Moiré plane wave expansion model for scanning tunneling microscopy simulations of incommensurate two-dimensional materials

Incommensurate heterostructures of two-dimensional (2D) materials, despite their attractive electronic behavior, are challenging to simulate because of the absence of translation symmetry. Experimental investigations of these structures often employ scanning tunneling microscopy (STM); however, there is to date no comprehensive theory to simulate an STM image in such systems. In this paper, we present a geometry-based method to generate STM images in incommensurate van der Waals (vdW) heterostructures, which we call the moiré plane wave expansion model (MPWEM). We generate the STM images using a weighted sum of three image terms: the noninteracting STM images of (1) the substrate layer, (2) the adsorbate layer, and (3) a semi-empirical Fourier expansion of the moiré wave vectors obtained analytically which results from the interaction of (1) and (2). We illustrate and benchmark the model using selected vdW 2D systems composed of structurally and electronically distinct crystals, and show that the method reproduces experimental STM images down to angstrom-scale details. The MPWEM, thanks to its simplicity, can serve as an image generation tool prior to more computationally expensive calculations which are often limited by the number of atoms and the requirement of periodic supercells, and thus offers a promising class of computationally friendly STM simulations, of high relevance in the growing field of twistronics. Published by the American Physical Society 2024