Aberration-corrected STEM has become a powerful technique for materials characterisation of complex nanostructures. Recent progress in the development of quantitative methods allows us to extract reliable structural and chemical information from experimental images in 2D as well as in 3D. In quantitative STEM, images are treated as datasets from which structure parameters are determined by comparison with image simulations or by using parameter estimation-based methods [1]. So-called scattering cross-sections, measuring the total scattered intensity for each atomic column, are useful values for quantification [2, 3]. Their high sensitivity and robustness for imaging parameters in combination with a statistical analysis enables us to count atoms with single-atom sensitivity [4].  An example is shown in Figure 1, where atom-counting from a single image combined with an energy minimisation approach [5] is used to reconstruct the 3D atomic structure of a Au nanorod. The close match with the 3D tomography reconstruction resulting from images recorded along 3 viewing directions [6] demonstrates the accuracy of the method.

Reducing the number of images by avoiding tilt tomography will be of great help when studying beam-sensitive nanostructures. However, the tolerable electron dose is often still orders of magnitude lower than what is typically used for atomic resolution imaging. Therefore, the question arises: how to optimise the experiment design in order to reduce the electron dose? To investigate this, we have developed a statistical framework that allows us to study the effect of electron shot noise, scan noise, and radiation damage on the atom-counting precision. Figure 2 shows that even for low-dose image acquisitions, statistical parameter estimation theory is a powerful tool to refine structure parameters of an incoherent imaging model and to measure scattering cross-sections. However, the presence of electron shot noise in this dose regime is limiting the atom-counting precision. Severe overlap in the distributions of scattering cross-sections related to columns of different thicknesses hampers one to achieve single-atom sensitivity. We will show how the precision improves with increasing electron dose until scan noise, followed by radiation damage, become the main limiting factors. This analysis allows one to balance atom-counting reliability and structural damage as a function of electron dose.

Finally, it will be shown how quantitative ADF STEM may greatly benefit from statistical detection theory in order to optimise the detector settings [7]. This is illustrated in Figure 3, where the number of atoms of a beam-sensitive Pt particle is determined from a STEM image acquired under the computed optimal detector settings. In addition, use is made of a novel hybrid method to count the number of Pt atoms, in which the benefits of a statistics-based and image simulations-based method are efficiently combined in one framework. In conclusion, new developments in the field of quantitative STEM will be presented enabling one to quantify atomic structures in their native state with the highest possible precision.

[1] S. Van Aert et al., IUCrJ 3 (2016) 71-83.

[2] S. Van Aert et al., Ultramicroscopy 109 (2009) 1236-1244.

[3] H. E et al., Ultramicroscopy 133 (2013) 109-119.

[4] S. Van Aert et al., Physical Review B 87 (2013) 064107.

[5] L. Jones et al., Nano Letters 14 (2014) 6336-6341.

[6] B. Goris et al., Nature Materials 11 (2012) 930-935.

[7] A. De Backer et al., Ultramicroscopy 151 (2015) 46-55.

The authors acknowledge financial support from the Research Foundation Flanders (FWO,Belgium) through project fundings (G.0374.13N, G.0369.15N and G.0368.15N) and postdoc grants to A.D.B. and B.G. S.B. and A.B. acknowledge funding from the European Research Council (Starting Grant No. COLOURATOMS 335078 and No. VORTEX 278510). The research leading to these results has also received funding from the European Union Seventh Framework Programme [FP7/2007- 2013] under Grant agreement no. 312483 (ESTEEM2).

Figures:

Figure 1. a) ADF STEM images of a Au nanorod taken from 3 directions. The number of Au atoms along [110] follows from the statistical decomposition of scattering cross-sections illustrated in b. c) 3D reconstructions when using compressed sensing tomography or atom-counting combined with atomistic modelling.

Figure 2. a) Low-dose ADF STEM image of a Pt wedge for an electron dose of 770 electrons per squared Å. b) Refined parametric model using statistical parameter estimation theory. c) Histogram of scattering cross-sections of all Pt columns.

Figure 3. a) Probability of error for counting the number of Pt atoms as a function of the inner detector radius. b) ADF STEM image under near-optimal ADF detector settings (35-190 mrad) together with the number of Pt atoms using a hybrid statistics-simulations based method.

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