Flat bands and gaps in twisted double trilayer graphene

We study the electronic band structure of twisted double trilayer graphene. This system consists of two regularly stacked graphene trilayers rotated by a given angle. We consider two different arrangements for the trilayer units following their stackings: ABA-ABA and ABC-ABC. First-principles calculations at an angle of $13.{1}^{\ensuremath{\circ}}$ show that these two systems are semiconductors with narrow gaps, which result from a reduced symmetry due to the presence of the second trilayer. Next, we use these results to build a tight-binding model to study lower twist angles, which are not accessible to first-principles methods. In both cases, we find that, as the angle decreases, the magnitude of the gap decreases and eventually closes. At the same time, the low-energy bands develop into flat bands. The corresponding magic angle is greater than the values found in twisted bilayer graphene and twisted double bilayer graphene, thus supporting a hypothesis that it should increase as a function of the number of stacked layers in the rotating unit. We also discuss the spatial separation of the electronic states and their hybrization as a function of the twist angle, as well as the behavior of the systems under an external electric field. We find patterns that are reminiscent of the behaviors found in gated ABA and ABC trilayers, such as tunable overlaps or band gaps, respectively.