## 1 Introduction
Based on this technique, a scanning near-field optical microscope with the aperture radius much smaller than the wavelength of incident light can be used. The near-field evanescent components of light coming out from the microscope provide the required in-plane momenta. Two independent research groups performed experiments on graphene plasmonics using a similar concept [5,6] where they detected graphene surface plasmon modes. Moreover, monolayer graphene supports transverse-electric modes, which are absent in normal metals. The reason that we have this mode in doped graphene lies in the fact that the imaginary part of the conductivity is negative due to the interband transitions. Moreover, the transverse-electric modes' dispersion does not depart much from the light line, which means poor confinement, and thus, its effect is negligible. We briefly look at another physical concept, which is the energy of the electro-magnetic field inside dispersive media, since the dielectric function is a complex-valued function. Since the amount of field localization is often quantified in terms of the electromagnetic energy distribution, a careful consideration of the effects of dispersion is necessary. In metals, the dielectric function is complex and frequency-dependent owing to the dispersion. For a field consisting of monochromatic components, [PERSON] and [PERSON] have shown that the conservation law can be well described by an effective electric energy density \(u_{\mathrm{eff}}\), defined as [13]:  
\[u_{\mathrm{eff}}=\frac{\epsilon_{0}}{4}(\varepsilon_{1}+\frac{2\omega \varepsilon_{2}}{\gamma})|\mathbf{E}|^{2} \tag{4}\]  
where \(\gamma=1/\tau\) in which the relaxation time of the free electron gas is \(\tau\) and the dielectric function is given by \(\varepsilon=\varepsilon_{1}+i\varepsilon_{2}\). It is worth mentioning that the real part of the dielectric function or the imaginary part of the conductivity describes the reflection of light (an elastic process); however, the imaginary part of the dielectric function describes the absorption of light (inelastic process). Basically, plasmons are observed when the real part of the dielectric function is negative (metallic behavior). In order to give some numbers, the plasmon modes occur at ultraviolet frequencies for aluminum and other materials, at ultraviolet frequencies for zero- and one-dimensional carbon structures, at visible-near infrared frequencies for noble materials (Ag, Cu, Au), at terahertz and mid-infrared frequencies for two-dimensional (2D) carbon structures (graphene) and at amplitude modulation radio frequencies for the ionosphere. With these general properties, let us now discuss in more detail the organization of the article. The scope of this paper is collective modes in pristine doped two-dimensional crystalline systems, where the real part of their dielectric functions is basically negative, with the emphasis on fundamental physics from theoretical and experimental viewpoints. Details of the band structure properties are covered in some stand stemming from the density functional theory. Phonon scattering, the effect of impurity or strain and corrugation and optical conductivity in those materials are not covered. Detailed reviews of the plasmon modes in general are available in [14; 15] and in particular for graphene in [16; 17; 18; 19; 20]. Our ultimate goal is to facilitate the reader's independent study of original papers on the plasmonics of crystalline two-dimensional materials using the density functional theory. In this article, we are using a recently-proposed theoretical formulation based on ab initio density functional theory (DFT) together with the random-phase approximation (RPA) to investigate the electronic excitation spectrum of doped graphene, MoS\({}_{2}\) and phosphorene in monolayer and bilayer structures. To commence, the electronic ground-state of the periodically-repeated slab of each material is first determined, and then a Dyson-like equation is solved within the RPA to calculate the density-density response function.