Thermal diffuse scattering in sub-angstrom quantitative electron microscopy—phenomenon, effects and approaches

Abstract

This paper reviews the recent progress in the following areas.(1) In quantitative high-resolution transmission electron microscopy, the theoretically calculated images usually give better contrast than the experimentally observed ones although all of the factors have been accounted for. This discrepancy is suggested due to thermal diffusely scattered (TDS) electrons, which were not included in the image calculation. The contribution from TDS electrons is especially important if the image resolution is approaching 0.1 nm and beyond with the introduction of Cs corrected microscopes. A more rigorous multislice theory has been developed to account for this effect. (2) We proved that the off-axis holography is an ideal energy filter that even filters away the contribution made by TDS electrons in the electron wave function, but conventional high-resolution microscopy do contain the contribution made by phonon scattered electrons. (3) In electron scattering, most of the existing dynamical theories have been developed under the first order diffuse scattering approximation, thus, they are restricted to cases where the lattice distortion is small. A formal dynamical theory is presented for calculating diffuse scattering with the inclusion of multiple diffuse scattering. By inclusion of a complex potential in dynamical calculation, a rigorous proof is given to show that the high order diffuse scattering are fully recovered in the calculations using the equation derived under the distorted wave Born approximation, and more importantly, the statistical time and structure averages over the distorted crystal lattices are evaluated analytically prior numerical calculation. This conclusion establishes the basis for expanding the applications of the existing theories. (4) The 'frozen lattice' model is a semi-classical approach for calculating electron diffuse scattering in crystals arisen from thermal vibration of crystal atoms. Based on a rigorous quantum mechanical phonon excitation theory, we have proved that the frozen lattice mode is an excellent approximation and no detectable error would be possible under normal experimental conditions. q

High-resolution imaging of zeolite with aberration-corrected transmission electron microscopy

“…Image simulation in this study did not take thermal diffuse scattering (TDS) into consideration. Temperature factors of oxygen atoms in the monoclinic crystal system are comparatively large 13 so that the decreases in the amplitude of the exit wave function due to TDS 26 may not be negligible in the NCS imaging mode, which utilizes additional amplitude contrast. Moreover, amplitude contrast also affected the contrast inside the nano-channels.…”

Nanoscale phase separation in a fcc-based CoCrCuFeNiAl0.5 high-entropy alloy

“…While the diffuse spots, indicated by arrows in Fig. 2c and d, are crossovers from strongly scattered electrons from Brillouin zone boundaries, which may be associated with thermally scattered electrons [19]. The L1 2 ordering reflections in the as-spun sample (Fig.…”

STEMSIM—a New Software Tool for Simulation of STEM HAADF Z-Contrast Imaging

“…An exact solution which is computationally much more demanding has been given by Wang [2]. Figure 3a shows the TDS intensity in the object exit plane at a specimen thickness of 20 nm obtained for InSb by the FP method by computing 500 wave functions corresponding to different thermal displacement configurations, excluding Bragg reflections from the Fourier-transformed wave functions, inverse Fourier transform, and finally summing the resulting TDS intensities.…”

“…3c. In our approach, we apply the MS method and use the TDS scattered intensity generated at position R in the exit surface of slice m as given by the integrand in equation (2). A point source with intensity one is put at position R which is assumed incoherent with respect to all other point sources generating TDS intensity.…”

“…To this end we programmed a MATLAB software tool for simulation of Z-contrast images using Weickenmeier-Kohl absorptive scattering amplitudes [1]. We implemented the frozen phonon (FP) approximation which has the highest accuracy as it approaches the exact solution if a large number of atomic thermal displacement configurations are used [2]. Additionally, we included the TDS absorptive potential method suggested by Allen et al [3] and Ishizuka [4].…”