Moiré 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) is stacked on top of each other with two twist angles [J. M. Park et al., Nature (London) 590, 249 (2021); Z. Hao et al., Science 371, 1133 (2021); X. Zhang et al., Phys. Rev. Lett. 127, 166802 (2021)]. When some of the MLGs in the double twisted TLG are replaced by bilayer graphene (BLG), we get a family of double twisted moiré heterostructure, namely, double twisted few layer graphene (DTFLG). In this paper, we theoretically investigate the moiré 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 bandwidth less than 5 meV. The influence of the electric field and the topological features of the moiré bands have been studied as well. Our paper indicates that the DTFLGs, especially (X + 1 + Z)-ATFLG and (X + 2 + Z)-CTFLG, are promising platforms to study the moiré flat band induced correlation and topological effects.

Figure 1

Schematic of the the DTFLG. (a) is the (1 $+$ 2 $+$ 1)-ATFLG where ${\theta }_{12}$ and ${\theta }_{23}$ have the opposite rotation direction. (b) is the (1 $+$ 2 $+$ 1)-CTFLG where ${\theta }_{12}$ and ${\theta }_{23}$ have the same rotation direction. (c) is the BZ of ATFLG and (d) is the BZ of CTFLG. Red, blue, and green represent the bottom, middle, and top vdW layers, respectively.

Figure 2

Classification of the DTFLGs, according to their different kinds of moiré band structures. All the possible arrangements of the MLG and BLG for each category of DTFLG are given as well.

Figure 3

Schematic of all the possible arrangements of the DTFLG. (a) is for the (1 $+$ 1 $+$ 2)-DTFLG, (b) is for the (2 $+$ 1 $+$ 2)-DTFLG, (c) is for the (1 $+$ 2 $+$ 1)-DTFLG, (d) is for the (1 $+$ 2 $+$ 2)-DTFLG, (e) is for the (2 $+$ 2 $+$ 2)-DTFLG. Here, the ATFLG and CTFLG have the same arrangements. Green, blue, and red represent the top, middle, and bottom vdW layers, respectively.

Figure 4

The electronic structure of (X $+$ 1 $+$ Z)-ATFLG. (a)–(d), (e)–(h), and (i)–(l) are the moiré bands of the (A $+$ A $+$ AB)-, (CA $+$ A $+$ AB)-, and (BA $+$ A $+$ AB)-ATFLG, respectively. (a), (e), (i) is for $\theta =2.{88}^{\circ }$; (b), (f), (j) is for $\theta =1.{54}^{\circ }$; (c), (g), (k) is for $\theta =1.{08}^{\circ }$. (d), (h), (l) show the influence of electric field at $\theta =1.{54}^{\circ }$, and the valley Chern numbers are labeled. The electric field: (d) $\mathrm{\Delta }=5$ meV, (h) $\mathrm{\Delta }=25$ meV, and (l) $\mathrm{\Delta }=20$ meV.

Figure 5

The electronic structure of (1 $+$ 2 $+$ Z)-ATFLG. (a)–(d), (e)–(h), and (i)–(l) are the moiré bands of the (A $+$ AB $+$ B)-, (A $+$ AB $+$ BC)-, and (A $+$ AB $+$ BA)-ATFLG, respectively. (a), (e), (i) is for $\theta =2.{88}^{\circ }$; (b), (f), (j) is for $\theta =1.{54}^{\circ }$; (c), (g), (k) is for $\theta =1.{08}^{\circ }$; (d), (h), (l) show the influence of electric field at $\theta =1.{54}^{\circ }$, and the valley Chern numbers are labeled. The electric field: $\mathrm{\Delta }=20$ meV.

Figure 6

The electronic structure of (2 $+$ 2 $+$ 2)-ATFLG. (a)–(d), (e)–(h), and (i)–(l) are the moiré bands of the (CA $+$ AB $+$ BC)-, (CA $+$ AB $+$ BA)-, and (BA $+$ AB $+$ BA)-ATFLG, respectively. (a), (e), (i) is for $\theta =2.{88}^{\circ }$; (b), (f), (j) is for $\theta =1.{54}^{\circ }$; (c), (g), (k) is for $\theta =1.{08}^{\circ }$; (d), (h), (l) show the influence of electric field at $\theta =1.{54}^{\circ }$, and the valley Chern numbers are labeled. The electric field: (d) $\mathrm{\Delta }=20$ meV, (h) $\mathrm{\Delta }=20$ meV, and (l) $\mathrm{\Delta }=40$ meV.

Figure 7

The electronic structure of (X $+$ 1 $+$ Z)-CTFLG. (a)–(d), (e)–(h), and (i)–(l) are the moiré bands of the (A $+$ A $+$ AB)-, (BA $+$ A $+$ AB)-, and (CA $+$ A $+$ AB)-CTFLG, respectively. (a), (e), (i) is for $\theta =2.{88}^{\circ }$; (b), (f), (j) is for $\theta =1.{70}^{\circ }$; (c), (g), (k) is for $\theta =0.{82}^{\circ }$; (d), (h), (l) show the influence of electric field at $\theta =1.{70}^{\circ }$, and the valley Chern numbers are labeled. The electric field: $\mathrm{\Delta }=20$ meV.

Figure 8

The electronic structure of (1 $+$ 2 $+$ Z)-CTFLG. (a)–(d), (e)–(h), and (i)–(l) are the moiré bands of the (A $+$ AB $+$ B)-, (A $+$ AB $+$ BC)-, and (A $+$ AB $+$ BA)-CTFLG, respectively. (a), (e), (i) is for $\theta =2.{88}^{\circ }$; (b), (f), (j) is for $\theta =1.7^{\circ }$; (c), (g), (k) is for $\theta =0.{82}^{\circ }$; (d), (h), (l) show the influence of electric field at $\theta =1.{70}^{\circ }$, and the valley Chern numbers are labeled. The electric field: (d) $\mathrm{\Delta }=20$ meV, (h) $\mathrm{\Delta }=10$ meV, and (l) $\mathrm{\Delta }=20$ meV.

Figure 9

The electronic structure of (2 $+$ 2 $+$ 2)-CTFLG. (a)–(d), (e)–(h), and (i)–(l) are the Moiré Bands of the (CA $+$ AB $+$ BC)-, (CA $+$ AB $+$ BA)-, and (BA $+$ AB $+$ BA)-CTFLG, respectively. (a), (e), (i) is for $\theta =2.{88}^{\circ }$; (b), (f), (j) is for $\theta =1.{70}^{\circ }$; (c), (g), (k) is for $\theta =0.{82}^{\circ }$; (d), (h), (l) show the influence of electric field at $\theta =1.{70}^{\circ }$, and the valley Chern numbers are labeled. The electric field: $\mathrm{\Delta }=5$ meV.

Figure 10

(a) The moiré bands of the TBG, TDBG, and (CA $+$ AB $+$ BC)-CTFLG at $\theta =1.{70}^{\circ }$. (b) The bandwidth of the the central moiré bands at $E_f$ as a function of $\theta$.