Near-Isotropic Sub-Ångstrom 3D Resolution Phase Contrast Imaging Achieved by End-to-End Ptychographic Electron Tomography

TL;DR

The paper tackles achieving near-isotropic sub-Ångström 3D resolution in transmission electron tomography by introducing an end-to-end multislice ptychographic tomography framework. It reconstructs the electrostatic potential volume directly from unaligned 4D-STEM tilt-series data, jointly recovering the volume, mixed-state probes, and tomographic alignments. The method combines a multi-slice forward model, partial coherence modeling, and end-to-end optimization to compensate for the missing wedge without extra hardware. Simulation with a Pt@Al2O3 core-shell nanoparticle and experimental imaging of a Te nanoparticle on a carbon nanotube demonstrate sub-Å accuracy in depth and robust performance under reduced-dose conditions. The work highlights significant potential for beam-sensitive materials and provides a pathway toward end-to-end atomic-scale 3D structure determination in electron tomography.

Abstract

Three-dimensional atomic resolution imaging using transmission electron microscopes is a unique capability that requires challenging experiments. Linear electron tomography methods are limited by the missing wedge effect, requiring a high tilt range. Multislice ptychography can achieve deep sub-Ångstrom resolution in the transverse direction, but the depth resolution is limited to 2 to 3 nanometers. In this paper, we propose and demonstrate an end-to-end approach to reconstructing the electrostatic potential volume of the sample directly from the 4D-STEM datasets. End-to-end multi-slice ptychographic tomography recovers several slices at each tomography tilt angle and compensates for the missing wedge effect. The algorithm is initially tested in simulation with a Pt@$\mathrm{Al_2O_3}$ core-shell nanoparticle, where both heavy and light atoms are recovered in 3D from an unaligned 4D-STEM tilt series with a restricted tilt range of 90 degrees. We also demonstrate the algorithm experimentally, recovering a Te nanoparticle with sub-Ångstrom resolution.

Figures (3)

• Figure 1: Workflow of the end-to-end reconstruction. (a) Bright-field reconstruction and aberration calibration. The weak-phase images are reconstructed using the SSB method on un-calibrated datasets. The optimal defocus for the bright-field reconstruction is used for probe initialization in (b). (b) Sequential reconstruction and joint alignment. After the 4D-STEM datasets are calibrated, multi-slice ptychography reconstruction produces several slice images for each tilt angle. The phase of the slices is summed into 2D phase projections, which are used in joint reconstruction and alignment. (c) The end-to-end joint reconstruction. The initial guess of the volume is affine transformed, batch cropped, and scanned by a mixed-state probe in a multi-slice manner to produce the model diffraction patterns. The model diffraction patterns are updated from the measured datasets and then back-propagated to update the volume for all tilt angles
• Figure 2: Simulation results of alignment reconstruction and joint reconstruction. (a) The reconstructed and ground truth of the alignment. The recovered alignment is linked to the corresponding ground truth value. (b) The slice image of the ground truth of Pt@$\mathrm{Al_2O_3}$ potential volume. (c) The reconstructed slice image of the Pt@$\mathrm{Al_2O_3}$ at the same depth. The scale bar is 0.4. Some oxygen, aluminum, and platinum atoms at the same positions are highlighted. The depth section extracted for (d) is indicated by the red dashed rectangle. (d) Depth section plot at the same area with different tomographic tilt ranges of 90, 80, 60, and 40 degrees, as indicated at the top. Inset: 3D atomic model of the core-shell nanocube, with cut-out atoms to reveal the Pt core.
• Figure 3: Experimental demonstration of the joint reconstruction and resolution analysis. (a) Volumetric rendering of the reconstructed potential volume. The Te nanoparticle is shown in blue and the carbon nanotube in green. Two slices along near-perpendicular zone axes are shown to the right. The scale box in the back right corner is 1nm$^3$. (b) The reconstructed result in the Fourier domain, the missing wedge direction is indicated by green lines. In this direction, the Bragg peaks prove the existence of recovered information and compensate for the missing wedge artifacts. (c) The radially integrated power spectrum of the reconstructed results was obtained using different hyperparameter tuning. The most obvious peak in the figure corresponds to a resolution of 3.28. The right-most peak shows a resolution of 1.24. The relaxed and strict Butterworth kernels have the cut-off frequency of 1.51^-1, 0.49^-1 and the orders of the filter are 2 and 8, respectively. (d) FSC analysis of the reconstructions using 2-fold, 12-fold, and 20-fold subsampling. The reconstruction is successful by using only 5% of the diffraction patterns for each angle, with a resolution of 2.84