A holographic method to measure the source size broadening in STEM

Exact inversion of partially coherent dynamical electron scattering for picometric structure retrieval

The greatly nonlinear diffraction of high-energy electron probes focused to subatomic diameters frustrates the direct inversion of ptychographic data sets to decipher the atomic structure. Several iterative algorithms have been proposed to yield atomically-resolved phase distributions within slices of a 3D specimen, corresponding to the scattering centers of the electron wave. By pixelwise phase retrieval, current approaches do not only involve orders of magnitude more free parameters than necessary, but also neglect essential details of scattering physics such as the atomistic nature of the specimen and thermal effects. Here, we introduce a parametrized, fully differentiable scheme employing neural network concepts which allows the inversion of ptychographic data by means of entirely physical quantities. Omnipresent thermal diffuse scattering in thick specimens is treated accurately using frozen phonons, and atom types, positions and partial coherence are accounted for in the inverse model as relativistic scattering theory demands. Our approach exploits 4D experimental data collected in an aberration-corrected momentum-resolved scanning transmission electron microscopy setup. Atom positions in a 20 nm thick PbZr0.2Ti0.8O3 ferroelectric are measured with picometer precision, including the discrimination of different atom types and positions in mixed columns.

Reliable phase quantification in focused probe electron ptychography of thin materials

Electron ptychography provides highly sensitive, dose efficient phase images which can be corrected for aberrations after the data has been acquired. This is crucial when very precise quantification is required, such as with sensitivity to charge transfer due to bonding. Drift can now be essentially eliminated as a major impediment to focused probe ptychography, which benefits from the availability of easily interpretable simultaneous Z-contrast imaging. However challenges have remained when quantifying the ptychographic phases of atomic sites. The phase response of a single atom has a negative halo which can cause atoms to reduce in phase when brought closer together. When unaccounted for, as in integrating methods of quantification, this effect can completely obscure the effects of charge transfer. Here we provide a new method of quantification that overcomes this challenge, at least for 2D materials, and is robust to experimental parameters such as noise, sample tilt.

Cost and Capability Compromises in STEM Instrumentation for Low-Voltage Imaging

Low-voltage transmission electron microscopy (≤80 kV) has many applications in imaging beam-sensitive samples, such as metallic nanoparticles, which may become damaged at higher voltages. To improve resolution, spherical aberration can be corrected for in a scanning transmission electron microscope (STEM); however, chromatic aberration may then dominate, limiting the ultimate resolution of the microscope. Using image simulations, we examine how a chromatic aberration corrector, different objective lenses, and different beam energy spreads each affect the image quality of a gold nanoparticle imaged at low voltages in a spherical aberration-corrected STEM. A quantitative analysis of the simulated examples can inform the choice of instrumentation for low-voltage imaging. We here demonstrate a methodology whereby the optimum energy spread to operate a specific STEM can be deduced. This methodology can then be adapted to the specific sample and instrument of the reader, enabling them to make an informed economical choice as to what would be most beneficial for their STEM in the cost-conscious landscape of scientific infrastructure.

Prismatic 2.0 - Simulation software for scanning and high resolution transmission electron microscopy (STEM and HRTEM)

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.

The maximum a posteriori probability rule for atom column detection from HAADF STEM images

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.

Atomic resolution elemental mapping using energy-filtered imaging scanning transmission electron microscopy with chromatic aberration correction

This paper addresses a novel approach to atomic resolution elemental mapping, demonstrating a method that produces elemental maps with a similar resolution to the established method of electron energy-loss spectroscopy in scanning transmission electron microscopy. Dubbed energy-filtered imaging scanning transmission electron microscopy (EFISTEM) this mode of imaging is, by the quantum mechanical principle of reciprocity, equivalent to tilting the probe in energy-filtered transmission electron microscopy (EFTEM) through a cone and incoherently averaging the results. In this paper we present a proof-of-principle EFISTEM experimental study on strontium titanate. The present approach, made possible by chromatic aberration correction, has the advantage that it provides elemental maps which are immune to spatial incoherence in the electron source, coherent aberrations in the probe-forming lens and probe jitter. The veracity of the experiment is supported by quantum mechanical image simulations, which provide an insight into the image-forming process. Elemental maps obtained in EFTEM suffer from the effect known as preservation of elastic contrast, which, for example, can lead to a given atomic species appearing to be in atomic columns where it is not to be found. EFISTEM very substantially reduces the preservation of elastic contrast and yields images which show stability of contrast with changing thickness. The experimental application is demonstrated in a proof-of-principle study on strontium titanate.

Ultra-high resolution electron microscopy

The last two decades have seen dramatic advances in the resolution of the electron microscope brought about by the successful correction of lens aberrations that previously limited resolution for most of its history. We briefly review these advances, the achievement of sub-Ångstrom resolution and the ability to identify individual atoms, their bonding configurations and even their dynamics and diffusion pathways. We then present a review of the basic physics of electron scattering, lens aberrations and their correction, and an approximate imaging theory for thin crystals which provides physical insight into the various different imaging modes. Then we proceed to describe a more exact imaging theory starting from Yoshioka's formulation and covering full image simulation methods using Bloch waves, the multislice formulation and the frozen phonon/quantum excitation of phonons models. Delocalization of inelastic scattering has become an important limiting factor at atomic resolution. We therefore discuss this issue extensively, showing how the full-width-half-maximum is the appropriate measure for predicting image contrast, but the diameter containing 50% of the excitation is an important measure of the range of the interaction. These two measures can differ by a factor of 5, are not a simple function of binding energy, and full image simulations are required to match to experiment. The Z-dependence of annular dark field images is also discussed extensively, both for single atoms and for crystals, and we show that temporal incoherence must be included accurately if atomic species are to be identified through matching experimental intensities to simulations. Finally we mention a few promising directions for future investigation.

Influence of spatial and temporal coherences on atomic resolution high angle annular dark field imaging

Aberration-corrected (scanning) transmission electron microscopy ((S)TEM) has become a widely used technique when information on the chemical composition is sought on an atomic scale. To extract the desired information, complementary simulations of the scattering process are inevitable. Often the partial spatial and temporal coherences are neglected in the simulations, although they can have a huge influence on the high resolution images. With the example of binary gallium phosphide (GaP) we elucidate the influence of the source size and shape as well as the chromatic aberration on the high angle annular dark field (HAADF) intensity. We achieve a very good quantitative agreement between the frozen phonon simulation and experiment for different sample thicknesses when a Lorentzian source distribution is assumed and the effect of the chromatic aberration is considered. Additionally the influence of amorphous layers introduced by the preparation of the TEM samples is discussed. Taking into account these parameters, the intensity in the whole unit cell of GaP, i.e. at the positions of the different atomic columns and in the region between them, is described correctly. With the knowledge of the decisive parameters, the determination of the chemical composition of more complex, multinary materials becomes feasible.

Interfacial Atomic Structure of Twisted Few-Layer Graphene

A twist in bi- or few-layer graphene breaks the local symmetry, introducing a number of intriguing physical properties such as opening new bandgaps. Therefore, determining the twisted atomic structure is critical to understanding and controlling the functional properties of graphene. Combining low-angle annular dark-field electron microscopy with image simulations, we directly determine the atomic structure of twisted few-layer graphene in terms of a moiré superstructure which is parameterized by a single twist angle and lattice constant. This method is shown to be a powerful tool for accurately determining the atomic structure of two-dimensional materials such as graphene, even in the presence of experimental errors. Using coincidence-site-lattice and displacement-shift-complete theories, we show that the in-plane translation state between layers is not a significant structure parameter, explaining why the present method is adequate not only for bilayer graphene but also a few-layered twisted graphene.

Quantitative annular dark-field imaging of single-layer graphene-II: atomic-resolution image contrast

We have investigated how accurately atomic-resolution annular dark-field (ADF) images match between experiments and simulations to conduct more reliable crystal structure analyses. Quantitative ADF imaging, in which the ADF intensity at each pixel represents the fraction of the incident probe current, allows us to perform direct comparisons with simulations without the use of fitting parameters. Although the conventional comparison suffers from experimental uncertainties such as an amorphous surface layer and specimen thickness, in this study we eliminated such uncertainties by using a single-layer graphene as a specimen. Furthermore, to reduce image distortion and shot noises in experimental images, multiple acquisitions with drift correction were performed, and the atomic ADF contrast was quantitatively acquired. To reproduce the experimental ADF contrast, we used three distribution functions as the effective source distribution in simulations. The optimum distribution function and its full-width at half-maximum were evaluated by measuring the residuals between the experimental and simulated images. It was found that the experimental images could be explained well by a linear combination of a Gaussian function and a Lorentzian function with a longer tail than the Gaussian function.

Quantitative STEM normalisation: The importance of the electron flux

Annular dark-field (ADF) scanning transmission electron microscopy (STEM) has become widely used in quantitative studies based on the opportunity to directly compare experimental and simulated images. This comparison merely requires the experimental data to be normalised and expressed in units of 'fractional beam-current'. However, inhomogeneities in the response of electron detectors can complicate this normalisation. The quantification procedure becomes both experiment and instrument specific, requiring new simulations for the particular response of each instrument's detector, and for every camera-length used. This not only impedes the comparison between different instruments and research groups, but can also be computationally very time consuming. Furthermore, not all image simulation methods allow for the inclusion of an inhomogeneous detector response. In this work, we propose an alternative method for normalising experimental data in order to compare these with simulations that consider a homogeneous detector response. To achieve this, we determine the electron flux distribution reaching the detector by means of a camera-length series or a so-called atomic column cross-section averaged convergent beam electron diffraction (XSACBED) pattern. The result is then used to determine the relative weighting of the detector response. Here we show that the results obtained by this new electron flux weighted (EFW) method are comparable to the currently used method, while considerably simplifying the needed simulation libraries. The proposed method also allows one to obtain a metric that describes the quality of the detector response in comparison with the 'ideal' detector response.

Local observation of the site occupancy of Mn in a MnFePSi compound

MnFePSi compounds are promising materials for magnetic refrigeration as they exhibit a giant magnetocaloric effect. From first principles calculations and experiments on bulk materials, it has been proposed that this is due to the Mn and Fe atoms preferentially occupying two different sites within the atomic lattice. A recently developed technique was used to deconvolve the obscuring effects of both multiple elastic scattering and thermal diffuse scattering of the probe in an atomic resolution electron energy-loss spectroscopy investigation of a MnFePSi compound. This reveals, unambiguously, that the Mn atoms preferentially occupy the 3g site in a hexagonal crystal structure, confirming the theoretical predictions. After deconvolution, the data exhibit a difference in the Fe L_{2,3} ratio between the 3f and 3g sites consistent with differences in magnetic moments calculated from first principles, which are also not observed in the raw data.

Conventional transmission electron microscopy imaging beyond the diffraction and information limits

There are mainly two complementary imaging modes in transmission electron microscopy (TEM): Conventional TEM (CTEM) and scanning TEM (STEM). In the CTEM mode the specimen is illuminated with a plane electron wave, and the direct image formed by the objective lens is recorded in the image plane. STEM is based on scanning the specimen surface with a focused electron beam and collecting scattered electrons with an extended disk or ring-shaped detector. Here we show that combination of CTEM imaging with STEM illumination generally allows extending the point resolution of CTEM imaging beyond the diffraction limit. This new imaging mode improves imaging characteristics, is more robust against chromatic aberration, exhibits direct structural imaging with superior precision, visualizes light elements with excellent contrast, and even allows us to overcome the conventional information limit of a microscope.

Practical aspects of removing the effects of elastic and thermal diffuse scattering from spectroscopic data for single crystals

A method to remove the effects of elastic and thermal diffuse scattering (TDS) of the incident electron probe from electron energy-loss and energy-dispersive X-ray spectroscopy data for atomically resolved spectrum images of single crystals of known thickness is presented. By calculating the distribution of the probe within a specimen of known structure, it is possible to deconvolve the channeling of the probe and TDS from experimental data by reformulating the inelastic cross-section as an inverse problem. In electron energy-loss spectroscopy this allows valid comparisons with first principles fine-structure calculations to be made. In energy-dispersive X-ray spectroscopy, direct compositional analyses such as ζ-factor and Cliff-Lorimer k-factor analysis can be performed without the complications of channeling and TDS. We explore in detail how this method can be incorporated into existing multislice programs, and demonstrate practical considerations in implementing this method using a simulated test specimen. We show the importance of taking into account the scattering of the probe in k-factor analysis in a zone axis orientation. The applicability and limitations of the method are discussed.

Three-dimensional optical transfer functions in the aberration-corrected scanning transmission electron microscope

In the scanning transmission electron microscope, hardware aberration correctors can now correct for the positive spherical aberration of round electron lenses. These correctors make use of nonround optics such as hexapoles or octupoles, leading to the limiting aberrations often being of a nonround type. Here we explore the effect of a number of potential limiting aberrations on the imaging performance of the scanning transmission electron microscope through their resulting optical transfer functions. In particular, the response of the optical transfer function to changes in defocus are examined, given that this is the final aberration to be tuned just before image acquisition. The resulting three-dimensional optical transfer functions also allow an assessment of the performance of a system for focal-series experiments or optical sectioning applications.