Fig. 1. The energy dispersion of TBG.

a The original Brillouin zone is folded into the mini-Brillouin zone for the moiré superlattice. The solid and dashed green lines defining the large hexagons represent the original Brillouin zones of the top and bottom graphene layers respectively. b The flat bands energy dispersion from the ${\mathbf{K}}_{+}$ valley with the twist angle $\theta =1.2^{\circ }$. The dashed and solid lines are the cases with strain $\epsilon =0$ and $\epsilon =0.1\%$, respectively. The red and blue bands represent the conduction $\left(\nu =\mathrm{c}\right)$ and valence $\left(\nu =\mathrm{v}\right)$ bands, respectively. The conduction flat band energy dispersion in the mini-Brillouin zone with strain $\epsilon =0$ for c and $\epsilon =0.1\%$ for d, respectively. The energy dispersion from the other valley ${\mathbf{K}}_{-}$ can be mapped by the time-reversal symmetry as $E_{s,+,\nu }\left(\mathbf{q}\right)=E_{s,-,\nu }\left(-\mathbf{q}\right)$.