Merged Four Dirac Points at the Critical Interlayer Distance in Commensurately Twisted Bilayer Graphene: the Origin of the Zero Velocity

We study the commensurately-tilted bilayer graphene in the tight-binding model with changing the interlayer distance, which can be tuned by pressure. We find that at the commensurately-tilted bilayer graphene with moderate rotating angles, when the energy gap at K point is not negligible, the other Dirac points within the upper two bands move along the $\Gamma$-K-M line in the bilayer Brillouzone, when the interlayer distance is changed. The velocity at K point becomes zero due to the merging of the four Dirac points within upper two bands. This mechanism of zero velocity at K point is expected to be the origin of the magic angle with flat band at ambient pressure, at which the upper two bands are almost degenerate and the band gap at K can be neglected.
Comment: 10 pages 13 figures