Unified fast reconstruction algorithm for conventional, phase-contrast, and diffraction tomography.

A unified method for three-dimensional reconstruction of objects from transmission images collected at multiple illumination directions is described. The method may be applicable to experimental conditions relevant to absorption-based, phase-contrast, or diffraction imaging using x rays, electrons, and other forms of penetrating radiation or matter waves. Both the phase retrieval (also known as contrast transfer function correction) and the effect of Ewald sphere curvature (in the cases with a shallow depth of field and significant in-object diffraction) are incorporated in the proposed algorithm and can be taken into account. Multiple scattering is not treated explicitly but can be mitigated as a result of angular averaging that constitutes an essential feature of the method. The corresponding numerical algorithm is based on three-dimensional gridding which allows for fast computational implementation, including a straightforward parallelization. The algorithm can be used with any scanning geometry involving plane-wave illumination. A software code implementing the proposed algorithm has been developed, tested on simulated and experimental image data, and made publicly available.

Method for virtual optical sectioning and tomography utilizing shallow depth of field.

A method is proposed for high-resolution, three-dimensional reconstruction of internal structures of objects from planar transmission images. The described approach can be used with any form of radiation or matter waves, in principle, provided that the depth of field is smaller than the thickness of the sample. The physical optics basis for the method is elucidated, and the reconstruction algorithm is presented in detail. A simulated example demonstrates an application of the method to three-dimensional electron transmission imaging of a nanoparticle under realistic radiation dose and spatial resolution constraints. It is envisaged that the method can be applicable in high-resolution transmission electron microscopy, soft x-ray microscopy, ultrasound imaging, and other areas.

A Method for High-Resolution Three-Dimensional Reconstruction with Ewald Sphere Curvature Correction from Transmission Electron Images.

A method for three-dimensional reconstruction of objects from defocused images collected at multiple illumination directions in high-resolution transmission electron microscopy is presented. The method effectively corrects for the Ewald sphere curvature by taking into account the in-particle propagation of the electron beam. Numerical simulations demonstrate that the proposed method is capable of accurately reconstructing biological molecules or nanoparticles from high-resolution defocused images under conditions achievable in single-particle electron cryo-microscopy or electron tomography with realistic radiation doses, non-trivial aberrations, multiple scattering, and other experimentally relevant factors. The physics of the method is based on the well-known Diffraction Tomography formalism, but with the phase-retrieval step modified to include a conjugation of the phase (i.e., multiplication of the phase by a negative constant). At each illumination direction, numerically backpropagating the beam with the conjugated phase produces maximum contrast at the location of individual atoms in the molecule or nanoparticle. The resultant algorithm, Conjugated Holographic Reconstruction, can potentially be incorporated into established software tools for single-particle analysis, such as, for example, RELION or FREALIGN, in place of the conventional contrast transfer function correction procedure, in order to account for the Ewald sphere curvature and improve the spatial resolution of the three-dimensional reconstruction.

Scattering Matrix Determination in Crystalline Materials from 4D Scanning Transmission Electron Microscopy at a Single Defocus Value.

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.

Optimizing Experimental Conditions for Accurate Quantitative Energy-Dispersive X-ray Analysis of Interfaces at the Atomic Scale.

The invention of silicon drift detectors has resulted in an unprecedented improvement in detection efficiency for energy-dispersive X-ray (EDX) spectroscopy in the scanning transmission electron microscope. The result is numerous beautiful atomic-scale maps, which provide insights into the internal structure of a variety of materials. However, the task still remains to understand exactly where the X-ray signal comes from and how accurately it can be quantified. Unfortunately, when crystals are aligned with a low-order zone axis parallel to the incident beam direction, as is necessary for atomic-resolution imaging, the electron beam channels. When the beam becomes localized in this way, the relationship between the concentration of a particular element and its spectroscopic X-ray signal is generally nonlinear. Here, we discuss the combined effect of both spatial integration and sample tilt for ameliorating the effects of channeling and improving the accuracy of EDX quantification. Both simulations and experimental results will be presented for a perovskite-based oxide interface. We examine how the scattering and spreading of the electron beam can lead to erroneous interpretation of interface compositions, and what approaches can be made to improve our understanding of the underlying atomic structure.

Inelastic Scattering in Electron Backscatter Diffraction and Electron Channeling Contrast Imaging.

Electron backscatter diffraction (EBSD) and electron channeling contrast imaging (ECCI) are used to extract crystallographic information from bulk samples, such as their crystal structure and orientation as well as the presence of any dislocation and grain boundary defects. These techniques rely on the backscattered electron signal, which has a large distribution in electron energy. Here, the influence of plasmon excitations on EBSD patterns and ECCI dislocation images is uncovered by multislice simulations including inelastic scattering. It is shown that the Kikuchi band contrast in an EBSD pattern for silicon is maximum at small energy loss (i.e., few plasmon scattering events following backscattering), consistent with previous energy-filtered EBSD measurements. On the other hand, plasmon excitation has very little effect on the ECCI image of a dislocation. These results are explained by examining the role of the characteristic plasmon scattering angle on the intrinsic contrast mechanisms in EBSD and ECCI.

Multi-modal and multi-scale non-local means method to analyze spectroscopic datasets.

A multi-modal and multi-scale non-local means (M3S-NLM) method is proposed to extract atomically resolved spectroscopic maps from low signal-to-noise (SNR) datasets recorded with a transmission electron microscope. This method improves upon previously tested denoising techniques as it takes into account the correlation between the dark-field signal recorded simultaneously with the spectroscopic dataset without compromising on the spatial resolution. The M3S-NLM method was applied to electron energy dispersive X-ray and electron-energy-loss spectroscopy (EELS) datasets. We illustrate the retrieval of the atomic scale diffusion process in an Al1-xInxN alloy grown on GaN and the surface oxidation state of perovskite nanocatalysts. The improved SNR of the EELS dataset also allows the retrieval of atomically resolved oxidation maps considering the fine structure absorption edge of LaMnO3 nanoparticles.

Probing the effect of electron channelling on atomic resolution energy dispersive X-ray quantification.

Advances in microscope stability, aberration correction and detector design now make it readily possible to achieve atomic resolution energy dispersive X-ray mapping for dose resilient samples. These maps show impressive atomic-scale qualitative detail as to where the elements reside within a given sample. Unfortunately, while electron channelling is exploited to provide atomic resolution data, this very process makes the images rather more complex to interpret quantitatively than if no electron channelling occurred. Here we propose small sample tilt as a means for suppressing channelling and improving quantification of composition, whilst maintaining atomic-scale resolution. Only by knowing composition and thickness of the sample is it possible to determine the atomic configuration within each column. The effects of neighbouring atomic columns with differing composition and of residual channelling on our ability to extract exact column-by-column composition are also discussed.

Accurate Nanoscale Crystallography in Real-Space Using Scanning Transmission Electron Microscopy.

Here, we report reproducible and accurate measurement of crystallographic parameters using scanning transmission electron microscopy. This is made possible by removing drift and residual scan distortion. We demonstrate real-space lattice parameter measurements with <0.1% error for complex-layered chalcogenides Bi2Te3, Bi2Se3, and a Bi2Te2.7Se0.3 nanostructured alloy. Pairing the technique with atomic resolution spectroscopy, we connect local structure with chemistry and bonding. Combining these results with density functional theory, we show that the incorporation of Se into Bi2Te3 causes charge redistribution that anomalously increases the van der Waals gap between building blocks of the layered structure. The results show that atomic resolution imaging with electrons can accurately and robustly quantify crystallography at the nanoscale.

Fast deterministic ptychographic imaging using X-rays.

We present a deterministic approach to the ptychographic retrieval of the wave at the exit surface of a specimen of condensed matter illuminated by X-rays. The method is based on the solution of an overdetermined set of linear equations, and is robust to measurement noise. The set of linear equations is efficiently solved using the conjugate gradient least-squares method implemented using fast Fourier transforms. The method is demonstrated using a data set obtained from a gold-chromium nanostructured test object. It is shown that the transmission function retrieved by this linear method is quantitatively comparable with established methods of ptychography, with a large decrease in computational time, and is thus a good candidate for real-time reconstruction.

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.

Contrast in atomically resolved EF-SCEM imaging.

Energy-filtered scanning confocal electron microscopy (EF-SCEM) is a technique that uses the reduced depth of field of an aberration-corrected transmission electron microscope to provide three-dimensional (3D) compositional information. Using a silicon sample in the <110> orientation, we show that EF-SCEM image data can be recorded that shows lattice resolution in the plane perpendicular to the incident beam direction. The confocal effect is demonstrated through the reduction of the mean intensity as the confocal plane is displaced from the sample mid-plane, unlike optical sectioning in high-angle annular dark-field scanning transmission electron microscopy (STEM). Simulations of the EF-SCEM data show agreement with the experimental data, and allow the interpretability of the data to be explored. The effects of channelling, absorption and delocalisation complicate the quantitative and qualitative interpretation of the data, highlighting the need for matching to simulations. Finally the effects of the finite detector pin-hole aperture size are explored, and we show that the EF-SCEM contrast in the plane perpendicular to the beam direction starts to resemble that of a STEM spectrum imaging experiment as the aperture size increases.

Three-dimensional imaging of individual dopant atoms in SrTiO3.

We report on three-dimensional (3D) imaging of individual Gd dopant atoms in a thin (∼2.3 nm) foil of SrTiO3, using quantitative scanning transmission electron microscopy. Uncertainties in the depth positions of individual dopants are less than 1 unit cell. The overall dopant concentration measured from atom column intensities agrees quantitatively with electrical measurements. The method is applied to analyze the 3D arrangement of dopants within small clusters containing 4-5 Gd atoms.

Atom-scale ptychographic electron diffractive imaging of boron nitride cones.

Ptychographic coherent diffractive imaging (CDI) has been extensively applied using both x rays and electrons. The extension to atomic resolution has been elusive. This Letter demonstrates ptychographic electron diffractive imaging at atomic resolution, permitting identification of structure in a boron nitride helical cone at a resolution of order 1 Å, beyond that of comparative Z-contrast images. A scanning transmission electron microscope is used to create a diverging illumination in a defocused Fresnel CDI geometry, providing a robust strategy leading to a unique solution.

Annular electron energy-loss spectroscopy in the scanning transmission electron microscope.

We study atomic-resolution annular electron energy-loss spectroscopy (AEELS) in scanning transmission electron microscopy (STEM) imaging with experiments and numerical simulations. In this technique the central part of the bright field disk is blocked by a beam stop, forming an annular entry aperture to the spectrometer. The EELS signal thus arises only from electrons scattered inelastically to angles defined by the aperture. It will be shown that this method is more robust than conventional EELS imaging to variations in specimen thickness and can also provide higher spatial resolution. This raises the possibility of lattice resolution imaging of lighter elements or ionization edges previously considered unsuitable for EELS imaging.

Bright-field scanning confocal electron microscopy using a double aberration-corrected transmission electron microscope.

Scanning confocal electron microscopy (SCEM) offers a mechanism for three-dimensional imaging of materials, which makes use of the reduced depth of field in an aberration-corrected transmission electron microscope. The simplest configuration of SCEM is the bright-field mode. In this paper we present experimental data and simulations showing the form of bright-field SCEM images. We show that the depth dependence of the three-dimensional image can be explained in terms of two-dimensional images formed in the detector plane. For a crystalline sample, this so-called probe image is shown to be similar to a conventional diffraction pattern. Experimental results and simulations show how the diffracted probes in this image are elongated in thicker crystals and the use of this elongation to estimate sample thickness is explored.

Standardless atom counting in scanning transmission electron microscopy.

We demonstrate that high-angle annular dark-field imaging in scanning transmission electron microscopy allows for quantification of the number and location of all atoms in a three-dimensional, crystalline, arbitrarily shaped specimen without the need for a calibration standard. We show that the method also provides for an approach to directly measure the finite effective source size of a scanning transmission electron microscope.