Moire Band Structures of the Double twisted Few Layer Graphene

Very recently, unconventional superconductivity has been observed in the double twisted trilayer graphene (TLG), where three monolayer graphene (MLG) are stacked on top of each other with two twist angles [J. M. Park, et al., Nature 590, 249 (2021); Z. Hao, et al., Science 371, 1133 (2021); X. Zhang, et al., Phys. Rev. Lett.127, 166802 (2021)]. When some of MLGs in the double twisted TLG are replaced by bilayer graphene (BLG), we get a new family of double twisted moire heterostructure, namely double twisted few layer graphene (DTFLG). In this work, we theoretically investigate the moire band structures of the DTFLGs with diverse arrangements of MLG and BLG. We find that, depending on the relative rotation direction of the two twist angles (alternate or chiral twist) and the middle van der Waals (vdW) layer (MLG or BLG), a general (X+Y+Z)-DTFLG can be classified into four categories, i.e. (X+1+Z)-ATFLG, (X+2+Z)-ATFLG, (X+1+Z)-CTFLG and (X+2+Z)-CTFLG, each of which has its own unique band structure. Here, X, Y, Z denote the three vdW layers, i.e. MLG or BLG. Interestingly, the (X+1+Z)-ATFLGs have a pair of perfect flat bands at the magic angle about $1.54^\circ$ coexisting with a pair of linear or parabolic bands, which is quite like the double twisted TLG. Meanwhile, when the twist angle is smaller than a "magic angle" $1.70^\circ$, the (X+2+Z)-CTFLGs can have two isolated narrow bands at $E_f$ with band width less than 5 meV. The influence of electric field and the topological features of the moire bands have been studied as well. Our work indicates that the DTFLGs, especially the (X+1+Z)-ATFLG and (X+2+Z)-CTFLG, are promising platforms to study the moire flat band induced novel correlation and topological effects.