Recent work has revived interest in the scattering matrix formulation of electron scattering in transmission electron microscopy as a stepping stone toward atomic-resolution structure determination in the presence of multiple scattering. We discuss ways of visualizing the scattering matrix that make its properties clear. Through a simulation-based case study incorporating shot noise, we shown how regularizing on this continuity enables the scattering matrix to be reconstructed from 4D scanning transmission electron microscopy (STEM) measurements from a single defocus value. Intriguingly, for crystalline samples, this process also yields the sample thickness to nanometer accuracy with no a priori knowledge about the sample structure. The reconstruction quality is gauged by using the reconstructed scattering matrix to simulate STEM images at defocus values different from that of the data from which it was reconstructed.

The arrival of direct electron detectors (DED) with high frame-rates in the field of scanning transmission electron microscopy has enabled many experimental techniques that require collection of a full diffraction pattern at each scan position, a field which is subsumed under the name four dimensional-scanning transmission electron microscopy (4D-STEM). DED frame rates approaching 100 kHz require data transmission rates and data storage capabilities that exceed commonly available computing infrastructure. Current commercial DEDs allow the user to make compromises in pixel bit depth, detector binning or windowing to reduce the per-frame file size and allow higher frame rates. This change in detector specifications requires decisions to be made before data acquisition that may reduce or lose information that could have been advantageous during data analysis. The 4D Camera, a DED with 87 kHz frame-rate developed at Lawrence Berkeley National Laboratory, reduces the raw data to a linear-index encoded electron event representation (EER). Here we show with experimental data from the 4D Camera that linear-index encoded EER and its direct use in 4D-STEM phase contrast imaging methods enables real-time, interactive phase-contrast from large-area 4D-STEM datasets. We detail the computational complexity advantages of the EER and the necessary computational steps to achieve real-time interactive ptychography and center-of-mass differential phase contrast using commonly available hardware accelerators.

The arrival of direct electron detectors (DEDs) with high frame rates in the field of scanning transmission electron microscopy (TEM) has enabled many experimental techniques that require collection of a full diffraction pattern at each scan position, a field which is subsumed under the name four-dimensional scanning transmission electron microscopy (4D-STEM). DED frame rates approaching 100 kHz require data transmission rates and data storage capabilities that exceed those of the commonly available computing infrastructures. Current commercial DEDs allow the user to make compromises in pixel bit depth, detector binning, or windowing to reduce the per-frame file size and allow higher frame rates. This change in detector specifications requires decisions to be made before data acquisition that may reduce or lose information that could have been advantageous during data analysis.

Scanning transmission electron microscopy (STEM) is an extremely versatile method for studying materials on the atomic scale. Many STEM experiments are supported or validated with electron scattering simulations. However, using the conventional multislice algorithm to perform these simulations can require extremely large calculation times, particularly for experiments with millions of probe positions as each probe position must be simulated independently. Recently, the PRISM algorithm was developed to reduce calculation times for large STEM simulations. Here, we introduce a new method for STEM simulation: partitioning of the STEM probe into "beamlets," given by a natural neighbor interpolation of the parent beams. This idea is compatible with PRISM simulations and can lead to even larger improvements in simulation time, as well requiring significantly less computer RAM. We have performed various simulations to demonstrate the advantages and disadvantages of partitioned PRISM STEM simulations. We find that this new algorithm is particularly useful for 4D-STEM simulations of large fields of view. We also provide a reference implementation of the multislice, PRISM and partitioned PRISM algorithms.

Scanning transmission electron microscopy (STEM) is an extremely versatile method for studying materials on the atomic scale. Many STEM experiments are supported or validated with electron scattering simulations. However, using the conventional multislice algorithm to perform these simulations can require extremely large calculation times, particularly for experiments with millions of probe positions as each probe position must be simulated independently. Recently, the plane-wave reciprocal-space interpolated scattering matrix (PRISM) algorithm was developed to reduce calculation times for large STEM simulations. Here, we introduce a new method for STEM simulation: partitioning of the STEM probe into “beamlets,” given by a natural neighbor interpolation of the parent beams. This idea is compatible with PRISM simulations and can lead to even larger improvements in simulation time, as well requiring significantly less computer random access memory (RAM). We have performed various simulations to demonstrate the advantages and disadvantages of partitioned PRISM STEM simulations. We find that this new algorithm is particularly useful for 4D-STEM simulations of large fields of view. We also provide a reference implementation of the multislice, PRISM, and partitioned PRISM algorithms.

Transmission electron microscopy (TEM) is a potent technique for the determination of three-dimensional atomic scale structure of samples in structural biology and materials science. In structural biology, three-dimensional structures of proteins are routinely determined using phase-contrast single-particle cryo-electron microscopy from thousands of identical proteins, and reconstructions have reached atomic resolution for specific proteins. In materials science, three-dimensional atomic structures of complex nanomaterials have been determined using a combination of annular dark field (ADF) scanning transmission electron microscopic (STEM) tomography and subpixel localization of atomic peaks, in a method termed atomic electron tomography (AET). However, neither of these methods can determine the three-dimensional atomic structure of heterogeneous nanomaterials containing light elements. Here, we perform mixed-state electron ptychography from 34.5 million diffraction patterns to reconstruct a high-resolution tilt series of a double wall-carbon nanotube (DW-CNT), encapsulating a complex $\mathrm{ZrTe}$ sandwich structure. Class averaging of the resulting reconstructions and subpixel localization of the atomic peaks in the reconstructed volume reveals the complex three-dimensional atomic structure of the core-shell heterostructure with 17 picometer precision. From these measurements, we solve the full $\mathrm{Zr_{11}Te_{50}}$ structure, which contains a previously unobserved $\mathrm{ZrTe_{2}}$ phase in the core. The experimental realization of ptychographic atomic electron tomography (PAET) will allow for structural determination of a wide range of nanomaterials which are beam-sensitive or contain light elements.

Three-dimensional phase contrast imaging of multiply-scattering samples in X-ray and electron microscopy is extremely challenging, due to small numerical apertures, the unavailability of wavefront shaping optics, and the highly nonlinear inversion required from intensity-only measurements. In this work, we present a new algorithm using the scattering matrix formalism to solve the scattering from a non-crystalline medium from scanning diffraction measurements, and recover the illumination aberrations. Our method will enable 3D imaging and materials characterization at high resolution for a wide range of materials.

Metal-organic layers (MOLs) are highly attractive for application in catalysis, separation, sensing and biomedicine, owing to their tunable framework structure. However, it is challenging to obtain comprehensive information about the formation and local structures of MOLs using standard electron microscopy methods due to serious damage under electron beam irradiation. Here, we investigate the growth processes and local structures of MOLs utilizing a combination of liquid-phase transmission electron microscopy, cryogenic electron microscopy and electron ptychography. Our results show a multistep formation process, where precursor clusters first form in solution, then they are complexed with ligands to form non-crystalline solids, followed by the arrangement of the cluster-ligand complex into crystalline sheets, with additional possible growth by the addition of clusters to surface edges. Moreover, high-resolution imaging allows us to identify missing clusters, dislocations, loop and flat surface terminations and ligand connectors in the MOLs. Our observations provide insights into controllable MOL crystal morphology, defect engineering, and surface modification, thus assisting novel MOL design and synthesis.

Three-dimensional phase-contrast imaging of multiply-scattering samples in x-ray and electron microscopy is challenging due to small numerical apertures, the unavailability of wave front shaping optics, and the highly nonlinear inversion required from intensity-only measurements. In this work, we present an algorithm using the scattering matrix formalism to solve the scattering from a noncrystalline medium from scanning diffraction measurements and simultaneously recover the illumination aberrations. We demonstrate our method experimentally in a scanning transmission electron microscope, recovering the scattering matrix of a heterogeneous sample with two layers of multiwall carbon nanotubes filled with TaTe2 core-shell structures, spaced 10nm apart in the axial direction. Our work enables phase contrast imaging and materials characterization in multiply-scattering samples at high resolution for a wide range of materials.

Scanning transmission electron microscopy (STEM), where a converged electron probe is scanned over a sample's surface and an imaging, diffraction, or spectroscopic signal is measured as a function of probe position, is an extremely powerful tool for materials characterization. The widespread adoption of hardware aberration correction, direct electron detectors, and computational imaging methods have made STEM one of the most important tools for atomic-resolution materials science. Many of these imaging methods rely on accurate imaging and diffraction simulations in order to interpret experimental results. However, STEM simulations have traditionally required large calculation times, as modeling the electron scattering requires a separate simulation for each of the typically millions of probe positions. We have created the Prismatic simulation code for fast simulation of STEM experiments with support for multi-CPU and multi-GPU (graphics processing unit) systems, using both the conventional multislice and our recently-introduced PRISM method. In this paper, we introduce Prismatic version 2.0, which adds many new algorithmic improvements, an updated graphical user interface (GUI), post-processing of simulation data, and additional operating modes such as plane-wave TEM. We review various aspects of the simulation methods and codes in detail and provide various simulation examples. Prismatic 2.0 is freely available both as an open-source package that can be run using a C++ or Python command line interface, or GUI, as well within a Docker container environment.