## 2 Dyadic Green's Function for the Electric Field
### Renormalized Polarizability of an Isotropic Quantum Emitter Near a Continuous Graphene Sheet
Some insight on the previous expressions can be obtained by estimating them in the electrostatic limit, valid for \(k_{1}z_{0}\ll 1\). In this limit, the main contribution is due to the \(r_{p}\) reflection coefficient. Approximating \(\sqrt{1-s^{2}}\simeq\sqrt{\epsilon_{2}/\epsilon_{1}-s^{2}}\simeq\) is we obtain  
\[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]0  
with the reflection coefficient being approximated by  
\[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]1  
with \(\text{Ei}(z)\) the exponential integral function, which for a real positive argument is written as \(\text{Ei}(x)=-\int_{-x}^{\infty}dte^{-t}/t.\) However, we point out that Equation (67) is valid even in the presence of finite broadening \(\gamma\) in graphene. We shall consider both metallic and polar semiconductor nanoparticles, with the relative dielectric function described, respectively, by the Drude and Lorentz models. The Drude model for the dielectric function reads  
\[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]2  
where \(\omega_{p}\) is the metal's plasma frequency and \(\Gamma_{0}\) is the relaxation rate, while the Lorentz model for the dielectric function of a polar material is given by  
\[r_{s} =\frac{\beta_{1}-\beta_{2}-\mu_{0}\omega\sigma_{T}(\omega)}{\beta _{1}+\beta_{2}+\mu_{0}\omega\sigma_{T}(\omega)}, \tag{55}\]3  
where \(\omega_{\text{TO}}\) and \(\omega_{\text{LO}}\) are the frequencies of the transverse and longitudinal optical phonons, \(\Gamma_{\text{TO}}\) is a phonon decay rate, and \(\epsilon_{\infty}\) is the high frequency limit of the dielectric function. As examples of commonly used materials for the production of nanoparticles, we consider gold (metallic) and CdSe (polar semiconducting) nanoparticles. Typical values of the polarizability for different substances are given in [55]. The used values for the Lorentz model of CdSe are taken from [56]. In Figure 3 we depict the real and imaginary parts of the polarizability of a gold nanoparticle near a doped graphene sheet on a substrate with \(\epsilon_{2}=2\). The figure clearly shows the strong renormalization of the polarizability of the nanoparticle relative to its value in the presence of the interface without graphene (blue dashed line). This is due to the close proximity of the nanoparticle to the graphene sheet, \(z_{0}=151\,\text{nm}\) (compared to the typical free-space wavelengths in the THz range). Nowadays, with the ubiquitous use of hexagonal boron nitride (h-BN) for encapsulating graphene, together with the possibility of controlling the number of layers of h-BN, it poses no difficulty to routinely produce structures where nanoparticles are positioned very close to the graphene sheet, that is, at distances much smaller than their radius. Also, the \(zz\)-component of the polarizability tensor (black dotted line) is renormalized differently from the \(xx\)-component (red solid line). This is a consequence of breaking the translation symmetry along the \(z\)-direction introduced by the graphene sheet and the dielectric change as we cross the \(z=0\) plane. We have verified that the broadband resonance seen in the imaginary part of the polarizability tensor is due to the excitation of surface plasmon polaritons in graphene. This was assessed studying the dispersion of the resonance as a function of the Fermi energy (more on this below). Given the close proximity of the nanoparticle to the graphene sheet, the question of the necessity of a nonlocal description of the graphene conductivity arises. In order to check the correctness of our local description, we have performed simulations (results not shown) using the nonlocal Drude-like conductivity [18] of graphene.