## 2 Interlayer Interactions
### AB Stacked Bilayer Graphene  
Since the unit cell of Bernal AB stacked 2 LG is the same as that of graphite (four atoms per unit cell), we can model the 2 LG electronic structure using the tight-binding model for graphite [35] by adapting the Slonczewski-Weiss-McClure (SWM) parametrization [36; 37], using \(\gamma_{0}\) and the interlayer (IL) nearest-neighbor hopping parameters, \(\gamma_{1}\), \(\gamma_{3}\) and \(\gamma_{4}\) (shown in Figure 4c). After solving the \(4\times 4\) tight-binding Hamiltonian [38], we find that the electronic structure of 2 LG has two valence bands (\(\pi_{1}\) and \(\pi_{2}\)) and two conduction bands \(\pi_{1}\)* and \(\pi_{2}\)*, as can be seen in Figure 4a. One valence band touch one conduction band at the K point and the other two bands have a gap of \(2\gamma_{1}\). 2 LG is also a zero gap semiconductor but the electronic dispersion is no longer linear around the K point, but it has a hyperbolic dependence with \(k\)[38]. The IL interaction in 2 LG with regular AB stacking modifies the linear dispersion of 1 LG into the quadratic dispersion, where an electron behaves now as a massive particle [39]. The \(\gamma_{3}\) and \(\gamma_{4}\) IL parameters have interesting consequences in the electronic structure. \(\gamma_{3}\) gives rise to a trigonal warping effect in the low energy spectrum, where the equienergies curves have triangular shape, while \(\gamma_{4}\) is related to the electron-hole asymmetry in 2 LG [38]. Figure 4a,b show, respectively, the 2 LG band structure near to the K point with \(\gamma_{3}\) and \(\gamma_{4}\) equal to zero and with \(\gamma_{3}=0.3\) eV and \(\gamma_{4}=0.15\) eV. Regarding the vibrational structure, the micro-mechanical cleavage of bulk graphite (so-called Scotch-tape method) gives 2 LG graphene samples with mostly \({}^{12/12}\)C AB-stacked structures and the strongest IL interactions. For this reason, the 2 LG system obtained from exfoliation is, so far, the best system to provide the fundamental physics behind the IL related phonon self-energy renormalizations. Another commonly used method is the chemical vapor deposition (CVD), and has been proven to grow 2 LG in which top and bottom layers are compose solely of either \({}^{13}\)C or \({}^{12}\)C resulting in a hybrid \({}^{12/13}\)C 2 LG. The method also grows 2 LG in which both layers are composed of \({}^{12}\)C, namely \({}^{12/12}\)C 2 LG [11; 41; 42]. Through the CVD method, in general, the 2 LG systems result in twisted layers (see Figure 3c,e for the twisted t-2 LG Raman spectra). Recently, however, selected area electrondiffraction (SAED) and Raman scattering have been used to show that the resultant 2 LG systems grown by CVD methods can also be \({}^{12/12}\)C and \({}^{12/13}\)C AB-2 LG [11, 41, 42]. Figure 3d,f show the Raman spectra for the AB-2 LG systems. Indeed, the Raman G\({}^{\prime}\)(2D)/G intensity ratio (\(\approx\)1) for the CVD grown \({}^{12/12}\)C AB-2 LG system agrees well with the Scotch-tape sample and also the G\({}^{\prime}\)-band linewidth (\(\approx\)60  cm\({}^{-1}\)) is similar. In fact, the lineshape for the G\({}^{\prime}\) band can be interpreted in terms of four peaks whose frequencies (ranging from 2640 to 2715  cm\({}^{-1}\)) and linewidths (\(\approx\)30  cm\({}^{-1}\)) are basically the same as the ones found for the Scotch-tape sample. However, the G\({}^{\prime}\) lineshape, which is determined by the Raman cross-sections of each of the four peaks, is slightly different mainly because the little shoulder around 2645  cm\({}^{-1}\) observed for the Scotch-tape samples [43] is not evident in the CVD grown samples. This suggests that the IL interactions from AB-stacked CVD samples might be different from that in AB-stacked scotch-tape samples.