Atomap: a new software tool for the automated analysis of atomic resolution images using two-dimensional Gaussian fitting

Initial positions and refinements

To exemplify this, we show the procedure to find the positions of all sublattices in an STO crystal projected along the [1\(\overline{1}\)0]-direction. While this demonstrates the use of this method on a specific crystal structure along a specific projection, the software should work for any kind of atomic structure or projection, as long as the atomic columns are clearly resolved. Comments on how to adapt this for other structures and projections are outlined in "Adapting for other structures and projections".

Finding distances between atomic columns

Having an accurate position for all the atomic columns is the first step toward making measurements of distances between columns or interplanar spacings. Having already grouped the atom columns into atomic planes, it is trivial to find the spacings in the (001)- and (110)-planes. The distances between neighboring atomic columns in the (001)-planes correspond to the (110) interplanar spacing as these are orthogonal (Fig. 2a). Similarly, the interplanar distances for the (110)-planes are found using the distances between the atomic columns in the (001)-planes. The case is less straightforward for (111)-planes, as neighboring atomic columns (of the same cation type) along the orthogonal (11\(\overline{2}\))-plane will be three monolayers apart. The interplanar spacing is the distance between one monolayer and its neighbor. To find this, a line is interpolated through the atomic columns in a (111)-plane. From an atomic column in the neighboring (111)-plane, the shortest distance from the atomic column to this line is found. This is the (111) interplanar spacing at this point, as shown to the bottom right in Fig. 2a. Repeating this for every atomic column and its neighbor atom plane gives a 2-D map of the monolayer distances.

In the perovskites, a common structural change is unit cell doubling along a specific direction as a result of oxygen octahedral tilting [20]. Depending on the tilt pattern, this will result in the oxygen columns shifting. An example of this is shown in Fig. 2b, where there is an obvious oxygen superstructure along the black line signified by alternating distances between the oxygen columns in the [001]-direction. One useful way of quantifying this is finding how much the oxygen atom deviates from the centrosymmetric position in a cubic perovskite structure. This displacement, D, is found by determining the distance from an atom to the next (N) and the previous (P) atoms in the atom plane, and divide the difference (N−P) by 4. This gives the distance the oxygen atom deviates from a centrosymmetric position, and is shown in the inset in Fig. 2b. For a cubic perovskite structure D is 0, while for bulk LFO D is 0.39 Å.

Atomic column shape

The software implementation

This program is implemented in the Python (3.x) programming language, and both the source code and instructions on how to install it is available at http://atomap.org. It relies heavily on the fitting and modelling routines implemented in HyperSpy [39]. Currently, the program is optimized for analysing STEM-images of perovskite oxide materials projected along a \(\langle110\rangle\) direction. However, it is trivial to adapt it for any structure as discussed below in "Adapting for other structures and projections". Extending the code should be easy, and requests for both new features and assistance in adapting for other structures are welcome on the issue tracker (linked from http://atomap.org/) or by e-mail to the corresponding author. The software and source code are distributed under the free and open source GNU General Public License v3.0.

The program is sorted into several classes: atom_position, atom_plane, sublattice, and atom_lattice. atom_position is the position of a single atomic column, and contains variables like position, \(\sigma\), \(\theta\), and other information about the shape of the atomic column. atom_plane contains all the atom_positions which belong to the same atomic plane. sublattice contains all the atom_positions and atom_planes belonging to the same sublattice, like the A-cations in Fig. 1. atom_lattice contains all the sublattices, so in Fig. 1, this would include the A-cations, B-cations, and oxygen sublattices. The atom_lattice class can be saved and loaded, saving all the atom_position parameters.

One current limitation is that the whole image given to the program must have a similar crystal structure. For example, a perovskite heterostructure shown in Figs. 4, 5, and 6 works fine, due to the structures being sufficiently similar. A perovskite oxide film grown on Si would, however, not work. Similarly, if there are any amorphous parts in the image, local bright features could be identified as atomic positions by the peak finding function. One simple solution is to crop the images, so only the same crystal structure is within the image given to the program. This could probably be automated, which would allow for automatic determination of regions with different structures. For example, in an aluminum alloy, one would be able to automatically figure out which regions are aluminum matrix and which are precipitates.

When fitting a single sublattice, the fitting is done on individual atomic column using a single 2D Gaussian. One obvious improvement would be to make a model containing all the atomic columns, and fitting them all simultaneously. While this is more computationally demanding, it could reduce the effects from neighboring atomic columns and, therefore, increase the accuracy of the fitting. The amount of improvement would be related to the degree of overlap between the atomic columns. With a very large separation between the atomic columns, this improvement would be zero or very small. With a very small separation, this improvement would be substantial. Quantifying the limits of fitting a single atomic column vs. including the neighboring atomic columns is interesting, but outside the scope of this work as the separation between the atomic columns was sufficiently large.

Atomap implements a version of this by removing the most intense sublattice before fitting the less intense sublattices. This is equivalent to having a model where the most intense atomic columns are fitted first, then locked, and afterwards fitting the second most intense atomic columns. Fitting 2D Gaussians to all atomic columns in a data set simultaneously will be experimented with in Atomap in the near future, using the aforementioned process to find reasonable initial values first.

Adapting for other structures and projections

The first step in using the program is finding a value for the smallest atomic separation for the first sublattice. It is important to find a good value for this variable, as having either too many or too few atoms in the first sublattice will lead to the wrong repeating unit being identified. This then results in incorrect identification of the sublattice. In the example, discussed in "Initial positions and refinements", this value is half the separation between the A-cations in the [1\(\overline{1}\)0]-direction. The projected (110) spacing is 2.76 Å for STO. However, half the value (1.38 Å) did not work very well, which is caused by the scanning distortions and sample drift during image acquisition. By trial and error, a value slightly less than half the smallest separation was found to work reliably. For the STO in Fig. 1: approximately 1.3 Å. This value is used for the atom column separation in the peak finding function. For only fitting a single sublattice, this will be enough for the program to work.

Finding a second sublattice requires some a priori knowledge. For a perovskite oxide in the [1\(\overline{1}\)0]-projection (Fig. 1), one would specify that the atoms in the second sublattice are found between the atom columns in the (110)-atomic planes for the first sublattice.

These procedures are documented on the webpage (http://atomap.org/).

Uncertainty and reducing scanning distortions

It is clear that the reliability and the uncertainty of the peak fitting technique will be connected to how clearly the different atomic columns can be resolved. An important aspect of this is scanning distortions, as the electron probe is susceptible to both mechanical, electrical, and magnetic disturbances, both from noise within the microscope system, as well as extraneous disturbances from the surrounding environment, and these will have some influence, even in the best designed microscope rooms [40]. One way to reduce the effects of these distortions is to acquire several fast images of the same area, register them, and sum them afterwards [40]. In earlier studies, this was simply performed by rigid registration of the images, which takes out the effect of any drift, and simply averages out any local distortions (at the cost of a slight deterioration of the resolution) [3]. A better approach is to perform rigid registration to remove the effects of drift, followed by non-rigid registration to remove the effects of short period instabilities in the scanning system [41, 42]. To investigate the effect of short acquisition and alignment, as opposed to recording a single scan at a longer dwell time, we analysed two different STEM-ADF images from the same session taken along the [1\(\overline{1}\)0] cubic direction of a sample of STO: (i) acquired as a single image using a pixel dwell time of 38.5 \({\upmu }\mathrm{s}\) (line synced) with the image shown in Fig. 3a. (ii) Acquired as an image stack of 20 images with dwell time of 2 \({\upmu }\mathrm{s}\) per pixel (i.e., almost the same total acquisition time per pixel) which is then processed using Smart Align [42]. This aligned stack is shown in Fig. 3b. Clearly, the aligned stack appears a little sharper and less noisy than the single long acquisition image, but to quantify the effects of this difference on atom spacing measurements, the images were quantified as set out above in "Finding distances between atomic columns", and some comparative results are shown in Fig. 3c–h. A distribution of distances between the atoms parallel to the [111] (vertical in image) and [\(11\overline{2}\)] (horizontal in image) crystallographic directions of STO is shown in Fig. 3c, d: these directions are parallel to the slow and fast scan directions, respectively. As these data are from a region of a pure STO sample far away from any interfaces, there should be no variations in the distance between the planes. Therefore, these plots should be essentially a measure of the uncertainty in the measurements, which we define as one standard deviation in the spread of the distances. The difference between the single scanned image and the Smart Align image is very obvious, especially for the slow scan direction. In this case, the uncertainty is reduced from \(\approx\)16 p.m. to 7 p.m. For the fast scan direction, the reduction in uncertainty is much less: from 7.5 to 5.9 p.m.

Similar measurements for distances between the atomic planes in the slow and fast scan directions are shown in Fig. 3e and f, respectively. Compared to c and d, this is in effect averaging over many atomic columns. The uncertainty in the slow scan direction for the single scanned image is clearly much larger, and comparing the normal and Smart Align data sets, there is a massive reduction in the uncertainty from about 7 p.m. to 1 p.m. For the fast scan direction, the uncertainty in the single scan image is less, as might be expected, although there is a big negative spike at one point, which is typical for the kind of local disturbance that often arises in atomic resolution images. In this case, the uncertainty is reduced from about 5 p.m. to 1 p.m. Thus, it should be possible using non-rigid registration to make measurements of A-site (or other heavy, bright atom) plane spacings with about 1 p.m. precision.

One interesting effect is the decrease of uncertainty between the individual atomic columns and atomic planes for the single scanned image and the Smart Align image. For the fast scan atomic column uncertainty there is a small improvement for the aligned stack (7.5 to 5.9 p.m.), while for the atomic plane uncertainty there is a much larger improvement (5 to 1 p.m.). This is most likely due to the scanning distortions being correlated when acquiring a single scanned image using line synchronization. When acquiring an image stack with a short pixel dwell time of 2 \({\upmu }\mathrm{s},\) these scanning distortions are much less correlated. Since finding the distances between the atom planes consists of averaging the positions of many atomic columns, these uncorrelated local distortions in the image stack are averaged out.

Finally, the interplanar spacings for the different sublattices are shown in Fig. 3g, h. Here, a similar STO Smart Align ADF data set has been used to estimate the A- and B-cation uncertainty levels, and an ABF data set has been used to estimate the oxygen uncertainty levels. The average distances between the interplanar spacings are calculated for the A-cations, B-cations, and oxygen. For all atom types in both fast and slow scan directions, the uncertainty is about 1.0–1.4 p.m.

In view of the fact that the non-rigid registration method gives clearly superior results, all data sets in the following sections are acquired as image stacks and processed using Smart Align, and then processed using the method outlined in "Initial positions and refinements".

Strain analysis: comparison to GPA

As stated in the introduction, GPA is a commonly used technique for characterizing distances between atomic columns in atomic resolution (S)TEM images [21]. Figure 4 shows a comparison between GPA and the method explained in this work, for an epitaxial LSMO/STO-(111) heterostructure. Figure 4a shows a STEM-ADF image and FFTs from the LSMO (top inset) and STO (bottom inset). The fast Fourier transform (FFT) of the LSMO region has extra \(\left\{ \frac{1}{2}\frac{1}{2}\frac{1}{2}\right\}\) spots (circled in red) which are not present in the STO region. These superstructure spots indicate doubling of the unit cell along the [111]- and [11\(\overline{1}\)]-directions in the LSMO. GPA is performed using the {111} and {11\(\overline{2}\)} FFT spots utilizing the STO substrate far away from the interface as a reference. These data are then summed in the [111]-direction, giving the average out-of-plane distance as a function of distance from the LSMO/STO interface (Fig. 4b, green dashed line). Using the method explained in "Finding distances between atomic columns", the average distance between the A-cation monolayers in the [111]-direction is calculated as a function of the distance from the interface (Fig. 4b, blue line). Comparing the two line profiles, they both show the same general trends: larger lattice size in the STO, and smaller in the LSMO, with a slight lattice size increase at about 8 nm into the film. However, the unit cell doubling observed in the FFT is only visible in the real space method. This is due to the choice of the mask in the GPA, since the one used to get the data in Fig. 4 did not include the superstructure spots. It could be possible to use the \(\left\{ \frac{1}{2}\frac{1}{2}\frac{1}{2}\right\}\) spots to do the GPA, instead of the {111} and {11\(\overline{2}\)}. This could show the presence of the unit cell doubling, but the regions where these peaks are not present would not yield any information, for example, the STO substrate and the regions in the LSMO film without the unit cell doubling.

Mapping oxygen octahedral tilting

Figure 5a shows an ABF image of the LSMO/LFO/STO-(111) heterostructure, where there is a “zig-zag” pattern of the oxygen columns along the [110]-direction in the LFO layer. This displacement from the centrosymmetric position can be quantified using the method discussed in "Finding distances between atomic columns". An example of this is shown in Fig. 5b, where the [001]-direction displacement is calculated for all the oxygen columns. The displacement in the LFO layer takes the form of an alternating long and short displacement in the [001]-direction. To give a better visualization, the displacement is set to zero at the A-cation positions, leading to the checkerboard pattern shown in Fig. 5b. An average of the displacements as a function of distance from the LFO/STO interface is shown in Fig. 5c. This shows both a clear superstructure and the variation in the displacement across the film. This can then be used to infer parameters, such as the octahedral tilting pattern and bond angles.

One important caveat with using ABF imaging is the relationship between real and measured atom column positions is not always intuitive [43]. For example, if the sample is slightly tilted off the zone axis, the atomic positions can shift. This effect varies with respect to thickness, atomic number, collection angle, and convergence angle [43]. Thus, to properly analyse the atomic positions, image simulations coupled with careful considerations of the experimental parameters are required. However, the measured atomic positions are still useful as the first approximations, even if further work with image simulations is required to ensure that the conclusions are robust.

Ellipticity

As discussed in "Atomic column shape", one can also use the shape of the atomic columns to extract structural information [12]. An example of this is seen in Fig. 6a, which shows A-cation ellipticity in the LSMO/LFO/STO-(111) heterostructure imaged using STEM-ADF. In the LFO layer, there is a clear elongation of the A-cation sites, annotated with the blue ellipses. By fitting elliptical 2-D Gaussians (Eq. 1) to every A-cation, this elongation can be quantified with \(\epsilon\) (Eq. 2) and \(\rho\) (Eq. 3). Where \(\epsilon\) is \(\frac{\sigma _{\rm Longest}}{\sigma _{\rm Shortest}}\), and \(\rho\) is the angle between the positive x-axis and \(\sigma _{\rm Longest}\). Having these values for every A-cation, one can visualize these in a vector plot (Fig. 6b). Here, the length of the arrows is given by \(\epsilon\), and the direction by \(\rho\). There is a clear difference in the LFO layer, which has an elongation close to the [111]-direction. This is even more apparent with the vector plot in Fig. 6c and the \(\epsilon\) plot in Fig. 6d, which takes the average of the data as a function of distance from the interface. The ellipticity is constant in the STO substrate until the LSMO/STO interface, where it increases to its maximum about 2 nm into the film, and decays gradually in the LSMO film to the same level as in the STO substrate.

As discussed in "Atomic column shape", there are many variables which can affect ellipticity of the atomic columns. Therefore, for the vector plots in Fig. 6b, c, a systematic average \(\epsilon\) and \(\rho\) “background” have been subtracted. This noise level was determined from the STO substrate (\(\approx\)0.12 \(\epsilon\) in the [111]-direction), where the ellipticity should be 1. This systematic background is most likely due to a variety of factors: sample drift, astigmatism, misalignment, having the sample slight off-axis, and residual scanning distortions. Especially, the latter is present in this data set, as horizontal “streaks” visible in some of the atomic columns (Fig. 6a). These are still present, even though the image was made by averaging an image stack. This is due to the fast scan being in the same direction for all the images in the image stack. One solution is rotating the fast scan direction by 90° in every second image in the image stack, which will average out this scanning distortion. Another effect can be the 2-D Gaussian fitting itself, where overlap from the neighboring atomic columns can influence the fitting. For example, in Fig. 6, the B-cations might add a slight component towards the [001]-direction.