# Experimental observation of vortex rings in a bulk magnet

## Abstract

Vortex rings are remarkably stable structures that occur in a large variety of systems, such as in turbulent gases (where they are at the origin of weather phenomena)1, fluids (with implications for biology)2, electromagnetic discharges3 and plasmas4. Although vortex rings have also been predicted to exist in ferromagnets5, they have not yet been observed. Using X-ray magnetic nanotomography6, we imaged three-dimensional structures forming closed vortex loops in a bulk micromagnet. The cross-section of these loops consists of a vortex–antivortex pair and, on the basis of magnetic vorticity (a quantity analogous to hydrodynamic vorticity), we identify these configurations as magnetic vortex rings. Although such structures have been predicted to exist as transient states in exchange ferromagnets5, the vortex rings we observe exist as static configurations, and we attribute their stability to the dipolar interaction. In addition, we observe stable vortex loops intersected by point singularities7 at which the magnetization within the vortex and antivortex cores reverses. We gain insight into the stability of these states through field and thermal equilibration protocols. The observation of stable magnetic vortex rings opens up possibilities for further studies of complex three-dimensional solitons in bulk magnets, enabling the development of applications based on three-dimensional magnetic structures.

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## Data availability

Experimental data and analysis codes used for this manuscript can be found at https://doi.org/10.5281/zenodo.4041745.

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## Acknowledgements

X-ray magnetic tomography measurements were performed at the cSAXS beamline at the Swiss Light Source, Paul Scherrer Institute (PSI), Switzerland, and X-ray microcrystallography measurements at the X06DA beamline at the Swiss Light Source, PSI, Switzerland. We thank A. Bogatyrëv for his careful reading of the manuscript and valuable remarks, R. Cowburn for discussions and V. Olieric for microcrystallography measurements. We thank R. M. Galera for providing and performing magnetic characterizations of the GdCo2 nugget, S. Stutz for the sample fabrication and E. Müller from the Electron Microscopy Facility at PSI for the focused ion beam preparation of the pillar samples. C.D. is supported by the Leverhulme Trust (ECF-2018-016), the Isaac Newton Trust (18-08) and the L’Oréal-UNESCO UK and Ireland Fellowship for Women in Science. S.G. was funded by the Swiss National Science Foundation, Spark project no. 190736. K.L.M. acknowledges the support of the Russian Science Foundation under project no. RSF 16-11-10349. N.R.C. was supported by EPSRC grant EP/P034616/1 and by a Simons Investigator Award.

## Author information

Authors

### Contributions

The study of topological magnetic features in three dimensions was conceived by S.G., C.D. and K.L.M., and originated from a larger project on 3D magnetic systems conceived by L.J.H. and J.R. C.D., M.G.-S., S.G., V.S., M.H. and J.R. performed the experiments. Magnetometry measurements of the material were performed by N.S.B. and V.S. C.D. performed the magnetic reconstruction with support from M.G.-S. and V.S. C.D. analysed the data and N.R.C. conceived the calculation of the magnetic vorticity. C.D., K.L.M., N.R.C. and S.G. interpreted the magnetic configuration. K.L.M. developed the analytical model. C.D., K.L.M., N.R.C. and S.G. wrote the manuscript with contributions from all authors.

### Corresponding authors

Correspondence to Claire Donnelly or Konstantin L. Metlov or Sebastian Gliga.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Paul Sutcliffe and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Detailed overview of the vortex ring with circulating magnetic vorticity (presented in Fig. 2), shown in successive slices through the loop.

The magnetization within each slice is represented by the streamlines. The colourscale in the top row indicates the $$\hat{x}$$ component of the magnetization, while the colour scale in the bottom row indicates the $$\hat{x}$$ component of the vorticity. The vorticity associated with the vortex structure extending throughout the pillar changes sign in slice d due to the presence of a Bloch point, while the vortex–antivortex pair conserves its vorticity throughout. In slices b and c, the magnetization forms a structure similar to that of a cross-tie wall, which dissolves as the pair unwinds, at slices a and d, resulting in a single vortex.

### Extended Data Fig. 2 Analytical models of vortex loops with different magnetization structures.

Top, middle and bottom rows: Magnetization, pre-images and vorticity distribution for the different 2+1 dimensional analytical models. The magnetization plots (top row) only include the projection of the magnetization onto the shown plane, while the rings correspond to the positions of the vortex and antivortex centres. The colour indicates the mz component of the magnetization. The pre-images are shown as volumes where the magnetization vectors deviate only slightly from certain directions di, indicated by the colour-coded arrows on each corresponding sphere. The opacity and colour on the vorticity plots indicate the magnitude of local vorticity vectors. The structure in c is comparable to the vortex rings in Fig. 2, while the structure in d is comparable to that in Fig. 3.

### Extended Data Fig. 3 Detailed overview of the magnetic state of the vortex loop containing Bloch points (presented in Fig. 3), shown in successive slices through the loop.

The magnetization within each slice is represented by the streamlines. The colour scale in the top row indicates the $$\hat{x}$$ component of the magnetization, while the colourscale in the bottom row indicates the $$\hat{x}$$ component of the vorticity. The vorticity along the vortex core reverses between slices b and c, while the vorticity along the antivortex core reverses between slices c and d. f, The white isosurface, plotted along with the vortex loop, corresponds to mx=0 and separates regions of mx=+1 and mx=−1, thus highlighting the presence of a complicated domain wall structure. The Bloch points are located at the intersection of the loop with this isosurface (locations indicated by the dashed circles).

### Extended Data Fig. 4 The vortex loop containing magnetization singularities (presented in Fig. 3) seen from multiple directions.

The vortex loop containing Bloch points is plotted using the isosurfaces mx= ± 1 (a,c) and pre-images (b,d). In a and b, the vortex loop and its pre-images have the same spatial orientation as in Fig. 3a. In c and d, the loop and pre-images are presented with the same orientation as in Fig. 3g.

### Extended Data Fig. 5 Effect of different field and thermal protocols on the presence and distribution of regions of high magnetic vorticity, and magnetization singularities.

a,c, Vorticity distribution following the application of a 7 T saturating field (a) and following saturation and field cooling (c). b, Regions of high divergence of the magnetic vorticity indicate the presence of Bloch points (red) and anti-Bloch points (blue) at remanence, following saturation. d, In the same way, singularities are identified after heating at 400 K and field cooling in a 7 T field. Noticeably fewer magnetic structures with high vorticity are present following the field-cooling procedure.

### Extended Data Fig. 6 A diffraction pattern from the GdCo2 pillar.

The substructure of the Bragg peaks, magnified in the inset to the right, indicates the polycrystalline nature of the material.

### Extended Data Fig. 7 Location of the central vortex following the two different protocols.

The position of the central vortex core is plotted using red and blue isosurfaces for the remanent magnetic structure after (red) the application of a 7 T magnetic field, and after (blue) the application of the field-cooling protocol. After both protocols, the vortex core occupies almost the same position.

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Donnelly, C., Metlov, K.L., Scagnoli, V. et al. Experimental observation of vortex rings in a bulk magnet. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-01057-3

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• ### Stable magnetic vortices

• Ankita Anirban

Nature Reviews Physics (2020)