Continuum Model of the Twisted Bilayer

TL;DR

This paper extends the continuum model of twisted bilayer graphene to include all commensurate structures, analyzing the Fourier components of hopping amplitudes and their impact on electronic properties at low twist angles.

Contribution

The authors analytically calculate Fourier components for all commensurate structures and explore their effects on electronic gaps and Dirac cone disappearance at small angles.

Findings

Fourier components vanish linearly with angle in certain structures.

Amplitudes saturate to finite values as angle approaches zero.

Dirac cone structure disappears below 1 degree twist angle.

Abstract

The continuum model of the twisted graphene bilayer (Phys. Rev. Lett. 99, 256802, 2007) is extended to include all types of commensurate structures. The essential ingredient of the model, the Fourier components of the spatially modulated hopping amplitudes, can be calculated analytically, for any type of commensurate structures in the low twist angle limit. We show that the Fourier components that could give rise to a gap in the SE-even structures discussed by Mele (Phys. Rev. B 81, 161405 2010) vanish linearly with angle, whereas the amplitudes that saturate to finite values, as $\theta\to0$, ensure that all low angle structures share essentially the same physics. We extend our previous calculations beyond the validity of perturbation theory, to discuss the disappearance of Dirac cone structure at angles below 1^{\circ}.