Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Lorentz electron ptychography for imaging magnetic textures beyond the diffraction limit

Abstract

Nanoscale spin textures, especially magnetic skyrmions, have attracted intense interest as candidate high-density and power-efficient information carriers for spintronic devices1,2. Facilitating a deeper understanding of sub-hundred-nanometre to atomic-scale spin textures requires more advanced magnetic imaging techniques3,4,5. Here we demonstrate a Lorentz electron ptychography method that can enable high-resolution, high-sensitivity magnetic field imaging for widely available electron microscopes. The resolution of Lorentz electron ptychography is not limited by the usual diffraction limit of lens optics, but instead is determined by the maximum scattering angle at which a statistically meaningful dose can still be recorded—this can be an improvement of up to 2–6 times depending on the allowable dose. Using FeGe as the model system, we realize a more accurate magnetic field measurement of skyrmions with an improved spatial resolution and sensitivity by also correcting the probe-damping effect from the imaging optics via Lorentz electron ptychography. This allows us to directly resolve subtle internal structures of magnetic skyrmions near the skyrmion cores, boundaries and dislocations in an FeGe single crystal. Our study establishes a quantitative, high-resolution magnetic microscopy technique that can reveal nanoscale spin textures, especially magnetization discontinuities and topological defects in nanomagnets6. The technique’s high-dose efficiency should also make it well suited for the exploration of magnetic textures in electron radiation-sensitive materials such as organic or molecular magnets7.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: Workflow for LEP.
Fig. 2: Measured lateral magnetic induction field of a skyrmion lattice in FeGe.
Fig. 3: Skyrmion-lattice edge dislocation in FeGe.
Fig. 4: Quantification of field measurements from LEP.

Data availability

The full raw experimental data is available via Zenodo at https://doi.org/10.5281/zenodo.6684163 (ref. 47).

Code availability

The source code for LEP is available via Zenodo at https://doi.org/10.5281/zenodo.4659690 (ref. 48).

References

  1. Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nat. Nanotechnol. 8, 152–156 (2013).

    Article  CAS  Google Scholar 

  2. Mathur, N., Stolt, M. J. & Jin, S. Magnetic skyrmions in nanostructures of non-centrosymmetric materials. APL Mater. 7, 120703 (2019).

    Article  Google Scholar 

  3. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).

    Article  CAS  Google Scholar 

  4. Milde, P. et al. Unwinding of a skyrmion lattice by magnetic monopoles. Science 340, 1076–1080 (2013).

    Article  CAS  Google Scholar 

  5. Park, H. S. et al. Observation of the magnetic flux and three-dimensional structure of skyrmion lattices by electron holography. Nat. Nanotechnol. 9, 337–342 (2014).

    Article  CAS  Google Scholar 

  6. Braun, H.-B. Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons. Adv. Phys. 61, 1–116 (2012).

    Article  CAS  Google Scholar 

  7. Coronado, E. Molecular magnetism: from chemical design to spin control in molecules, materials and devices. Nat. Rev. Mater. 5, 87–104 (2020).

    Article  Google Scholar 

  8. Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).

    Article  CAS  Google Scholar 

  9. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).

    Article  Google Scholar 

  10. Shibata, K. et al. Large anisotropic deformation of skyrmions in strained crystal. Nat. Nanotechnol. 10, 589–592 (2015).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  12. Rana, A. et al. Direct observation of 3D topological spin textures and their interactions using soft X-ray vector ptychography. Preprint at https://doi.org/10.48550/arXiv.2104.12933 (2021).

  13. Bode, M. et al. Atomic spin structure of antiferromagnetic domain walls. Nat. Mater. 5, 477–481 (2006).

    Article  CAS  Google Scholar 

  14. McVitie, S. & Cushley, M. Quantitative Fresnel Lorentz microscopy and the transport of intensity equation. Ultramicroscopy 106, 423–431 (2006).

    Article  CAS  Google Scholar 

  15. Lichte, H. Performance limits of electron holography. Ultramicroscopy 108, 256–262 (2008).

    Article  CAS  Google Scholar 

  16. McVitie, S. et al. Aberration corrected Lorentz scanning transmission electron microscopy. Ultramicroscopy 152, 57–62 (2015).

    Article  CAS  Google Scholar 

  17. Tate, M. W. et al. High dynamic range pixel array detector for scanning transmission electron microscopy. Microsc. Microanal. 22, 237–249 (2016).

    Article  CAS  Google Scholar 

  18. Ophus, C. Four-dimensional scanning transmission electron microscopy (4D-STEM): from scanning nanodiffraction to ptychography and beyond. Microsc. Microanal. 25, 563–582 (2019).

    Article  CAS  Google Scholar 

  19. Chapman, J., Batson, P., Waddell, E. & Ferrier, R. The direct determination of magnetic domain wall profiles by differential phase contrast electron microscopy. Ultramicroscopy 3, 203–214 (1978).

    Article  CAS  Google Scholar 

  20. Nguyen, K. X. et al. Disentangling magnetic and grain contrast in polycrystalline FeGe thin films using four-dimensional Lorentz scanning transmission electron microscopy. Phys. Rev. Appl. 17, 034066 (2022).

    Article  CAS  Google Scholar 

  21. Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959).

    Article  Google Scholar 

  22. Yu, X. Z. et al. Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe. Nat. Mater. 10, 106–109 (2011).

    Article  CAS  Google Scholar 

  23. Kovács, A. et al. Mapping the magnetization fine structure of a lattice of Bloch-type skyrmions in an FeGe thin film. Appl. Phys. Lett. 111, 192410 (2017).

    Article  Google Scholar 

  24. McGrouther, D. et al. Internal structure of hexagonal skyrmion lattices in cubic helimagnets. New J. Phys. 18, 095004 (2016).

    Article  Google Scholar 

  25. Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).

    Article  CAS  Google Scholar 

  26. Milnor, J. W. Topology From the Differentiable Viewpoint (Press Of Virginia, 1965).

  27. Chen, Z. et al. Mixed-state electron ptychography enables sub-angstrom resolution imaging with picometer precision at low dose. Nat. Commun. 11, 2994 (2020).

    Article  CAS  Google Scholar 

  28. Suzuki, T. et al. Improvement of the accuracy of phase observation by modification of phase-shifting electron holography. Ultramicroscopy 118, 21–25 (2012).

    Article  CAS  Google Scholar 

  29. Boureau, V. et al. High-sensitivity mapping of magnetic induction fields with nanometer-scale resolution: comparison of off-axis electron holography and pixelated differential phase contrast. J. Phys. D: Appl. Phys. 54, 085001 (2020).

    Article  Google Scholar 

  30. Close, R., Chen, Z., Shibata, N. & Findlay, S. D. Towards quantitative, atomic-resolution reconstruction of the electrostatic potential via differential phase contrast using electrons. Ultramicroscopy 159, 124–137 (2015).

    Article  CAS  Google Scholar 

  31. Isaacson, M. Electron beam induced damage of organic solids: implications for analytical electron microscopy. Ultramicroscopy 4, 193–199 (1979).

    Article  CAS  Google Scholar 

  32. Thibault, P. & Menzel, A. Reconstructing state mixtures from diffraction measurements. Nature 494, 68–71 (2013).

    Article  CAS  Google Scholar 

  33. Chen, Z. et al. Electron ptychography achieves atomic-resolution limits set by lattice vibrations. Science 372, 826–831 (2021).

    Article  CAS  Google Scholar 

  34. Stolt, M. J., Sigelko, X., Mathur, N. & Jin, S. Chemical pressure stabilization of the cubic B20 structure in skyrmion hosting Fe1–xCoxGe alloys. Chem. Mater. 30, 1146–1154 (2018).

    Article  CAS  Google Scholar 

  35. Maiden, A. M. & Rodenburg, J. M. An improved ptychographical phase retrieval algorithm for diffractive imaging. Ultramicroscopy 109, 1256–1262 (2009).

    Article  CAS  Google Scholar 

  36. Rodenburg, J. M. & Bates, R. H. T. The theory of super-resolution electron microscopy via Wigner-distribution deconvolution. Philos. Trans. R Soc. A 339, 521–553 (1992).

    Google Scholar 

  37. Thibault, P. & Guizar-Sicairos, M. Maximum-likelihood refinement for coherent diffractive imaging. New J. Phys. 14, 063004 (2012).

    Article  Google Scholar 

  38. Odstrcil, M., Menzel, A. & Guizar-Sicairos, M. Iterative least-squares solver for generalized maximum-likelihood ptychography. Opt. Express 26, 3108–3123 (2018).

    Article  Google Scholar 

  39. Odstrcil, M. et al. Ptychographic coherent diffractive imaging with orthogonal probe relaxation. Opt. Express 24, 8360–8369 (2016).

    Article  CAS  Google Scholar 

  40. Reimer, L. Transmission Electron Microscopy (Springer, 1989).

  41. Lubk, A. et al. Nanoscale three-dimensional reconstruction of elastic and inelastic mean free path lengths by electron holographic tomography. Appl. Phys. Lett. 105, 173101 (2014).

    Article  Google Scholar 

  42. Egerton, R. F. & Cheng, S. C. Measurement of local thickness by electron energy-loss spectroscopy. Ultramicroscopy 21, 231–244 (1987).

    Article  Google Scholar 

  43. Bechhoefer, J. Curve fits in the presence of random and systematic error. Am. J. Phys. 68, 424–429 (2000).

    Article  Google Scholar 

  44. Song, D. et al. Quantification of magnetic surface and edge states in an FeGe nanostripe by off-axis electron holography. Phys. Rev. Lett. 120, 167204 (2018).

    Article  Google Scholar 

  45. Mochizuki, M. et al. Thermally driven ratchet motion of a skyrmion microcrystal and topological magnon Hall effect. Nat. Mater. 13, 241–246 (2014).

    Article  CAS  Google Scholar 

  46. Vansteenkiste, A. et al. The design and verification of MuMax3. AIP Adv. 4, 107133 (2014).

    Article  Google Scholar 

  47. Chen, Z. & Muller, D. A. Datasets for Lorentz electron ptychography for imaging magnetic textures beyond the diffraction limit. Zenodo https://doi.org/10.5281/zenodo.6684163 (2022).

  48. Chen, Z., Jiang, Y., Muller, D. A. & Odstrcil, M. PtychoShelves_EM, source code for multislice electron ptychography. Zenodo https://doi.org/10.5281/zenodo.4659690 (2021).

Download references

Acknowledgements

This work is supported by the DARPA TEE-D18AC00009. Z.C. was partly supported by the PARADIM Materials Innovation Platform in-house program by NSF grant DMR-2039380. M.J.S. and S.J. were supported by NSF ECCS-1609585. This work made use of the Cornell Center for Materials Research facility supported by NSF grant DMR-1719875.

Author information

Authors and Affiliations

Authors

Contributions

Z.C., D.A.M., G.D.F. and D.C.R. conceived the project. Z.C. performed the experiments and data analysis under the supervision of D.A.M. E.T. carried out the micromagnetic simulations. Y.J. contributed to the LEP. E.T. and K.X.N. performed the magnetic field calibration of the electron microscope. M.J.S. synthesized the FeGe single crystal under the supervision of S.J. Z.C. wrote the draft with revisions from D.A.M. All the authors discussed the results and implications throughout the investigation and have given approval to the final version of the manuscript.

Corresponding authors

Correspondence to Zhen Chen or David A. Muller.

Ethics declarations

Competing interests

Cornell University has licensed the EMPAD hardware to Thermo Scientific (D.M. and K.X.N. each receives 4.4% of the license fee).

Peer review

Peer review information

Nature Nanotechnology thanks Jianwei (John) Miao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

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

Extended data

Extended Data Fig. 1 Synthesized image modes from the 4D LSTEM dataset used in Fig. 1.

a, Center-of-mass (CoM) along horizontal direction (x-axis) ; b, CoM along vertical direction (y-axis); c, Annular dark-field (ADF) image; d, Thickness determined from the diffraction. Scale bar is 50 nm.

Extended Data Fig. 2 Discontinuity near skyrmion boundaries.

a, b, Magnitude and direction of lateral magnetic field of skyrmion lattice in FeGe, respectively. Scale bar is 50 nm. c, d, Magnitude (c) and direction (d) of the magnetic field along the blue dashed line marked on (a) and (b). e & f, Magnitude (e) and direction (f) of the magnetic field along the black dashed line marked on (a) and (b). In order to show the reversal of the magnetic field direction across the boundaries, the line for (d) and (f) is slightly away the skyrmion boundary. The inset in (e) is a cropped region of (a) with arrows indicating the local maxima along the skyrmion boundary.

Extended Data Fig. 3 Comparison of measured and simulated magnetization of skyrmion lattice.

a, Experimental measurements of magnetization vector map; b-d, Enlarged magnetization vector maps near the singular points labeled on (a). e-h, Micromagnetic simulations of magnetization of skyrmions lattice and singular points. Scale bar for (a) and (e) is 50 nm, for (b)-(d) and (f)-(h) is 2 nm.

Extended Data Fig. 4 Resolution improvements of ptychography compared to center-of-mass (CoM).

Magnetic field (Bx) from CoM (a) and ptychography (b). c, Line profiles from the position marked by the red line on (a) and (b). The profile from CoM has a further broadening of 12.4 nm (Gaussian, FWHM) compared to that from ptychography (pty_blur).

Extended Data Fig. 5 Simulations for sub-nanometer spatial resolution of ptychography.

a–c, Phase images from iCoM and e–g, from ptychography at different doses; d, The model phase structure used to generate diffraction patterns; h, Radial distribution function of the Fourier intensity of phase images in (a), (c), (e), (g). The model structure contains varying peak distances vertically and the arrows on (d) mark two rows with the distance of 5.2 nm. Scale bar is 50 nm.

Extended Data Fig. 6 Model phase image for ptychography simulations.

a, Original phase image for ptychography simulations generated from arrays of two-dimensional Gaussian functions; b, Field strength along horizontal direction from (a). c, d, Phase and field retrieved from ptychography with an illuminating dose of 1 e Å−2, respectively. e, f, Phase and field retrieved from ptychography with an illuminating dose of 10 e Å−2, respectively. Scale bar is 30 nm.

Supplementary information

Supplementary Information

Supplementary Figs. 1–8.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, Z., Turgut, E., Jiang, Y. et al. Lorentz electron ptychography for imaging magnetic textures beyond the diffraction limit. Nat. Nanotechnol. 17, 1165–1170 (2022). https://doi.org/10.1038/s41565-022-01224-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41565-022-01224-y

This article is cited by

Search

Quick links

Find nanotechnology articles, nanomaterial data and patents all in one place. Visit Nano by Nature Research