38 Figure 2-5: Performance of atom finder on test data with different atomic column width, (a)(b) graphs show the percentage of identified atomic columns (red line) and missed (blue line) along with false positives (black line) ); (c)(d) graphs show the accuracy of the found atomic pillar positions with algorithm error defined as the average deviation from the perfect lattice. 40 Figure 2-7: Performance of atom finder on test data with different atomic column intensities, (a)(b) graphs show the percentage of identified atomic columns (red line) and missed (blue line) together with false positives (black line) ); (c)(d) graphs show the accuracy of the found atomic pillar positions with algorithm error defined as the average deviation from the perfect lattice.
Density Functional Theory (DFT)
A more elaborate approximation is the Generalized-Gradient Approximation (GGA), which depends not only on the local density but also on its local gradient [4], [5]. They are usually included as a semi-empirical correction to the total energy. where 𝐸𝑇 is the total energy, 𝐸𝐷𝐹𝑇 is the DFT total energy as in Eq. 1.
Scanning Transmission Electron Microcopy (STEM)
The intensity of the atoms is proportional to Z2, where Z is the total atomic number in the atom's column (Figure 1-3). However, high energy electrons can damage the sample and cause the sample to undergo changes in structure.
Scanning Tunneling Microcopy (STM)
By measuring the resonant frequency of the tip at each point on the surface, an image of the surface can be reconstructed. It has been used for many applications, including using the tip to arrange atoms in patterns on the surface of a material, to measure the lifetime of surface electrons, and to image the electron orbitals of molecules[13].
Defect Identification
For a single defect, this intensity change may not be detectable due to the small percentage change in the intensity of the atomic column. Alex Belianinov [17] showed that by mapping the location of the adjacent atomic columns it is possible to detect defects.
Atomic column isolation and localization
This image is then used to find the atoms through methods such as determining the intensity of the image [23]. The results of K-means clustering are then reclustered to find the locations of the atomic columns (Figure 1-10).
Image filtering
It then fills this parameter space using the inverse of a line function. In Hough space, a circle is transformed into multiple circles that overlap the center of the original circle when the correct radius RH =R is chosen, as illustrated in Fig. Since the computation of the Hough space for a circle preserves the previous parts of the sum, not much more computation is needed to generate the Hough space of a cylinder than that of a simple circle.
However, since a cylinder contains smaller solid circles within it, the Hough transform does not return a single point per center, but rather a large central area of the cylinder. In Hough space, these 7 points from real space become 7 circles with RH < R. c) the Hough space circles of 7 points from real space all overlap at the center of the real space circle when RH = R. This is achieved by using a very rigid circular weighting in which the width of the Gaussian is RH.
Thesis overview
Using the first few loadings corresponding to the eigenvectors of greatest interest (and thus containing the majority of data), the parameter space is transformed into {x, y, l} with l being only loadings of significantly high variance and as such much smaller dimensionally, providing a path to statistically weighted data compression and denoising. The number of loading values required is determined by looking at the eigenvalues as a function of the component number (so-called "scree plot") to select the upper cutoff above which the rest of the data is most likely noise. This grid is then optimized using DFT and the relaxed structured used to simulate an ideal STEM image which is compared to the original.
Any differences in the equation can be isolated and then tested for possible reasons to identify defects in the grid.
Hough atomic-column finder
- Introduction
- Methods
- Results
- Discussion
- Conclusions
As a simple approximation of the true atomic column PSF, we decided to use a cylinder/Gaussian in the Hough transform. The Hough transform changes the image parameter space from {x, y} to {x, y, r}, where r represents the radius of the PSF. The above method can be further optimized to detect atomic columns in any image by changing the PSF.
In all cases, the performance of the algorithm was determined by defining the number of correctly detected atomic columns, not found atomic columns and false positives. To test the effect of the image noise on the atomic finding, various degrees of random (Gaussian?) noise were added to the test image, in the range from 0 to 300% of the maximum value in the original image, namely the amplitude at the center of a Gaussian (Figure 2-2). It should be noted that intensity variation Figure 2-5: Atomic locator performance on test data with different atomic column widths, (a)(b) graphs show the percentage of atomic columns identified (red line) and missed (blue line) along with false positives (black line); (c)(d) graphs show the accuracy of the found atomic column positions, where the algorithm error is defined as the average deviation from the perfect lattice.
Atomic-column recovery and defect identification aided by DFT
- Introduction
- Methods and materials
- Results and discussion
- Outlook
- Conclusions
DFT is then used to optimize the atomic coordinates in the reference patch under the constraint of the known xy atomic coordinates. After evaporation of the IPA, the graphene was pulled from the copper surface and strongly adhered to the TEM grid. Due to the nature of the moiré interference, atomic columns can only be distinguished around the center of.
The atomic coordinates in this idealized bilayer graphene patch are then replaced by the coordinates of the atomic columns determined from the experimental image. This procedure necessitated the interpolation of the simulated image to the same coordinate system as the experimental one. First, however, the coordinates of the atoms causing the low correlation must be determined by splitting the 2D cross-correlation map into a cross-correlation map for each layer (Figure 3.4 b-c).
Rippling in the z direction in bilayer graphene with a small angle of relative rotation causes a change in the interlayer spacing that coincides with a change in the intensity of the moiré pattern [82]. To measure these waves in the current data, a series of line profiles were recorded along the path between the centers of the primary moiré nodes in the patch.
Defect detection and grouping
Introduction
Using cross-correlation between a STEM image and a simulated STEM image based on coordinates obtained by relaxing the model structure with density functional theory (DFT) calculations and then detecting errors through regions of low correlation [91]. With these methods, they achieved the detection of defects that would not be possible with the human eye or would be extremely time-consuming. In this chapter, we describe the development of a method that uses graph theory for the positions of atomic column centers and is capable of detecting a wide range of defects in STEM images without prior knowledge of the material.
For this work, a special type of loop is created with the following conditions: no vertex can be repeated, no connecting line can cross another connecting line, the loop must include a reference atomic column, the loop must not include any column other atomic. and, finally, the cycle must be the shortest path connecting the vertices. Based on the number of peaks and the area of the cycles, it is possible to detect and categorize the defects in the STEM image. In bismuth-doped bulk silicon, we demonstrate the ability of cycles to detect Bi dopants in atomic columns and compare with Z contrast.
Methods and materials
Starting with the largest loops in the library, loops overlap at points. If every point in the cycle coincides, within some uncertainty, with a point in the image, it is chosen as the correct cycle. The first method of detecting defects is by looking at deviations in the number of points in the cycle.
To detect a defect, we mark as acceptable any atomic column that has a cycle with the same number of points in it as a cycle in the perfect crystal. These groups are then grouped based on the number of atomic columns in the group. Another method of using cycles to find faults is by looking at changes in the cycle area.
Results
A possible explanation for the presence of S-vacancies near Re could be due to the nature of the defects. S-vacancies act as electron traps for the excess electron of the Re atom, making them complementary defects. In Mo–V–M–O we can see the accumulation of vacancies or Nb atoms beneath the surface of the material (Figure 4.4 c).
These areas are not visible in the Z-contrast image due to the presence of surface contamination of the sample, which obscures slight changes that may be present in the intensity of the atomic columns. Using the cycle size, only the approximate location of about 80% of the Bi dopants could be identified (Figure 4-5 e-f). In regions with more than one Bi dopant in close proximity, it is difficult to determine the number and exact location of the dopant.
Conclusions
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Castro et al., “Uniaxial bilayer graphene: semiconductor with a gap tunable by electric field effect,” Phys. Bai et al., “Creation of one-dimensional nanoscale periodic ripples in a continuous layer of mosaic graphene,” Phys. Yang et al., “Direct cation exchange in monolayer MoS2 via “explosive” recombination–enhanced migration (under review).
