Abstract: We report an innovative method to explore the optimal experimental settings to detect light atoms from scanning transmission electron microscopy (STEM) images. Since light elements play a key role in many technologically important materials, such as lithium-battery devices or hydrogen storage applications, much effort has been made to optimize the STEM technique in order to detect light elements. Therefore, classical performance criteria, such as contrast or signal-to-noise ratio, are often discussed hereby aiming at improvements of the direct visual interpretability. However, when images are interpreted quantitatively, one needs an alternative criterion, which we derive based on statistical detection theory. Using realistic simulations of technologically important materials, we demonstrate the benefits of the proposed method and compare the results with existing approaches.

Abstract: Oriented attachment of PbSe nanocubes can result in the formation of two-dimensional (2D) superstructures with long-range nanoscale and atomic order. This questions the applicability of classic models in which the superlattice grows by first forming a nucleus, followed by sequential irreversible attachment of nanocrystals, as one misaligned attachment would disrupt the 2D order beyond repair. Here, we demonstrate the formation mechanism of 2D PbSe superstructures with square geometry by using in situ grazing-incidence X-ray scattering (small angle and wide angle), ex situ electron microscopy, and Monte Carlo simulations. We observed nanocrystal adsorption at the liquid/gas interface, followed by the formation of a hexagonal nanocrystal monolayer. The hexagonal geometry transforms gradually through a pseudo-hexagonal phase into a phase with square order, driven by attractive interactions between the {100} planes perpendicular to the liquid substrate, which maximize facet-to-facet overlap. The nanocrystals then attach atomically via a necking process, resulting in 2D square superlattices.

Abstract: We present a computational imaging mode for large scale electron microscopy data, which retrieves a complex wave from noisy/sparse intensity recordings using a deep learning approach and subsequently reconstructs an image of the specimen from the Convolutional Neural Network (CNN) predicted exit waves. We demonstrate that an appropriate forward model in combination with open data frameworks can be used to generate large synthetic datasets for training. In combination with augmenting the data with Poisson noise corresponding to varying dose-values, we effectively eliminate overfitting issues. The U-NET[1] based architecture of the CNN is adapted to the task at hand and performs well while maintaining a relatively small size and fast performance. The validity of the approach is confirmed by comparing the reconstruction to well-established methods using simulated, as well as real electron microscopy data. The proposed method is shown to be effective particularly in the low dose range, evident by strong suppression of noise, good spatial resolution, and sensitivity to different atom types, enabling the simultaneous visualisation of light and heavy elements and making different atomic species distinguishable. Since the method acts on a very local scale and is comparatively fast it bears the potential to be used for near-real-time reconstruction during data acquisition.

Abstract: Recently, the maximum a posteriori (MAP) probability rule has been proposed as an objective and quantitative method to detect atom columns and even single atoms from high-resolution high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) images. The method combines statistical parameter estimation and model-order selection using a Bayesian framework and has been shown to be especially useful for the analysis of the structure of beam-sensitive nanomaterials. In order to avoid beam damage, images of such materials are usually acquired using a limited incoming electron dose resulting in a low contrast-to-noise ratio (CNR) which makes visual inspection unreliable. This creates a need for an objective and quantitative approach. The present paper describes the methodology of the MAP probability rule, gives its step-by-step derivation and discusses its algorithmic implementation for atom column detection. In addition, simulation results are presented showing that the performance of the MAP probability rule to detect the correct number of atomic columns from HAADF STEM images is superior to that of other model-order selection criteria, including the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). Moreover, the MAP probability rule is used as a tool to evaluate the relation between STEM image quality measures and atom detectability resulting in the introduction of the so-called integrated CNR (ICNR) as a new image quality measure that better correlates with atom detectability than conventional measures such as signal-to-noise ratio (SNR) and CNR.

Abstract: In electron microscopy, the maximum a posteriori (MAP) probability rule has been introduced as a tool to determine the most probable atomic structure from high-resolution annular dark-field (ADF) scanning transmission electron microscopy (STEM) images exhibiting low contrast-to-noise ratio (CNR). Besides ADF imaging, STEM can also be applied in the annular bright-field (ABF) regime. The ABF STEM mode allows to directly visualize light-element atomic columns in the presence of heavy columns. Typically, light-element nanomaterials are sensitive to the electron beam, limiting the incoming electron dose in order to avoid beam damage and leading to images exhibiting low CNR. Therefore, it is of interest to apply the MAP probability rule not only to ADF STEM images, but to ABF STEM images as well. In this work, the methodology of the MAP rule, which combines statistical parameter estimation theory and model-order selection, is extended to be applied to simultaneously acquired ABF and ADF STEM images. For this, an extension of the commonly used parametric models in STEM is proposed. Hereby, the effect of specimen tilt has been taken into account, since small tilts from the crystal zone axis affect, especially, ABF STEM intensities. Using simulations as well as experimental data, it is shown that the proposed methodology can be successfully used to detect light elements in the presence of heavy elements.

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: Atomically resolved electron energy-loss spectroscopy experiments are commonplace in modern aberration-corrected transmission electron microscopes. Energy resolution has also been increasing steadily with the continuous improvement of electron monochromators. Electronic excitations however are known to be delocalized due to the long range interaction of the charged accelerated electrons with the electrons in a sample. This has made several scientists question the value of combined high spatial and energy resolution for mapping interband transitions and possibly phonon excitation in crystals. In this paper we demonstrate experimentally that atomic resolution information is indeed available at very low energy losses around 100 meV expressed as a modulation of the broadening of the zero loss peak. Careful data analysis allows us to get a glimpse of what are likely phonon excitations with both an energy loss and gain part. These experiments confirm recent theoretical predictions on the strong localization of phonon excitations as opposed to electronic excitations and show that a combination of atomic resolution and recent developments in increased energy resolution will offer great benefit for mapping phonon modes in real space.

Abstract: Statistical parameter estimation theory is proposed as a quantitative method to measure unknown structure parameters from electron microscopy images. Images are then purely considered as data planes from which structure parameters have to be determined as accurately and precisely as possible using a parametric statistical model of the observations. For this purpose, an efficient algorithm is proposed for the estimation of atomic column positions and intensities from high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) images. Furthermore, the so-called CramérRao lower bound (CRLB) is reviewed to determine the limits to the precision with which continuous parameters such as atomic column positions and intensities can be estimated. Since this lower bound can only be derived for continuous parameters, alternative measures using the principles of detection theory are introduced for problems concerning the estimation of discrete parameters such as atomic numbers. An experimental case study is presented to show the practical use of these measures for the optimization of the experiment design if the purpose is to decide between the presence of specific atom types using STEM images.

Abstract: A decade ago, a statistics-based method was introduced to count the number of atoms from annular dark-field scanning transmission electron microscopy (ADF STEM) images. In the past years, this method was successfully applied to nanocrystals of arbitrary shape, size, and composition (and its high accuracy and precision has been demonstrated). However, the counting results obtained from this statistical framework are so far presented without a visualization of the actual uncertainty about this estimate. In this paper, we present three approaches that can be used to represent counting results together with their statistical error, and discuss which approach is most suited for further use based on simulations and an experimental ADF STEM image.

Abstract: A small sample tilt away from a main zone axis orientation results in an elongation of the atomic columns in ADF STEM images. An often posed research question is therefore whether the ADF STEM image intensities of tilted nanomaterials should be quantified using a parametric imaging model consisting of elliptical rather than the currently used symmetrical peaks. To this purpose, simulated ADF STEM images corresponding to different amounts of sample tilt are studied using a parametric imaging model that consists of superimposed 2D elliptical Gaussian peaks on the one hand and symmetrical Gaussian peaks on the other hand. We investigate the quantification of structural parameters such as atomic column positions and scattering cross sections using both parametric imaging models. In this manner, we quantitatively study what can be gained from this elliptical model for quantitative ADF STEM, despite the increased parameter space and computational effort. Although a qualitative improvement can be achieved, no significant quantitative improvement in the estimated structure parameters is achieved by the elliptical model as compared to the symmetrical model. The decrease in scattering cross sections with increasing sample tilt is even identical for both types of parametric imaging models. This impedes direct comparison with zone axis image simulations. Nonetheless, we demonstrate how reliable atom-counting can still be achieved in the presence of small sample tilt.

Abstract: A hybrid statistics-simulations based method for atom-counting from annular dark field scanning transmission electron microscopy (ADF STEM) images of monotype crystalline nanostructures is presented. Different atom-counting methods already exist for model-like systems. However, the increasing relevance of radiation damage in the study of nanostructures demands a method that allows atom-counting from low dose images with a low signal-to-noise ratio. Therefore, the hybrid method directly includes prior knowledge from image simulations into the existing statistics-based method for atom-counting, and accounts in this manner for possible discrepancies between actual and simulated experimental conditions. It is shown by means of simulations and experiments that this hybrid method outperforms the statistics-based method, especially for low electron doses and small nanoparticles. The analysis of a simulated low dose image of a small nanoparticle suggests that this method allows for far more reliable quantitative analysis of beam-sensitive materials.

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 neighbouring columns, enabling the analysis of a large field of view. For this algorithm, the accuracy and precision with which measurements for the atomic column positions and scattering cross-sections from annular dark field (ADF) STEM images can be estimated, has been investigated. The highest attainable precision is reached even for low dose images. Furthermore, the advantages of the model-based approach taking into account overlap between neighbouring columns are highlighted. This is done for the estimation of the distance between two neighbouring columns as a function of their distance and for the estimation of the scattering cross-section which is compared to the integrated intensity from a Voronoi cell. To provide end-users this well-established quantification method, a user friendly program, StatSTEM, is developed which is freely available under a GNU public license.

Abstract: In this paper, it has been investigated how to measure atom column positions as accurately and precisely as possible using a focal series of images. In theory, it is expected that the precision would considerably improve using a maximum likelihood estimator based on the full series of focal images. As such, the theoretical lower bound on the variances of the unknown atom column positions can be attained. However, this approach is numerically demanding. Therefore, maximum likelihood estimation has been compared with the results obtained by fitting a model to a reconstructed exit wave rather than to the full series of focal images. Hence, a real space model-based exit wave reconstruction technique based on the channelling theory is introduced. Simulations show that the reconstructed complex exit wave contains the same amount of information concerning the atom column positions as the full series of focal images. Only for thin samples, which act as weak phase objects, this information can be retrieved from the phase of the reconstructed complex exit wave.

Abstract: In the present paper, a statistical model-based method to count the number of atoms of monotype crystalline nanostructures from high resolution high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) images is discussed in detail together with a thorough study on the possibilities and inherent limitations. In order to count the number of atoms, it is assumed that the total scattered intensity scales with the number of atoms per atom column. These intensities are quantitatively determined using model-based statistical parameter estimation theory. The distribution describing the probability that intensity values are generated by atomic columns containing a specific number of atoms is inferred on the basis of the experimental scattered intensities. Finally, the number of atoms per atom column is quantified using this estimated probability distribution. The number of atom columns available in the observed STEM image, the number of components in the estimated probability distribution, the width of the components of the probability distribution, and the typical shape of a criterion to assess the number of components in the probability distribution directly affect the accuracy and precision with which the number of atoms in a particular atom column can be estimated. It is shown that single atom sensitivity is feasible taking the latter aspects into consideration.