## 4 Plasmon Dispersion  
The plasmon dispersion of graphene sandwiched between air and a substrate of a constant permittivity \(\varepsilon_{\text{sub}}\) is well known [40, 43, 45]. The physical structure is illustrated in Figure 3a. Using [PERSON]'s equations with appropriate boundary conditions, the plasmon dispersion can be obtained by solving the equation:  
\[\varepsilon_{\text{ave}}+\text{i}\,\frac{q}{2\omega}\sigma\big{(}q,\omega\big{)} =0 \tag{12}\]  
where \(\varepsilon_{\text{ave}}=\big{(}\varepsilon_{\text{sub}}+\varepsilon_{{}_{0}} \big{)}/\,2\). Because of the finite scattering rate, \(q=q_{{}_{1}}+\text{i}q_{{}_{2}}\) has to be a complex number with \(q_{{}_{2}}\
eq 0\) for Equation (12) to be valid. In Equation (12) and below, we have ignored the retardation effect in view of the fact that the speed of light \(c\) in vacuum is 300 time higher than the Fermi velocity of graphene. As a result, \(\omega/q_{{}_{1}}<\!<\!<\text{c}\) is generally true as we shall see in the following figures for the plasmon dispersion of graphene on various substrates. The region where the retardation effect is important, _i.e._, \(\omega/\,q_{{}_{1}}\) is comparable to \(c\), is indistinguishable from the \(\hbar\omega\) axis, and thus is not considered in this paper. [PERSON] _et al._[43] have shown that as long as \(q_{{}_{2}}<\!<\!q_{{}_{1}}\) and \(\hbar\omega<\!<\!2E_{\text{F}}\), the Drude model is sufficient to determine the plasmon dispersion described by Equation (12) without the need of the RPA-RT approach. By plugging Equation (7) in Equation (12), we obtain  
\[q_{{}_{1}}^{{}^{-1}} =\frac{e^{2}k_{{}_{\text{B}}}T}{\varepsilon_{\text{ave}}\omega^{ 2}\pi\hbar^{{}^{2}}}\ln\!\left[2\cosh\!\left(\frac{\mu}{2 kT}\right)\right] \tag{13}\] \[q_{{}_{2}}^{{}^{-1}} =\frac{e^{2}k_{{}_{\text{B}}}T\tau}{\varepsilon_{\text{ave}} \omega\pi\hbar^{{}^{2}}}\ln\!\left[2\cosh\!\left(\frac{\mu}{2 kT}\right)\right] \tag{14}\]  
where \(\tau\) is the inverse of \(\tau^{{}^{-1}}\big{(}E_{{}_{\text{F}}}\big{)}\). Because \(2\pi q_{{}_{1}}^{{}^{-1}}\) is the wavelength of the plasmons and \(q_{{}_{2}}^{{}^{-1}}\) represents the distance of propagation for the plasmonic field amplitude to decrease to \(1/\,e\) of its  
Figure 2: Optical conductivity of graphene for (a) \(\hbar\omega\leq\mu\), and (b) \(\hbar\omega\geq\mu\). The solid curves are derived from the Drude model (Equation (7)), and the dashed curves are obtained using the RPA-RT approach (Equation (11)). The real part and the imaginary part of the optical conductivity are plotted in red and black colors, respectively. For both figures, the parameters used are \(T=300\) K, \(\mu=100\) meV, and \(\tau^{{}^{-1}}\big{(}E_{{}_{\text{F}}}\big{)}=10\) THz. original amplitude, it is meaningful to define \({q_{2}}^{-1}/{q_{1}}^{-1}=\pi\omega\) as a figure of merit. Clearly, regarding surface plasmons in terms of \({q_{2}}^{-1}/{q_{1}}^{-1}\), a high frequency is preferred for graphene, whereas for a metal, \({q_{2}}^{-1}/{q_{1}}^{-1}\) decreases with increasing frequency in the optical region [46]. Note that \(\tau\)  is also a function of frequency [33]. For a frequency \(\omega\)  that is too large, above either 48 THz or \(\hbar\omega>2E_{\mathrm{F}}\), the optical phonon scattering or the interband scattering becomes important; then, \(\tau\) decreases and therefore \({q_{2}}^{-1}/{q_{1}}^{-1}\)  might decrease with increasing frequency. The plasmon dispersion of graphene can be greatly altered by coupling graphene plasmons with other quasiparticles. Both theoretically and experimentally, it has been shown that graphene plasmons can couple with surface phonons of polar substrates [3, 12], and with plasmon modes of metal particles or a metal surface [7, 10, 15]. Coupled plasmon modes resulted from the coupling of two graphene layers have also been studied [16, 17, 18, 19, 47].