Quasi-crystalline phase of twisted bilayer graphene - extracting twist angles from LEED data by a high-resolution moiré spot analysis

The moiré analysis of low-energy electron diffraction (LEED) data is outlined to determine the twist angle of stacked graphene layers in twisted bilayer graphene (BLG). The analysis relies on the asymmetric motion of 2nd-order moiré diffraction spots as the twist angle in BLG varies. The data treatment is robust to potential distortions of the LEED pattern and can detect the twist angle with a precision of up to 0.03°. The concept was applied to two data sets. The first data set was recorded in a low-energy electron microscope and uses the area-selective LEED pattern taken from a twisted BLG island in a single-layer graphene film synthesized by chemical vapor deposition (CVD) on a (111)-textured Cu foil. The second data set was acquired using a commercial laboratory LEED apparatus from twisted BLG after synthesis on SiC(0001) and subsequent decoupling from the substrate by hydrogen intercalation. The reported stacking of 30° rotated graphene layers on SiC(0001) was confirmed, showing that the detected twist angle deviates by less than ±0.1°, verifying the synthesis of a quasi-crystalline 2-dimensional material. The stacking of 30° rotated graphene layers is also reported for CVD-grown BLG on (111)-textured Cu foils. However, on Cu, the rotation angle between the stacked graphene lattices in BLG was shown to deviate from 30° by about (0.75 ± 0.05)°. In addition, the single graphene lattice was found to be misaligned with respect to the underlying Cu(111) lattice by about 2.5°. We attribute both rotational misalignments to wrinkles in the graphene layer. Postulating that during CVD synthesis, perfect alignment of graphene on Cu(111) and the stacking of exactly 30° rotated graphene layers in BLG are met, wrinkle formation during sample cooling induces local delamination of stacked graphene layers in BLG and the detachment of graphene from the Cu support, leading to the observed deviations.