## 3 Result and Discussion
### Monolayer MoS\({}_{2}\)  
The lack of a natural band gap makes graphene unsuitable for developing optoelectronic and photovoltaic devices [62]. A particularly interesting class of 2D materials is the transition metal dichalcogenides (TMDC) whose electronic properties range from semiconducting to metallic and even superconducting. Among them, molybdenum disulfide (MoS\({}_{2}\)) with a natural band gap is gaining increasing interest [63]. MoS\({}_{2}\) has a hexagonal structure that consists of two planes of hexagonally-arranged S atoms bonded through covalent bonds to central layer Mo atoms. In agreement with previous reports [64; 65; 66; 67], the MoS\({}_{2}\) monolayer is a direct band gap semiconductor with a maximum of the valence and the minimum of the conduction band located at the Kpoint of the Brillouin zone (Figure 8). The bands around the energy gap are relatively flat, which are as expected from the \(d\) character of the electron states at these energies. The states around the Fermi energy are mainly due to the \(d\) orbitals of Mo, while strong hybridization between \(d\) orbitals of Mo and \(p\) orbitals of \(S\) atoms below the Fermi energy has been observed [30]. Figure 8: **(a,b)** Top and side view of the atomic structure of molybdenum disulfide (MoS\({}_{2}\)). **(c)** The band structure of MoS\({}_{2}\) along the high symmetry \(\Gamma-M-K-\Gamma\) directions and the associated Brillouin zone. The Fermi level is set at 0 eV. The blue balls are molybdenum atoms, and the yellow ones are sulfur atoms. Figure 7: The gate voltage dependence of the plasmon mode in bilayer graphene. The values of the gate voltages are 1.1 V/nm and 2.5 V/nm related to band gaps of about 0.1 and 0.2 eV, respectively. The GW approximation, is an approximation made in order to calculate the self-energy of a many-body system of electrons, has been shown to provide very reliable descriptions of the electronic and dielectric properties for many semiconductors and insulators [68]. The recent quasiparticle self-consistent GW calculations have reported that MoS\({}_{2}\) is a direct gap semiconductor at both the LDA and GW levels, and the GW gap is 2.78 eV at both the \(K\) and \(K^{\prime}\) points [69]. In the following, we will explore the dispersion behavior of plasmon modes of monolayer MoS\({}_{2}\) by using our DFT-RPA code. The intraband plasmons in metallic single-layer transition metal dichalcogenides (TMDCs) have been studied using density functional theory in the random phase approximation [70]. They have found that at very small momentum transfer, the plasmon energy follows the classical \(\sqrt{q}\) behavior of free electrons in 2D. For larger momentum transfer, the plasmon energy is significantly red shifted due to screening by interband transitions. The non-interacting response function of MoS\({}_{2}\) with \(n=5.6\times 10^{13}\) cm\({}^{-2}\) and for \(q=0.069\) A\({}^{-1}\) is plotted in Figure 9 and shows that the intraband term of \(\chi^{0}(q,\omega)\) is dominated at energies below 1.0 eV, and the interband term is inconsiderable. In Figure 10a, the EEL function of MoS\({}_{2}\) for different values of \(q\) with electron doping of \(n=5.6\times 10^{13}\) cm\({}^{-2}\) is depicted and the plasmon dispersion of system plotted in Figure 10b (black dots). In the following, we compare the resulting plasmon dispersion calculated by our ab initio model to that obtained by a model Hamiltonian. Figure 10: _Cont._  
Figure 9: The interband and intraband of non-interacting response function (\(\chi^{0}(q,\omega)\)) of doped MoS\({}_{2}\) in units of the Fermi-level density of states as a function of \(\hbar\omega\) for \(q=0.069\) Å\({}^{-1}\). We use Eq.11 in the paper by [PERSON], et al.