Theory for the electronic structure of incommensurate twisted bilayer graphene

The experimental control over the twist angle in twisted bilayer graphene has not been reported and its realistic structure is most likely incommensurate. In this paper, we develop a tight-binding virtual crystal approximation theory to study the electronic properties in incommensurate twisted bilayer graphene. The theory yields the electronic band structure and the local density of states for any incommensurate twist angle {\theta} between the graphene sheets. Angle dependent Van Hove singularities are observed in the numerically calculated local density of states. In accord with observations in scanning tunneling microscopy and spectroscopy, our theoretical calculation indicates that the rotation angle between graphene sheets does not result in a significant reduction in the Fermi velocity in comparison with monolayer graphene. The developed theory is quite general and can be applied to investigate the electronic properties in any incommensurate multilayer heterostructures.