Determining the precise atomic arrangement in complex nanostructures is essential for understanding material properties. However, materials like metal-organic frameworks and organic perovskites are radiation-sensitive, making high-resolution imaging challenging. Existing atomic-resolution STEM techniques, such as annular dark-field (ADF) and coherent bright-field (cBF) imaging, are inherently dose-inefficient, utilizing only a small fraction of scattered electrons. This limitation prevents sub-angstrom resolution imaging of radiation-sensitive materials. High-precision measurement of atomic positions is also hampered by low signal-to-noise ratios in ADF images from weakly scattering samples. Electron ptychography offers a potential solution, leveraging the entire diffraction pattern for improved dose efficiency and resolution. While previous work has demonstrated the potential of electron ptychography, the impact of the inherent partial coherence of the electron beam on reconstruction quality has remained under-explored. This study aims to address this gap by employing mixed-state electron ptychography, which explicitly accounts for partial coherence, to achieve sub-angstrom resolution imaging with high precision and low dose.

Electron ptychography has emerged as a powerful phase-contrast imaging technique with benefits such as high dose efficiency, high resolution, and high contrast. It has even surpassed the resolution of the best physical lenses, reaching deep sub-angstrom resolutions. Simulations suggest extremely low-dose imaging capabilities, surpassing other electron imaging approaches. While Wigner-distribution deconvolution (WDD)-based electron ptychography has shown advantages over conventional high-resolution TEM, the impact of partial coherence of the illumination probe on the reconstruction quality is still largely unexplored. Previous attempts to address partial coherence using approaches like Gaussian blind deconvolution or modal decomposition have not yielded sub-angstrom resolution reconstructions.

This research utilizes defocused electron ptychography, employing a pixel-array detector (EMPAD) for diffraction pattern acquisition. Instead of a focused probe, a defocused probe broadens the illumination area, enabling a larger scan step size while maintaining sufficient overlap between adjacent scan positions. The EMPAD's high dynamic range allows capturing both bright-field and dark-field signals. To account for partial coherence, a mixed-state model representing the illumination as a linear combination of mutually incoherent probe modes is implemented. The modal decomposition approach expands the probe into eigenmodes of the density matrix, with the total intensity normalized to measured diffraction pattern intensity. The ptychographic reconstruction utilizes a generalized maximum-likelihood algorithm that employs preconditioned gradient descent and a mini-batch optimization scheme to improve robustness and speed. The algorithm also includes illumination wavefront correction to account for minor time variations in the probe.

Mixed-state electron ptychography achieved sub-angstrom resolution (0.69 Å) and picometer precision in imaging a monolayer WS₂ sample with bilayer WS₂/MoSe₂ islands. The Moiré pattern resulting from the lattice mismatch served as a resolution test. The technique demonstrated significantly improved contrast and signal-to-noise ratio compared to conventional ADF imaging. Low-dose imaging was achieved with doses up to 50 times lower than conventional STEM, reaching atomic resolution even at a dose of 375 e Å⁻². The use of a defocused probe allowed for a much larger scan step size (up to 5.08 Å with 72% probe overlap), resulting in a four-times-faster acquisition time compared to conventional ADF imaging. The mixed-state approach relaxed the real-space overlap constraints, making large field-of-view atomic-resolution imaging feasible. High-precision atomic position measurements were obtained, with standard deviations of 5.8 pm (S-S) and 5.2 pm (W-W) for a dose of 1.25 × 10⁵ e Å⁻². Even at low doses (10⁴ e Å⁻²), a precision of around 10 pm was maintained, surpassing the precision of conventional ADF imaging.

The successful implementation of mixed-state electron ptychography demonstrates the importance of accurately modeling partial coherence in electron microscopy. This approach offers significant improvements over conventional STEM methods in terms of resolution, speed, and dose efficiency, particularly for radiation-sensitive samples. The ability to achieve sub-angstrom resolution with picometer precision at low doses opens up new possibilities for studying a wide range of materials, including those previously inaccessible to high-resolution imaging techniques. The results highlight the potential of mixed-state ptychography for various applications, such as in situ imaging of dynamic processes and three-dimensional structural reconstruction.

Mixed-state electron ptychography provides a powerful new tool for atomic-scale imaging, combining high resolution, large field of view, low dose, high contrast, and high precision. The ability to account for partial coherence is crucial for achieving optimal performance. Future work could explore further improvements in algorithm design and experimental setups to further enhance dose efficiency and resolution, pushing the limits of what is possible in electron microscopy. The high dose efficiency makes this method highly promising for beam-sensitive materials and could potentially be beneficial for cryo-electron microscopy applications.

The current mixed-state ptychography method is limited to relatively thin samples due to the use of the generalized strong phase approximation, which neglects multiple scattering effects. The thickness limit depends on the maximum scattering angle and electron wavelength, and becomes more restrictive for samples containing high atomic number elements. While the algorithm is relatively fast with GPU acceleration, further optimization may be necessary for real-time applications. The reported precision in atomic position measurements might be slightly overestimated due to the presence of residual polymer residue in the sample.