Abstract text (incl. figure legends and references)

To understand the structure-property relationship of nanostructures, reliably quantifying parameters, such as the number of atoms, is important. Advanced statistical methodologies have made it possible to count the number of atoms for monotype crystalline nanoparticles from a single ADF STEM image [1,2]. However, recent developments enable one to simultaneously acquire multiple ADF STEM images. Therefore, we extended the statistics-based atom-counting method in order to enhance the accuracy and precision of the atom counts.

So-called scattering cross-sections have been introduced corresponding to the total fraction of scattered intensity attributed to each atomic column to count atoms [3]. Within the univariate approach, the distribution of scattering cross sections (SCSs) is modelled by a one-dimensional Gaussian mixture model (GMM) (Fig 1b) [4]. The number of components often corresponds to a local minimum in the evaluation of an order selection criterion (ICL – Fig 1d). When having two independent sets of SCSs, the distribution can be represented in a scatter plot (Fig 1c). A multivariate approach is then required to estimate the locations and number of components for this 2D GMM. By comparing the ICL criteria from the 1D and 2D GMM analysis, it is clear that the second set of SCSs improves the selection of the correct number of components in the GMM.

In a more extensive simulation study, we evaluated quantitatively the performance of the 1D vs 2D GMM analysis. For this purpose, a more realistic distribution of SCSs is simulated corresponding to a nanoparticle having a thickness up to 20 atoms. The results of this study are presented in Table 1 for different amounts of overlap of the Gaussian components (ơ/δ). Typically, this overlap increases when lowering the electron dose. The low performance of the 1D GMM analysis for the low detector angles is the result of an underestimation of the number of components by the ICL criterion leading to missing components in the thickness range where SCSs of columns with different number of atoms have a similar magnitude. From this table, it is clear that the percentage of correctly determined number of components significantly increases when evaluating the 2D dataset.

In conclusion, we present a statistics-based method for atom-counting from independent multiple STEM images which improves the accuracy and allows one to retrieve precise information, especially from low-dose images [5].

Figure 1: Illustration of the concept. (a) Input structure. (b) Histogram of the SCSs obtained from a single image (115-157 mrad) together with the 1D GMM. (c) Distribution of the SCSs extracted from two images (35-45 mrad and 115-157 mrad). (d) ICL criterion evaluated as a function of the number of components of the GMM.

Table1: Percentage of correctly determined number of components from simulated data by using 1D and 2D GMM analysis for different amounts of overlap of the Gaussian components (ơ/δ as indicated in Figure 1b).

REFERENCES

[1] A. De Backer et al., Nanoscale 9 (2017) 8791-8798

[2] T. Altantzis et al., Nanoletters 19 (2019) 477-481

[3] A. De Backer et al., Ultramicroscopy 134 (2013) 23-33

[4] G. McLachlan, D. Peel, Finite Mixture Models, Wiley Series in Probability and Statistics, John Wiley and Sons, Inc, 2000

[5] This work was supported by the European Research Council (Grant 770887 PICOMETRICS to SVA and Grant 823717 ESTEEM3). The authors acknowledge financial support from the Research Foundation Flanders (FWO Belgium) through project fundings and a postdoctoral grant to ADB.