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Electron ptychography of 2D materials to deep sub-ångström resolution

Naturevolume 559pages343349 (2018) | Download Citation


Aberration-corrected optics have made electron microscopy at atomic resolution a widespread and often essential tool for characterizing nanoscale structures. Image resolution has traditionally been improved by increasing the numerical aperture of the lens (α) and the beam energy, with the state-of-the-art at 300 kiloelectronvolts just entering the deep sub-ångström (that is, less than 0.5 ångström) regime. Two-dimensional (2D) materials are imaged at lower beam energies to avoid displacement damage from large momenta transfers, limiting spatial resolution to about 1 ångström. Here, by combining an electron microscope pixel-array detector with the dynamic range necessary to record the complete distribution of transmitted electrons and full-field ptychography to recover phase information from the full phase space, we increase the spatial resolution well beyond the traditional numerical-aperture-limited resolution. At a beam energy of 80 kiloelectronvolts, our ptychographic reconstruction improves the image contrast of single-atom defects in MoS2 substantially, reaching an information limit close to 5α, which corresponds to an Abbe diffraction-limited resolution of 0.39 ångström, at the electron dose and imaging conditions for which conventional imaging methods reach only 0.98 ångström.

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  1. 1.

    Meyer, J. C., Girit, C. O., Crommie, M. F. & Zettl, A. Imaging and dynamics of light atoms and molecules on graphene. Nature 454, 319–322 (2008).

  2. 2.

    Krivanek, O. L. et al. Atom-by-atom structural and chemical analysis by annular dark-field electron microscopy. Nature 464, 571–574 (2010).

  3. 3.

    Huang, P. Y. et al. Grains and grain boundaries in single-layer graphene atomic patchwork quilts. Nature 469, 389–392 (2011).

  4. 4.

    Sparrow, C. M. On spectroscopic resolving power. Astrophys. J. 44, 76–86 (1916).

  5. 5.

    Black, G. & Linfoot, E. H. Spherical aberration and the information content of optical images. Proc. R. Soc. Lond. A 239, 522–540 (1957).

  6. 6.

    Haider, M. et al. Electron microscopy image enhanced. Nature 392, 768–769 (1998).

  7. 7.

    Batson, P. E., Dellby, N. & Krivanek, O. L. Sub-ångström resolution using aberration corrected electron optics. Nature 418, 617–620 (2002).

  8. 8.

    Erni, R., Rossell, M. D., Kisielowski, C. & Dahmen, U. Atomic-resolution imaging with a sub-50-pm electron probe. Phys. Rev. Lett. 102, 096101 (2009).

  9. 9.

    Sawada, H. et al. STEM imaging of 47-pm-separated atomic columns by a spherical aberration-corrected electron microscope with a 300-kV cold field emission gun. J. Electron Microsc. 58, 357–361 (2009).

  10. 10.

    Kaiser, U. et al. Transmission electron microscopy at 20kV for imaging and spectroscopy. Ultramicroscopy 111, 1239–1246 (2011).

  11. 11.

    Meyer, J. C. et al. Accurate measurement of electron beam induced displacement cross sections for single-layer graphene. Phys. Rev. Lett. 108, 196102 (2012).

  12. 12.

    Sawada, H., Sasaki, T., Hosokawa, F. & Suenaga, K. Atomic-resolution STEM imaging of graphene at low voltage of 30 kV with resolution enhancement by using large convergence angle. Phys. Rev. Lett. 114, 166102 (2015).

  13. 13.

    Linck, M. et al. Chromatic aberration correction for atomic resolution TEM imaging from 20 to 80 kV. Phys. Rev. Lett. 117, 076101 (2016).

  14. 14.

    Henderson, R. The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of unstained biological molecules. Q. Rev. Biophys. 28, 171–193 (1995).

  15. 15.

    Gabor, D. A new microscopic principle. Nature 161, 777–778 (1948).

  16. 16.

    Hoppe, W. Beugung im inhomogenen Primärstrahlwellenfeld. I. Prinzip einer Phasenmessung von Elektronenbeungungsinterferenzen. Acta Crystallogr. A 25, 495–501 (1969).

  17. 17.

    Nellist, P. D., McCallum, B. C. & Rodenburg, J. M. Resolution beyond the ‘information limit’ in transmission electron microscopy. Nature 374, 630–632 (1995).

  18. 18.

    Nellist, P. D. & Rodenburg, J. M. Electron ptychography. I. Experimental demonstration beyond the conventional resolution limits. Acta Crystallogr. A 54, 49–60 (1998).

  19. 19.

    Thibault, P. et al. High-resolution scanning X-ray diffraction microscopy. Science 321, 379–382 (2008).

  20. 20.

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

  21. 21.

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

  22. 22.

    Li, P., Edo, T. B. & Rodenburg, J. M. Ptychographic inversion via Wigner distribution deconvolution: noise suppression and probe design. Ultramicroscopy 147, 106–113 (2014).

  23. 23.

    Yang, H. et al. Simultaneous atomic-resolution electron ptychography and Z-contrast imaging of light and heavy elements in complex nanostructures. Nat. Commun. 7, 12532 (2016).

  24. 24.

    Pelz, P. M., Qiu, W. X., Bücker, R., Kassier, G. & Miller, R. J. D. Low-dose cryo electron ptychography via non-convex Bayesian optimization. Sci. Rep. 7, 9883 (2017).

  25. 25.

    Maiden, A. M., Humphry, M. J. & Rodenburg, J. M. Ptychographic transmission microscopy in three dimensions using a multi-slice approach. J. Opt. Soc. Am. A 29, 1606–1614 (2012).

  26. 26.

    Gao, S. et al. Electron ptychographic microscopy for three-dimensional imaging. Nat. Commun. 8, 163 (2017).

  27. 27.

    Rodenburg, J. M., Hurst, A. C. & Cullis, A. G. Transmission microscopy without lenses for objects of unlimited size. Ultramicroscopy 107, 227–231 (2007).

  28. 28.

    Rodenburg, J. M. et al. Hard-X-ray lensless imaging of extended objects. Phys. Rev. Lett. 98, 034801 (2007).

  29. 29.

    Hüe, F., Rodenburg, J. M., Maiden, A. M., Sweeney, F. & Midgley, P. A. Wave-front phase retrieval in transmission electron microscopy via ptychography. Phys. Rev. B 82, 121415 (2010).

  30. 30.

    Hüe, F., Rodenburg, J. M., Maiden, A. M. & Midgley, P. A. Extended ptychography in the transmission electron microscope: possibilities and limitations. Ultramicroscopy 111, 1117–1123 (2011).

  31. 31.

    Putkunz, C. T. et al. Atom-scale ptychographic electron diffractive imaging of boron nitride cones. Phys. Rev. Lett. 108, 073901 (2012).

  32. 32.

    D’Alfonso, A. J. et al. Deterministic electron ptychography at atomic resolution. Phys. Rev. B 89, 064101 (2014).

  33. 33.

    Pennycook, T. J. et al. Efficient phase contrast imaging in STEM using a pixelated detector. Part 1: experimental demonstration at atomic resolution. Ultramicroscopy 151, 160–167 (2015).

  34. 34.

    Yang, H. et al. Electron ptychographic phase imaging of light elements in crystalline materials using Wigner distribution deconvolution. Ultramicroscopy 180, 173–179 (2017).

  35. 35.

    Wang, P., Zhang, F., Gao, S., Zhang, M. & Kirkland, A. I. Electron ptychographic diffractive imaging of boron atoms in LaB6 crystals. Sci. Rep. 7, 2857 (2017).

  36. 36.

    Humphry, M. J., Kraus, B., Hurst, A. C., Maiden, A. M. & Rodenburg, J. M. Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging. Nat. Commun. 3, 730 (2012).

  37. 37.

    Frojdh, E. et al. Count rate linearity and spectral response of the Medipix3RX chip coupled to a 300μm silicon sensor under high flux conditions. J. Instrum. 9, C04028 (2014).

  38. 38.

    Rose, A. Vision Human and Electronic Ch. 1 (Plenum Press, New York, 1949).

  39. 39.

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

  40. 40.

    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).

  41. 41.

    Lazić, I., Bosch, E. G. T. & Lazar, S. Phase contrast STEM for thin samples: integrated differential phase contrast. Ultramicroscopy 160, 265–280 (2016).

  42. 42.

    Maiden, A. M., Humphry, M. J., Zhang, F. & Rodenburg, J. M. Superresolution imaging via ptychography. J. Opt. Soc. Am. A 28, 604–612 (2011).

  43. 43.

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

  44. 44.

    Lee, J. & Barbastathis, G. Denoised Wigner distribution deconvolution via low-rank matrix completion. Opt. Express 24, 20069–20079 (2016).

  45. 45.

    Abbe, E. The relation of aperture and power in the microscope. J. R. Microsc. Soc. 2, 300–309 (1882).

  46. 46.

    van der Zande, A. M. et al. Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist. Nano Lett. 14, 3869–3875 (2014).

  47. 47.

    Hovden, R. & Muller, D. A. Efficient elastic imaging of single atoms on ultrathin supports in a scanning transmission electron microscope. Ultramicroscopy 123, 59–65 (2012).

  48. 48.

    Zuo, C., Sun, J. & Chen, Q. Adaptive step-size strategy for noise-robust Fourier ptychographic microscopy. Opt. Express 24, 20724–20744 (2016).

  49. 49.

    Maiden, A., Johnson, D. & Li, P. Further improvements to the ptychographical iterative engine. Optica 4, 736–745 (2017).

  50. 50.

    Suzuki, A. et al. High-resolution multislice X-ray ptychography of extended thick objects. Phys. Rev. Lett. 112, 053903 (2014).

  51. 51.

    Hovden, R., Jiang, Y., Xin, H. L. & Kourkoutis, L. F. Periodic artifact reduction in Fourier transforms of full field atomic resolution images. Microsc. Microanal. 21, 436–441 (2015).

  52. 52.

    Allen, L. J., D’Alfonso, A. J. & Findlay, S. D. Modelling the inelastic scattering of fast electrons. Ultramicroscopy 151, 11–22 (2015).

  53. 53.

    Waasmaier, D. & Kirfel, A. New analytical scattering-factor functions for free atoms and ions. Acta Crystallogr. A 51, 416–431 (1995).

  54. 54.

    Nellist, P. D. & Rodenburg, J. M. Beyond the conventional information limit: the relevant coherence function. Ultramicroscopy 54, 61–74 (1994).

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Y.J. and V.E. acknowledge support from DOE grant DE-SC0005827. Z.C. and D.A.M. are supported by the PARADIM Materials Innovation Platform in-house programme by NSF grant DMR-1539918. We acknowledge the electron microscopy facility and support for Y.H. from the NSF MRSEC programme (DMR 1719875) and NSF MRI grant DMR-1429155. H.G., S.X. and J.P. acknowledge additional support from AFOSR MURI (FA9550-16-1-003) and UChicago NSF MRSEC programme (DMR 1420709). Detector development at Cornell was supported by the Kavli Institute at Cornell for Nanoscale Science and DOE grant DE-SC0017631 to S.M.G. We thank K. Nguyen, P. Huang, M. Humphry and P. Nellist for discussions and B. Jiang from Thermo Scientific for help during the initial experiments.

Reviewer information

Nature thanks J. Rodenburg and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information

Author notes

  1. These authors contributed equally: Yi Jiang, Zhen Chen.


  1. Department of Physics, Cornell University, Ithaca, NY, USA

    • Yi Jiang
    • , Pratiti Deb
    • , Prafull Purohit
    • , Mark W. Tate
    • , Sol M. Gruner
    •  & Veit Elser
  2. School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA

    • Zhen Chen
    • , Yimo Han
    • , Pratiti Deb
    • , Saien Xie
    •  & David A. Muller
  3. Department of Chemistry, Institute for Molecular Engineering, James Franck Institute, University of Chicago, Chicago, IL, USA

    • Hui Gao
    • , Saien Xie
    •  & Jiwoong Park
  4. Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA

    • Hui Gao
  5. Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA

    • Sol M. Gruner
    •  & David A. Muller


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Experiments were performed and designed by Z.C., Y.H. and D.A.M. Y.J. contributed to data analysis and ptychographic reconstruction, with support from V.E. Sample preparation was done by P.D. and Y.H., from MoS2 thin films synthesized by H.G., S.X. and J.P. EMPAD was optimized by P.P., M.W.T. and S.M.G. All authors discussed the results and implications throughout the investigation. All authors have approved the final version of the manuscript.

Competing interests

Cornell University has licensed the EMPAD hardware to Thermo Scientific.

Corresponding author

Correspondence to David A. Muller.

Extended data figures and tables

  1. Extended Data Fig. 1 Position-averaged diffraction pattern of the 4D dataset from monolayer MoS2.

    a, Position-averaged convergent beam electron diffraction (CBED) pattern from the 4D dataset from monolayer MoS2. b, Radially averaged intensity distribution (on a logarithmic scale) of the CBED pattern, showing the dynamic range spanned by the scattering distribution.

  2. Extended Data Fig. 2 Line profiles through atom pairs in the twisted bilayer MoS2.

    Line profiles are from atom pairs in Fig. 3, with the respective subregions shown on the left. ac, The measured peak–peak separations between two atoms are 0.42 ± 0.02 Å (a), 0.61 ± 0.02 Å (b) and 0.85 ± 0.02 Å (c).

  3. Extended Data Fig. 3 ADF image and line profiles through atom pairs in the twisted bilayer MoS2.

    a, ADF image synthesized from the 4D diffraction dataset. b, Phase of the transmission function reconstructed by ptychography. The yellow marker indicates a pair of atoms that is predicted to have a separation of 0.2 Å on the basis of the structural model, but cannot be resolved explicitly in our reconstruction. For a more detailed comparison, a red box is placed over corresponding regions in a and b. c, Enlarged image of the red boxed region in b, with the false colour scale of Fig. 3. d, Line profiles across the atom pairs labelled with dashed lines in c. The peak–peak separations are overlaid near the line profiles.

  4. Extended Data Fig. 4 Reconstructed amplitude and phase of monolayer MoS2 at different cutoff angles.

    Both the amplitude (left panels) and phase (right panels) of the reconstructed transmission function show the atomic structure of monolayer MoS2. Image resolution improves as the cutoff angle increases. Amplitude modulations are relatively weak, deviating by only a few per cent from a pure phase object (that is, an object function with unit amplitude).

  5. Extended Data Fig. 5 Comparison between ptychography techniques and low-angle ADF imaging of graphene.

    a, b, Ptychographic reconstructions of simulated data with an in-focused probe, using the WDD (a) and ePIE (b) methods. c, Low-angle ADF (integrating from 1α to 4α) reconstruction using the same simulated datasets. Both ptychographic methods show similar reconstructions and are about 10 times more dose-efficient than the low-angle ADF technique. Beam energy, 80 keV; aperture size (α), 21.4 mrad.

  6. Extended Data Fig. 6 Influence of scanning drift and contamination.

    ac, ADF image (a), iCoM image (b) and phase of transmission reconstructed by full-field ptychography (c) using 128 × 128 diffraction patterns, covering a field of view of 2.7 nm × 2.7 nm. The ADF and iCoM reconstructions both suffer from stripe artefacts and large contrast variations. In the ptychographic reconstruction, scanning drift distorts and blurs reconstructed atoms in the vicinity of the scan distortion, but the overall resolution away from the distortion remains higher than the other imaging modes.

  7. Extended Data Fig. 7 Effect of dose and cutoff angles on ptychographic reconstructions of monolayer MoS2 using simulated diffraction patterns.

    At high beam current, the resolution of the ptychography reconstruction is fundamentally determined by the collection angle of the detector. As the beam current decreases, the resolution becomes dose-limited and noise artefacts start to appear in the ePIE reconstruction. Beam energy, 80 keV; aperture size (α), 21.4 mrad.

  8. Extended Data Fig. 8 Effect of chromatic aberrations at different electron doses for ptychographic reconstructions of monolayer MoS2 using simulated datasets at 80 keV.

    Two convergence semi-angles are shown, 21.4 mrad (left two columns) and 35 mrad (right two columns), representing conditions under which chromatic aberrations have moderate and large effects on the incident probe shape, respectively (Cc = 1.72 mm, ΔE = 1.1 eV). 21.4 mrad is also the experimental convergence angle. The incident electron dose levels are an infinite dose (top row), the experimental dose of 1.16 × 105 electrons per Å2 (middle row) and a low dose of 104 electrons per Å2 (bottom row). In the presence of noise, chromatic aberrations degrade the phase range of the reconstruction compared with the achromatic data. The data for the larger convergence semi-angle are more strongly affected. At infinite and experimental doses, ptychographic reconstructions with and without chromatic aberration are visually similar for both convergence angles. At low dose and with chromatic aberration, the reconstructed atoms are broadened, and distinct artefacts appear for a convergence angle of 35 mrad.

Supplementary information

  1. Video 1: Evolution of unit-cell averaged diffraction patterns at various scan positions.

    The left panel shows a synthesized ADF image from a unit-cell averaged 4D dataset, and the right panels show the averaged diffraction patterns from the scan positions marked with the red circle on the left ADF image. The intensity of the diffraction patterns are displayed on a linear scale to show variations in the centre disk (upper right panel) and a logarithmic scale to show variations in the weaker the dark field disks (lower right panel), respectively.

  2. Video 2: Evolution of raw diffraction patterns at various scan positions.

    The left panel shows a synthesized ADF image from a raw 4D dataset, and the right panels show the diffraction patterns from the scan positions marked with the circle on the left ADF image. The intensity of the diffraction patterns are displayed on a linear scale to show variations in the centre disk (upper right panel) and a logarithmic scale to show variations in the weaker the dark field disks (lower right panel), respectively.

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