Holographic reconstruction of the interlayer distance of bilayer two-dimensional crystal samples from their convergent beam electron diffraction patterns

TL;DR

This paper presents a holographic method to reconstruct the interlayer distance in twisted bilayer 2D crystals from CBED patterns, achieving sub-angstrom accuracy and analyzing the effects of twist angle on reconstruction quality.

Contribution

It introduces a detailed protocol and an exact formula for interlayer distance reconstruction from CBED interference patterns, accounting for chemical differences and twist angles.

Findings

Reconstruction accuracy of 0.5 Å with holography.

Interference fringes suffice for distance measurement.

Smaller twist angles yield more precise reconstructions.

Abstract

The convergent beam electron diffraction (CBED) patterns of twisted bilayer samples exhibit interference patterns in their CBED spots. Such interference patterns can be treated as off-axis holograms and the phase of the scattered waves, meaning the interlayer distance can be reconstructed. A detailed protocol of the reconstruction procedure is provided in this study. In addition, we derive an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account the different chemical compositions of the individual monolayers. It is shown that one interference fringe in a CBED spot is sufficient to reconstruct the distance between the layers, which can be practical for imaging samples with a relatively small twist angle or when probing small sample regions. The quality of the reconstructed interlayer distance is studied as a function of the twist angle. At smaller twist angles, the reconstructed interlayer distance distribution is more precise and artefact free. At larger twist angles, artefacts due to the moir\'e structure appear in the reconstruction. A method for the reconstruction of the average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at an accuracy of 0.5 A, which is a few hundred times better than the intrinsic z-resolution of diffraction limited resolution, as expressed through the spread of the measured k-values. Moreover, we show that holographic CBED imaging can detect variations as small as 0.1 A in the interlayer distance, though the quantitative reconstruction of such variations suffers from large errors.