Abstract
Characterizing magnetic structures down to atomic dimensions is central to the design and control of nanoscale magnetism in materials and devices. However, real-space visualization of magnetic fields at such dimensions has been extremely challenging. In recent years, atomic-resolution differential phase contrast scanning transmission electron microscopy (DPC STEM)1 has enabled direct imaging of electric field distribution even inside single atoms2. Here we show real-space visualization of magnetic field distribution inside antiferromagnetic haematite (α-Fe2O3) using atomic-resolution DPC STEM in a magnetic-field-free environment3. After removing the phase-shift component due to atomic electric fields and improving the signal-to-noise ratio by unit-cell averaging, real-space visualization of the intrinsic magnetic fields in α-Fe2O3 is realized. These results open a new possibility for real-space characterization of many magnetic structures.
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Data availability
The data supporting the findings of this study are available within the paper and its Supplementary Information. The raw DPC images after 25-image averaging but before kernel filtering are available at https://github.com/sigma-users/kernel-filter.
Code availability
The custom-designed code used for kernel image filtering is available at https://github.com/sigma-users/kernel-filter.
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Acknowledgements
This work was supported by the JST SENTAN grant number JPMJSN14A1, Japan. A part of this work was supported by the JSPS KAKENHI grant numbers 20H05659, 19H05788 and 17H06094. A part of this work was supported by the Research Hub for Advanced Nano Characterization, The University of Tokyo, under the support of ‘Nanotechnology Platform’ by MEXT, Japan (grant number JPMXP09A21UT0259). We also thank the Corporate Sponsored Research Program ‘Next generation electron microscopy’, School of Engineering, The University of Tokyo. T.S. acknowledges support from JST-PRESTO (JPMJPR21AA), JSPS KAKENHI grant number 20K15014, and the Kazato Research Foundation. This research was partly supported under the Discovery Projects funding scheme of the Australian Research Council (project numbers DP160102338 and FT190100619).
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Y.K., T.S. and N.S. designed the study and wrote the paper. Y.K. performed the STEM experiments and image analysis. T.S. performed image simulation, theoretical analysis and prepared the TEM specimen. S.D.F. and Y.I. contributed to the discussion and comments. N.S. directed the study.
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Extended data figures and tables
Extended Data Fig. 1 Kernel filter for subtracting the electric field signal component.
Schematic illustration of the kernel filter. In this filter, DPC signals of four nearby points (denoted 2, 4, 6 and 8) are averaged and then subtracted from that of the centre point (denoted 0). This operation is illustrated in the right schematic. The numbers in the schematic show the weighting factors at each point in this kernel filter. The distance between the points corresponds to the distance between the Fe atomic dumbbells. This filter subtracts the electric field component and reinforces the magnetic field component at the centre Fe atomic columns.
Extended Data Fig. 2 Comparison between experimental DPC(x, y) images obtained using different kernel filters.
a, Kernel filter (raw) for obtaining the same DPC images as the original DPC images. b, Kernel filter (E-field) for subtracting the magnetic field component. Therefore, the obtained DPC images should correspond to the electric field images. It is seen that the original DPC image in a and the electric field image in b appear to be identical. This is because the main signal component of the original DPC is the atomic electric field. c, Kernel filter (B-field) for subtracting the electric field component as already shown in Extended Data Fig. 1. d, Kernel filter (null) for subtracting both the electric and magnetic field components. In the field vector colour map, the field vectors are plotted for the beam-deflection angle range 0–0.6 mrad in the raw and E-field kernel filtered images, whereas those in the B-field and null kernel filtered images are plotted for the range 0–50 μrad. In the DPC(x, y) images, the grey scale beam-deflection angle ranges are shown on the colour bars. In the fast Fourier transforms (FFTs), the scale bar corresponds to (0.4 nm)−1. The intensity scales are the same in all the FFTs. The inset in the FFT of DPC(y) shows the weak double periodicity spots related to the antiferromagnetic structure.
Extended Data Fig. 3 The standard-error maps for raw, E-field-filtered and B-field-filtered DPC(x, y) images (after binning down to 128 × 128 px).
Here, the propagation of uncertainty from the raw DPC images are fully considered for the kernel-filtered images.
Extended Data Fig. 4 The average/standard error maps for B-field-filtered DPC(x, y) images.
After binning down to 128 × 128 px.
Extended Data Fig. 5 Nearby-unit-averaged B-field-filtered DPC images.
The kernels used to average nearby units are shown schematically in the right panels. The centre panel shows the corresponding simulated B-filtered DPC images including the effect of finite electron dose used in the experiment. In the field vector colour maps (left), the field vectors are plotted with beam-deflection angle ranges shown by the inset (top right). Note especially that the structural regularity that emerges in the experimental images is similar across regions further apart than the width of the averaging region. Scale bars are 1 nm.
Extended Data Fig. 6 Statistical image analysis of the high-S/N kernel-filtered DPC(x, y) images.
Plot of the averages and the standard errors of the A-site and B-site averaged beam-deflection angles for B-field, E-field and null-filtered high-S/N DPC images. The standard errors are shown as the error bars. The inset shows a magnified portion of the plot around (0, 0).
Extended Data Fig. 7 Amplitudes of Fourier components of the unit-cell-averaged DPC images.
Only the Fourier components corresponding to the double periodicity due to the magnetic structure are plotted. The amplitudes that are non-zero with statistical significance (the error bars are shown by ±2SE) are plotted in blue, whereas the remainder are plotted in red. The statistically significant non-zero amplitudes of Fourier components with the highest spatial frequency can be found at 6.83 nm−1 = 1/(1.46 Å), indicated by the vertical arrow. Thus, the unit-cell-averaged magnetic field reconstruction contains statistically reliable spatial frequency information up to around (1.46 Å)−1.
Extended Data Fig. 8 Spin orientations.
a–c, Red arrows show the \([\bar{1}\bar{1}20]\)- projected spin orientations assuming that they have the tilt angle θ0 to the (0001) plane and perpendicular to the \([\bar{1}\bar{1}20]\) (a), \([\bar{1}2\bar{1}0]\) (b) and \([2\bar{1}\bar{1}0]\) (c) directions. The orientation parameters used in equation (7) become sinτ = 1. In a, η = θ0; in b and c, \(\sin \,\tau =\sqrt{{\sin }^{2}{\theta }_{0}+\frac{1}{4}{\cos }^{2}{\theta }_{0}}\) and η = arctan(2tanθ0). In the present \([\bar{1}\bar{1}20]\) projection, b and c cannot be distinguished.
Extended Data Fig. 9 Simulated magnetic phase-shift images (infinite dose case) by systematically changing the out-of-plane component (in angle) of the spin direction from the (0001) basal plane.
a–e, The out-of-plane components are 0° (a), 10° (b), 20° (c), 30° (d) and 90° (e), for the antiferromagnetic domain shown in Extended Data Fig. 8b. It is seen that the magnetic phase-shift images sensitively change depending on the out-of-plane components of the spin direction. f–j, The out-of-plane components are 0° (f), 10° (g), 20° (h), 30° (i) and 90° (j), for the antiferromagnetic domain shown in Extended Data Fig. 8a. The best match with the experimental phase-shift image shown in Fig. 3 is c.
Extended Data Fig. 10 Comparison between simulated magnetic DPC image approximations and a simulated magnetic DPC image using the multislice method.
a, Simulated, approximate magnetic field vector colour map, in which a purely magnetic potential is placed at each Fe atomic position and averaged over a simulated probe profile as it would evolve in the absence of the sample (that is, without scattering). b, Simulated, approximate magnetic field vector colour map, in which a purely magnetic potential is placed at each Fe atomic position and averaged over the simulated probe profile resulting from assuming scattering from the electrostatic potential only. c, Residual electric field component calculated by the first term in equation (12). d, Simulated, approximate magnetic DPC image formed by adding c to the magnetic DPC image shown in b; that is, as per both terms in equation (12). e, Magnetic field vector colour map obtained by applying B-field kernel filter to the multislice image simulation (infinite dose version of Fig. 3e). All these images assume infinite dose imaging condition.
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Kohno, Y., Seki, T., Findlay, S.D. et al. Real-space visualization of intrinsic magnetic fields of an antiferromagnet. Nature 602, 234–239 (2022). https://doi.org/10.1038/s41586-021-04254-z
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DOI: https://doi.org/10.1038/s41586-021-04254-z
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