Band structure of the rippled twisted bilayer graphene

The twisted bilayer graphene has attracted great interest due to its ability to manipulate new and rich electronic properties. It is known that 2D crystals may have some out-of-plane deformations to stabilize their atomic structures. Here, we studied how ripples may affect the electronic band structure of the twisted bilayer graphene using classical force fields and tight-binding models validated by density functional theory. We generated ripples by applying strain and found that the ripples may induce electric dipole moments perpendicular to the 2D plane, which may significantly affect the band structure of the twisted bilayer graphene close to the first magic angle. Our study suggests that small ripples may not obviously affect electronic properties of monolayer graphene, whereas they cannot be ignored when studying electronic properties of the twisted bilayer graphene.