Is Twisted Bilayer Graphene Stable under Shear?

In twisted bilayer graphene (TBLG), extremely small deviations from the magic twist angle $\theta_m{\approx}1.08^\circ$ change its electronic structure near the Fermi level drastically, causing a meV-wide flat band to appear or disappear. In view of such sensitivity to minute structural deformations, we investigate the combined effect of shear and atomic relaxation on the electronic structure. Using precise experimental data for monolayer and bilayer graphene as input in a simplified formalism for the electronic structure and elastic energy, we find TBLG near $\theta_m$ to be unstable with respect to global shear by the angle $\alpha{\approx}0.08^\circ$. In TBLG, the effect of shear on the electronic structure is as important as that of atomic relaxation. Under optimum global shear, calculated $\theta_m$ is reduced by $0.04^\circ$ and agrees with the observed value.