## 3 Result and Discussion
### Monolayer Phosphorene  
Although two-dimensional materials such as monolayer graphene and monolayer transition metal dichalcogenide, the most common component MoS\({}_{2}\), have attracted intensive research interest owing to their fascinating electronic, mechanical, optical and thermal properties, the lack of the band gap in graphene and low carrier mobility in MoS\({}_{2}\) limits its wide applications in electronic devices and nanophotonics [82; 83]. Recently, isolated two-dimensional black phosphorus (BP), known as phosphorene, with a nearly direct band gap and high carrier mobility has received enormous interest owing to its extraordinary electronic and optical properties in engineering application [84; 85]. Figure 14: The damping parameter of plasmon modes in bilayer MoS\({}_{2}\) as a function of the momentum for \(n=6.9\times 10^{13}\) cm\({}^{-2}\). Note that in order to have a well-defined plasmon mode, \(\gamma(q)/\omega_{p}(q)<<1\). Our DFT calculations for mono- and bi-layer phosphorene were carried out using the Perdew-Burke-Ernzerhof exchange-correlation functional [86] coupled with the DFT-vdw method. Figure 15a,b shows that phosphorene is a puckered honeycomb structure with each phosphorus atom covalently bonded with three neighboring atoms within a rectangular unit cell, and the two lattice constants are \(a_{x}=4.62\) A and \(a_{y}=3.30\) A along the armchair and zigzag directions, respectively [87]. As a result of the puckered structure, each single-layer phosphorene contains two atomic layers where the distances between the two nearest atoms (2.22 A) and the distance between the top and bottom atoms (2.24 A) are slightly different. It should be mentioned that monolayer phosphorene is a nearly direct band gap semiconductor because the exact top of the valence band is slightly away from the \(\Gamma\) point. However, they are sufficiently close to be considered as a direct band gap as previous first-principles calculations mention [88]. Our DFT calculations predict that the band gap of monolayer phosphorene is 0.98 eV, and the self-energy correction enlarges the band gap to 2.0 eV, which is ideal for potentially broad electronic applications [89]. The optical gap of phosphorene was reported to be 1.6 eV by using the GW and Bethe-Salpeter equation method, and it is much lower than the electronic band gap of 2.15 eV, which shows significant excitonic effects in phosphorene [90]. The electronic band structure of monolayer phosphorene is plotted in Figure 15c, and it shows the high anisotropic band structure with very different effective masses along the armchair and zigzag directions for both electrons and holes. This anisotropic electronic structure is useful for thermoelectric materials, because for the direction with a smaller effective mass, the carrier mobility and thus the electrical conductivity can be high, while the larger effective mass along the other direction contributes to an overall large density of states that improves the Seebeck coefficient [84; 91]. In Figure 16, the plasmon spectrum of monolayer phosphorene with \(E_{\rm F}=0.07\) eV along the armchair direction based on our DFT simulations is compared to the result recently reported in [92], and they show an acceptable agreement. Figure 15: **(a,b)** Top and side view of the atomic structure of monolayer phosphorene. **(c)** The band structure of monolayer phosphorene along the high symmetry \(X-\Gamma-Y\) directions and the associated Brillouin zone. The Fermi level is set at 0 eV. To know the origin of the electronic excitations in the monolayer phosphorene, we obtain the different contributions (interband and intraband) of the non-interacting response function, and they are shown in Figure 17 for \(q=0.046\) A\({}^{-1}\) and \(n=3.9\times 10^{13}\) cm\({}^{-2}\) along the armchair direction. These results show that the intraband term of this quantity plays an important role, and its interband term is negligible.