In thin plates of the Sc-doped M-type hexaferrite BaFe12-x-0.05ScxMg0.05O19, Lorentz transmission electron microscopy revealed stripe domains with random reversals of wall helicity, a triangular skyrmion-bubble lattice with random chiralities, and multi-ring ('triple-ring') skyrmions with regular internal helicity reversals, all stabilized by magnetodipolar interactions rather than by the Dzyaloshinskii-Moriya interaction. The originating study left open the microscopic origin of the random helicity selection, the role of Bloch lines in mediating helicity changes, the exact conditions (film thickness, anisotropy, field range) stabilizing the triple-ring skyrmion, the dynamics of these textures under external perturbations, and their extension to other materials for helicity-based spintronics. Later work has identified a thermally activated, Bloch-line (kink-soliton) mechanism for skyrmion helicity reversal, imaged Bloch lines directly in the same material, demonstrated helicity switching by currents, electric fields, and ultrafast laser pulses in related centrosymmetric magnets, and realized skyrmion-bubble lattices in many other dipolar systems. What remains open is a quantitative microscopic theory of the statistics of helicity selection (random chirality and its short-range correlations in the bubble lattice, energetic versus topological-defect origins in the stripe state) together with a systematic mapping of the parameter window that stabilizes multi-ring skyrmion structures.

Background

A magnetic skyrmion is a nanoscale, particle-like, topologically nontrivial spin texture in which the magnetization wraps the unit sphere once; its integer topological charge protects it against smooth unwinding. A magnetic bubble is a cylindrical reversed domain in a film with perpendicular anisotropy, topologically equivalent to a skyrmion when its domain wall is of Bloch type. In a Bloch domain wall the magnetization rotates in the film plane as one crosses the wall; helicity (also called vector spin chirality in this context) is the sense of this in-plane rotation (clockwise or counterclockwise), and a Bloch line is a localized defect in a Bloch wall where the wall helicity switches. In noncentrosymmetric helimagnets such as MnSi, Fe1-xCoxSi, or FeGe, skyrmion lattices are stabilized by the Dzyaloshinskii-Moriya interaction (DMI), a relativistic exchange interaction whose sign fixes a single helicity. By contrast, in uniaxial ferromagnetic thin films the competing short-range exchange and long-range magnetodipolar interactions stabilize stripes and bubbles roughly 100 nanometers to 1 micrometer in size when the quality factor QQ (the ratio of uniaxial magnetocrystalline anisotropy energy to magnetostatic demagnetizing energy) is close to 1, and the two opposite helicities are then nearly degenerate: helicity becomes an extra internal degree of freedom. Lorentz transmission electron microscopy (LTEM) images in-plane magnetization via electron-beam deflection, and the transport-of-intensity equation (TIE) analysis of through-focus image series reconstructs quantitative in-plane magnetization maps.

Yu, Mostovoy, Tokunaga, Zhang, Kimoto, Matsui, Kaneko, Nagaosa, and Tokura (PNAS 109, 8856 (2012)) used Lorentz TEM and TIE analysis on a 30-nanometer (001) plate of the Sc-doped M-type hexaferrite BaFe12-x-0.05ScxMg0.05O19 (x = 0.16; the M-type hexaferrite is the magnetoplumbite-structure form of barium ferrite BaFe12O19) at room temperature. In zero field they observed a stripe domain state with numerous random helicity reversals between neighboring Bloch walls. Under a perpendicular field of 80-180 mT the stripes pinched off into a triangular lattice of skyrmion bubbles with random chiralities (random bright/dark LTEM contrast), in which short-range one-dimensional chirality correlations along the lattice basis vectors were noticed but not explained. Each bubble had an unusual triple-ring structure: three concentric rings whose winding directions alternate, i.e. two regular helicity reversals inside a single skyrmion, with topological charge -1 as in ordinary skyrmions. Tilting the field by only 1.5-2.3 degrees converted the skyrmions into so-called type II bubbles, pairs of counter-winding Bloch walls connected by Bloch lines, with topological charge 0. The authors explained qualitatively, via an extension of the Garel-Doniach Ginzburg-Landau theory of dipolar stripe and bubble phases to include in-plane magnetization, that the energy gain from helicity reversals decays exponentially with the ratio of stripe width to domain wall thickness and may therefore not dominate helicity selection; they conjectured that topological defects of the stripe pattern — disclinations (line defects at which the local stripe direction terminates or rotates abruptly) and grain boundaries (interfaces between differently oriented stripe domains) — control the observed stripe helicities, and that the random bubble chiralities reflect the absence of long-range interactions between the toroidal moments (the net in-plane swirling magnetization of a wall or bubble, whose sign is opposite for the two helicities) associated with wall helicity. Bloch lines, by contrast, were not observed in the zero-field stripe state and appear only in the tilted-field type II bubbles. The multi-ring skyrmion was qualitatively reproduced in numerical minimization of the extended model, but the conditions selecting it were not systematically mapped.

Problem Statement

For dipolar (centrosymmetric, DMI-free) ferromagnetic films exemplified by the Sc-doped M-type hexaferrite BaFe12-x-0.05ScxMg0.05O19 (x = 0.16): (1) establish the microscopic mechanism that selects wall helicity in the random-helicity stripe domain state and the random chiralities of the triangular skyrmion-bubble lattice — in particular determine quantitatively the relative roles of energy minimization (whose helicity gain decays exponentially with the stripe-width-to-wall-thickness ratio), thermally activated kink-soliton (Bloch-line pair) nucleation, and topological defects of the stripe pattern (disclinations, i.e. line defects at which the local stripe direction terminates or rotates abruptly, and grain boundaries, i.e. interfaces between differently oriented stripe domains), and account for the observed short-range one-dimensional chirality correlations along the bubble-lattice basis vectors; (2) determine the role of Bloch lines in mediating helicity changes, including why they are absent in the zero-field stripe state but appear in tilted-field type II bubbles; and (3) map the conditions — film thickness, magnetic anisotropy (quality factor QQ, the ratio of uniaxial anisotropy energy to magnetostatic demagnetizing energy), and magnetic-field range and direction — under which the multi-ring (triple-ring) skyrmion structure is stabilized over single-ring skyrmions, plain bubbles, and type II bubbles. Parts (1)–(3) together form the research target of this problem. The common objective is a predictive, quantitative account of helicity-reversal spin textures in dipolar ferromagnets, from which the controllability of the helicity degree of freedom (by field, current, electric field, or temperature) can be derived; the 2012 paper's complementary questions about the dynamics of these textures under external perturbations and their realization in other materials have since been addressed in the follow-up literature surveyed in refs 2-20 and are therefore treated there rather than as part of the open core.

References

  1. Yu, Xiuzhen; Mostovoy, Maxim; Tokunaga, Yusuke; Zhang, Weizhu; Kimoto, Koji; Matsui, Yoshio; Kaneko, Yoshio; Nagaosa, Naoto; Tokura, Yoshinori. 'Magnetic stripes and skyrmions with helicity reversals.' Proc. Natl. Acad. Sci. USA 109, 8856-8860 (2012). DOI: 10.1073/pnas.1118496109. https://doi.org/10.1073/pnas.1118496109
  2. Yu, X. Z.; Shibata, K.; Koshibae, W.; Tokunaga, Y.; Kaneko, Y.; Nagai, T.; Kimoto, K.; Taguchi, Y.; Nagaosa, N.; Tokura, Y. 'Thermally activated helicity reversals of skyrmions.' Phys. Rev. B 93, 134417 (2016). DOI: 10.1103/PhysRevB.93.134417. https://doi.org/10.1103/PhysRevB.93.134417
  3. Kurushima, K.; Tanaka, K.; Nakajima, H.; Mochizuki, M.; Mori, S. 'Microscopic magnetization distribution of Bloch lines in a uniaxial magnet.' J. Appl. Phys. 125, 053902 (2019). DOI: 10.1063/1.5042678. https://doi.org/10.1063/1.5042678
  4. Kotani, Atsuhiro; Harada, Ken; Malac, Marek; Nakajima, Hiroshi; Kurushima, Kosuke; Mori, Shigeo. 'Magnetic textures in a hexaferrite thin film and their response to magnetic fields revealed by phase microscopy.' Jpn. J. Appl. Phys. 58, 065004 (2019). DOI: 10.7567/1347-4065/ab1b87. https://doi.org/10.7567/1347-4065/ab1b87
  5. Kotani, Atsuhiro; Nakajima, Hiroshi; Kawaguchi, Atsushi; Fujibayashi, Yukihiro; Uchihashi, Kento; Shimada, Keiko; Harada, Ken; Mori, Shigeo. 'Magnetic bubbles in an M-type hexagonal ferrite observed by hollow-cone Foucault imaging and small-angle electron diffraction.' Jpn. J. Appl. Phys. 59, 095003 (2020). DOI: 10.35848/1347-4065/aba9e9. https://doi.org/10.35848/1347-4065/aba9e9
  6. Hou, Zhipeng; Ren, Weijun; Ding, Bei; Xu, Guizhou; Wang, Yue; Yang, Bing; Zhang, Qiang; Zhang, Ying; Liu, Enke; Xu, Feng; Wang, Wenhong; Wu, Guangheng; Zhang, Xixiang; Shen, Baogen; Zhang, Zhidong. 'Observation of Various and Spontaneous Magnetic Skyrmionic Bubbles at Room Temperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy.' Adv. Mater. 29, 1701144 (2017). DOI: 10.1002/adma.201701144. https://doi.org/10.1002/adma.201701144
  7. Zhang, Xichao; Xia, Jing; Zhou, Yan; Liu, Xiaoxi; Zhang, Han; Ezawa, Motohiko. 'Skyrmion dynamics in a frustrated ferromagnetic film and current-induced helicity locking-unlocking transition.' Nat. Commun. 8, 1717 (2017). DOI: 10.1038/s41467-017-01785-w. https://doi.org/10.1038/s41467-017-01785-w
  8. Montoya, S. A.; Couture, S.; Chess, J. J.; Lee, J. C. T.; Kent, N.; Henze, D.; Sinha, S. K.; Im, M. Y.; Kevan, S. D.; Fischer, P.; McMorran, B. J.; Lomakin, V.; Roy, S.; Fullerton, E. E. 'Tailoring magnetic energies to form dipole skyrmions and skyrmion lattices.' Phys. Rev. B 95, 024415 (2017). DOI: 10.1103/PhysRevB.95.024415. https://doi.org/10.1103/PhysRevB.95.024415
  9. Hou, Zhipeng; Zhang, Qiang; Zhang, Xichao; Xu, Guizhou; Xia, Jing; Ding, Bei; Li, Hang; Zhang, Senfu; Batra, Nitin M.; Costa, Pedro M. F. J.; Liu, Enke; Wu, Guangheng; Ezawa, Motohiko; Liu, Xiaoxi; Zhou, Yan; Zhang, Xixiang; Wang, Wenhong. 'Current-Induced Helicity Reversal of a Single Skyrmionic Bubble Chain in a Nanostructured Frustrated Magnet.' Adv. Mater. 32, 1904815 (2019). DOI: 10.1002/adma.201904815. https://doi.org/10.1002/adma.201904815
  10. Ding, Bei; Cui, Jie; Xu, Guizhou; Hou, Zhipeng; Li, Hang; Liu, Enke; Wu, Guangheng; Yao, Yuan; Wang, Wenhong. 'Manipulating Spin Chirality of Magnetic Skyrmion Bubbles by In-Plane Reversed Magnetic Fields in (Mn1-xNix)65Ga35 (x = 0.45) Magnet.' Phys. Rev. Applied 12, 054060 (2019). DOI: 10.1103/PhysRevApplied.12.054060. https://doi.org/10.1103/PhysRevApplied.12.054060
  11. Ding, Bei; Li, Zefang; Xu, Guizhou; Li, Hang; Hou, Zhipeng; Liu, Enke; Xi, Xuekui; Xu, Feng; Yao, Yuan; Wang, Wenhong. 'Observation of Magnetic Skyrmion Bubbles in a van der Waals Ferromagnet Fe3GeTe2.' Nano Lett. 20, 868-873 (2019). DOI: 10.1021/acs.nanolett.9b03453. https://doi.org/10.1021/acs.nanolett.9b03453
  12. Zhang, Junwei; Zhang, Xiaomin; Chen, Huanjian; Guang, Yao; Zeng, Xue; Yu, Guoqiang; Zhang, Senfu; Liu, Yizhou; Feng, Jiafeng; Zhao, Yuelei; Zhou, Yan; Qiu, Xuepeng; Han, Xiufeng; Peng, Yong; Zhang, Xixiang. 'Formation and magnetic-field stability of magnetic dipole skyrmions and bubbles in a ferrimagnet.' Appl. Phys. Lett. 116, 142404 (2020). DOI: 10.1063/1.5142562. https://doi.org/10.1063/1.5142562
  13. Yao, Yuan; Ding, Bei; Liang, Jinjing; Li, Hang; Shen, Xi; Yu, Richeng; Wang, Wenhong. 'Chirality flips of skyrmion bubbles.' Nat. Commun. 13, 5991 (2022). DOI: 10.1038/s41467-022-33700-3. https://doi.org/10.1038/s41467-022-33700-3
  14. Tang, Jin; Wu, Yaodong; Jiang, Jialiang; Kong, Lingyao; Wang, Shouguo; Tian, Mingliang; Du, Haifeng. 'Skyrmion-Bubble Bundles in an X-Type Sr2Co2Fe28O46 Hexaferrite above Room Temperature.' Adv. Mater. 35, 2306117 (2023). DOI: 10.1002/adma.202306117. https://doi.org/10.1002/adma.202306117 (preprint: arXiv:2311.17416, https://arxiv.org/abs/2311.17416)
  15. Wu, Kai; Zhao, Yuelei; Hao, Hongyuan; Yang, Sheng; Li, Shuang; Liu, Qingfang; Zhang, Senfu; Zhang, Xixiang; Akerman, Johan; Xi, Li; Zhang, Ying; Cai, Kaiming; Zhou, Yan. 'Topological transformation of synthetic ferromagnetic skyrmions: thermal assisted switching of helicity by spin-orbit torque.' Nat. Commun. 15, 10463 (2024). DOI: 10.1038/s41467-024-54851-5. https://doi.org/10.1038/s41467-024-54851-5
  16. El Gazzah, Mohamed; Yasin, Fehmi Sami; Jamaluddin, Sk; Bhandari, Hari; Regmi, Resham Babu; Yu, Xiuzhen; Ghimire, Nirmal J. 'Skyrmion Bubbles by Design in a Centrosymmetric Kagome Magnet.' arXiv:2504.19045 (2025). https://arxiv.org/abs/2504.19045
  17. Muto, Tatsuki; Mochizuki, Masahito. 'Switching magnetic texture via in-plane magnetic field in noncentrosymmetric dipolar magnets: From skyrmions to antiskyrmions and nontopological magnetic bubbles.' Phys. Rev. B 112, 104409 (2025). DOI: 10.1103/6ryj-gfw1. https://doi.org/10.1103/6ryj-gfw1
  18. Han, Myung-Geun; Thomsen, Joachim Dahl; Philbin, John P.; Mun, Junsik; Park, Eugene; Camino, Fernando; Dekanovsky, Lukas; Liu, Chuhang; Sofer, Zdenek; Narang, Prineha; Ross, Frances M.; Zhu, Yimei. 'Electric Field Control of Magnetic Skyrmion Helicity in a Centrosymmetric 2D van der Waals Magnet.' Nano Lett. 25, 5174-5180 (2025). DOI: 10.1021/acs.nanolett.4c06200. https://doi.org/10.1021/acs.nanolett.4c06200
  19. Liu, Can; Li, Zefang; Hu, Xuange; Guo, Jiangteng; Hu, Yue; Deng, Ying; Wang, Haixue; Ji, Shaozheng; Gao, Cuntao; Liu, Fang; Zhang, Huai; He, Wei; Guo, Tengyu; Chen, Shaohui; Lu, Peng-Han; Wu, Jinxiong; Hu, Yangfan; Liao, Zhi-Min; Liu, Jun-Ming; Yu, Dapeng; Hou, Zhipeng; Dunin-Borkowski, Rafal E.; Zhu, Yimei; Fu, Xuewen. 'Picosecond-scale coherent toggle switching of topological spin helicity.' Nat. Nanotechnol. 21, 655-662 (2026). DOI: 10.1038/s41565-026-02142-z. https://doi.org/10.1038/s41565-026-02142-z
  20. Rosch, Achim; Pfleiderer, Christian; Fiebig, Manfred; Yu, Xiuzhen; Tokura, Yoshinori; Nagaosa, Naoto. 'Topology, imaging, and emergent phenomena of spin textures.' Sci. Adv. 12, eaeg8564 (2026). DOI: 10.1126/sciadv.aeg8564. https://doi.org/10.1126/sciadv.aeg8564