Published: July 3rd, 2021
This work presents a workflow for atomic position tracking in atomic resolution transmission electron microscopy imaging. This workflow is performed using an open-source Matlab app (EASY-STEM).
The modern aberration-corrected scanning transmission electron microscopes (AC-STEM) have successfully achieved direct visualization of atomic columns with sub-angstrom resolution. With this significant progress, advanced image quantification and analysis are still at the early stages. In this work, we present the complete pathway for the metrology of atomic resolution scanning transmission electron microscopy (STEM) images. This includes (1) tips for acquiring high-quality STEM images; (2) denoising and drift-correction for enhancing measurement accuracy; (3) obtaining initial atomic positions; (4) indexing the atoms based on unit cell vectors; (5) quantifying the atom column positions with either 2D-Gaussian single peak fitting or (6) multi-peak fitting routines for slightly overlapping atomic columns; (7) quantification of lattice distortion/strain within the crystal structures or at the defects/interfaces where the lattice periodicity is disrupted; and (8) some common methods to visualize and present the analysis.
Furthermore, a simple self-developed free MATLAB app (EASY-STEM) with a graphical user interface (GUI) will be presented. The GUI can assist in the analysis of STEM images without the need for writing dedicated analysis code or software. The advanced data analysis methods presented here can be applied for the local quantification of defect relaxations, local structural distortions, local phase transformations, and non-centrosymmetry in a wide range of materials.
The development of spherical aberration correction in the modern scanning transmission electron microscope (STEM) has enabled microscopists to probe crystals with sub-angstrom sized electron beams1,2. This has allowed imaging of individual atomic columns in a wide variety of crystals with interpretable atomic resolution images for both heavy and light elements3,4. Recent developments in pixelated direct electron detectors and data analysis algorithms have enabled phase reconstruction imaging techniques, such as ptychography, with further improvements in spatial resolution to around 30 pm5,6,7. Additionally, the recent progress in STEM tomography has even enabled three-dimensional atomic resolution reconstruction of the single nano-particle8. The electron microscope has thus become an extraordinarily powerful tool for quantifying structural properties in materials with both high precision and site-specificity.
With the ultra-high resolution STEM images as the data input, direct measurements of structural distortions were performed to extract physical information from crystals at the atomic scale9,10. For example, the defect coupling between a Mo dopant in the WS2 monolayer and a single S vacancy was directly visualized by measuring the atomic positions and then calculating projected bond lengths11. Furthermore, the measurement on crystal interfaces, such as the coalesced grain boundaries in monolayer WS2, can exhibit the local atomic arrangement12. The interfacial analysis performed on the ferroelectric domain walls in LiNbO3 revealed the domain wall to be a combination of Ising and Neel states13. Another example is the visualization of the polar vortex structures achieved in the SrTiO3-PbTiO3 superlattices, achieved through calculation of the titanium atomic column displacements with respect to the strontium and lead column positions14. Finally, the advances in computer vision algorithms, such as image denoising with non-local principle component analysis15, Richardson and Lucy deconvolution16, drift-correction with non-linear registration17, and pattern recognition with deep learning, have significantly strengthened the accuracy of the measurement to sub-picometer precision18. One such example is the alignment and image registration of multiple fast-scan cryogenic-STEM images to enhance the signal-to-noise ratio. Subsequently, the Fourier-masking technique was applied to analyze the charge density waves in crystals by directly visualizing the periodic lattice distortion19. Even though the incredible aberration-corrected STEM instrumentation is increasingly accessible to researchers around the globe, the advanced data analysis procedures and methods remain uncommon and an enormous barrier for one without experience in data analysis.
In the present work, we showcase the complete pathway for the metrology of atomic resolution STEM images. This process includes firstly acquiring the STEM images with an aberration-corrected microscope followed by performing post-acquisition denoising/drift-correction for enhanced measurement accuracy. We will then further discuss the existing methods to clearly resolve and accurately quantify the atom column positions with either 2D-Gaussian single peak fitting or multi-peak fitting routines for slightly overlapping atomic columns20,21. Lastly, this tutorial will discuss methods for the quantification of lattice distortion/strain within the crystal structures or at the defects/interfaces where the lattice periodicity is disrupted. We will also introduce a simple self-developed free MATLAB app (EASY-STEM) with a graphical user interface (GUI) that can help with the analysis of STEM images without the need for writing dedicated analysis code or software. The advanced data analysis methods presented here can be applied for the local quantification of defect relaxations, local structural distortions, local phase transformations, and non-centrosymmetry in a wide range of materials.
NOTE: The flow chart in Figure 1 shows the general procedure of the atomic position quantification.
Figure 1: The workflow of the atomic position quantification and structural measurement. Please click here to view a larger version of this figure.
1. STEM image drift-correction and denoising
Figure 2: The graphical user interface (GUI) of the Matlab app EASY-STEM. All steps described in the protocol section are labeled accordingly. Please click here to view a larger version of this figure.
Figure 3: Example results of atomic position tracking. (i) An example of the position refining with the mp-fit algorithm. The results of regular 2D-Gaussian fitting and mpfit algorithm are shown with red and green circles respectively. The yellow arrows highlight the failure of regular 2D-Gaussian fitting due to the intensity from neighboring atoms. (a) The drift-corrected ADF-STEM image showing a typical unit cell of the ABO3 Perovskite. (b) The 3D plot of the intensity in (a). (c) The same image denoised with a Gaussian filter. (d) The 3D plot of the intensity in (c). (e) The contour plot of the intensity in (c) with the initial atomic positions (yellow circles) overlaid. (f) An example of the unit cell vector indexing system showing the index of the atomic positions in the image. (g) The contour plot of the intensity in (c) with the initial atomic positions (yellow circles) and refined atomic positions (red circles) overlaid, and (h) the 3D plot of the intensity with initial and refined atomic positions shown with yellow and red circles. Please click here to view a larger version of this figure.
NOTE: This technique uses a filter that averages the intensity of the nearby pixels in the images. The effect of the Gaussian filtering is presented in Figure 3a-d.
2. Finding and refining the atom position
3. Physical information extraction
4. Data Visualization
Figure 3 shows the example results of atomic position tracking by following steps 1 and 2 in the protocol. A raw ADF-STEM image of a unit cell of the ABO3 perovskite is shown in Figure 3a, and its intensity profile is plotted in 3-D in Figure 3b. Figure 3c shows the result after Gaussian filtering is applied to the STEM image in Figure 3a, and the intensity profile is plotted in Figure 3d. The initial positions are determined by finding the local maxima in the image and the positions are indicated by yellow circles in Figure 3e. The atomic positions are indexed based on the unit cell vector and shown in Figure 3f. After the initial position is found and indexed, 2D-Gaussian fitting is applied to further refine the measurement. In Figure 3g and Figure 3h, the fitted positions are indicated as red circles, the measurement precision is improved as the refined positions are closer to the center compared to the initial positions (yellow circles). Lastly, the advantage of applying the mpfit algorithm on the overlapping intensities is showcased in an ADF-STEM image of BaMnSb2 crystal (Figure 3i). The regular 2D-Gaussian fitting (red circles) fails severely on the Mn columns as highlighted by yellow arrows, while the mpfit algorithm can determine the positions much more accurately (green circles).
Figure 4: HAADF-STEM image of the Ca3Ru2O7 (CRO). (a) The magnified image of the ADF-STEM image of the Ca3Ru2O7 (CRO) crystal with the crystal structure schematic superimposed. The relative displacement of the Ca atom in the perovskite layer is highlighted with the yellow arrow. (b) Drift-corrected and denoised ADF-STEM image of CRO and (c) with overlaying refined atomic positions (red dots). (d) An example of using an indexing system to identify the upper (red), center (blue) and lower (yellow) Ca atoms in the perovskite layer. Please click here to view a larger version of this figure.
The HAADF-STEM image of the Ca3Ru2O7 (CRO) is shown in Figure 4a and Figure 4b (with the overlaid crystal structure). CRO is a Ruddlesden-Popper phase perovskite crystal with the polar space group A21am. ADF-STEM imaging shows the contrast from the heavier elements well (Ca and Ru), but the O atoms are not visualized as lighter atoms do not scatter the beam strong enough to become visible with HAADF detectors. The non-centrosymmetry of the crystal structure is caused by the tilting of O octahedra and can be visualized in ADF-STEM images by analyzing the displacement of the Ca atom in the center of the double perovskite layer. By following the steps listed in the Protocol section, all atomic positions in this image can be located by finding the centers of the fitted 2D-Gaussian peaks, as shown in Figure 4c. Furthermore, using the indexing system, in step 3.2, each type of atom in the unit cell can be identified and used for further processing. For example, the Ca atoms at the upper, center, and lower side of the perovskite double layer can be easily identified and their positions are presented with circles filled with different colors, as shown in Figure 4d.
Figure 5: Physical information. (a) An example of the implementation of the vector map showing the polarization obtained from the center Ca displacement pattern. The arrows are colored based on the orientation (red to the right, blue to the left). The vertical 90° head-to-head and head-to-tail domain walls are indicated with blue arrows and a horizontal 180° domain wall is indicated with a red arrow. (b) An example of the implementation of the false-colored map showing the polarization. The color indicates the magnitude in left (yellow) and right (purple) directions. Reduced magnitude results in faded color. (c) An example of implementation of the false-colored map showing the εxx strain in the image. The color indicates the value of tensile (red) and compressive (blue) strain. Please click here to view a larger version of this figure.
After positioning and indexing the atoms in the STEM images, the physical information can be extracted and visualized via various types of plots, as shown in Figure 5. The vector map that shows the polarization direction is shown in Figure 5a. The arrows point towards the projected polarization direction, and by coloring the arrows based on their orientation, a vertical head-to-head 90° domain wall (labeled with blue arrows), and a horizontal 180° domain wall (labeled with red arrows) are shown at the top of the image. By constructing the false-colored map as shown in Figure 5b, a decreasing polar displacement magnitude can be observed via the fading color in the center, and thus the head-to-tail domain wall can be visualized. By combining the vector map and false-color map, the T-junction formed by three domain walls is shown in the ADF-STEM image. Additionally, with the dimension of each unit cell in the image measured, an εxx strain map can be constructed, as shown in Figure 5c.
When working on the post-acquisition processing, some caution needs to be taken as well. To start with, during the image drift-correction, the algorithm assumes that the 0° image has the horizontal fast scanning direction, so double-check the direction before the calculation. If the scanning direction is not correctly set, the drift-correction algorithm will fail and may even introduce artifacts in the output17. Then during the imaging denoising, certain methods may introduce an artifact; for example, the Fourier filtering may create atom column contrast at the vacancy sites or remove fine features in the images, if the spatial resolution is not limited properly. As a result, it is vitally important to verify if the denoised images closely resemble the original raw input images.
Next, when determining the initial atomic positions based on local maximum/minimum, try to adjust the restriction minimum distance between peaks to avoid creating redundant positions between atomic columns. These redundant positions are artifacts generated due to the algorithm mistakenly recognizing the local maxima/minima in the image as atomic columns. Additionally, one can adjust the threshold value to find most of the positions if there are large contrast differences between various atomic species in the image (e.g., in ADF-STEM images of WS2). After obtaining most of the initial atomic positions in the image, try to manually add missing ones or remove extra ones with best effort. Moreover, the method for the indexing of the atoms is the most effective when there are not large interruptions in the periodicities within the image. When there are interruptions such as grain boundaries or phase boundaries presented in the image, the indexing may fail. The solution to this problem is to define the areas of interest in the image (by clicking the Define Area of Interest button in EASY-STEM app), and then indexing and refining the positions within each area separately. Afterward, one can easily combine datasets of different areas in the same image into one set of data and work on the analysis.
Finally, after applying 2D-Gaussian peak fittings, scatter the refined positions points onto the input image to verify the fitting results to see if the refined positions deviate from the atomic columns. The accuracy provided by the single Gaussian fitting algorithm is sufficient in most of the STEM experiments; however, if the position deviates due to the intensity from a neighboring atom, use the multi-peak fitting (mpfit) algorithm instead to isolate the intensity from adjacent atomic columns21. Otherwise, if the position deviates due to the image quality issue or the low intensity from the specific atom columns, it is suggested to discard the fitted position at that location.
There are several existing and specialized algorithms for the atomic position measurement, for example, the oxygen octahedra picker software22, Atomap python package23, and StatSTEM Matlab package24. However, these algorithms have some limitations in certain aspects. For instance, the oxygen octahedra picker requires the input of STEM images to contain only clearly resolved atomic columns and thus failed to address the issue in the images with atomic columns overlapping intensities21. On the other hand, although Atomap can calculate the positions of "dumbbell-like" atomic columns, the process is not very straightforward. Additionally, the StatSTEM is a great algorithm for quantifying the overlapping intensities, but its iterative model-based fitting process is computationally expensive21. In contrast, our approach, introduced in this work along with the Matlab app EASY-STEM, which is integrated with the advanced mpfit algorithm, can address the problem of the overlapping intensity and is less computationally expensive than StatSTEM, while offering competitive measurement precision. Furthermore, the analysis from Atomap and the oxygen octahedra picker software packages are designed and specialized for analyzing the data from ABO3 perovskite crystals, while the indexing system shown in this work is much more flexible about different materials systems. With the method in this work, users can fully design and customize the data analysis for their unique material systems based on the output results that contain both refined atomic positions and the unit cell vector indexing.
Figure 6: Statistical quantification of the atomic position finding. (a) The distribution of the perovskite A-site to A-site distance presented in a histogram. The normal distribution fitting is plotted and overlaid as the red dashed line showing the mean of 300.5 pm and the standard deviation of 4.8 pm. (b) The statistical quantification of the perovskite unit cell vector angle measurement is presented as a histogram. The normal distribution fitting is plotted and overlaid as the red dashed line showing the mean of 90.0° and the standard deviation of 1.3°. (c) The statistical quantification of the polar displacement measurement in Ca3Ru2O7 (CRO) is presented as histogram. The normal distribution fitting is plotted and overlaid as the red dashed line showing the mean of 25.6 pm and the standard deviation of 7.7 pm. Please click here to view a larger version of this figure.
The method introduced here provides picometer-level precision and simplicity for implementation. To demonstrate the measurement precision, the statistical quantification of the atomic position finding is presented in Figure 6. The measurements of cubic ABO3 perovskite A-site distance distribution and unit cell vector angle distribution are plotted using histogram in Figure 6a and Figure 6b, respectively. By fitting the normal distribution curve to the distributions, the A-site distance distribution shows a mean of 300.5 pm and standard deviation of 4.8 pm and unit cell vector angle distribution shows a mean of 90.0° and standard deviation of 1.3°. The statistical quantification indicates the method proposed here enables picometer-level precision and can greatly alleviate the distortion due to drift during imaging. This result suggests that this measurement is trustworthy when the physical information to be measured is greater or equal to roughly 10 pm. For example, in the case of aforementioned CRO crystals, the measurement of the magnitude of the polar displacement is presented in Figure 6c. The measurement shows a mean of 25.6 pm, a standard deviation of 7.7 pm, and it shows that the polar displacement measurement in CRO STEM images is solid. Additionally, more caution needs to be taken in case of experimental limitations such as low signal-to-noise ratio when imaging beam-sensitive samples. In those cases, the measured atomic positions need to be closely examined against the raw images to ensure the validity of the measurement. Consequently, the analysis method introduced here has limitations to the measurement precision when compared to more recent and advanced algorithms. Our method is insufficient when the precision is required at the sub-picometer level, so a more advanced analysis routine is necessary if the feature to be extracted in the image is below a certain threshold. For example, the non-rigid registration algorithm has shown sub-picometer precision measurement on silicon and it enables accurate measurement of bond length variation on a single Pt nanoparticle25. Most recently, the deep learning algorithm was employed to identify various types of point defects in 2-D transition metal dichalcogenides monolayers from a huge amount of STEM image data. Later, the measurement was conducted on the averaged image of different types of defects and this method also demonstrated sub-picometer level precision on the distortion around those defects18. Consequently, as a future plan for increasing the analysis capacity, we are in the progress of developing and implementing more advanced algorithms such as deep learning. We will also try to integrate them into the future data analysis tool updates.
The authors have nothing to disclose.
L.M. and N.A.'s work supported by the Penn State Center for Nanoscale Sciences, an NSF MRSEC under the grant number DMR-2011839 (2020 - 2026). D.M. was supported by ORNL's Laboratory Directed Research and Development (LDRD) Program, which is managed by UT-Battelle, LLC, for the U.S. Department of Energy (DOE). A.C. and N.A. acknowledge the Air Force Office of Scientific Research (AFOSR) program FA9550-18-1-0277 as well as GAME MURI, 10059059-PENN for support.
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