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

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, and hence the interlayer distance can be reconstructed. A detail protocol of the reconstruction procedure is provided. In addition, we derived an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account different chemical composition of the individual monolayers. It is shown that one interference fringe in 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 artifact-free. At larger twist angles, artifacts due to the moiré structure appear in the reconstruction. A method for reconstruction of average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at accuracy of ±0.5 Å, which is a few hundreds 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 the holographic CBED imaging can pick as small as 0.1 Å variations in the interlayer distance, though the quantitative reconstruction of such variations suffers from a large error.