## 3 Result and Discussion  
### Monolayer Graphene  
Graphene is a 2D layer of carbon atoms arranged in a honeycomb lattice with Dirac cones, i.e., massless Dirac fermion at \(K\) point, where the \(\pi\) and \(\pi^{*}\) bands show a linear energy dispersion [36; 37; 38]. In Figure 1, we illustrate the atomic structure and also the electronic energy band structure of graphene based on our DFT simulations. Research on collective electronic excitations (plasmons) in graphene has attracted enormous interest both from theoretical and experimental viewpoints [4; 7; 39; 40; 41; 42; 43; 44; 45; 46]. Three kinds of the collective excitations of electrons can be considered in graphene that unroll on a wide range of energy. The first kind is attributed to finite electron doping, originating from the intraband transitions of Dirac fermions in the vicinity of the \(K\) and \(K^{\prime}\) points of the BZ at low energies (0-2 eV), and it can be regarded as intraband plasmon modes. The second kind of plasmons in the monolayer graphene is the intrinsic \(\pi\) plasmons, arising from the collective excitations of electrons from the \(\pi\) to \(\pi^{*}\) bands at energies of about 4-15 eV. At higher energies, \(\sigma\) bands start to contribute, and the mixture of the \(\pi\rightarrow\pi^{*}\) and \(\sigma\) transitions leads to another kind of plasmon excitation of graphene, usually denoted as \(\pi+\sigma\) plasmons. It is worthwhile mentioning here that the calculation of the \(\pi+\sigma\) plasmon would require including high-lying bands; hence, the ab initio approach would be the more appropriate way to capture them [47]. The plasmon spectrum of a pristine single-layer graphene was investigated in [48] using ab initio calculations. They observed the \(\pi\) and \(\pi+\sigma\) plasmon modes in freestanding single sheets at 4.7 and 14.6 eV, which were red shifted in comparison to the corresponding modes in the bulk graphite. Our attention is now focused on the intraband and \(\pi\) plasmons of the spectrum (below 10 eV), and it appears to be owed to the intraband and interband transitions, respectively, as mentioned above. In order to better perceive the electronic excitations of monolayer graphene, we calculate the non-interacting density-density response function, \(\chi^{0}(q,\omega)\). The non-interacting density-density response function can be decomposed into two parts where the first part can be considered by only including transitions within a band and the second part contains all transitions between separate bands. An example of the non-interacting density-density response function is shown in Figure 2, for \(q=0.076\) A\({}^{-1}\) and \(E_{\rm F}=0.8\) eV (the Fermi level is shifted upward by 0.8 eV above the Dirac cone, and this corresponds to an electron doping level of \(7.4\times 10^{13}\) cm\({}^{-2}\)). The \(\lim_{q\to 0}\chi^{0}({\bf q},\omega=0)\) is finite and equal to the density of states at the Fermi energy, \(N(0)\) a measure of the number of excited states. This figure shows that intraband transitions contribute more at low-energy, while the interband transitions dominate at high-energy. Figure 1: **(a,b)** Top and side view of monolayer graphene. **(c)** The band structure of graphene along the high symmetry \(\Gamma-M-K-\Gamma\) directions and the associated Brillouin zone. The zero in the energy axis is set at the Fermi level as shown by the solid line. In Figure 3, the EEL functions for various momentum transfers and the plasmon spectrum of electron doped monolayer graphene (\(n=7.4\times 10^{13}\,\mathrm{cm}^{-2}\)) are computed using the linear response DFT-RPA approach. The effect of doping on the band structures of 2D materials is ignorable [49]. In the long-wavelength limit, plasmons can be viewed as a center-of-mass oscillation of the electron gas as a whole. The physical origin of plasmons is described as follows. When electrons in free space move to screen a charge inhomogeneity, they tend to overshoot the mark.