## 3 Result and Discussion
### Monolayer MoS\({}_{2}\)
[71], where the long wavelength behavior of the plasmon frequency of monolayer MoS\({}_{2}\) can be obtained from:  
\[\omega_{q}^{0}=\sqrt{\frac{e^{2}q}{2\pi\epsilon_{0}\epsilon_{r}}}\sqrt{\frac{(2 E_{F}-\Delta)[E_{F}(\Delta+2E_{F})-\lambda^{2}]}{4E_{F}^{2}-\lambda^{2}}}\]  
Here, \(\Delta\) is the energy gap and \(\lambda=80\) meV. In this case, we consider \(\epsilon_{r}=1\) because it can be compared to our ab initio results. It is obvious that the plasmon mode obtained within DFT-RPA approach for monolayer MoS\({}_{2}\) differs significantly from with that calculated using the low-energy model Hamiltonian. The similar comparison was performed for hole doping in the case of MoS\({}_{2}\) in [72], and they observed strongly reduced plasmon energies and concluded that neglecting the material-specific dielectric function \(\epsilon_{\alpha\beta}(\mathbf{q})\) within the minimal three-band model is a severe approximation leading to unrealistic plasmonic properties. This discrepancy could be understood by looking at both the bare potential and the Lindhard function of the MoS\({}_{2}\). In order to perceive the aforementioned discrepancy, we look at the structure of the MoS\({}_{2}\) again, which basically consists of three atomic layers. In ordinary 3D materials, the effect of lattice screening is simply a re-scaling of the interaction strength by a dielectric constant. In quasi-2D materials, however, the interaction is modified by the polarizability of the crystalline material originated from the induced charge, \(n_{ind}=-\
abla\cdot\mathbf{P}\), where \(\mathbf{P}\) is the polarizability [73]. Therefore, the Poisson equation for the potential of the external point charge takes the form:  
\[\
abla^{2}V(\mathbf{r},a)=-4\pi e\delta(z)-4\pi a\
abla_{\rho}^{2}V(\rho,z=0,a )\delta(z) \tag{19}\]  
where \(\mathbf{r}=(\rho,z)\) and \(\rho=(x,y)\), and we use the fact that \(\mathbf{P}=-a\
abla_{\rho}V(\rho,z=0,a)\delta(z)\). In order to solve this equation, it is convenient to take the Fourier transformation of the equation to obtain \(V(q,a)\). Afterwards, by taking the inverse Fourier transformation of \(V(q,a)\), we can find the effective potential in real space, which is no longer \(e^{2}/r\), and it yields:  
\[V(\mathbf{r},a)=\frac{e^{2}\pi[-Y_{0}(r/a)+H_{0}(r/a)]}{2a} \tag{20}\]  
Figure 10: (**a**) The EEL functions of MoS\({}_{2}\) for different \(q\) and \(n{\sim}5.6\times 10^{13}\,\mathrm{cm}^{-2}\) for \(\epsilon=1\). (**b**) Our optical plasmon mode of MoS\({}_{2}\) (dots), obtained from the peaks in the EEL functions in the (**a**), in comparison with a model Hamiltonian by [PERSON] et al. [71] (blue line). The red line is obtained using the modified bare potential given by Equation (22) by a model Hamiltonian. The black line shows our ab initio calculation with the modified bare potential (see the text for more details). where the Bessel function of the second kind is defined by:  
\[Y_{n}(x)=\frac{J_{n}(x)\cos(nx)-J_{-n}(x)}{\sin(nx)} \tag{21}\]  
where \(J_{n}(x)\) is the Bessel function of the first kind. For \(n\) an integer, this formula should be understood as a limit. The Struve function, \(H_{n}(x)\), solves the inhomogeneous Bessel equation. It would be worthwhile to mention that this potential was proposed in the Keldysh model [74], which is based on a slab of constant dielectric value, the potential between two charges in a slab of thickness \(d\). In this model, \(a=d\epsilon_{\parallel}/2\) where \(\epsilon_{\parallel}\) is the in-plane dielectric constant of the bulk material. Let us look at the Fourier transformation of the modified bare potential, which is given by:  
\[V(q,a)=\frac{2\pi e^{2}}{\epsilon(q+aq^{2})}=\frac{2\pi e^{2}}{eq}\frac{1}{ \epsilon_{lattice}(q)} \tag{22}\]  
where \(\epsilon\) is the average dielectric constant of the environment, \(a\) might depend on the thickness of the 2D crystalline material and \(\epsilon_{lattice}(q)\) plays the role as a lattice local field factor. In the case of MoS\({}_{2}\), [PERSON] et al.