Phase Object Approximation Research Articles

Dose-efficient phase-contrast imaging of thick weak phase objects via OBF STEM using a pixelated detector.

Optimum bright-field scanning transmission electron microscopy (OBF STEM) is a recently developed low-dose imaging technique that uses a segmented or pixelated detector. While we previously reported that OBF STEM with a segmented detector has a higher efficiency than conventional STEM techniques such as annular bright field (ABF), the imaging efficiency is expected to be further improved by using a pixelated detector. In this study, we adopted a pixelated detector for the OBF technique and investigated the imaging characteristics. Because OBF imaging is based on the thick weak phase object approximation (tWPOA), a non-zero crystalline sample thickness is considered in addition to the conventional WPOA, where the pixelated OBF method can be regarded as the theoretical extension of single side band (SSB) ptychography. Thus, we compared these two techniques via signal-to-noise ratio transfer functions (SNRTFs), multi-slice image simulations, and experiments, showing how the OBF technique can improve dose efficiency from the conventional WPOA-based ptychographic imaging.

Differential phase contrast from electrons that cause inner shell ionization

Differential Phase Contrast (DPC) imaging, in which deviations in the bright field beam are in proportion to the electric field, has been extensively studied in the context of pure elastic scattering. Here we discuss differential phase contrast formed from core-loss scattered electrons, i.e. those that have caused inner shell ionization of atoms in the specimen, using a transition potential approach for which we study the number of final states needed for a converged calculation. In the phase object approximation, we show formally that differential phase contrast formed from core-loss scattered electrons is mainly a result of preservation of elastic contrast. Through simulation we demonstrate that whether the inelastic DPC images show element selective contrast depends on the spatial range of the ionization interaction, and specifically that when the energy loss is low the delocalisation can lead to contributions to the contrast from atoms other than that ionized. We further show that inelastic DPC images remain robustly interpretable to larger thicknesses than is the case for elastic DPC images, owing to the incoherence of the inelastic wavefields, though subtleties due to channelling remain. Lastly, we show that while a very high dose will be needed for sufficient counting statistics to discern differential phase contrast from core-loss scattered electrons, there is some enhancement of the signal-to-noise ratio with thickness that makes inelastic DPC imaging more achievable for thicker samples.

Dynamical scattering in ice-embedded proteins in conventional and scanning transmission electron microscopy.

Structure determination of biological macromolecules using cryogenic electron microscopy is based on applying the phase object (PO) assumption and the weak phase object (WPO) approximation to reconstruct the 3D potential density of the molecule. To enhance the understanding of image formation of protein complexes embedded in glass-like ice in a transmission electron microscope, this study addresses multiple scattering in tobacco mosaic virus (TMV) specimens. This includes the propagation inside the molecule while also accounting for the effect of structural noise. The atoms in biological macromolecules are light but are distributed over several nanometres. Commonly, PO and WPO approximations are used in most simulations and reconstruction models. Therefore, dynamical multislice simulations of TMV specimens embedded in glass-like ice were performed based on fully atomistic molecular-dynamics simulations. In the first part, the impact of multiple scattering is studied using different numbers of slices. In the second part, different sample thicknesses of the ice-embedded TMV are considered in terms of additional ice layers. It is found that single-slice models yield full frequency transfer up to a resolution of 2.5 Å, followed by attenuation up to 1.4 Å. Three slices are sufficient to reach an information transfer up to 1.0 Å. In the third part, ptychographic reconstructions based on scanning transmission electron microscopy (STEM) and single-slice models are compared with conventional TEM simulations. The ptychographic reconstructions do not need the deliberate introduction of aberrations, are capable of post-acquisition aberration correction and promise benefits for information transfer, especially at resolutions beyond 1.8 Å.

Phase Object Reconstruction for 4D-STEM using Deep Learning

Abstract In this study, we explore the possibility to use deep learning for the reconstruction of phase images from 4D scanning transmission electron microscopy (4D-STEM) data. The process can be divided into two main steps. First, the complex electron wave function is recovered for a convergent beam electron diffraction pattern (CBED) using a convolutional neural network (CNN). Subsequently, a corresponding patch of the phase object is recovered using the phase object approximation. Repeating this for each scan position in a 4D-STEM dataset and combining the patches by complex summation yields the full-phase object. Each patch is recovered from a kernel of 3×3 adjacent CBEDs only, which eliminates common, large memory requirements and enables live processing during an experiment. The machine learning pipeline, data generation, and the reconstruction algorithm are presented. We demonstrate that the CNN can retrieve phase information beyond the aperture angle, enabling super-resolution imaging. The image contrast formation is evaluated showing a dependence on the thickness and atomic column type. Columns containing light and heavy elements can be imaged simultaneously and are distinguishable. The combination of super-resolution, good noise robustness, and intuitive image contrast characteristics makes the approach unique among live imaging methods in 4D-STEM.

The Effect of Dynamical Scattering on Single-plane Phase Retrieval in Electron Ptychography

Abstract Segmented and pixelated detectors on scanning transmission electron microscopes enable the complex specimen transmission function to be reconstructed. Imaging the transmission function is key to interpreting the electric and magnetic properties of the specimen, and as such four-dimensional scanning transmission electron microscopy (4D-STEM) imaging techniques are crucial for our understanding of functional materials. Many of the algorithms used in the reconstruction of the transmission function rely on the multiplicative approximation and the (weak) phase object approximation, which are not valid for many materials, particularly at high resolution. Herein, we study the breakdown of simple phase imaging in thicker samples. We demonstrate the behavior of integrated center of mass imaging, single-side band ptychography, and Wigner distribution deconvolution over a thickness series of simulated GaN 4D-STEM datasets. We further give guidance as to the optimal focal conditions for obtaining a more interpretable dataset using these algorithms.

Potentials of individual atoms by convergent beam electron diffraction

In convergent beam electron diffraction (CBED) on two-dimensional (2D) materials, the intensity distributions within the individual CBED spots map the local atomic arrangements within the probed region. In this study we demonstrate that the average intensities within the CBED spots essentially depend on the scattering parameters by a single atom, thus, offering the possibility of the direct measurement of such parameters. Scattering potential of an individual atom can be approximated by the Gaussian function and its parameters, the standard deviation and the maximal phase shift, can be recovered from the ratio of the intensities in the zero- and higher orders spots in CBED pattern. In order to demonstrate this, we simulated CBED patterns and extracted the atomic scattering parameters from such patterns. In the simulated examples, no weak phase object approximation is applied and the proposed method provides accurate results even for materials with large phase shifts up to 1.5 rad, as for example tungsten. The scattering parameters recovered from the experimental CBED patterns of graphene and twisted graphene-hBN structure show good agreement with the theoretically obtained values.

Toward complex observation in electron microscopy using two-dimensional electron detector coupled with phase plate STEM.

A two-dimensional (2D) detector was used to construct phase plate STEM (P-STEM) images. Phase-contrast can be enhanced by the electron intensity inside the hole region of a thin film phase plate. The electron intensity outside the hole region also provides a dark image contrast, which is inconsistent with the weak phase object approximation. We consider that both images have scattering effects that provide a dark contrast. Therefore, scattering contrast was derived by summing these two images, and scattering effects were subtracted from each image to display negative and positive phase contrast. The resultant images are consistent with the weak phase object approximation. These results propose separating scattering (electron amplitude) and phase-contrast (electron phase) using P-STEM, along with a two-dimensional electron detector.

Ultra-high contrast STEM imaging for segmented/pixelated detectors by maximizing the signal-to-noise ratio

Atomic-resolution low-dose imaging for beam-sensitive materials is one of the most challenging topics in electron microscopy research. In this study, we theoretically developed a new scanning transmission electron microscopy (STEM) imaging technique by maximizing the signal-to-noise ratio of an obtainable image under the weak phase object approximation (WPOA), which we will call optimum bright-field (OBF) imaging. OBF images are obtained by processing multiple images acquired by segmented/pixelated detectors through complex frequency filtering. This method has been confirmed through a systematic image simulation to be highly dose-efficient. Furthermore, we experimentally demonstrate the high dose efficiency of the OBF technique by visualizing the atomic structure in a lithium-ion battery material using a high-speed segmented detector. Furthermore, it was shown that OBF imaging is usable for real-time imaging, which makes low-dose observations of beam-sensitive materials much easier to achieve.

Asymmetry and 4D-STEM: When the Phase Object Approximation Is Qualitatively Incorrect

Asymmetry and 4D-STEM: When the Phase Object Approximation Is Qualitatively Incorrect

What Do You Get If You Cross a Phase Object Approximation with a Dynamically Scattering Sample?

What Do You Get If You Cross a Phase Object Approximation with a Dynamically Scattering Sample?

Comparison of first moment STEM with conventional differential phase contrast and the dependence on electron dose

This study addresses the comparison of scanning transmission electron microscopy (STEM) measurements of momentum transfers using the first moment approach and the established method that uses segmented annular detectors. Using an ultrafast pixelated detector to acquire four-dimensional, momentum-resolved STEM signals, both the first moment calculation and the calculation of the differential phase contrast (DPC) signals are done for the same experimental data. In particular, we investigate the ability to correct the segment-based signal to yield a suitable approximation of the first moment for cases beyond the weak phase object approximation. It is found that the measurement of momentum transfers using segmented detectors can approach the first moment measurement as close as 0.13 h/nm in terms of a root mean square (rms) difference in 10 nm thick SrTiO3 for a detector with 16 segments. This amounts to 35% of the rms of the momentum transfers. In addition, we present a statistical analysis of the precision of first moment STEM as a function of dose. For typical experimental settings with recent hardware such as a Medipix3 Merlin camera attached to a probe-corrected STEM, we find that the precision of the measurement of momentum transfers stagnates above certain doses. This means that other instabilities such as specimen drift or scan noise have to be taken into account seriously for measurements that target, e.g., the detection of bonding effects in the charge density.

Large angle illumination enabling accurate structure reconstruction from thick samples in scanning transmission electron microscopy

Most reconstructions of the electrostatic potential of a specimen at atomic resolution assume a thin and weakly scattering sample, restricting accurate quantification to specimens only tens of Ångströms thick. We demonstrate that using large-angle-illumination scanning transmission electron microscopy (STEM)—a probe forming aperture with convergence angle larger than about 50 mrad—allows us to better meet the weak phase object approximation and thereby accurately reconstruct the electrostatic potential in samples thicker than the order of 100 Å.

Grating-based holographic diffraction methods for X-rays and neutrons: phase object approximation and dynamical theory

A method dubbed grating-based holography was recently used to determine the structure of colloidal fluids in the rectangular grooves of a diffraction grating from X-ray scattering measurements. Similar grating-based measurements have also been recently made with neutrons using a technique called spin-echo small-angle neutron scattering. The analysis of the X-ray diffraction data was done using an approximation that treats the X-ray phase change caused by the colloidal structure as a small perturbation to the overall phase pattern generated by the grating. In this paper, the adequacy of this weak phase approximation is explored for both X-ray and neutron grating holography. It is found that there are several approximations hidden within the weak phase approximation that can lead to incorrect conclusions from experiments. In particular, the phase contrast for the empty grating is a critical parameter. While the approximation is found to be perfectly adequate for X-ray grating holography experiments performed to date, it cannot be applied to similar neutron experiments because the latter technique requires much deeper grating channels.

Quantitative electric field mapping in thin specimens using a segmented detector: Revisiting the transfer function for differential phase contrast

Differential phase contrast in scanning transmission electron microscopy can visualize local electromagnetic fields inside specimens. The contrast derived from first moments, the so-called center of mass, of the diffraction patterns for each probe position can be quantitatively related to the local electromagnetic field under the phase object approximation. While only approximate first moments can be obtained with a segmented detector, in weak phase objects the fields can be accurately quantified on the basis of a phase contrast transfer function. Through systematic image simulations we further show that the quantification based on the approximated first moment is a good approximation also for strong phase objects.

Limitations of the linear and the projection approximations in three-dimensional transmission electron microscopy of fully hydrated proteins.

We establish expressions for the linear and quadratic terms in the series expansion of the phase and the phase and amplitude object description of imaging thin specimens by transmission electron microscopy. Based on these expressions we simulate the corresponding contributions to images of unstained protein complexes of varying thickness and arrive at an estimate for how much each term contributes to the contrast of the image. From this we can estimate a maximum specimen thickness for which the weak phase and the weak amplitude and phase object approximation (and therefore linear imaging) is still reasonably accurate. When discussing thick specimens it is also necessary to consider limitations due to describing the image as a filtered projection of the specimen, since the different layers of the specimen are not imaged with the same defocus value. We therefore compared simulations based on the projection approximation with the more accurate multislice model of image formation. However, we find that the errors due to nonlinear image contributions are greater than those due to the defocus gradient for the defocus values chosen for the simulations. Finally, we study how the discussed nonlinear image contributions and the defocus gradient affect the quality of three-dimensional reconstructions. We find that three-dimensional reconstructions reach high resolution when at the same time exhibiting localized systematic structural errors.

Atomic configurations of dislocation cores in a small-angle grain boundary of 3C-SiC film

[110] images are taken for 3C-SiC/(001)Si hetero epitaxial films containing small-angle grain boundaries by using a 200 kV LaB6 filament high-resolution transmission electron microscope. Deconvolution processing is performed to transform the experimental images which do not represent intuitively the projected crystal structure into structure images. First, Si and C atomic columns with a distance of 0.109 nm are resolved in a perfect structure image region, and then recognized from each other by analyzing the image contrast change with sample thickness based on the pseudo-weak phase object approximation. Subsequently, two complex dislocation cores located in the vicinity of small-angle grain boundaries are obtained at an atomic level, and the atomic structure models are constructed and confirmed by matching the experimental images with the simulated ones. Hence, the atomic configurations of dislocation cores are derived from only a single experimental image with the average structure of perfect crystal known in advance. The formation of small-angle grain boundaries in 3C-SiC/Si with the occurence of complex dislocations in their vicinity is discussed.

High-resolution STEM imaging with a quadrant detector--conditions for differential phase contrast microscopy in the weak phase object approximation.

Differential phase contrast is a contrast mechanism that can be utilized in the scanning transmission electron microscope (STEM) to determine the distribution of magnetic or electric fields. In practice, several different detector geometries can be used to obtain differential phase contrast. As recent high resolution differential phase contrast experiments with the STEM are focused on ring quadrant detectors, we evaluate the contrast transfer characteristics of different quadrant detector geometries, namely two ring quadrant detectors with different inner detector angles and a conventional quadrant detector, by calculating the corresponding phase gradient transfer functions. For an ideal microscope and a weak phase object, this can be done analytically. The calculated phase gradient transfer functions indicate that the barely illuminated ring quadrant detector setup used for imaging magnetic fields in the specimen reduces the resolution limit to about 2.5Å for an aberration corrected STEM. Our results show that the resolution can be drastically improved by using a conventional quadrant detector instead.

Random vs realistic amorphous carbon models for high resolution microscopy and electron diffraction

Amorphous carbon and amorphous materials in general are of particular importance for high resolution electron microscopy, either for bulk materials, generally covered with an amorphous layer when prepared by ion milling techniques, or for nanoscale objects deposited on amorphous substrates. In order to quantify the information of the high resolution images at the atomic scale, a structural modeling of the sample is necessary prior to the calculation of the electron wave function propagation. It is thus essential to be able to reproduce the carbon structure as close as possible to the real one. The approach we propose here is to simulate a realistic carbon from an energetic model based on the tight-binding approximation in order to reproduce the important structural properties of amorphous carbon. At first, we compare this carbon with the carbon obtained by randomly generating the carbon atom positions. In both cases, we discuss the limit thickness of the phase object approximation. In a second step, we show the influence of both carbons models on (i) the contrast of Cu, Ag, and Au single atoms deposited on carbon and (ii) the determination of the long-range order parameter in CoPt bimetallic nanoalloys.

Direct acquisition of interferogram by stage scanning in electron interferometry

We present an electron holography technique to acquire an interferogram, that is, cosine image of phase distribution. The interferogram is constructed by shifting the specimen in one direction with a stage-scanning system and acquiring line intensities of holograms. Taking line intensities eliminates the carrier fringes in the holograms and yields the interferogram. Under phase object approximation, the object phase can be readily obtained from the interferogram without any reconstruction procedure. The spatial resolution of phase is determined independently of the fringe spacing, overcoming the limitation of conventional techniques based on the Fourier transformation method.

Determining polarity and dislocation core structures at atomic level for epitaxial AlN/(0 0 0 1)6H-SiC from a single image in HRTEM

The polarity of epitaxial AlN film grown on (0001)6H-SiC and dislocation core structures in the film have been studied using a 200 kV LaB6 high-resolution transmission electron microscope of point resolution about 0.2 nm. A posterior image processing technique, the image deconvolution, was utilized to transform a single [21¯1¯0] image that does not intuitively represent the structure into the projected structure map. The adjacent Al and N projected atomic columns with the interatomic distance 0.109 nm can be distinguished from each other by analyzing the image contrast change with the sample thickness based on the pseudo-weak phase object approximation. This makes possible to derive the polarity and core structures of partial dislocations in the epitaxial AlN film at atomic level from a single image without relying on any other additional structure information. The atomic configurations for two partial dislocations containing a 10-atom ring and a 12-atom ring, respectively, have been attained. The method is available for II-VI and other III-V compounds. Its principle and procedure are briefly introduced.