Imaging of twisted monolayers in three-dimensional nanoporous graphene

The investigation of intricate alignments in two-dimensional materials has garnered significant attention, fueled by the growing ability to precisely manipulate and engineer electronic properties through controlled stacking and orientation. Recent observations of spontaneous arrangements in multilayer materials with stochastic twist angles offer new opportunities to elucidate the unique physics governing moir\'e superstructures, and the critical role of the electronic coupling between layers. In this context, the present study focuses on the atomic-scale characterization of nanoporous graphene, with particular emphasis on regions formed by misoriented graphene layers that assemble into crumpled, continuous patterns. Despite the absence of flat domains at the submicron scale, scanning tunneling microscopy is employed to identify dominant twisted multilayer structures in the samples. To complement the experimental observations, a simplified, yet, effective exponential parametric model is introduced, merging independent analytic electron densities to estimate the number of misaligned layers in the scanned areas. This approach is further refined through density-functional theory calculations of projected electron densities from graphene, which are rotated and stacked at adjusted interlayer distances. The synergy between the two models provides a robust framework for distinguishing between twisted bilayer and twisted trilayer domains, as observed in constant-current imaging. Finally, a full density-functional theory analysis is conducted on simple few-layer graphene structures, assessing the role of interlayer correlations. The present study provides an atomic-level description of porous graphene, while also offering accessible tools for simulating twisted layered materials beyond graphene, which may be beneficial for the STM community.