Flat bands in slightly twisted bilayer graphene: Tight-binding calculations

The presence of flat bands near Fermi level has been proposed as an explanation for high transition temperature superconductors. The bands of graphite are extremely sensitive to topological defects which modify the electronic structure. In this Rapid Communication, we found nondispersive flat bands no farther than 10 meV of the Fermi energy in slightly twisted bilayer graphene as a signature of a transition from a parabolic dispersion of bilayer graphene to the characteristic linear dispersion of graphene. This transition occurs for relative rotation angles of layers around $1.5°$ and is related to a process of layer decoupling. We have performed ab initio calculations to develop a tight-binding model with an interaction Hamiltonian between layers that include the $\pi$ orbitals of all atoms and takes into account interactions up to third nearest neighbors within a layer.

Figure 1

(Color online) Energy band dispersion for six different structures. Twisted BLG (straight blue line), single-layer graphene (red dotted), and DFT results (green diamonds). At the Fermi level, around the K point a linear behavior of the bands is observed for all structures. In (e) and (f) the Fermi velocity is renormalized.

Figure 2

(Color online) A flattening of the bands is observed with a depletion in $v_F$ for the slightly twisted bilayer (blue line). The $AB$ stacked BLG (green dashed line) and single-layer graphene (red dotted line) are included for comparison.

Figure 3

(Color online) Localized states in the band structure of twisted BLG for angles around $1.5°$. Flat bands are not present for $1°$, instead a parabolic dispersion appears. Insets in (a) and (b) are obtained near the K point. They show a change in the convexity of the band from $\Gamma$ to K point and, in (b), a near zero Fermi velocity is revealed.

Figure 4

(Color online) (a) Dependence of Fermi velocity in twisted BLG $v_F^{\left(T\right)}$ and in single-layer graphene $v_F^{\left(G\right)}$ with the rotating angle. (b) $\Delta E$ at $\Gamma$ point versus the rotation angle.