Type of presentation: Oral

IT-16-O-2356 Measuring structure parameters from electron microscopy images : what are the limits?

Van Aert S.1, Gonnissen J.1, De Backer A.1, Sijbers J.2, den Dekker A.2,3
1Electron Microscopy for Materials Science (EMAT), University of Antwerp, Antwerp, Belgium, 2iMinds-Vision Lab, University of Antwerp, Antwerp, Belgium, 3Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands
sandra.vanaert@uantwerpen.be

State-of-the-art electron microscopy combined with advanced model-based methods can provide reliable numbers for unknown structure parameters. Aberration correction greatly improves the quality of experimental images, new STEM data collection geometries allow one to visualise light atoms, and detectors start to behave as ideal quantum detectors. In combination with statistical parameter estimation theory to analyse experimental data, electron microscopy then performs at its ultimate limits. For example, atomic column positions can be measured down to picometer scale precision [1], differences in averaged atomic number of only 3 can be detected from HAADF STEM images [2], and the number of atoms in an atomic column can be counted with single atom sensitivity [3,4] (see fig. 1). The question then arises: how far can we go?

Ultimately, the attainable precision with which unknown structure parameters can be estimated is set by the unavoidable presence of electron counting noise. For continuous parameters such as atom positions, this limit is expressed by means of the Cramér–Rao lower bound (CRLB). For discrete parameters such as the number of atoms, the probability of error can be derived (defining e.g. the probability of reporting an atom when there is none or reporting no atom when there is one) [5].

Using these expressions, we show that the precision of the 3D atom positions estimated from depth sectioning data is poor under realistic exposure times. However, when simplifying the problem to the estimation of the vertical position of each atomic column with known number of atoms, picometer precision can be reached. The performance of depth sectioning can now be compared with HAADF STEM tomography. Furthermore, evaluating the probability of error helps us to determine STEM detector settings resulting into the highest detectability of light atoms (see fig. 2). Even so, we can define the minimally required electron dose in order to attain a maximum allowable error for miscounting atoms. This is of great importance when studying beam sensitive structures.

In conclusion, statistical parameter estimation theory is used to explore fundamental limits with which structure parameters can be estimated. The CRLB and the probability of error not only outperform classical performance criteria (including resolution, contrast or SNR), they also allow us to predict attainable limits under given experimental conditions and to explore the optimal experimental settings.

[1] S. Van Aert et al., Advanced Materials 24 (2012), p.523

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

[3] S. Van Aert et al., Nature 470 (2011), p.374

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

[5] A.J. den Dekker et al., Ultramicroscopy 134 (2013), p.34

The authors kindly acknowledge funding from the Fund for Scientific Research, Flanders (FWO).

Fig. 1: Examples of quantitative analysis using statistical parameter estimation theory. (a)Measurement of Ti displacements of ca. 5 pm inside a twin boundary in CaTiO3 [1]. (b)Quantitative characterisation of a La0.7Sr0.3MnO3-SrTiO3 interface from an HAADF STEM image [2]. Counting results of Ag (c) and Au atoms (d) with single atom sensitivity [3,4].

Fig. 2: Probability of error as a function of the inner radius of an annular detector for an aberration corrected microscope (300kV acceleration voltage, 21.7mrad probe forming angle, 100mrad outer detector radius, 12000 incident electrons/Å2). Green, red, and blue correspond to the problem of detecting Li, H, and Al/Ti in LiV2O4, YH2, and SrTiO3/LaAlO3.