Picometer-Precision Atomic Position Tracking through Electron Microscopy

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

1. Acquire high-quality annular dark-field (ADF)/annular bright-field (ABF) STEM images.
   NOTE: The quality of the input data is key to ensuring the accuracy of data analysis, so we start the protocol with a few tips for acquiring good image data.
   1. Ensure a high-quality TEM sample. The sample quality is extremely crucial. Use thin and clean TEM samples with no beam damage for imaging. Avoid touching the sample during handling and loading as this can cause sample contamination.
   2. Clean the sample before insertion (if possible). Clean the sample by using plasma cleaner, baking in a vacuum, or irradiating the region of interest in the sample at low magnifications by spreading the electron beam after sample insertion into the microscope ('beam shower'). Avoid damaged or contaminated areas when imaging.
   3. Align the microscope and tune the aberration correctors to minimize the lens aberrations as much as possible. Test the resolution by acquiring a few STEM images on a standard sample to confirm that the spatial resolution can resolve the specific crystal structures and further fine-tune the aberrations in the image.
   4. Tilt the sample until the optical axis is aligned with the specific zone axis of the crystal. For certain crystals, make observations from a required zone axis. For instance, align the viewing axis with the planes of the domain walls in ferroelectric crystals for the measurement.
   5. Optimize the electron dose while limiting electron beam damage and the sample drift during imaging. If the sample is stable under the electron beam and does not show drift or damage during the acquisition, it may be possible to try a higher electron dose or acquire multiple images of the same region to boost the signal-to-noise ratio. The goal here is to have a higher signal-to-noise ratio without beam damage or image artifacts.
   6. Acquire STEM images with different scanning directions to correct for potential drift during acquisition. First, acquire an image and then take the second one from the same region immediately after rotating the scan direction by 90°.
      1. Take images using the same imaging condition except for the scan directions. The purpose of this step is to feed the rotated images to the drift correction algorithm developed recently¹⁷.
         NOTE: One can also input more than two images with more varying scanning directions (with arbitrary angles) into the algorithm. However, successive scanning of the same region may lead to lattice damage or drift in that area. Additionally, it is recommended that the scan direction and the low index lattice planes not maintain parallel or perpendicular directions with each other and instead maintain oblique angles. If the scan direction coincides with certain horizontal or vertical features (lattice planes, interfaces, etc.), the drift along the direction of the strong vertically/laterally varying features may cause artifacts during image registration.
2. Perform drift-correction with a non-linear correction algorithm.
   NOTE: The non-linear drift correction algorithm was proposed and constructed by C. Ophus et al.¹⁷, and the open-source Matlab code can be found in the paper. Two or more images with different scanning directions are fed into the correction algorithm, and the algorithm will output the drift corrected STEM images. The downloaded code package includes a detailed yet simple procedure for the implementation. A more detailed algorithm and description of the process can be found in the original paper.
3. Apply various image denoising techniques.
   ​NOTE: After the drift correction, perform image denoising to enhance the accuracy of future analysis. Some of the common denoising techniques are listed here. Furthermore, we introduce a free interactive Matlab app named EASY-STEM with a graphical user interface to help with the analysis. The interface is shown in Figure 2, with all the steps labeled on the corresponding buttons.

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.

1. Apply the Gaussian filtering. In the EASY-STEM app, find a tab called Gaussian on the bottom left. Use the slider to select how many nearby pixels to average. Move the slider to apply the Gaussian filter to the image.

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. Apply Fourier filtering. In the EASY-STEM app, find a tab called FFT on the bottom left. There is a slider to restrict the spatial frequency to reduce high-frequency noise. Move the slider to apply the Fourier filter to the image.
   NOTE: This technique limits the spatial frequency of the image to remove the high-frequency noise in the image.
3. Apply the Richardson-Lucy deconvolution. In the EASY-STEM app, find a tab called Deconvolution on the bottom left, where there are two input boxes for the iterations of blind deconvolution and Richardson-Lucy deconvolution, respectively. Change the value and apply this denoising algorithm by clicking the button.
   ​NOTE: This technique is a deconvolution algorithm for effectively removing the noise in the image by calculating the point spread function.

2. Finding and refining the atom position

1. Find the initial atomic positions.
   NOTE: After the post-acquisition image processing, the initial atomic positions can be simply extracted as the local intensity maximum or minimum for the ADF or ABF STEM images respectively. A minimum distance between the neighboring atomic columns needs to be defined to remove the extra positions.
   1. Define the minimum distance (in pixels) by changing the value in the input box that determines the distance between the neighboring peaks.
   2. Click the Find Initial Positions button in the EASY-STEM app. The result is shown in Figure 3e.
      NOTE: Frequently, extra positions or missing positions are observed with a simple local max/min finding algorithm. Thus, a manual correction mode is created in the EASY-STEM app to further refine the atomic positions (Add Missing/Remove Extra Points buttons). This feature enables the addition and removal of the initial positions by using the mouse cursor.
2. Index the initial atomic positions with a unit-cell vector-based system.
   1. Define an origin point in the image. In the EASY-STEM app, click on the Find Origin button. After clicking the button, drag the pointer to one of the initial atomic positions to define it as the origin.
   2. Define the 2D unit cell u and v vectors and the unit cell fractions.
      1. Click the Find U/V button and drag the pointer to the end of the unit cells.
      2. Define the lattice fraction value by changing the value in the Lat Frac U and Lat Frac V input boxes.
         NOTE: This value determines the lattice fraction value along the unit cell vector. For example, in the ABO₃ perovskite unit cell, the unit cell can be divided equally into two halves along the two perpendicular unit cell vector directions. Consequently, there are two fractions along each unit cell vector direction, so the unit cell fraction values are 2 and 2 for u and v directions, respectively. The example result of the indexing and the corresponding u and v unit cell vectors are demonstrated in Figure 3f. For example, in Figure 3f, we will index the atoms on the corners as (0, 0), (1, 0), (0, 1), (1, 1); and we will index the atom in the center as (1/2, 1/2). This indexing system helps with information extraction in the following steps.
      3. Click on the Calculate Lattice button to index all the atoms.
3. Click on the Refine Positions button in the EASY-STEM app to refine atomic positions with 2D-Gaussian fitting.
   NOTE: After obtaining the initial atomic positions and indexing the atoms in the image, a 2D-Gaussian fitting around each atomic column needs to be applied to achieve the sub-pixel level precision in the analysis. With this algorithm, it is possible to first crop an area in the image around each initial atomic position in the image and then fit a 2D-Gaussian peak in the cropped image. We then use the centers of the fitted 2D-Gaussian peaks as the refined atomic positions. This algorithm fits the 2D-Gaussian function to each atomic column in the image and the center of the fitted peak will be plotted after fitting. The result of the 2D-Gaussian fitting is shown in Figure 3g,h.
4. (Optional) Click the mpfit Overlaps button in EASY-STEM to refine atomic positions with 2D-Gaussian multi-peak fitting (mp-fit).
   NOTE: Refine the atomic positions using the mp-fit algorithm when the intensities from adjacent atomic columns are overlapping with each other. The mp-fit algorithm and its effectiveness are discussed in detail by D. Mukherjee et al.²¹. The EASY-STEM app has incorporated this algorithm and can be used to separate neighboring atoms with overlapping intensities. The result of mp-fit is shown in Figure 3i.
5. Save the results by clicking the Save Atomic positions button.
   NOTE: The app will prompt the user for location saving and file name. All saved results are included in the variable called "atom_pos".

3. Physical information extraction

1. Measure the atomic displacements based on the unit cell vector indexing and atomic positions.
   1. Define a unit cell center.
      NOTE: For example, for an ABO₃ perovskite unit cell looking from its [100] axis, the unit cell centers can be defined as the average position of the four A-site atoms. In the first unit cell, those A-site atoms have been previously indexed as (0, 0), (1, 0), (0, 1), (1, 1).
   2. Find the position of the displaced atoms.
      NOTE: In the case of the ABO₃ perovskite unit cell, the displaced atom is the B-site atom, which was previously labeled as (1/2, 1/2).
   3. Iteratively find the position of the reference unit cell centers and displacement atoms for all the complete unit cells in the image.
      NOTE: Unit cells may be incomplete near the edge of the TEM image. The atomic positions in those unit cells are discarded.
   4. Measure the displacement vector by entering the following command:
      d = pos(B) – mean(pos(A))
2. Quantify the lattice strain.
   1. Extract the unit cell vectors from each unit cell based on the atomic positions.
      NOTE: Extract vector matrix "C", which is a 2×2 matrix consisting of u-vector and v-vector for each unit cell in x and y directions.
   2. Define a reference vector, "C₀".
      NOTE: C₀ can be defined as the average unit cell vectors from the part of the image (recommended) or the theoretically calculated unit cell vector value.
   3. Calculate the 2×2 transformation matrix "T" using the following equation:
      or (1)
   4. Calculate the distortion matrix "D":
      D = T – I (2)
      where the "I" is the identity matrix.
   5. Decompose the distortion "D" to symmetric strain matrix "ε" and anti-symmetric rotation matrix "ω":
      (3)
      NOTE: Strain matrix "ε" and rotation matrix "ω" can be extracted by using the equations:
      ε =  (4) And ω = (5).
   6. Iteratively calculate strains for all unit cells.
   7. In the EASY-STEM app, click on the Calculate Strain based on the atomic positions button under the Quantify tab on the top-left of the interface.
      ​NOTE: The users can customize the displayed range of the strain map by changing the value within the Strain Upper/Lower limit input box.

4. Data Visualization

1. Create colored line maps.
   NOTE: Colored line mapping of the atomic bonds is a straightforward way to present the distance between nearby atoms. In Matlab, the command to draw a line between two points is: Line([x1 x2],[y1 y2],'Color',[r g b]). The inputs [x1 x2]and [y1 y2] are the coordinate values of the first and the second position. The distance variation can be presented with varying colors in the line map, which is defined by the [r g b] value. The [r g b] values stand for the red, green, and blue color values, each ranging from 0 to 1. Then iteratively connect all nearby atoms with colored lines.
   1. Generate colored line maps in the EASY-STEM app.
      NOTE: In the EASY-STEM app, line maps can be generated by a simple button clicking, which is under the Quantity tab on the top right of the interface.
      1. Adjust the value (in pm) in the Mean Distance input box and Measurement Range input box in EASY-STEM. These two values define the average distance of the projected atom distance and the distance range of the measurement.
      2. In the EASY-STEM app, click on the Calculate Bond Length Based on Near Neighbor button.
         NOTE: The line maps will be generated automatically. The users can adjust the colormap, line style, and line width for better visualization.
2. Create vector maps.
   NOTE: Vector maps can present atomic displacements in an area of the crystal. Since the displacement analysis is unique to individual systems, we have not integrated the code into the EASY-STEM app, but instead, here, we will introduce the Matlab commands for such analysis based on the standard ABO₃ perovskite unit cells.
   1. Calculate the reference position for displacement measurement.
      NOTE: In the example of ABO₃ perovskite, we have indexed the atoms on the corners (A-site) as (0, 0), (1, 0), (0, 1), (1, 1), and the atom in the center (B-site) as (1/2, 1/2). To compute the displacement with respect to the unit cell center, we first calculate the reference position as the averaged position of the corner (A-site) atoms. The Matlab command for this calculation is:
      ref_center=(positionA1+positionA2+positionA3+
      PostionA4)/4
   2. Calculate the displacement by entering the command:
      [displace_x displace_y] = PositionB – ref_center
   3. Implement the vector map:
      quiver(x,y,displace_x,displace_y)
      NOTE: The input x and y are the positions of the displaced atom. The variables displace_x and displace_y are the displacement magnitudes in x and y directions. The vector maps can be uniformly colored (e.g., yellow, white, red…) or shaded based on the displacement magnitude.
3. Create false-colored maps.
   1. Generate the false-colored maps by upsampling to estimate the measured value (displacement, strain, etc.) for each pixel in the image:
      ImageSize = Size(Image);
      [xi,yi] = meshgrid(1:1:ImageSize(1),1:1:ImageSize(2));
      Upsampled_Data = griddata(x,y,YourData,xi,yi,'v4');
      NOTE: The "griddata" function upsamples the data at position (x,y) to estimate the value for each pixel in the entire image. The inputs xi and yi are the grid coordinates, and the 'v4' is the bicubic upsampling method.
   2. Plot the upsampled data using user-defined color scale.