Accounting for Correlated Thermal Vibrations in Quantitative STEM Simulations

Modeling the scattering of fast electrons by a crystal populated by phonons is an essential aspect of quantitative scanning transmission electron microscopy (STEM) analyses. Specifically, direct quantitative comparisons between experiments and simulations require that the redistribution of electron scattering due to the atomic thermal motion be taken into account. [1] This thermal vibration is typically incorporated in high-angle annular dark-field STEM using the Einstein approximation (uncorrelated thermal motion) via multi-slice methods. [2] Measurable error, however, has been found for scattering at low angles using these uncorrelated simulations, including diffraction patterns and images. [3, 4] Accurately modeling the correlated thermal displacements of the atoms that accounts for this error requires knowledge of the atomic potentials/force constants, and hence the phonon band structure.<br/>In this presentation, we use simulations of electron scattering in Si as a test to demonstrate the sensitivity of the electron scattering to the phonon band structure by comparing thermal displacements incorporated using DFT-based methods with molecular dynamics using empirical and machine-learned (ML) interatomic potentials. With the correlated thermal motion included, we will show how the different models influence the electron scattering as a function of angle, thickness, and projection direction, even though the mean square displacements are nearly the same. Further we will provide insights into sensitivity to this correlated thermal motion using angle-resolved STEM images, pointing to the possibility of robust quantification of correlated thermal displacements at the atomic scale. We emphasize that the influence of correlated thermal vibrations also depends on orientation of the crystal. For example, the [111] orientation shows enhanced sensitivity to the different interatomic potentials, and hence phonon band structure. Overall, we find that the ML potentials more accurately reproduce the electron scattering than the empirical potentials when compared to using the phonon band structures determined with inelastic neutron diffraction.<br/>References:<br/>[1] Wang, Zhong Lin. Micron 34.3–5 (2003): 141–155.<br/>[2] Kirkland, Earl J. Advanced computing in electron microscopy. Vol. 12. New York: Plenum Press, 1998.<br/>[3] Muller, David A., et al. Ultramicroscopy 86.3–4 (2001): 371-380.<br/>[4] Grieb, Tim, et al. Ultramicroscopy 221 (2021): 113175.<br/>[5] We acknowledge support for this work from the Air Force Office of Scientific Research (FA9550–20–0066).