## 4 Plasmon Dispersion
### Graphene on a Polar Substrate  
Consider a graphene layer at a distance \(d_{{}_{2}}\) above a polar substrate, as shown in Figure 3b. The polar substrate is characterized by a permittivity given by  
\[\varepsilon_{{}_{\mathrm{sub}}}=\varepsilon_{{}_{\mathrm{high}}}+\left(\varepsilon _{{}_{\mathrm{int}}}-\varepsilon_{{}_{\mathrm{high}}}\right)\frac{\omega_{{}_{ \mathrm{IO2}}}{{}^{2}}}{\omega_{{}_{\mathrm{IO2}}}{{}^{2}}-\omega^{2}}+\left( \varepsilon_{{}_{\mathrm{low}}}-\varepsilon_{{}_{\mathrm{int}}}\right)\frac{ \omega_{{}_{\mathrm{IO1}}}{{}^{2}}}{\omega_{{}_{\mathrm{IO1}}}{{}^{2}}- \omega^{2}} \tag{16}\]  
where \(\omega_{{}_{\mathrm{IO1}}}\) and \(\omega_{{}_{\mathrm{IO2}}}\) are the two lowest TO frequencies of the substrate with \(\omega_{{}_{\mathrm{IO2}}}>\omega_{{}_{\mathrm{IO1}}}\), and \(\varepsilon_{{}_{\mathrm{int}}}\) is the intermediate permittivity for frequency \(\omega\) between the two TO frequencies, \(\omega_{{}_{\mathrm{IO2}}}>\omega>\omega_{{}_{\mathrm{IO1}}}\). These parameters for different materials can be found in Reference [32]. Using Equations (7), (15), and (16), the coupled plasmon-phonon dispersion of graphene at a distance \(d_{{}_{2}}\) above a SiO\({}_{2}\) substrate is shown in Figure 4a, where \(q\) is normalized to \(k_{{}_{\mathrm{F}}}=E_{{}_{\mathrm{F}}}/\hbar v_{{}_{\mathrm{F}}}\), and \(E_{{}_{\mathrm{F}}}\equiv\mu=100\) meV is used; the scattering rare is chosen to be \(10\) THz  as a reasonable value given in Figure 1b. As can be seen, the dispersion of the coupled plasmon mode deviates from the decoupled mode given by Equation (12) with \(\varepsilon_{{}_{\mathrm{ave}}}=\left(\varepsilon_{{}_{2}}+\varepsilon_{{}_{0} }\right)/2\)  (dashed curve). The deviation is especially strong around \(\hbar\omega=\hbar\omega_{{}_{\mathrm{I}}}\equiv 59\) meV where the surface phonon mode is located and the coupling of plasmons and phonons is the strongest. As the distance \(d_{{}_{2}}\) between graphene and the substrate increases, the dispersion of the coupled plasmon mode (red curve) approaches that of the decoupled mode. As the thickness of the substrate \(d_{{}_{\mathrm{sub}}}\) increases, the effective permittivity \(\varepsilon_{{}_{\mathrm{ave}}}\) increases as the space otherwise filled by air \(\varepsilon_{{}_{\mathrm{I}}}=\varepsilon_{{}_{0}}\) is now filled with SiO\({}_{2}\). As a consequence, \(q_{{}_{\mathrm{I}}}\) also increases for the same \(\hbar\omega\) (blue curve), as can be told from their relation in Equation (13). Therefore, we see that \(d_{{}_{\mathrm{sub}}}\) and \(d_{{}_{2}}\) can serve as tuning parameters for shaping the plasmon dispersion. Figure 4: Plasmon dispersion of graphene on (**a**) SiO\({}_{2}\) and (**b**) various semi-infinite polar substrates ( \(d_{{}_{\mathrm{sub}}}\rightarrow\infty\) ). The structure considered is shown in Figure 3b. In all figures, \(\varepsilon_{{}_{\mathrm{I}}}=\varepsilon_{{}_{0}}\), \(\varepsilon_{{}_{\mathrm{2}}}=2\varepsilon_{{}_{0}}\), \(\mu=100\) meV, and \(\tau^{-1}=10\) THz are used. In (**a**), unless otherwise specified, \(d_{{}_{\mathrm{sub}}}=50\) nm and \(d_{{}_{2}}=30\) nm are used for the calculation. In (**b**), different materials for the polar substrate are represented by different colors as specified in the legend; \(d_{{}_{2}}=30\) nm is used for the solid curves, and \(d_{{}_{2}}\rightarrow\infty\) is assumed for the dashed curve. The black curves in (**a**) calculated using the Drude conductivity are plotted again in (**c**) to compare with the dotted curve obtained using the RPA-RT approach. The parameters used for the RPA-RT calculation are \(d_{{}_{\mathrm{sub}}}=50\) nm  and \(d_{{}_{2}}=30\) nm. In Figure 4b, we plot the dispersion of the coupled plasmon-phonon mode for graphene at a distance \(d_{2}\) above a semi-infinite substrate of various materials. As can be seen, some dispersion relations are characterized by two peaks because they are associated with substrates that have two surface phonon modes of energies within the plotted range.