Atom counting in HAADF STEM using a statistical model-based approach: Methodology, possibilities, and inherent limitations

Backer, Annick De; Martinez, G.T.; Rosenauer, A.; Aert, Sandra Van

doi:10.1016/j.ultramic.2013.05.003

Cited by 100 publications

(136 citation statements)

References 44 publications

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“…The intensities of atomic columns of the precipitate examined were estimated using a statistical analysis [19,20]. The HAADF-STEM intensity I(R) can be approximated as a convolution of an object function and a probe function as follows [18]:…”

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

“…On the other hand, the object function describes the scattering, related to the projected potential of atomic columns. This function is peaked at the atomic column positions and can be modelled as a superposition of Gaussian peaks [19,20]. The scattered intensity for an atomic column is expected to be identical for columns with the same chemical composition.…”

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

“…In this manner, histograms of the estimated peak intensity of the atomic columns can be drawn [19]. This methodology [19,20] has been developed to count the atoms in nano-sized particles from HAADF-STEM images. We here applied the method to identify the atom types in the various columns, assuming the same number of atoms in each column of a precipitate.…”

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

“…Following the methodology proposed by van Aert et al [19,20], integrated column intensities were measured for each atomic column in the experimental images (i.e. without any filtering).…”

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

“…without any filtering). Histograms of column intensities were then fitted with a Gaussian "mixture model", using a statistical analysis in which the amounts of Gaussian components [19,20] are estimated from the variation of the ICL (integrated classification likelihood) criterion. This approach enables the intensity distribution of each atomic column to be estimated and discussed.…”

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

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HAADF-STEM and DFT investigations of the Zn-containing β″ phase in Al–Mg–Si alloys

et al. 2014

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The Zn-containing β" phase in Al-Mg-Si alloys has been investigated by aberration corrected high angle annular dark-field scanning transmission electron microscopy (HAADF-STEM), combined with density functional theory (DFT) calculations. The mean intensity of one Si site of the β" phase is higher than the other Si sites, suggesting partial Zn occupancy. DFT studies support that this Si site is competitive for Zn incorporation. While HAADF-STEM image simulations show an influence of the Zn distribution along the β" main growth direction, total energy calculations predict a weak Zn-Zn interaction. This suggests that Zn atoms are not clustering, but uniformly distributed along the atomic columns. The Zn incorporation has a weak influence on the β" phase where Zn is admitted as a "defect" according to the DFT studies.

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Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

“…Following the methodology proposed by van Aert et al [19,20], integrated column intensities were measured for each atomic column in the experimental images (i.e. without any filtering).…”

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

Section: Analysis Of Haadf-stem Intensity By a Statistical Approachmentioning

confidence: 99%

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HAADF-STEM and DFT investigations of the Zn-containing β″ phase in Al–Mg–Si alloys

et al. 2014

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Atom‐counting in a non‐probe corrected STEM

Schowalter

Gerken

Krause

et al. 2016

European Microscopy Congress 2016: Proceedings

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In recent years scanning transmission electron microscopy (STEM) has attracted great attention due to its high sensitivity with respect to atomic number and specimen thickness. The great advantage of this technique is to determine the number of atoms in single atomic columns in the specimen [1,2]. For instance, the reconstruction of the atomic structure of Ag nanoparticles was realized by acquiring a small series of high resolution STEM images in different zone axis orientations. Such a reconstruction enables to study facets of nanoparticles. Basically two different atom counting techniques exist: (1) In simulation‐based techniques typically the mean intensity in a well‐known material within a certain region around each atom column position, such as a Voronoi cell [3] is measured and compared with appropriate simulations. (2) Statistics‐based techniques, as introduced by Van Aert et al. [2] fit a parametric model consisting of Gaussians at the positions of the atom columns to the image intensities. Then volumes below the Gaussians are computed. Gaussian mixture models are fitted to the distribution of the volumes using the expectation maximization algorithm (see e.g. Fig. 1a) as a function of the number of Gaussian components. For each fit an order selection criterion is computed and plotted as shown in Fig. 1b. The minimum of the order selection criterion then determines the number of components in the applicable mixture model [2,4]. Finally, the number of atoms (Fig. 1c) in each column is determined by maximizing the probability that a column's intensity volume belongs to a certain component of the selected mixture model. The component with smallest volume is assumed to belong to a column with one atom. Usually, atom counting is performed using probe corrected STEM. We performed a simulation test, whether atom counting is possible in a non‐probe corrected STEM on a hypothetical Au wedge used by De Backer et al. [4]. In the respective model the number of atoms increases from 1 to 7 and then decreases again from 7 to one. From Fig. 1 it becomes clear that atoms were correctly counted from the simulated image. In general it can be expected that the column with mimimum number of atoms in an image does not necessarily contain only one atom. We studied the effect of an offset in the number of atoms on the counting result by adding additional layers of Au on top of the model. In order to account for the offset in the statistics‐based measurement positions of the Gaussian components were linearly fitted and the respective offset was derived from the parameters of the fit. We found that for small offsets the method worked reasonable, however for an offset of 7 atoms the number of atoms was overestimated by one atom, due to small deviations from the linear behaviour of the volumes. For a larger offset of 13 atoms severe errors were observed and a combination with simulations is needed to retrieve the offset value [5]. In a further test we added an amorphous carbon layer to an InAs cleavage wedge (Fig. 2a). Fig. 2b shows that the statistics‐based method well identifies the number of atoms despite the amorphous carbon layer. Errors can be observed for simulation‐based atom counting due to the additional intensity arising in the amorphous carbon layers. Fig. 3a shows an experimental HRSTEM image of a twinned Pt nanoparticle taken in our non‐probe corrected Titan 80‐300ST. Atoms have been counted in one of the twins using the simulation‐based atom counting technique (Fig 3b). For a statistics‐based atom counting evaluation the same limits have been found. For the evaluations instrumental imperfections such as reported in ref. [6] were taken care.

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Understanding the use of scattering cross‐sections in quantitative ADF STEM

Martínez

Bos

Alania

et al. 2016

European Microscopy Congress 2016: Proceedings

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Quantitative scanning transmission electron microscopy (STEM) using an annular dark field (ADF) detector has become a widely used technique for the characterization of materials at the atomic level. The quantification process involves the comparison of experimental data with image simulations, the use of statistical tools in a parameter estimation framework or a combination of both [1]. These methods have been developed using different measures for comparison, like peak intensities at the atom column position [2], image contrast variations [3] or so‐called scattering cross‐sections [4, 5]. The latter correspond to the total scattered intensity integrated over the atom column area. They have been shown to be very sensitive to the number of atoms in a column and its composition [1, 4, 6, 7]. Figure 1a shows the increase in peak intensity (green axis) and cross‐section (black axis) versus increase in number of atoms for a Pt column in [110] zone axis. As it can be observed, the peak intensity saturates after around 8 atoms meanwhile the cross‐section monotonically increases. In this work, we perform an analysis of how the electron wave propagates inside the crystal for the probe positions that conform the scattering cross‐section. With this, we analyse how the signal is generated for different detector collection angle regimes. Then, the analysis allows to identify the origin of the scattered signal and why scattering cross‐sections are more sensitive for composition and number of atoms as compared to peak intensities. In Figure 1b, we show a simulated image of a unit cell of a Pt crystal in [110] zone axis with a color‐edited version indicating the labels of the probe positions that form the scattering cross‐section and their respective distance to the atom column position. Figure 2 shows the probability amplitude of the electron wave as it propagates through the crystal for probe position r0 (a), which corresponds to the peak intensity, and for the sum of all the probe positions that conform the scattering cross‐section (b). From this, we observe that the atom column is excited deeper in the column when analyzing the cross‐sections. The off‐column probe positions carry more rich information about the scattering process for different thickness and collection regimes, which explains the increased sensitivity of this measure to the number of atoms and its composition. We then discuss the contribution to the scattered intensity for different detector collection angle regimes, such as LAADF, MAADF and HAADF.

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Atom counting in HAADF STEM using a statistical model-based approach: Methodology, possibilities, and inherent limitations

Cited by 100 publications

References 44 publications

HAADF-STEM and DFT investigations of the Zn-containing β″ phase in Al–Mg–Si alloys

HAADF-STEM and DFT investigations of the Zn-containing β″ phase in Al–Mg–Si alloys

Atom‐counting in a non‐probe corrected STEM

Understanding the use of scattering cross‐sections in quantitative ADF STEM

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