## 2 Interlayer Interactions
### Twisted Systems  
The misorientation of a few stacked graphene layers can affect the electronic structure of the carbon system. The rotation between the graphene layers has been receiving a lot of attention recently [44, 45, 46, 47, 48, 49, 50]. This has been stimulated by the observation of [PERSON] _et al._[47] that graphene grown epitaxially on SiC (\(000\bar{1}\)) contains a high density of twist boundary faults, _i.e._, layers with a relative rotation. The same authors [47] demonstrated that 2 LG structure with the relative rotation frequently observed in experiment (\(\theta=30^{\circ}\pm 2.20^{\circ}\)) manifests an apparent electronic decoupling. Namely, _ab initio_ calculations showed that both layers displayed the Dirac cone and Fermi velocity characteristic of 1 LG [47]. This remarkable result has inspired a number of subsequent theoretical and experimental works [44, 48, 49]. In Reference [44], bilayer and trilayer twist boundary systems were investigated by _ab initio_ calculations, and in both cases layers with a relative rotation were found to display apparent 1 LG behavior. On the other hand, Reference [48] considered the \(\theta\to 0\) limit of the rotated 2 LG in a continuum approximation, with the result that the layers exactly decoupled but with the Fermi velocity of the Dirac cone suppressed as compared to 1 LG. This latter result is in contrast to _ab initio_ calculations [44, 48] for finite rotation angles where a Fermi velocity exactly equal to that of 1 LG is found [50]. In a twisted (or turbostratic) bilayer graphene (t-2 LG) two layers are stacked with a random rotation angle (see Figure 5b). When stacking these two supercells, we obtain a bilayer structure with short-range incommensurability, then the unit-cell area of t-2 LG can be more than 1000 times as large as that of 1 LG, due to slightly misoriented lattice vectors of two layers. The electronic structure of t-2 LG shows  
Figure 5: (**a**) The 2D electronic band structures of Bernal AB 2 LG [44]; (**b**) Atomic structure of t-2 LG with rotation angle \(\theta=21.8^{\circ}\). Dashed (red) and solid (green) lines represent the lattices of layers 1 and 2, respectively [51]; (**c**) Velocity ratio \(V_{\rm 2 LG}/V_{\rm 1 LG}\) for a commensurate (n, m) bilayer cell versus rotation angle \(\theta\): circle, VASP; cross, tight-binding calculations. The line is the model of [PERSON] _et al._[48]: \(V_{\rm 2 LG}/V_{\rm 1 LG}=1-9[\tilde{t}/(V_{\rm 1 LG}Ksin(\theta/2))]^{2}\), with \(\tilde{t}=0.11\) eV and \(V_{\rm 1 LG}K=2\gamma_{0}\pi(3^{1/2})=9.8\) eV [45]. Band structure of t-2 LG with rotation angles (**d**) 9.43\({}^{\circ}\); (**e**) 3.89\({}^{\circ}\); (**f**) 2.65\({}^{\circ}\) and (**g**) 1.47\({}^{\circ}\). Dashed (red) slopes around the K point indicate the monolayer’s band dispersion. Note that the scale of wave number (horizontal axis) reduces as the rotation angle decreases. Dirac point energy is set to zero [51]. a linear band dispersion near Dirac points [44, 48, 50, 52] rather than the massive dispersion of AB-2 LG, and this is due to the relatively weak IL interaction. In t-2 LG the superposition of the two honeycomb lattices generates a Moire pattern with longer period. The two Dirac electron gases are then coupled by a periodic interaction, with a large supercell, which can restore a Dirac linear dispersion, but with a Fermi velocity substantially reduced with respect to 1 LG (see Figure 5c). In Figure 5d-g, we show examples of band structures of t-2 LG with rotation angles of 9.43\({}^{\circ}\), 3.89\({}^{\circ}\), 2.65\({}^{\circ}\) and 1.47\({}^{\circ}\), respectively [51]. Dashed (red) slopes around the K point indicate the monolayer's band dispersion. Note that the scale of wave number (horizontal axis) reduces as the rotation angle decreases. Dirac point energy is set to zero [51]. It is important to note that the changes in the electronic dispersion due to different twisting angles are expected to be maintained to \({}^{12/13}\)C t-2 LG systems.