simulated and experimental in-plane magnetic induction field Bxy mappings of skyrmion-antiskyrmion strings.The skyrmion-antiskyrmion strings can reveal various shapes, i.e., the rectangle with a long x-axis, the rectangle with a long y-axis, and the square.Moreover, the in-plane average magnetization mappings of skyrmion-antiskyrmion strings with rectangle shapes show the mixed characteristics of skyrmions and antiskyrmions, i.e., a weak Bloch-twisted skyrmion-like core encircled by twisted antiskyrmion-like configurations.Fresnel images of rectangle skyrmion-antiskyrmion strings also contain additional dotted contrasts in the center (Figs.S10 and S11).In contrast, Fresnel images of square skyrmion-antiskyrmion strings reveal no additional dotted contrasts in the center.When integrating the near-surface magnetic induction fields with Q = −1 of the rectangle skyrmion-antiskyrmion string, the Néel component cancels out but a weak residual Bloch-skyrmion-like configuration remains (Fig.S11

Fig. 2 .
Fig. 2. (Color online) (a) Magnetic phase diagram in the field-increasing process.We increased the field step by step with a field interval of 20 mT.(b) Magnetic phase diagram in the fielddecreasing process.We decreased the field step by step with a field interval of 20 mT.Here, mixed state means the coexisted skyrmion-antiskyrmion strings and stripe domains.(c) Reversible transformation between skyrmion-antiskyrmion and skyrmion-bubble strings by varying magnetic field at T = 350 K.The dots in (a) and (b) represent threshold magnetic fields between different phases.Defocused distance, −300 μm.
Fig. S2.Simulated 3D magnetic configurations of (a) the skyrmion string, (b) the antiskyrmion string, and (c) the skyrmion-antiskyrmion string.The dashed circles in b mark the locations of

Fig
Fig. S7.(a) Temperature dependence of saturated magnetization Ms and uniaxial magnetic anisotropy Ku.Ms is taken as the magnetization along the c-axis at 1.5 T. The uniaxial magnetic

Fig
Fig. S8.(a) Simulated 3D configurations and (b) average in-plane of magnetization m, demagnetization field B Dem , and total magnetic induction field B of the antiskyrmion string.

FigFig. S11 .Fig. S12 .
Fig. S9.(a) Simulated 3D configurations and (b) average in-plane of magnetization m, demagnetization field B Dem , and total magnetic induction field B of the antiskyrmionantiskyrmion string.
The energy is expressed by the differenced energy between each state and