## 2 Electronic Structure of Twist Bilayer Graphene  
When two layers of graphene are stacked up, they are in Bernal (AB) stacking or AA stacking when the crystal orientations of both layers are aligned, which is usually the case for grown crystals due to thermodynamics. However, when the bilayer structure is fabricated by transferring the exfoliated monolayers, the crystal orientations are mostly misaligned with a twist angle \(\theta\), which could be well controlled to the accuracy of 0.1\({}^{\circ}\) recently [25; 26]. Such a twisted bilayer forms a super structure called the moire pattern, with lattice constant \(\mathrm{d}=\mathrm{a}(2\sin\left(\frac{\theta}{2}\right))^{-1}\), where \(\mathrm{a}=0.246\) nm, i.e., the lattice constant of graphene. In each moire unit cell, there are three high symmetry spots, AA, AB and BA, named after the local stacking geometry. [PERSON] and coworkers [24] studied the electronic structure of the such unique geometry of the bilayer graphene as early as in 2007. Within the framework of the continuum approximation on a small twist angle model, they found that the linear dispersion at low energy and the Dirac cones structure remain in the twisted bilayer, as shown in Figure 1c, but with a significant reduction of the  
Figure 1: Schematic structure of twisted bilayer graphene (TBG) formed by transfer assembly (**a**) and epitaxial growth on SiC (**b**). The calculated band structure of TBG with \(\theta=3.9^{\circ}\) (**c**) and \(\theta=1.05^{\circ}\) (**d**), where flat bands (moire) close to the Dirac point appears. (**e**) The density of states corresponding to \(\theta=1.05^{\circ}\). Panel (**c**) adapted with permission from Ref. [24], Copyright American Physical Society 2007. Panels (**d**,**e**) are adapted with permission from Ref. [8], Copyright PNAS 2011. Fermi velocity for very small twist angles. Such results imply the twist angle could be a significant parameter that has a profound effect on the electronic structure of the bilayer graphene. A theoretical breakthrough was made by [PERSON] and [PERSON] in 2011 [8]. They unambiguously pointed out that the Fermi velocity at the Dirac point vanishes at a series of magic twist angles, e.g., \(\theta=1.05^{\circ}\), when the dispersionless flat bands (also called moire bands) are formed around the charge neutrality point, as shown in Figure 1d. Such a flat band contributes to a sharp peak to the density of state, which is shown in Figure 1e. They also remarked that the electron-electron interaction in the magic twist angle systems could give rise to strong correlation effects. [PERSON] et al. [7] mentioned the link between superconductivity and the flat bands of twisted graphene, although they give the critical angle of \(1.5^{\circ}\) that is a little deviated from the magic angle. From the view of the band structure, comparing with gapless monolayer graphene, it is two superlattice bandgaps that are next to the upper and bottom sub flat band due to the effect of interlayer hybridization in TBG. According to theoretical calculation [7; 9; 27], when the twist angle is increasing, and away from the first magic angle, the position of the gap would shift to higher energy until being disappeared as the two sub minibands become wider. Such twist angle tuned gaps are valuable for the application in the optoelectronic device, for example, [PERSON], et al. [28] demonstrated the gate-tunable photoresponse in the mid-infrared wavelength range of 5-12 um in 1.81\({}^{\circ}\) TBG. Additionally, a number of recent optoelectronic works based on bigger twist angle TBG were done [29; 30; 31; 32]. Another interesting feature in the electronic structure of the TBG is the doubled van Hove singularity in the density of states that generally exists at arbitrary angles TBG as a consequence of the saddle points at two emerging Dirac cones [6; 9]. Additionally, a number of theory works have investigated the band topology [27; 33; 34; 35; 36; 37; 38] in the magic angle TBG system recently.