Abstract
Topological magnetic monopoles (TMMs), also known as hedgehogs or Bloch points, are three-dimensional (3D) non-local spin textures that are robust to thermal and quantum fluctuations due to the topology protection1,2,3,4. Although TMMs have been observed in skyrmion lattices1,5, spinor Bose–Einstein condensates6,7, chiral magnets8, vortex rings2,9 and vortex cores10, it has been difficult to directly measure the 3D magnetization vector field of TMMs and probe their interactions at the nanoscale. Here we report the creation of 138 stable TMMs at the specific sites of a ferromagnetic meta-lattice at room temperature. We further develop soft X-ray vector ptycho-tomography to determine the magnetization vector and emergent magnetic field of the TMMs with a 3D spatial resolution of 10 nm. This spatial resolution is comparable to the magnetic exchange length of transition metals11, enabling us to probe monopole–monopole interactions. We find that the TMM and anti-TMM pairs are separated by 18.3 ± 1.6 nm, while the TMM and TMM, and anti-TMM and anti-TMM pairs are stabilized at comparatively longer distances of 36.1 ± 2.4 nm and 43.1 ± 2.0 nm, respectively. We also observe virtual TMMs created by magnetic voids in the meta-lattice. This work demonstrates that ferromagnetic meta-lattices could be used as a platform to create and investigate the interactions and dynamics of TMMs. Furthermore, we expect that soft X-ray vector ptycho-tomography can be broadly applied to quantitatively image 3D vector fields in magnetic and anisotropic materials at the nanoscale.
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Data availability
All the experimental data are available at https://doi.org/10.5281/zenodo.5450910.
Code availability
The MATLAB source codes for the scalar and vector tomography reconstruction algorithms and data analysis used in this work are available at https://doi.org/10.5281/zenodo.5450910.
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Acknowledgements
We thank R. Dunin-Borkowski and J. E. Han for stimulating discussions and Y. Yuan and Y. Yang for help with data analysis. This work was primarily supported by STROBE: a National Science Foundation Science and Technology Center under award DMR1548924. J.M. and A.R. acknowledge support by the US Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering under award number DE-SC0010378 for the contribution to the development of vector ptycho-tomography. J.M. thanks partial support by the Army Research Office MURI Program under grant number W911NF-18-1-0431. M.M.M. and H.C.K. acknowledge partial support by the US Department of Energy, Office of Science, Basic Energy Sciences X-Ray Scattering Program Award DE-SC0002002 and DARPA TEE Award D18AC00017 for the data acquisition and analysis. Y.T. and J.Z. were supported by the US Department of Energy, Office of Basic Energy Sciences under grant number DE-SC0012190. E.I. acknowledges the College of Letters, Arts, and Sciences at UCCS for start-up support. Soft X-ray ptycho-tomography experiments were performed at COSMIC and used resources of the Advanced Light Source, which is a US Department of Energy Office of Science User Facility under contract number DE-AC02- 05CH11231. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
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Contributions
J.M. directed the project. M.M.M. suggested the sample. A.J.G, J.V.B., P.M., T.E.M., C.-T.L. and S.Y. synthesized and fabricated the sample. A.R., C.-T.L., Y.H.L., E.-E.C.S., S.A.R., X.L., C.S.B., R.M.K., A.J.G., J.R., H.O., Y.-S.Y., D.A.S., H.C.K., M.M.M. and J.M. planned and/or performed the experiments. M.P., A.R., S.J.O. and J.M. developed the scalar and vector tomography algorithms. A.R. and J.M. reconstructed the 3D magnetization vector field. A.R., E.I., J.Z., X.L. and J.M. analysed the data with input from M.M.M., Y.T., C.-T.L., W.L. and V.H.C. J.H., T.O., E.I. and J.M. discussed and/or conducted the atomistic simulations. A.R., J.M., E.I. and J.Z. wrote the manuscript with input from M.M.M., Y.T., C.-T.L., S.Y., E.-E.C.S. and T.E.M.
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Extended data
Extended Data Fig. 1 Magnetic hysteresis measurements of a Ni meta-lattice and a Ni thin film.
5 T field sweep measurements of the hysteresis loops of the Ni meta-lattice and Ni thin film at 3 K and 305 K, where the inset shows the magnified hysteresis loops at 305 K. The experiments were performed with a Quantum Design MPMS SQUID magnetometer, and a diamagnetic background subtraction was implemented by subtracting off the average of two linear fits to the data at high positive and negative fields, beyond the saturation of the samples. The thickness of the meta-lattice varies from 400 to 450 nm, measured by a scanning electron microscope. As the meta-lattice has an fcc structure (Extended Data Fig. 6), the effective thickness of Ni in the meta-lattice was estimated to be 110.5 nm by considering the fcc packing efficiency of 74%. As a comparison, a pure Ni thin film of 200 nm thick was characterized by the same experimental procedure. The hysteresis loops of the Ni meta-lattice and the Ni thin film show similar remanent magnetization and saturation magnetization at both 3 K and 305 K. The slight differences of the remanent magnetization and saturation magnetization between the Ni meta-lattice and the Ni thin film are due to two factors: i) the thickness of the meta-lattice varies from 400 to 450 nm; and ii) the experimental packing efficiency of the sample may deviate from the theoretical value of 74%.
Extended Data Fig. 2 Sample preparation.
a, b, Optical microscopy images of the meta-lattice sample, prepared by FIB milling. The sample was mounted on a 3-mm transmission electron microscopy grid and glued on a copper ring (b). c-f, Scanning electron microscopy images of the sample. The mounting geometry of the sample is important for the soft x-ray vector ptycho-tomography experiment with three in-place rotation angles. The meta-lattice sample was thinned to 150 nm by a FIB (f), allowing the sample to be tilted to high angles. g, X-ray absorption spectroscopy of the Ni meta-lattice sample (red curve). For a comparison, the x-ray absorption spectra of a pure Ni film (blue curve) and a NiO film (green curve) are adapted from ref. 41. The three grey arrows indicate that the L3 peak position of the meta-lattice agrees well with that of pure Ni, while the NiO L3 peak is shifted to a higher energy. The black arrow shows that the absorption coefficients of the meta-lattice are in good agreement with those of pure Ni in the energy range from 885 eV to 870 eVs, but NiO has smaller values due to sp-hybridization. The purple arrow indicates that the L2 peak of the meta-lattice is more consistent with that of pure Ni than of NiO.
Extended Data Fig. 3 Improvement of the ptychography reconstruction.
a, The ptychography reconstruction of a representative projection with a small number of corrupted diffraction patterns, where reconstruction artifacts are clearly visible. The corrupted diffraction patterns were resulted from detector readout malfunction or unstable x-ray flux. b, The same reconstructed projection after the removal of the corrupted diffraction patterns. c, The ptychography reconstruction of a representative high tilt projection, in which artefacts were induced by phase unwrapping. d, The same reconstructed projection after phase unwrapping was enforced in the reconstruction. Scale bar, 200 nm.
Extended Data Fig. 4 3D structural characterization of the ferromagnetic meta-lattice.
a-c, The experimentally reconstructed 3D electron density of the meta-lattice is oriented along the [100], [110] and [111] directions with red, yellow and blue representing high, medium and low density, respectively. d-f, The corresponding 2D power spectrum of the projections along the [100], [110] and [111] directions, in which the Bragg peaks are clearly visible. Scale bar, 200 nm.
Extended Data Fig. 5 Structural characterization of the sample with scanning transmission electron microscopy (STEM).
a, An annular dark-field STEM image of the meta-lattice, where the rectangle with dashed lines represents the reconstruction region by soft x-ray vector ptycho-tomography and the square with solid lines shows a more ordered region. The circle indicates some imperfections in the sample. Scale bar, 200 nm. b, 2D power spectrum of the STEM image, where the sharp Bragg peaks indicate that the meta-lattice is ordered. c, Histograms of the nearest-neighbour distances between the TMM and anti-TMM, TMM and TMM, anti-TMM and anti-TMM pairs in the more ordered region (square with solid lines in (a)), which is consistent with Fig. 3d-f, obtained from the region with some imperfections (rectangle with dashed lines in (a)).
Extended Data Fig. 6 Difference of a left- and a right-circularly polarized projection of the ferromagnetic meta-lattice.
a, b, Representative left- and right-circularly polarized projections, respectively. c, The difference of the left- and right-circularly polarized projections, showing the comparable charge and magnetic contrast of the meta-lattice in our experiment. The colour bars are in arbitrary units and the values of the colour bars are consistent in (a-c). Scale bar, 100 nm.
Extended Data Fig. 7 Quantification of the 3D spatial resolution of the vector reconstruction.
a-f, FSC for |mx|, |my|, |mz|, |mxy|, |mxz| and |myz|, respectively, where mx, my, and mz are the x-, y-, and z-component of the unnormalized magnetization vector field with \(|m_{xy}| = \sqrt {m_x^2 + m_y^2}\), \(|m_{xz}| = \sqrt {m_x^2 + m_z^2}\) and \(|m_{yz}| = \sqrt {m_y^2 + m_z^2}\). The FSC curves were calculated from two independent vector reconstructions of the meta-lattice. According to the criterion of FSC = 0.143 (dashed lines), a 3D spatial resolution of 10 nm was achieved with soft x-ray vector ptycho-tomography, which corresponds to a spatial frequency of 0.1 nm−1. The FSC values for |mz| are slightly smaller than 0.143 at some high spatial frequency because only a half of the projections were used to perform each 3D vector reconstruction (Methods). Three TMM and anti-TMM pairs distributed along the x- (g-i), y- (j-l) and z-axis (m-o) in the 3D vector reconstruction. The net topological charge of each pair was calculated to be Q = 0, while the topological charge of the TMM and anti-TMM in each pair was computed to be Q = +1 (red dot) and −1 (green dot), respectively. The distance between the red and green dot in each pair is 2 voxels with a voxel size of 5 nm, demonstrating that a spatial resolution of 10 nm was achieved along the x-, y- and z-axis.
Extended Data Fig. 8 The emergent magnetic field of real and virtual TMMs.
a, b, The emergent magnetic field of the TMM and anti-TMM shown in Fig. 2c, e in the main text, respectively. The vector plots indicate that the TMM and anti-TMM form a source and a sink of the emergent magnetic field, respectively. c, d, The emergent magnetic field of the virtual TMM and anti-TMM shown in Fig. 4a, b, respectively. The red and blue cones represent outflow and inflow of the emergent magnetic field, respectively, while the cone size indicates the total emergent flux through the facet. Note that while there is both outflow and inflow of the emergent magnetic field in each case, the net flow corresponds to a source and sink, respectively. The scale bars, 5 nm (a) and 15 nm (c).
Extended Data Fig. 9 Effects of the experimental errors and statistical fluctuations on the analysis of TMMs.
a-c, Histograms of the topological charges calculated from equation (1) after adding random angular fluctuations to the experimentally measured magnetization vectors with a standard deviation of 2° (a), 15° (b) and 20° (c). d-l, Histograms of the nearest-neighbour distances of the TMM and anti-TMM, TMM and TMM, anti-TMM and anti-TMM pairs for the angular fluctuation of 2° (d-f), 15° (g-i) and 20° (j-l), which are consistent with those without the introduction of the angular fluctuation (Fig. 3d-f).
Extended Data Fig. 10 Atomistic simulations using the experimental data as direct input.
Four 15 × 15 × 15 nm3 volumes were extracted from the ferromagnetic meta-lattice, containing two TMMs and two anti-TMMs. The atomistic spins were fixed on the outer boundary of each volume, while all the other spins were allowed to relax to an equilibrium configuration. After 50 ps, a stable TMM or anti-TMM formed in each volume with a topological charge matching the experimental value. a-d, Two stable TMMs (red dots) and two anti-TMMs (blue dots) after relaxation, respectively, which are consistent with the experimental results. With the atomistic spins fixed on four of the six surfaces of each volume, the two TMMs and two anti-TMMs remained stable inside the volumes (e-h). Scale bar, 5 Å.
Supplementary information
Supplementary Video 1
3D scalar (green) and vector (arrow) reconstructions of the ferromagnetic meta-lattice. The global view of the 3D magnetization vector field zooms in to show a TMM and anti-TMM pair (Fig. 3a), a TMM and TMM pair (Fig. 3b), and an anti-TMM and anti-TMM pair (Fig. 3c), where TMMs and anti-TMMs are indicated by red and blue dots, respectively. In each magnified view, the global field fades away and the local magnetization vector field around each topological monopole is shown by grey arrows. The field lines follow the emergent magnetic field.
Supplementary Video 2
3D spatial distribution of 68 TMMs (red dots) and 70 anti-TMMs (blue dots) in the ferromagnetic meta-lattice, where 8 virtual TMMs and 11 virtual anti-TMMs are labelled with red and blue blobs (triangulated surfaces), respectively. The silica nanospheres are rendered as grey isosurfaces.53
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Rana, A., Liao, CT., Iacocca, E. et al. Three-dimensional topological magnetic monopoles and their interactions in a ferromagnetic meta-lattice. Nat. Nanotechnol. 18, 227–232 (2023). https://doi.org/10.1038/s41565-022-01311-0
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DOI: https://doi.org/10.1038/s41565-022-01311-0
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