Intrinsic DMI-free skyrmion formation and robust dynamic behaviors in magnetic hemispherical shells

We performed finite-element micromagnetic simulations to examine the formation of skyrmions without intrinsic Dzyaloshinskii–Moriya interaction (DMI) in magnetic hemispherical shells. We found that curvature-induced DM-like interaction allows for further stabilization of skyrmions without the DMI in curved-geometry hemispherical shells for a specific range of uniaxial perpendicular magnetic anisotropy (PMA) constant Ku. The larger the curvature of the shell, the higher the Ku value required for the formation of the skyrmions. With well-stabilized skyrmions, we also found in-plane gyration modes and azimuthal spin-wave modes as well as an out-of-plane breathing mode, similarly to previously found modes for planar geometries. Furthermore, additional higher-frequency hybrid modes were observed due to coupling between the gyration and azimuthal modes. This work provides further physical insight into the static and dynamic properties of intrinsic DMI-free skyrmions formed in curved-geometry systems.


Results
Model geometry. The model system employed was half-spherical shells of t = 1 nm thickness for different outer diameters in the range of 2R = 25-100 nm. Since thin magnetic films can have different interfacial magnetic anisotropy depending on their thickness 34 , we also varied the uniaxial magnetic anisotropy constant K u , even for a single thickness of t = 1 nm, in the present micromagnetic simulations. Figure 1a shows an example of a hemispherical shell of 2R = 100 nm. The 3D volume of the model was discretized into tetrahedrons with an average node distance of 1 nm. We also discretized the curved surfaces into triangles of roughly equal area using Hierarchical Triangular Mesh 35 , thereby preventing numerical errors incurred by irregularities. For all hemispherical shells used in the simulations, the height (h), the distance to the outer top surface along the z-axis from the base (see Fig. 1a) was set to 99.9% of the given radii R in order to initially place the core of a single skyrmion at the center. This geometry does not change the overall static and dynamic characteristics of skyrmions.
For the sake of convenience in the analysis of simulation data obtained from the hemispherical shells, we converted the numerical data in the frame of Cartesian coordinates (x, y, z) to the corresponding data in the frame of spherical coordinates (e r , e θ , e φ ) on every local surface (see Fig. 1b). The unit vector of local magnetizations is expressed as (m r , m θ , m ϕ ) = (m · e r , m · e θ , m · e ϕ ) , where r is the radial distance, θ is the polar angle, and φ is the azimuthal angle in the frame of local spherical reference, as shown in Fig. 1b. For better visualizations of numerical data in our analysis, the static and dynamic data for the 3D curved system (e.g., the magnetization profiles and the skyrmion numbers of single skyrmions in hemispherical shells) are presented in unrolled 2D planar views according to the transformation formula X = θ sin θ x , Y = θ sin θ y , and R ′ = π 2 R , where (X, Y) and R' are the in-plane position in the equivalent disk and its radius in the frame of Cartesian coordinates, respectively. (1/4π)m · (∂ X m × ∂ Y m)dXdY with A the entire area in the unrolled disk geometry converted from the 3D hemispherical shape. Figure 2b shows the variation of S with K u . The topological textures of the vortex, transient state, skyrmion, and perpendicular magnetization states were classified by ~ 0.5, 0.5 < S < 1, ~ 1, and 0, respectively. Note that S for the skyrmion state is not exactly an integer as in infinite flat thin films, but rather is  www.nature.com/scientificreports/ close to 1 due to its confined geometry. The local magnetizations tilted near the edge boundaries cause a noninteger skyrmion number (i.e., S = 0.94-0.99 for K u = 0.23-0.27 MJ/m 3 ) even for the stabilized skyrmion state.

Spin textures including skyrmion formation without intrinsic DMI in hemispherical shells.
With increasing K u , S is continuously variable from the vortex to the skyrmion state, but is suddenly changed from ~ 1 (skyrmion) to 0 (uniform perpendicular magnetizations) (see Fig. 2b). Such clear transformation of the corresponding spin texture occur around very specific values of K u , such as 0.211, 0.224, and 0.274 MJ/m 3 . These characteristic K u values can differently affect an effective magnetic anisotropy term in competition with curvedgeometry-induced DM-like interaction for the formation of stable skyrmions even without DMI. Since the vortex state represents in-plane curling magnetization around the perpendicular magnetization at the core while the skyrmion state represents out-of-plane magnetizations in most areas, we can expect that the sign of the effective www.nature.com/scientificreports/ magnetic anisotropy constant K eff affects the corresponding spin textures with the help of the curved-geometryinduced DM-like interaction. Roughly considering in-plane dipolar interaction on the local surfaces in such thin shells, we assume K eff = K u − 1 2 µ 0 M 2 s . Since K eff < 0 and > 0 prefer in-plane and out-of-plane collinear spin ordering, respectively, we can readily expect that the value of K u = 0.2114 MJ/m 3 leads to K eff = 0 with a given value of M s = 580 kA/m for Co. The estimated value of K u = 0.2114 MJ/m 3 agrees well with the simulation result of spin-texture evolution according to K u . In the simulation result, for K u < 0.2114 MJ/m 3 , the local magnetizations favor alignment in the surface plane (vortex states) except for the vortex core region, but for K u > 0.2114 MJ/ m 3 , they start to prefer alignment in the local surface normal (transient state, skyrmion, and perpendicular magnetization states).
For further details on the classified spin textures, we also plotted the three different magnetization components averaged over the entire area: radial <m r >, polar <m θ >, and azimuthal <m φ >, as shown in Fig. 2c, along with the exchange, magnetostatic, and magnetic anisotropy energy terms as a function of K u , as shown in Fig. 2d. The main features are described in detail below. In the vortex state (0 ≤ K u ≤ 0.21 MJ/m 3 ), the magnetization configuration consists of out-of-plane magnetizations at the core and in-plane curling magnetizations around the core. As K u increases to 0.21 MJ/m 3 , the size of the vortex core increases, because the higher the K u , the more favorable are the out-of-plane magnetizations. The vortex state is also modified by the magnitude of K u . In two regions of 0 < K u ≤ 0.06 MJ/m 3 and 0.16 ≤ K u ≤ 0.21 MJ/m 3 , <m θ > is larger than <m φ >, indicating dominant radial vortex structure. On the other hand, for 0.07 ≤ K u ≤ 0.14 MJ/m 3 , the circular component of local magnetizations around the core is more dominant, representing a circular vortex structure (see Supplementary Material S1 for details). This result is similar to those found for a 2D planar disk 37 where a circular vortex structure becomes a radial vortex as the interfacial DMI increases. The radial vortex state found in the hemispherical shells of nonintrinsic DMI suggests that such a curved geometry gives rise to effective DM-like interaction.
In the transient state from vortex to skyrmion (the two cases of K u = 0.215, 0.22 MJ/m 3 , as shown in Fig. 2a), the local magnetizations at the edge boundary are tilted from the in-plane to the perpendicular orientation with respect to the surface plane. In the skyrmion state (0.23 ≤ K u ≤ 0.27 MJ/m 3 ), the out-of-plane magnetizations are favorable in most areas, i.e., upward at the core and downward in the background. In this state, <m θ > is larger than <m φ >, indicating the existence of the Néel-type domain walls between the upward and downward local magnetizations. In the Néel-type domain walls, in-plane magnetization points towards the skyrmion core, suggesting that the curved-shell system allows for the curvature induced DM-like interaction of a positive DMI constant (see Supplementary Material S2 for details). Also, the size of the skyrmion core decreases with increasing K u . As the K u increases, the skyrmion width (the length of the domain wall) decreases, which makes the neighboring spins further non-collinear, thus making for higher exchange energy (see Fig. 2d, and Supplementary Material S2 for details). In the perpendicular magnetization state (K u ≥ 0.28 MJ/m 3 ), the higher K u values favor more perpendicular magnetizations at every local surface in the hemispherical shells, overcoming the in-plane dipolar interaction. Thus, when K u exceeds 0.28 MJ/m 3 , the skyrmion state totally disappears.
Phase diagram of spin textures. Next, in order to elucidate the curvature effect on the formation of skyrmions in hemispherical shells without intrinsic DMI, we constructed a phase diagram of the spin textures by calculating S versus both K u and 2R, as shown in Fig. 3a. The characteristic features of the phase diagram are very clear: the vortex, skyrmion, and uniform perpendicular magnetization states appeared in the green-, yellow-, and blue-color regions, respectively. The transient state also existed between the vortex and skyrmion states. It has been reported that in curved geometries, the effective anisotropy constant K eff can be modified by a curvature-induced anisotropy as

33
. Thus, in our system, the boundary between the transition and vortex states can be estimated, according to K u (K eff = 0) , to be the white dotted line shown in Fig. 3a. With decreasing 2R, the boundary line is slightly shifted to higher K u values owing to the curvatureinduced anisotropy 32,33 . Therefore, the vortex state is always formed in a range of K u below the boundary (white dashed line). The skyrmion and uniformly perpendicular magnetization states appeared separately depending on K u for the given 2R values. It is interesting that skyrmion formation occurs only in a narrow range of K u , i.e., ΔK u . As 2R becomes smaller, the K u value required for skyrmion formation becomes larger with a wider ΔK u range, while with increasing 2R, K u becomes smaller and ΔK u becomes narrower, as shown in Fig. 3a. Since PMA favors co-linear magnetizations while intrinsic DMI leads to non-collinear chiral magnetizations, the curvature of hemispherical shells plays the crucial role of curved-induced DM-like interaction in the formation of skyrmions in the DMI-free shell model. As 2R becomes smaller, the curvature-induced DM-like interaction becomes more dominant in the formation of skyrmions, even at higher K u values. In the hemispherical shell structure, the curvature-induced DM-like interaction plays a crucial role in the formation of skyrmions, just as intrinsic DMI does in planar-geometry structures 33 . Therefore, non-zero intrinsic DMI in such a curved system would further increase the stability of skyrmion formation.
In the phase diagram shown in Fig. 3a, there exists a threshold value K + th ( K − th ) above (below) which skyrmions no longer are formed in given shell diameters. Figure 3b further shows the 2R (curvature)-dependent K th for both the upper and lower limits. Since the estimation of K th is somewhat affected by the size of meshes used in finite-element simulations 38 , we kept a constant surface node distance of 1 nm at minimum for all of the shell diameters. The polynomial fitting to the numerical data (solid circles) in the phase diagram gives rise to K + th = 83 .757κ 2 − 0.100κ + 0.187 MJ/m 3 and K − th = 64.381κ 2 + 2.859κ + 0.188 MJ/m 3 , where κ (nm -1 ) is the curvature being defined as the reciprocal of radius, 1/R. It is worth noting that both K + th and K − th increase as the curvature increases, while at κ = 0, K + th and K − th both reach a single constant value of K u = 0.1875 (± 0.0012) MJ/m 3 . This means that skyrmions cannot be formed at κ = 0 that corresponds to a flat thin film. This also agrees well with the fact that curved-geometric confinements lead to DM-like contribution to the formation of skyrmions (even without intrinsic DMI) 26 and enhance the stability of skyrmion formation, as reported in one of our previous studies 27  www.nature.com/scientificreports/ Excited dynamic modes in skyrmion state. Next, in order to examine the dynamic characteristics of a single skyrmion formed in a hemispherical shell structure, we chose specific values of 2R = 100 nm and K u = 0.25 MJ/m 3 on the basis of the phase diagram of spin textures (see the 5th state from the left in Fig. 2a). To excite the dynamic modes of the skyrmion, we applied a sinc-function field H sinc to the whole shell structure along the x (in-plane) and z (out-of-plane) axes, where H sinc (t) = H 0 sin[2πf H (t − t 0 )]/(2πf H (t − t 0 )) with |H 0 | = 10 Oe, f H = 100 GHz, t 0 = 1 ns, and t = 100 ns. To obtain a better spectral resolution for all of the excited modes, we used a damping constant of α = 0.01. As reported in Ref. 39 , the damping constant can affect the width and amplitude of modes' resonance spectra, but does not change their resonance frequencies. Figure  To identify all of the excited modes, Fig. 4b displays the spatial distributions of the FFT power and -phase of the local m r oscillations at the corresponding resonance peaks identified in Fig. 4a. The power line profiles across the skyrmion center along the radial axis clearly inform the position and number of radial nodes n, as denoted by the black arrows, while the phase distributions inform the position and number of the azimuthal wave nodes, m. We note that one node nearby the shell edge in all of the excited modes (indicated by a gray arrow) was excluded in the numbering of the radial nodes n, because the node was caused by the shell edge effect and the curvature-induced DM-like interaction, not by the real eigenmode of the skyrmion. Recently, Mruczkiewicz et al. 13 formulated a classification of the fundamental eigenmodes of the magnetic skyrmions (e.g., gyration, azimuthal, and breathing modes) according to the radial (n) and azimuthal (m) indices. Accordingly, the excited modes shown in Fig. 4a can be indexed by both n and m: for the in-plane excitation eigenmodes, the two lower-frequency modes have n = 0, and the two higher ones have n = 1; each n = 0 and 1 has a phase change between − π and π, thus either a CCW or CW rotation sense, and thus is indexed as m = + 1 and m = − 1, www.nature.com/scientificreports/ respectively; or the out-of-plane excitation modes, the radial node number is either n = 0 or 1, but with the same azimuthal node of m = 0. To clarify the characteristic dynamic modes shown in Fig. 4, we again excited each mode at the corresponding resonance AC field, H(t) = H AC sin(2πf res t) , with f res the resonance frequency. Figure 5 plots the temporal variation of the spatial distributions of Δm r (t) = [m r (t) − m r (t = 0)] in one cycling period τ, where m r (t = 0) is the initial m r state. The characteristic motions of the six modes are clearly manifested in the respective temporal evolutions of their dynamic magnetization profiles. Also, the rotation senses of all of the modes are clearly depicted by the representative snapshot images over one cycle, as shown in Fig. 5. For the in-plane excitation modes, the first (0.05 GHz) and second (3.59 GHz) peaks both have n = 0, and thus the CCW and CW rotational senses are m = + 1 and − 1, respectively. The third (6.72 GHz) and fourth (9.36 GHz) peaks have an additional node in the radial direction (n = 1) with the CCW (m = + 1) and CW (m = − 1) rotational senses. Although the first and second peaks exhibit similar dynamic behaviors in the change of magnetization Δm r , their dynamic origins are completely different. The FFT power for the 0.05 GHz mode is localized inside the core, while that for the 3.59 GHz mode is spread throughout the background of the skyrmion state (see Supplementary Material S3 for details). Therefore, the 0.05 GHz mode is the gyrotropic mode of the skyrmion core, while the 3.59 GHz mode can be interpreted as an azimuthal spin-wave mode in the outer region of the skymion state 13 . The rotation sense of the gyrotropic motion is CCW for the up-core polarity of the skyrmion state. The rotation sense of the 3.59 GHz mode is opposite to that of the first mode, i.e., CW even with the up-core polarity. In addition, the radius of the core motion at f res = 0.05 GHz is ~ 8.5 nm, but is as small as ~ 0.077 nm at f res = 3.59 GHz, which is much smaller than at the 0.05 GHz mode. Thus, the first mode is the gyration mode (n = 0, m = + 1) and the second mode is the azimuthal mode (n = 0, m = − 1), as indexed in Fig. 4a.
The two remaining 6.72 and 9.36 GHz modes are both azimuthal spin-wave modes, with the radial quantization (n = 1) corresponding to the CCW (m = + 1) and CW (m = − 1) rotation senses, respectively. As can be seen from the boundary separated by the black dashed lines in Fig. 5, the rotation sense of magnetization fluctuation in the inner circle is the same as that in the outer ring region, but with a π phase difference. Here, we note that the core motion's frequencies coincide exactly with those of the azimuthal spin-wave modes (6.72 and 9.36 GHz). Thus, the rotation senses of the core motion at f res = 6.72 and 9.36 GHz are not the same as that of the gyrotropic mode, but rather is determined by the rotational sense of the azimuthal modes through their interaction with the given upward core, similarly to the vortex state reported in Ref. 40 . It has been reported that such frequency www.nature.com/scientificreports/ splitting between the CW (m < 0) and CCW (m > 0) azimuthal spin-wave modes in skyrmion-state nanodots is caused by DMI-induced spin-wave non-reciprocity 14 .
For the out-of-plane excitation modes, the lower-frequency resonance peak (0.89 GHz) corresponds to the breathing mode 15 of the skyrmion (n = 0, m = 0), where the skyrmion radially expands and shrinks with significant fluctuation around its core. The higher-frequency resonance peak (7.05 GHz) with weaker intensity can be viewed as a hybridization mode (n = 1, m = 0) that combines the breathing mode and the radial spin-wave mode 15 , where the additional node appears along the radial direction. The dynamic magnetization is oppositely oscillating between the core area and the shell edges, with π phase difference. The gyration/breathing mode observed in the sub-GHz region is a unique mode hitherto unseen in the uniform ferromagnetic state (see Supplementary Material S4). All of these modes represent a low-frequency (sub-GHz range) gyrotropic and breathing mode as well as high-frequency radially/azimuthally symmetric spin waves.

Discussion
Compared with 2D planar geometry, the hemispherical shell structure used in this study has an advantage in terms of choosing materials, because the present geometry structure allows for the formation of stable skyrmions without an intrinsic DMI. With the advance in the development of nanofabrication technologies, such halfspherical caps can be synthesized by depositing a few layered or multilayered films onto 2D arrays of curvedsurface dot templates (e.g., spherical polystyrene particles 41,42 and nanoporous alumina membranes 43 ). Perpendicular magnetic anisotropy in such films can also be realized using Co/Pt, Co/Ni, and Co/Pt interfaces. Thus, curvature-stabilized skyrmions can be realized in practical samples. Furthermore, magnetic imaging microscopy techniques such as Lorentz transmission electron microscopy, spin-polarized scanning tunneling microscopy, and high-resolution scanning transmission X-ray microscopy have been used to observe the real-space profiles of skyrmions 9 . Other electrical reading methods were also used to read and detect skyrmions in device applications via magnetic tunnel junction or skyrmion-induced Hall voltages 9 . In addition, it was reported that localized magnetic fields from a magnetic tip allow for direct writing and reading of isolated skyrmions 43 . With the help of advances in the development of 3D nanostructures [41][42][43] as well as high-frequency measurements 44 , thereby, the present results offer an alternative opportunity to design spintronic devices based on curved-geometry-induced skyrmion formation. Additionally, since skyrmions possess the eigenfrequencies of dynamic modes in the GHz range, they show promise for magnonics applications such as skyrmion-based tunable resonators 45 .
In summary, we explored curvature-induced DM-like interaction for stabilization of skyrmions in magnetic hemispherical shells and their intrinsic dynamic modes' excitation driven by AC-oscillating magnetic fields. A phase diagram showed the continuous transitions between three topological magnetization states (i.e., uniform ferromagnetic state, vortex, and skyrmion) according to different perpendicular anisotropy constants and shell curvatures. We made a classification of skyrmion eigenmodes into low-frequency gyrotropic modes and highfrequency spin-wave modes based on the mode symmetry and the number of nodes of the dynamical magnetization profiles in the radial and azimuthal directions. The skyrmion stability condition obtained from the phase diagram can serve as a benchmark for calculations of magnetic skyrmion dynamics in curved geometry as well as a guide to experimentalists for preparation of samples in the magnetic skyrmion ground state.