Abstract text (incl. figure legends and references)

Electron Tomography is able to reconstruct nanoscopic three dimensional structures via acquisition of a tilt series of projections [1].
While different types of signal have been exploited in the past (e.g. holographically obtained phase [2]), the tomographic reconstruction still relies to a great extent on the fulfillment of the projection requirement, requesting the signal to reassemble a linear integral of a 3D distribution along the projection direction.
For STEM Tomography in particular, the HAADF signal, among others, can be exploited to gain quantitative results when approximated as Lambert−Beer-exponentially attenuated with the coefficient
μα [3]. However, the often observed so-called cupping artifacts indicate that there is still considerable deviation from an ideal projection. Especially when dynamical scattering gets significant, as in case of close to zone axis orientations, the assumption of linearity gets problematic.
Recently, with the advent of new fast direct electron detectors, momentum-resolved STEM [4, 5] becomes increasingly more accessible, enabling users to shape the entity of the detected signal in dedicated new ways.
In this study, for a 10nm sized truncated gold nanocube (relaxation using Tersoff potentials, implemented in LAMMPS software), a whole tilt series has been simulated (+- 180° in 5° steps, 8mrad semi-convergence angle, 600x600 scan points, 15x15x15nm³ (with 15 slices), 4096x4096px² diffraction pattern, 8mrad ring detectors, 5 frozen phonon configurations) , using a highly efficient STEM multislice simulation code, running on several V100 GPUs. Gold was chosen, due to its high atomic number, leading to violation of linearity already at very low specimen thickness. Several STEM signals have been compared as to their linearity using the a priori known projected depth as reference. The second moment in momentum space (in this case from 8mrad to 64mrad) has been found to be almost linear (Fig. 1).
Fig. 2 shows the final second moment tomogram, reconstructed using a WSIRT algorithm with 5 iterations [2]. Even though it was not the aim of this study, at least under the ideal conditions of this simulation (high amount of projections along crystal axes, 180° tilt range), atomic resolution was easily achieved. Notably, there are virtually no cupping artifacts. Of course, experimental conditions like coherence, misalignment or noise would affect reconstruction quality. Nevertheless, second moment tomography should be highly beneficial e.g. at medium resolution.

Figure 1: Simulated tilt series of a truncated Au-Nanocube.

(a) - (d) Second moment projections at 0°, 15°, 30° and 45°, respectively.
(e) - (h) Profiles of (a) - (d) summed up in tilt direction (green). For each tilt the likewise obtained profiles of projected thickness (black), HAADF detector (orange) and from HAADF retrieved Lambert-Beer attenuation µα (blue) are given as reference.

Figure 2: Second Moment STEM Tomogram
(a) Volume rendering with indication of a (1,0,0) plane.
(b) Slice through the indicated (1,0,0) plane in (a).

References:
[1] M WEYLAND et al. Materials Today, 2004, 10.1016/S1369-7021(04)00569-3
[2] D WOLF et al. et al. Ultramicroscopy, 2010, 10.1016/j.ultramic.2009.12.015
[3] D WOLF et al. Nano Letters 2018, 10.1021/acs.nanolett.8b01270
[4] K MÜLLER, et al. Nature communications, 2014, 10.1038/ncomms6653
[5] H ROBERT, et al. Ultramicroscopy, 2022, 10.1016/j.ultramic.2021.113425
[6] K. M.-C. acknowledges funding from the DFG, contract EXC 2089/1 – 390776260 (e-conversion).