Type II magic angles and interlayer-coupling-dependence of magic angles in commensurately twisted bilayer graphene.

The magic angle, at which the velocity at K point in twisted bilayer graphene becomes zero, is studied theoretically. We study the commensurately-tilted bilayer graphene in the tight-binding model with changing the interlayer distance, which can be tuned by pressure. The almost doubly degeneracy of the bands at small rotating angles ($\lesssim 1^{\circ}$) and the fourfold degeneracy at the Dirac points are lifted as the rotating angle is large, and the finite gap becomes not negligible at the Dirac K points, although the twofold degeneracies at the Dirac points remain above and below the energy gap. Even at the commensurately-tilted bilayer graphene with moderate rotating angles, the velocity at K points becomes zero due to the merging of the four Dirac points at the critical values of interlayer distance, despite the finite gap at the Dirac points. We call this type of magic angle as Type II magic angle. We find that the crossover from the type II magic angle to the usual magic angle, at which the band gap can be neglected, depends on the choice of a parameter for numerical differentiation to calculate the velocity at K point. The crossover occurs around $\alpha \sim 3.48^{\circ}$ if we take the parameter to be 0.01, and the crossover angle becomes small as the parameter is small.