## 3 Result and Discussion
### Monolayer Graphene
They are then pulled back toward the charge disturbance and overshoot again, setting up a weakly-damped oscillation. The restoring force responsible for the oscillation is the average self-consistent field created by all the electrons. As expected, the dispersion behavior of the plasmon mode at the low-energies shows a standard \(\sqrt{q}\) dispersion predicted in the 2D electron gas system. To be more precise, the plasmon mode at the long wavelength limit in graphene is given by:  
\[\omega_{p}(q)=\sqrt{\frac{2D_{0}}{\epsilon}q}\left[1+\frac{12-N_{f}^{2}\alpha_{ ee}^{2}-8(1-\kappa_{0}/\kappa)}{16}\frac{q}{q_{TF}}+...\right] \tag{14}\]  
where \(D_{0}=v_{\mathrm{F}}k_{\mathrm{F}}e^{2}/\hbar\) is the Drude weight in graphene. For ordinary parabolic-band fermions with mass \(m_{b}\), the Drude weight is given by \(D_{0}=\tau ne^{2}/m_{b}\). Graphene's fine-structure constant is \(\alpha_{ee}=e^{2}/\hbar v_{\mathrm{F}}e\); \(N_{\mathrm{F}}=4\) is the flavor number in graphene; the Thomas-Fermi screening wave number is defined by \(q_{TF}=N_{\mathrm{F}}\alpha_{ee}k_{\mathrm{F}}\); and the electron compressibility of interacting and non-interacting graphene is \(\kappa\) and \(\kappa_{0}\), respectively [7]. The second term in the square brackets refers to the quantum non-local effects. Notice that in the classical picture, the long-wavelength plasmon mode behaves like \(\omega_{p}(q)=\sqrt{\frac{2\tau ne^{2}}{\epsilon m}q}\), which is totally different from the one we have in graphene. Our numerical results show that the general behavior of the plasmonic dispersion agrees with the results from [47; 49]. The dispersion relation of the \(\pi\) plasmon is presented, and it has a quasi-linear behavior, in good agreement with the earlier theoretical study [47]. In this figure, the electron-hole continuum is indicated with a black line, and it can be obtained at any specific momentum transfer \(\mathbf{q}\) by the difference between system energy at \(\mathbf{k}_{\mathrm{F}}\) and \(\mathbf{k}_{\mathrm{F}}+\mathbf{q}\). We would like to compare the plasmon mode obtained here with that calculated within a semiclassical approach. We suppose that the graphene sheet, along the \(x\) and \(y\) directions, is located between two semi-infinite dielectric media with the same dielectric constant, \(\epsilon\), and consider  
Figure 2: The real (solid line) and imaginary (dashed line) parts of the non-interacting density response function of monolayer graphene in units of the Fermi-level density of states as a function of \(\hbar\omega\) for \(q=0.076\) Å\({}^{-1}\). a solution of [PERSON]'s equations for a transverse magnetic wave. By using the proper boundary conditions, we arrive at [19]:  
\[2\frac{\epsilon}{\sqrt{q^{2}-\omega^{2}\epsilon/c^{2}}}=-i\frac{\sigma(\omega)}{ \omega\epsilon_{0}} \tag{15}\]  
which describes the plasmon mode, \(\omega(q)\), of graphene with conductivity \(\sigma(\omega)\). It should be noted that this expression implements every 2D crystalline material with its \(\sigma(\omega)\). This equation has solutions, if the imaginary part of the conductivity is positive and its real part vanishes. The conductivity of graphene from the terahertz to mid-infrared regime is dominated by a Drude term and given by:  
\[\sigma(\omega)=\frac{e^{2}}{\pi\hbar}\frac{iE_{\mathrm{F}}}{\hbar\omega} \tag{16}\]  
It turns out that at the long wavelength limit, the plasmon mode is given by:  
\[\hbar\omega_{p}=\sqrt{\frac{2\epsilon^{2}}{\epsilon_{0}}E_{\mathrm{F}}q} \tag{17}\]  
which is the same expression that we have in the quantum many-body framework. We also include the semiclassical plasmon mode in Figure 3b, and our numerical results show that they are the same at long wavelength regimes; however, at large momenta, the semiclassical plasmon mode deviates from that calculated by the many-body approach, especially at lower electron density. Figure 3: (**a**) The electron-energy loss (EEL) function for different momentum transfers and (**b**) plasmon dispersions of electron-doped graphene along the \(\Gamma-M\) direction.