Four Moiré materials at One Magic Angle in Helical Quadrilayer Graphene

Abstract

We introduce helical twisted quadrilayer graphene (HTQG), four graphene sheets rotated by the same small angle, as a versatile and experimentally accessible platform for correlated topological matter. HTQG consists of three moiré lattices, formed by interference between adjacent graphene layers, that are twisted relative to each other. Lattice relaxation produces four types of large-scale commensurate domains. The domains are characterized by the stacking of the three moiré lattices and come in two types: Type-I "Bernal" stacking and Type-II "rhombohedral" stacking. Domain walls between adjacent stackings often host topologically protected edge states, forming networks at the supermoiré and super-supermoiré scales. Remarkably, all four moiré substructures have narrow bands at the same magic angle $θ\approx 2.3^\circ$, allowing their correlated phases to be simultaneously targeted in device manufacturing. We argue that the Type-I domains are especially suitable for realizing robust superconductivity which emerges from doping topological insulators, and Chern insulators in $C = \pm 2$ bands.

Figures (8)

• Figure 1: (a) Schematic overview of the four graphene layers and their associated moiré scales. From left to right: the atomic lattice scale ( $\sim 0.2\,\mathrm{nm}$), the moiré scale $a^M$ (typical TBG period of $\sim6\,\mathrm{nm}$), the supermoiré scale $a^{MM}$ for HTTG ( $\sim0.1\,\mathrm{\mu m}$), and the super‐supermoiré scale $a^{MMM}$ for HTQG (up to a few micrometers). (b,c) Local domains in the XX super-supermoiré region are Type-II and carry valley Chern numbers $C_v=+4$ and $-4$ for $\alpha\beta\gamma$ and $\gamma\beta\alpha$ respectively at charge neutrality. XY includes four domains, $\alpha\beta\gamma$, $\gamma\beta\alpha$, $\alpha\beta\alpha$ and $\beta\alpha\beta$. The arrows indicate the topological chiral boundary modes associated with one of the valleys. The band structure for and (d)$\alpha\beta\alpha$ and (e)$\alpha\beta\gamma$ are depicted alongside the valley Chern numbers corresponding to the single particle gaps. The valence band of $\alpha \beta \gamma$ has $C=-2$ in the $K$ valley leading to a change in valley Chern number of $C_K-C_{K'}=-8$ when accounting for spin degeneracy. (f) The Chern-sublattice basis for the narrow bands in the Type-I domains. The single particle dispersion tunnels between symmetry-related Chern sectors, as in TBG. The real space charge density of the narrow bands is depicted for (g)TBG, (h) $\alpha\beta\alpha$ and (i)$\alpha\beta\gamma$.
• Figure 2: The moiré superlattice structure for XX is shown for (a) the non-relaxed and (b) the relaxed. The moiré structure of three layer out of (b) are depicted in (c) for layer 1, 2 and 3, and (d) for layer 2, 3 and 4. The blue/black/red points denotes $A_1A_2$/$A_2A_3$/$A_3A_4$. The shaded region represents $\alpha\beta$ while non-shaded region represents $\beta\alpha$. The blue/red is for layer 1, 2 and 3, and layer 2, 3 and 4, respectively. (e) is a schematic picture of arrangement of $\alpha\beta$ and $\beta\alpha$. We slide the blue region relatively for the red region for emphasizing the overlap of the red and the blue region. (f)-(j) is the corresponding plots for (a)-(e). In all figures, white scale bar indicates $25~\rm{nm}$.
• Figure 3: (a)The green/orange line represents $T$, deviation from ideal quantum geometry, as a function of $\kappa$ for TBG/$\alpha\beta\alpha$. (b)The bandwidth as a function of $\kappa$ and $\theta$ for $\alpha\beta\alpha$. The cyan dot indicates $(\theta,\kappa)=(2.25^\circ, 0.6)$ which we use in this paper.
• Figure 4: (a)The band structure of Eq.\ref{['eq_Hamiltonian_chiral_general']} for $\alpha\beta\gamma$ with $\alpha=0.322$. (b)The velocity of Dirac cone at $\kappa$, normalized by the velocity of $\alpha=0$, as a function of $\alpha$. (c) $\psi_{\kappa,3}(-{\bm{r}}_{BA})$ and $W_{\alpha\beta\gamma}^{B}$ [Eq.\ref{['eq_chiral_abc_1stMA_Bsub_W2']}] are plotted with blue and red lines, respectively.
• Figure 5: (a)The band structure of Eq.\ref{['eq_Hamiltonian_chiral_general']} for $\alpha\beta\alpha$ with $\alpha=0.320$. (b)The velocity of Dirac cone at $\kappa$, normalized by the velocity of $\alpha=0$, as a function of $\alpha$. (c) $\psi_{\kappa,3}(-{\bm{r}}_{BA})$ and $W_{\alpha\beta\alpha}^{A}$ [Eq.\ref{['eq_chiral_aba_1stMA_Asub_W2']}] is plotted with red line.