Graphene Moiré Superlattice: Flat-Band Electron Localization

Twist two graphene sheets by a fraction of a degree and their overlapping honeycomb lattices interfere into a moiré superlattice — a triangular pattern of alternating AA, AB and BA stacking regions with a period tens of nanometres across. Near the famous "magic angle" of about 1.1°, the Bistritzer–MacDonald continuum model predicts the low-energy electronic bands become nearly flat, and the corresponding wavefunctions stop spreading evenly across the lattice, instead pooling into puddles centred on the AA-stacked sites. This simulator renders that moiré pattern and a live electron-density proxy that sharpens into those puddles as the twist angle and interlayer coupling are tuned toward the flat-band condition, alongside a real atomic-lattice inset showing the two twisted honeycomb layers and live readouts of the moiré period, the dimensionless coupling ratio α, and the renormalized Dirac velocity v*/v_F that vanishes exactly at the magic angle.