## 3 Result and Discussion
### Bilayer Graphene
Our numerical results show that the non-interacting response function of bilayer graphene is similar to its monolayer, but there is an extra contribution of the interband part at low-energies that is obviously absent in monolayer graphene. In Figure 6, we illustrate the loss spectra for various amounts of \(q\) and also electronic excitations for bilayer graphene along the \(\Gamma-M\) direction for \(n=9.5\times 10^{13}\) cm\({}^{-2}\). The loss spectra involves both intraband and interband excitations at energies below 10 eV, but we neglect to show \(\pi\) plasmon in Figure 6b. Figure 5: The inter- and intra-band terms of the non-interacting response function (\(\chi^{0}(q,\omega)\)) of doped bilayer graphene in units of the Fermi-level density of states as a function of \(h\omega\) for \(q=0.084\) Å\({}^{-1}\) along the \(\Gamma-M\) direction. The real and imaginary parts of this function are indicated by solid and dashed lines, respectively. Figure 6: \(Cont.\)  
It is clear that the plasmon dispersion of bilayer graphene follows a general \(\sqrt{q}\) dispersion at low energies, which is also seen in monolayer graphene. These results are in reasonable agreement with earlier work reported by [PERSON] and coauthors [PERSON] and coauthors (2018). Our ab initio calculations show the acoustic plasmon mode, which is highly damped in bilayer graphene structures. Most importantly, by breaking the inversion symmetric of the two layers, a non-zero band gap can be induced. The potential of a continuously tunable band gap through a gate voltage applied perpendicularly to the sample is very interesting ([PERSON] et al., 2016; [PERSON] et al., 2017; [PERSON] et al., 2017). The realization of a widely tunable electronic band gap in electrically-gated bilayer graphene has been experimentally demonstrated ([PERSON] et al., 2017). They showed a gate-controlled, continuously tunable band gap of up to 250 meV by using dual-gate bilayer graphene field-effect transistor infrared micro-spectroscopy. Moreover, this electrostatic band gap control suggests nanoelectronic and nanophotonic device applications based on graphene. Notice that the band gap can be observed in photoemission, magneto transport, infrared spectroscopy and scanning tunneling spectroscopy. The low-energy effective model Hamiltonian for a gated bilayer graphene can be written as:  
\[H_{\text{eff}}=-\frac{1}{2m}(\vec{\sigma}\cdot\vec{p})\sigma_{x}(\vec{\sigma} \cdot\vec{p})+\Delta\sigma_{z} \tag{18}\]  
where \(\vec{\sigma}\) are Pauli matrices and \(\Delta\) is the gated energy. In bilayer graphene, changing the applied gate voltages will turn into controlling the electron density, \(n\), and the interlayer asymmetry different potential energies, \(\Delta\). In other words, the asymptotic energy \(\Delta V\) is related to layer density, and the layer densities depend on \(\Delta\), ultimately ([PERSON] et al., 2017). Here, we ignore this effect and assume that the band gap is independent of the electron density. We have examined first-principle calculations to investigate the plasmon modes of AB stacked bilayer graphene in the presence of a perpendicular applied electric field. Figure 7 shows the gate dependence of the optical plasmon modes of bilayer graphene for \(\Delta=1.1\) and 2.5 eV. Clearly, the gap reduces the plasmon mode and softens the collective excitation modes, especially at the long wavelength limit. In this regime, the plasmon mode behaves slightly different from that obtained for a system with \(\Delta=0\) given by \(\omega_{p}(q)=\sqrt{\frac{e^{2}}{4\epsilon_{0}}\frac{nq}{m}}\)([PERSON] et al., 2017). Moreover, the interband contribution decreases with the gated energy \(\Delta\); however, the intraband contribution increases with the bias owing to the fact that the energy dispersion leads to an enhancement of the density of states near the Fermi energy.