Flatbands and Perfect Metal in Trilayer Moiré Graphene

We investigate the electronic structure of a twisted multilayer graphene system forming a moiré pattern. We consider small twist angles separating the graphene sheets and develop a low-energy theory to describe the coupling of Dirac Bloch states close to the $K$ point in each individual plane. Extending beyond the bilayer case, we show that, when the ratio of the consecutive twist angles is rational, a periodicity emerges in quasimomentum space with moiré Bloch bands even when the system does not exhibit a crystalline lattice structure in real space. For a trilayer geometry, we find flatbands in the spectrum at certain rotation angles. Performing a symmetry analysis of the band model for the trilayer, we prove that the system is a perfect metal in the sense that it is gapless at all energies. This striking result originates from the three Dirac cones which can only gap in pairs and produce bands with an infinite connectivity. It also holds quite generally for multilayer graphene with an odd number of planes under the condition of $C_{2z}T$ symmetry.

Figure 1

Trilayer graphene with the same rotation angle ${\theta }_{12}={\theta }_{23}$ between consecutive layers. (a) Original Brillouin zones in each layer with their respective $K$ points. (b) $k$-space lattice generated by the vectors ${\mathbf{q}}_j^{ab}$ (for ${\mathbf{q}}_j^{12}={\mathbf{q}}_j^{23}$). Green, red, and blue sites belong, respectively, to the layers 1,2,3. (c) Moiré Brillouin zone with high-symmetry points $\mathrm{\Gamma }$, $K_M$, $K_M^{\prime }$, $M$ and high-symmetry lines (in black).

Figure 2

Moiré bands (a)–(d) and renormalized Dirac-point velocities (e) at the symmetric points $\mathrm{\Gamma }$ (red line) and $K_M$ (green line). The eight bands closest to zero energy are represented along the moiré Brillouin zone trajectory $-M\to \mathrm{\Gamma }\to M\to K_M\to \mathrm{\Gamma }$ for $\alpha =0.2$, 0.28, 0.42, 0.85 (a)–(d). The particle-hole symmetry discussed in the main text—sending $k$ to $-k$ and $E$ to $-E$—is clearly visible along the path $-M\to \mathrm{\Gamma }\to M$. (e) Velocities of the two Dirac cones at $\mathrm{\Gamma }$ and $K_M$ as function of $\alpha$.