Stacking, strain, and twist in 2D materials quantified by 3D electron diffraction

The field of two-dimensional (2D) materials has expanded to multilayered systems in which electronic, optical, and mechanical properties change---often dramatically---with stacking order, thickness, twist, and interlayer spacing. For transition metal dichalcogenides (TMDs), bond coordination within a single van der Waals layer changes the out-of-plane symmetry that can cause metal-insulator transitions or emergent quantum behavior. Discerning these structural order parameters is often difficult using real-space measurements; however, we show that 2D materials have distinct, conspicuous three-dimensional (3D) structure in reciprocal space described by nearly infinite oscillating Bragg rods. Combining electron diffraction and specimen tilt we probe Bragg rods in all three dimensions to identify multilayer structure with subangstrom precision across several 2D materials---including TMDs (${\mathrm{MoS}}_{2}$, ${\mathrm{TaSe}}_{2}$, ${\mathrm{TaS}}_{2}$) and multilayer graphene. We demonstrate quantitative determination of key structural parameters such as surface roughness, inter- and intralayer spacings, stacking order, and interlayer twist using a rudimentary transmission electron microscope. We accurately characterize the full interlayer stacking order of multilayer graphene (1, 2, 6, 12 layers) as well the intralayer structure of ${\mathrm{MoS}}_{2}$ and extract a chalcogen-chalcogen layer spacing of $3.07\ifmmode\pm\else\textpm\fi{}0.11$ \AA{}. Furthermore, we demonstrate quick identification of multilayer rhombohedral graphene.