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Lorentz electron ptychography for imaging magnetic textures beyond the diffraction limit


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.

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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 (ref. 47).

Code availability

The source code for LEP is available via Zenodo at (ref. 48).


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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



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.

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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).

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

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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.

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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).

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