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K. Müller, F. Krause, A. Béché, M. Schowalter, V. Galioit, S. Löffler, J. Verbeeck, J. Zweck, P. Schattschneider, A. Rosenauer:
"Quantum mechanical interpretation of electron picodiffraction reveals atomic electric fields";
Talk: Microscopy of Semiconducting Materials (MSM-XIX), Cambridge; 2015-03-29 - 2015-04-02.

A prominent scanning TEM (STEM) technique for studying atomic-scale electric fields is differential phase contrast (DPC) microscopy. Conventionally, segmented ring detectors are utilised to record portions of the ronchigram [1], which is assumed to be homogeneously filled and shifted as a whole in presence of electric fields in the specimen. Here we first act on the reliability of these assumptions by showing that electron ronchigrams exhibit rich intensity variations already for thinnest specimens (1-2nm), and that the dominant effect of atomic electric fields on aberration-corrected probes is a complex redistribution of intensity inside ronchigrams. This explains why segmented DPC detectors yield high contrast at atomic sites whereby a quantification of electric fields in terms of ronchigram shifts is prone to errors.

Second, we give a quantum mechanical interpretation of DPC: The ronchigram is recorded in the diffraction plane, showing the Fourier transform of the specimen exit wave. According the axioms of quantum mechanics, the local ronchigram intensity I(p,q) is thus proportional to the probability for the momentum (p,q) to occur. Hence the expectation value for the momentum is calculated by a centre-of-gravity-type summation, relating the rich details of the ronchigram to a single vector with fundamental physical meaning in a direct and simple manner. To put this into practice, we use a pixelated detector in experiment.

Third, Ehrenfest's theorem is applied to relate the electric field to the momentum transfer via a proportionality factor. In a comprehensive simulation study of GaN, we demonstrate the capability of our method by calculating momentum transfer, electric field and charge-/electron densities from simulated ronchigrams. We then prove the experimental applicability by mapping the electric field in an SrTiO3 unit cell quantitatively and verify these results by simulations [2].

[1] Shibata et al., Nat.Phys.8, p611 (2012)
[2] Müller et al., Nat.Comm. 5, p5653, (2014)