# Influence of Impurity Scattering on Surface Plasmons in Graphene in the Lindhard Approximation
## 2 Scattering-Dependent Optical Properties  
We consider a graphene sheet in between two media with the permittivities \(e_{1}\) and \(e_{2}\), respectively. The corresponding reflection and transmission coefficients are obtained by satisfying the boundary conditions for the longitudinal and perpendicular components of the TM field. The transmission coefficient is given by [15]  
\[t(q,\omega)=2\bigg{[}\frac{n_{2}}{n_{1}}+\frac{\cos\theta_{t}}{\cos\theta_{t} }+\frac{\sigma(q,\omega)\cos\theta_{t}}{n_{1}\epsilon_{0}c}\bigg{]}^{-1}. \tag{1}\]  
Here, \(\epsilon_{0}n_{t}^{2}=\epsilon_{1}\), \(i=1,2\), \(n_{i}\) is the refractive index, \(\sigma\) is the conductivity of graphene, and \(\theta_{i}\), \(\theta_{t}\) are the angles of incidence and transmission, respectively. In general, the index \(n\) depends on the momentum and frequency, but here we assume that it is constant. The conductivity can be evaluated as, e.g., in [15]. The result is  
\[\sigma(q,\omega)=ie^{2}\omega\chi(q,\omega)/q^{2}, \tag{2}\]  
where \(\chi\) is the polarization function also known as density-density response function. The polarization function can be evaluated in the absence or presence of the Coulomb interaction. Here, we employ the Lindhard approximation, in which the effects of electron-electron interactions and the local field factor are neglected. It is worth pointing out that including many-body effects such as electron-electron interaction or exchange effects, known as the random-phase approximation (RPA) or Hubbard approximation, respectively, provide a good insight about the system response but its evaluation becomes very complicated. For the sake of simplicity, we limit our calculation to the Lindhard approximation in which the computation time is much shorter than in the RPA or Hubbard approximation. The Lindhard polarization function in the presence of impurity scattering, within the van Hove limit, is given by [31]  
\[\chi^{0}(q,\omega)=\chi^{0}_{non}(q,\omega)+\chi^{0}_{imp}(q,\omega), \tag{3}\]  
where \(\chi^{0}_{non}\) and \(\chi^{0}_{imp}\) are the polarization functions in the absence [32] and presence [31] of impurity scattering, respectively. Since we are interested in SPs with lower and higher energies than the Fermi energy, which are known as intra- and inter-band SPs or SB and TB SPs, the polarization functions at zero temperature and in the long-wavelength limit [31; 32] are given by  
\[\chi^{0,SB}_{non}(q^{\prime},\omega^{\prime})=\frac{q^{\prime 2}}{\omega^{ \prime 2}},\ \chi^{0,SB}_{imp}(q^{\prime},\omega^{\prime})=C(\omega^{\prime}, \gamma^{\prime})/2, \tag{4}\]  
\[\chi^{0,TB}_{non}(q^{\prime},\omega^{\prime})=\frac{M(\omega^{\prime})q^{ \prime 2}}{2\omega^{\prime}},\ \chi^{0,TB}_{imp}(q^{\prime},\omega^{\prime})=C(\omega^{\prime}, \gamma^{\prime}), \tag{5}\]where we introduced the dimensionless parameters \(\chi^{{}^{\prime}0}\), \(q^{\prime}\), \(\omega^{\prime}\), and \(\gamma^{\prime}\) (\(\chi^{{}^{\prime}0}=\chi^{0}\chi\eta_{F}/k_{F}\), \(q^{\prime}=q/k_{F}\), \(\omega^{\prime}=\hbar\omega/E_{F}\), \(\gamma^{\prime}=\hbar\gamma/E_{F}\) to simplify all expressions. \(E_{F}\), \(k_{F}=E_{F}/\hbar v_{F}\), and \(v_{F}\) are the Fermi energy, wave vector, and velocity, respectively. We used \(v_{F}=9\times 10^{5}\) m/s. Further,  
\[C(\omega^{\prime},\gamma^{\prime})=\frac{1-\delta_{\gamma^{\prime},\beta}}{ \omega^{\prime}+i\gamma^{\prime}},\ M(\omega^{\prime})=\big{[}A(\omega^{ \prime})-i\frac{\gamma}{2}\Theta(\omega^{\prime}-2)\big{]}, \tag{6}\]  
and  
\[A(\omega^{\prime})=\frac{2}{\omega^{\prime}}+\frac{1}{2}\ln\Big{|}\frac{2- \omega^{\prime}}{2+\omega^{\prime}}\Big{|}. \tag{7}\]  
The form of the response function (3) is derived in Ref. [33] for _homogeneous systems_: \(\chi_{non}\) is independent of the scattering, while \(\chi_{im}\) depends on it but it was not explicitly evaluated in Refs. [33; 34]. Importantly, when the response is evaluated from the collisionless Boltzmann equation, one obtains _the term_\(\chi_{non}^{0,5B}\)[35] and the term \(\chi_{non}^{0,TB}\) is absent.