## 3 Result and Discussion
### Bilayer Graphene  
Bilayer graphene displaces the simplest possible system where graphene sheets are brought together to create a new nanostructure whose physical properties show remarkable similarities and differences as compared to monolayer graphene. Bilayer graphene, like single-layer graphene, is a zero-gap semimetal that consists of two coupled monolayers of carbon atoms stacked as in natural graphite (AB stacking or Bernal-stacked form where half of the atoms lie directly over the center of a hexagon in the lower graphene sheet, and half of the atoms lie over an atom) yielding a unit cell of four atoms (Figure 4) [50; 51; 52]. We calculate the band structure of bilayer graphene through DFT simulations, and this is illustrated in Figure 4. Bilayer graphene has four electronic bands (a pair of conduction bands and a pair of valence bands) with \(p_{z}\) symmetry, namely: \(\pi_{1}\), \(\pi_{2}\), \(\pi_{1}^{*}\) and \(\pi_{2}^{*}\). The dispersion of these bands is parabolic near the \(K\) point, and their occupation depends on the doping values. In undoped bilayer graphene, the \(\pi_{1}\) and \(\pi_{2}\) bands are fully occupied, while the \(\pi_{1}^{*}\) and \(\pi_{2}^{*}\) bands are fully unoccupied. Furthermore, an overlap of the \(\pi_{1}^{*}\) and \(\pi_{2}^{*}\) bands along \(K-M\) leads to an anisotropic dispersion at energies from 1-1.5 eV. Research on the collective electronic excitations (plasmons) in freestanding multilayer graphene [53; 54; 55] shows that, similar to single-layer graphene and graphite, the spectra of bilayer graphene feature two characteristic high-energy plasmon peaks, i.e., the \(\pi\) plasmon below 10 eV and the \(\pi+\sigma\) plasmon above 15 eV, but these plasmons are red shifted with respect to graphite. In the low-energy range, a conventional 2D plasmon was predicted to exist originating from the intraband transitions. It is noteworthy to mention that, as reported in [56], when the two layers are near each other (separated by a distance \(d\) in the \(z\) direction with the 2D layers in the \(x-y\) plane), the 2D plasmons are coupled by the interlayer Coulomb interaction leading to the formation of in-phase and out-of-phase interlayer density fluctuation modes: an in-phase optical plasmon mode, where the densities in the two layers fluctuate in phase with the usual 2D plasma dispersion (\(\omega\sim\sqrt{q}\)) and an out-of-phase acoustic plasmon mode, where the densities in the two layers fluctuate out-of-phase with a linear wavelength dispersion (\(\omega\sim q\)). In a recent study, the plasmon modes of undoped (intrinsic) and doped (extrinsic) bilayer graphene were calculated and analyzed carefully based on density-functional theory in an energy range from a few meV to \(\sim\)30 eV, along the inequivalent \(\Gamma-K\) and \(\Gamma-M\) directions [33]. In that paper, they found  
Figure 4: (**a,b**) Top and side view of the atomic structure of bilayer graphene (BLG). (**c**) The band structure of BLG along the high symmetry \(\Gamma-K-M-\Gamma\) directions in reciprocal space. The optimized interplane distance (\(d\)) is calculated to be 3.45 Å.  
an acoustic plasmon mode for momenta along the \(\Gamma-M\) direction for a positive shift in the Fermi level of 1 eV. Although they have claimed this acoustic plasmon is an undamped collective excitation, an overdamped acoustic plasmon was predicted by [PERSON] et al. [56]. To investigate the origin of different plasmon modes in bilayer graphene, we calculate the interband and intraband parts of the non-interacting response function of bilayer graphene, and we present our results in Figure 5 for only \(q=0.084\) A\({}^{-1}\). In this case, the Fermi energy is shifted upward by 0.7 eV towards the bottom of the conduction bands in bilayer graphene, corresponding to \(n=9.5\times 10^{13}\) cm\({}^{-2}\).