Magic Angles In Equal-Twist Trilayer Graphene

Abstract

We consider a configuration of three stacked graphene monolayers with equal consecutive twist angles θ. Remarkably, in the chiral limit when interlayer coupling terms between AA sites of the moiré pattern are neglected we find four perfectly flat bands (for each valley) at a sequence of magic angles which are exactly equal to the twisted bilayer graphene (TBG) magic angles divided by √ 2. Therefore, the first magic angle for equal-twist trilayer graphene (eTTG) in the chiral limit is θ * ≈ 1.05 • / √ 2 ≈ 0.74 • . We prove this relation analytically and show that the Bloch states of the eTTG's flat bands are non-linearly related to those of TBG's. Additionally, we show that at the magic angles, the upper and lower bands must touch the four exactly flat bands at the Dirac point of the middle graphene layer. Finally, we explore the eTTG's spectrum away from the chiral limit through numerical analysis.

Magic-angle helical trilayer graphene

“…The distinction between even and odd magic angles, together with ratios between magic α that do not match those of TBG, suggests that the magic angles here do not descend from those of TBG. This is in contrast to the chiral magic angles of twisted chirally stacked multilayers (4, 31), alternating twist multilayers (4), and the d = 0 periodic HTG (43), which can all be related to TBG. A detailed understanding of the mathematical structure of this model is an interesting subject beyond the scope of this work.…”

“…In the domain wall region and their intersection, A(r) > 0 and is much larger than in the unrelaxed structure, A 0 = 2|1 − cosθ| ≈ 0.68 × 10 −3 . This implies that these regions [which contain the previously studied d = 0 model (42,43)] actually relax away from the periodic structure and therefore appear more locally quasi-crystalline (57).…”

Pseudomagnetic fields in fully relaxed twisted bilayer and trilayer graphene

“…Lattice relaxation has been shown to play a pivotal role in the piezoelectric, ferroelectric and electronic properties of twisted TMDs [26][27][28][29][30][31][32][33][34][35][36][37][38][39] as well as in the charge density wave phases of twisted NbSe 2 [40,41]. In graphitic systems [42], atomic relaxation has been shown to determine the outof-plane polarization of twin boundaries [43] while in supermoiré systems such as the non-symmetric twisted trilayer graphene, lattice relaxation leads to a clear separation between the flat band and the highly dispersive Dirac cone [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60]. For sufficiently small twist angles, θ ≲ 0.5 • , a domain pattern, made of AB and BA regions separated by domain walls emerges, leading to a conducting network of topologically protected 1D channels [61][62][63][64][65][66].…”