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Three-dimensional magnetization structures revealed with X-ray vector nanotomography


In soft ferromagnetic materials, the smoothly varying magnetization leads to the formation of fundamental patterns such as domains, vortices and domain walls1. These have been studied extensively in thin films of thicknesses up to around 200 nanometres, in which the magnetization is accessible with current transmission imaging methods that make use of electrons or soft X-rays. In thicker samples, however, in which the magnetization structure varies throughout the thickness and is intrinsically three dimensional, determining the complex magnetic structure directly still represents a challenge1,3. We have developed hard-X-ray vector nanotomography with which to determine the three-dimensional magnetic configuration at the nanoscale within micrometre-sized samples. We imaged the structure of the magnetization within a soft magnetic pillar of diameter 5 micrometres with a spatial resolution of 100 nanometres and, within the bulk, observed a complex magnetic configuration that consists of vortices and antivortices that form cross-tie walls and vortex walls along intersecting planes. At the intersections of these structures, magnetic singularities—Bloch points—occur. These were predicted more than fifty years ago4 but have so far not been directly observed. Here we image the three-dimensional magnetic structure in the vicinity of the Bloch points, which until now has been accessible only through micromagnetic simulations, and identify two possible magnetization configurations: a circulating magnetization structure5 and a twisted state that appears to correspond to an ‘anti-Bloch point’. Our imaging method enables the nanoscale study of topological magnetic structures6 in systems with sizes of the order of tens of micrometres. Knowledge of internal nanomagnetic textures is critical for understanding macroscopic magnetic properties and for designing bulk magnets for technological applications7.

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Figure 1: X-ray magnetic tomography.
Figure 2: Axial tomographic slice of the reconstructed magnetization vector field.
Figure 3: Details of the reconstructed magnetization.


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X-ray measurements were performed at the cSAXS beamline at the Swiss Light Source, Paul Scherrer Institut, Switzerland. We thank R. M. Galera for providing and performing magnetic characterizations of the GdCo2 nugget, and S. Stutz and E. Müller for the sample fabrication, as well as K. Metlov, A. Arrott and G. Hrkac for discussions. S.G. was funded by the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement number 708674.

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Authors and Affiliations



J.R. and L.J.H. conceived the project. C.D., M.G.-S., V.S., M.H. and J.R. designed the experiment. M.H. and J.R. modified the instrument for the 30° tilt measurement. C.D., M.G.-S., V.S., M.H. and J.R. performed the experiment. C.D. and M.G.-S. developed the vector tomography reconstruction algorithm. C.D. performed the data analysis (with support from M.G.-S. and V.S.). C.D., S.G. and J.R. interpreted the magnetic results. C.D., M.G.-S. and S.G. wrote the manuscript with contributions from all authors.

Corresponding authors

Correspondence to Claire Donnelly, Manuel Guizar-Sicairos or Sebastian Gliga.

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The authors declare no competing financial interests.

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Extended data figures and tables

Extended Data Figure 1 Transmission spectrum obtained with spectroscopic ptychography.

a, Absorption part of a ptychographic reconstruction of the pillar measured at 7.203 keV. Regions of air and material are highlighted by the blue and red boxes, respectively. These regions were used to obtain the transmission of the sample at different energies. Such images were measured for a range of energies between 7.203 keV and 7.282 keV in steps of 1.5 eV. b, The transmission spectrum obtained from the range of images is shown, and the difference in absorption across the absorption edge, Δedge, is indicated.

Extended Data Figure 2 Validation of the magnetic reconstruction.

Single polarization images for different angles about the rotation axis (0°, 30° and 60°) in the 30° tilt geometry are shown in the left-hand column. In the central column, XMCD projections obtained by taking the difference between images measured with circular left and right polarization are given, which are the integrals of the magnetic component along the path of the X-ray beam. The equivalent projections obtained from the reconstructed magnetic tomogram are given in the right-hand column. When we compare the images in the central and right columns, the contrast and projection of the magnetization match well. This good agreement provides a validation for the magnetic reconstruction.

Extended Data Figure 3 Structure of the magnetization surrounding the singularity shown in Fig. 3f.

a, View parallel to the plane of the domain wall shown in Fig. 3c; b, view perpendicular to the plane of the domain wall shown in Fig. 3c. The observed magnetic structure does not show a one-to-one correspondence to any of the Bloch point configurations in Fig. 3g–i. While circulating Bloch points have been shown to minimize the magnetostatic energy as result of their swirling magnetization, this is not the case for other Bloch point configurations, where the magnetization in the vicinity of the singularity is predicted to deform in order to minimize the magnetostatic energy. We therefore attribute the structure above to the redistribution of the magnetization in order to screen the magnetostatic charge associated with the monopole generated by the diverging magnetization at the Bloch point.

Extended Data Figure 4 Estimation of the spatial resolution obtained by resolving nearby structures.

ac, Streamlines showing the direction of the magnetization for three xz slices at heights of 0 nm (midway through the pillar), +20 nm and +40 nm. A vortex–antivortex pair draw closer together as the height increases until they merge and can barely be resolved. df, The absolute value of the magnetization is shown for the same areas as ac, where the cores of the vortex and antivortex are minima. gi, The line profiles along the dotted lines in df, where we show that we are able to resolve features in the vector field that are separated by approximately 126 nm (h), the structure of which can be seen clearly in b, whereas in i the two structures can no longer be resolved.

Extended Data Figure 5 Estimation of the spatial resolution using FSC.

ac, The spatial resolution of the single components of the magnetization vector— mx (a), my (b) and mz (c)—are found to be 195 nm, 250 nm and 196 nm, respectively. The spatial resolution of my is lower because of lower sampling resulting from the experimental geometry, as explained in the text. df, To estimate the spatial resolution of the vector field, the FSC values are calculated for (d), (e) and (f) to be 125 nm, 97 nm and 127 nm, respectively. The x–y and y–z planes exhibit lower spatial resolution, owing to the low sampling of my caused by the experimental geometry.

Extended Data Figure 6 The correspondence of the three-dimensional region in Fig. 3c with the axial slice of Fig. 2a.

a, From above; b, from the side. The core of vortex (i) is mapped with an isosurface in b. The relative positions of the slice in Fig. 3a and the three-dimensional region in Fig. 3c are given in c.

Supplementary information

Circulating Bloch point

The magnetic configuration in the vicinity of an energetically stable circulating Bloch point. The magnetic structure is shown in a radius of 125 nm. (AVI 11026 kb)

Twisted magnetic structure surrounding an anti-Bloch point

The magnetic configuration in the vicinity of an energetically unstable anti-Bloch point. The magnetisation is twisted to shield the Bloch point structure at length scales greater than the exchange length. The magnetic structure is shown in a radius of 125 nm. (AVI 10087 kb)

Intersection of a magnetic vortex with a vortex domain wall

The intersection of a magnetic vortex (purple-orange tube) with a vortex domain wall (white surface with a blue-red core). Across the intersection points the direction of the magnetisation in the core of the vortex (indicated by the colour) reverses (from red to blue, and orange to purple). At these points, magnetic singularities occur. (AVI 16733 kb)

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Donnelly, C., Guizar-Sicairos, M., Scagnoli, V. et al. Three-dimensional magnetization structures revealed with X-ray vector nanotomography. Nature 547, 328–331 (2017).

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