The Origin of Magic Angle in Twisted Bilayer Graphene is Heisenberg’s Uncertainty Principle

Abstract

The magic-angle twisted bilayer graphene (MATBG) recently attracted intensive research attention because of its fascinating and unconventional electronic properties. Herein, we claim the magic-angle phenomenon originates from the Heisenberg uncertainty principle, which can provide intensive explanations on finite size effect and twist-dependent low energy band variations. We showed that flat bands could exist only near the AA stacking structure rather than AB. The finite-size effect gives the minimal size of graphene quantum dots (R ≳ 4 nm) for the emergence of the Dirac point, and the uncertainty relation provides the upper bound for moiré supercells (R ≲ 23.5 nm) in twisted bilayer graphene, which is the quantum mechanical boundary for the emergence of flat bands. Combining the twist dependence of moiré supercell size, we proved that there is only one possible magic angle in MATBG at θ ≈ 1.1°. Our result implies that the unconventional phenomena in MATBG originate from the fundamental feature of condensed matter physics.