Abstract: Over the last decade, scanning transmission electron microscopy has become one of the most powerful tools to characterise nanomaterials at the atomic scale. Often, the ultimate goal is to retrieve the three-dimensional structure, which is very challenging since small species are typically sensitive to electron irradiation. Nevertheless, measuring individual atomic positions is crucial to understand the relation between the structure and physicochemical properties of these (nano)materials. In this review, we highlight the latest approaches that are available to reveal the 3D atomic structure of small species. Finally, we will provide an outlook and will describe future challenges where the limits of electron microscopy will be pushed even further.

Abstract: By combining statistical parameter estimation and model-order selection using a Bayesian framework, the maximum a posteriori (MAP) probability rule is proposed in this chapter as an objective and quantitative method to detect atom columns from high-resolution scanning transmission electron microscopy (HRSTEM) images. The validity and usefulness of this approach is demonstrated to both simulated and experimental annular dark-field (ADF) STEM images, but also to simultaneously acquired annular bright-field (ABF) and ADF STEM image data.

Abstract: An efficient model-based estimation algorithm is introduced to quantify the atomic column positions and intensities from atomic-resolution (scanning) transmission electron microscopy ((S)TEM) images. This algorithm uses the least squares estimator on image segments containing individual columns fully accounting for overlap between neighboring columns, enabling the analysis of a large field of view. To provide end-users with this well-established quantification method, a user friendly program, StatSTEM, is developed which is freely available under a GNU public license. In this chapter, this efficient algorithm is applied to three different nanostructures for which the analysis of a large field of view is required.

Abstract: The maximum a posteriori (MAP) probability rule for atom column detection can also be used as a tool to evaluate the relation between scanning transmission electron microscopy (STEM) image quality and atom detectability. In this chapter, a new image-quality measure is proposed that correlates well with atom detectability, namely the integrated contrast-to-noise ratio (ICNR). Furthermore, the working principle of the MAP probability rule is described in detail showing a close relation to the principles of model-selection methods.

Abstract: In this chapter, the principles of detection theory are used to quantify the probability of error for atom counting from high-resolution scanning transmission electron microscopy (HRSTEM) images. Binary and multiple hypothesis testing have been investigated in order to determine the limits to the precision with which the number of atoms in a projected atomic column can be estimated. The probability of error has been calculated when using STEM images, scattering cross-sections or peak intensities as a criterion to count atoms. Based on this analysis, we conclude that scattering cross-sections perform almost equally well as images and perform better than peak intensities. Furthermore, the optimal STEM detector design can be derived for atom counting using the expression of the probability of error. We show that for very thin objects the low-angle annular dark-field (LAADF) regime is optimal and that for thicker objects the optimal inner detector angle increases.

Abstract: Determining the three-dimensional (3D) atomic structure of nanoparticles (NPs) is critical to understand their structure-dependent properties. It is hereby important to perform such analyses under conditions relevant for the envisioned application. Here, we investigate the 3D structure of supported Au NPs at high temperature, which is of importance to understand their behavior during catalytic reactions. To overcome limitations related to conventional high-resolution electron tomography at high temperature, 3D characterization of NPs with atomic resolution has been performed by applying atom-counting using atomic resolution annular darkfield scanning transmission electron microscopy (ADF STEM) images followed by structural relaxation. However, at high temperatures, thermal displacements, which affect the ADF STEM intensities, should be taken into account. Moreover, it is very likely that the structure of a NP investigated at elevated temperature deviates from a ground state configuration, which is difficult to determine using purely computational energy minimization approaches. In this paper, we therefore propose an optimized approach using an iterative local minima search algorithm followed by molecular dynamics (MD) structural relaxation of candidate structures associated with each local minimum. In this manner, it becomes possible to investigate the 3D atomic structure of supported NPs, which may deviate from their ground state configuration.

Abstract: We introduce a Bayesian genetic algorithm for reconstructing atomic models of monotype crystalline nanoparticles from a single projection using Z-contrast imaging. The number of atoms in a projected atomic column obtained from annular dark field scanning transmission electron microscopy images serves as an input for the initial three-dimensional model. The algorithm minimizes the energy of the structure while utilizing a priori information about the finite precision of the atom-counting results and neighbor-mass relations. The results show promising prospects for obtaining reliable reconstructions of beam-sensitive nanoparticles during dynamical processes from images acquired with sufficiently low incident electron doses.

Abstract: Functional interfaces are at the core of research in the emerging field of domain boundary engineering where polar, conducting, chiral, and other interfaces and twin boundaries have been discovered. Ferroelectricity was found in twin walls of paraelectric CaTiO3. We show that the effect of functional interfaces can be optimized if the number of twin boundaries is increased in densely twinned materials. Such materials can be produced by shear in the ferroelastic phase rather than by rapid quench from the paraelastic phase.

Abstract: The charge-coupled devices used in electron microscopy are coated with a scintillating crystal that gives rise to a severe modulation transfer function (MTF). Exact knowledge of the MTF is imperative for a good correspondence between image simulation and experiment. We present a practical method to measure the MTF above the Nyquist frequency from the beam blocker's shadow image. The image processing has been fully automated and the program is made public. The method is successfully tested on three cameras with various beam blocker shapes.

Abstract: Modern TEM enables the investigation of nanostructures at the atomic scale. However, TEM images are only two-dimensional (2D) projections of a three-dimensional (3D) object. Electron tomography can overcome this limitation. The technique is increasingly focused towards quantitative measurements and reaching atomic resolution in 3D has been the ultimate goal for many years. Therefore, one needs to optimize the acquisition of the data, the 3D reconstruction techniques as well as the quantification methods. Here, we will review a broad range of methodologies and examples. Finally, we will provide an outlook and will describe future challenges in the field of electron tomography.