# Influence of Impurity Scattering on Surface Plasmons in Graphene in the Lindhard Approximation
## 3 Transverse Magnetic Surface Plasmons  
The poles of the transmission or reflection coefficients give the SPs [15; 36]. For TM modes, we have  
\[q=i\omega e_{0}(n_{1}^{2}+n_{2}^{2})/\sigma(q,\omega). \tag{8}\]  
We emphasize that Equation (3) is valid only for SP momenta that are much larger than that of light, \(q\gg\omega/c\). Since the polarization function can be evaluated in the Lindhard, random-phase or Hubbard approximation, so can the SPs resulting from (8) due to the relation between \(\chi\) and conductivity, Equation (2). The simplest approximation on which we focus here is the Lindhard one that neglects the effects of the Coulomb interaction and local field factor [31]. Since the polarization function acquires different values for SB and TB transitions, the SPs would be different in each case. Notice that the real part of the transmission pole yields an SP mode [36], while its imaginary part is responsible for dissipation. For SB transitions, Equations (2), (4), and (8) give  
\[q^{\prime 2}-q^{\prime}\beta\omega^{\prime 2}+\frac{\omega^{\prime 2}}{2} \mathfrak{Re}C(\omega^{\prime},\gamma^{\prime})=0 \tag{9}\]  
with \(\beta=v_{F}(n_{1}^{2}+n_{2}^{2})/4\alpha c\) and \(\alpha\), \(v_{F}\), and \(c\) being the fine-structure constant, Fermi velocity, and light velocity, respectively. The physical solution is given by  
\[q^{\prime}=\frac{1}{2}\bigg{[}\beta\omega^{\prime 2}+\Big{[}(\beta\omega^{ \prime 2})^{2}-2\omega^{\prime 2}\mathfrak{Re}C(\omega^{\prime},\gamma^{ \prime})\Big{]}^{1/2}\bigg{]}. \tag{10}\]  
For TB transitions using Equations (2), (5) and (8) we obtain the physically acceptable solution  
\[q^{\prime}=\frac{\omega^{\prime}\beta}{A(\omega^{\prime})}\bigg{(}1+\Big{[}1- \frac{2A(\omega^{\prime})}{\omega^{\prime}\beta^{2}}\mathfrak{Re}C(\omega^{ \prime},\gamma^{\prime})\Big{]}^{1/2}\bigg{)}. \tag{11}\]  
Notice that there are other solutions for Equations (10) and (11) with the \(+\) sign in front of the square root \(+[..]^{1/2}\) replaced by a \(-\) sign. However, these solutions are unphysical and therefore rejected. Figure 3: 3D bar and contour plot of the real and imaginary parts of \(\chi\) for \(\gamma^{\prime}=0.06\). In Figures 4a and 5a, we show the TM SP dispersion relation, for several values of \(\gamma^{\prime}\), obtained with \(n_{1}=1\) and \(n_{2}=2\). To render its behaviour more clear, we replotted it in Figures 4b and 5b for a smaller range of \(q^{\prime}\). As seen, with decreasing \(\gamma^{\prime}\) the acceptable value of \(q^{\prime}\) increases and that for SB transitions is smaller than for the TB ones. Moreover, the SP frequency is larger in the TB case for typical values of \(q^{\prime}\). In either case, the group velocity \(\partial\omega/\partial q\) is nearly constant and independent of the impurity strength. The SP dispersion changes when the strength \(\gamma^{\prime}\) is varied. This can be understood as follows. A coherent ensemble of dipole moments (DMs) oscillating about their equilibrium configuration, under a phase matching condition, starts to generate an evanescent electromagnetic field that propagates at the interface of the metallic-dielectric medium. Damping effects due to defects, impurities, or electron-phonon interactions are expected to reduce the effective number of these coherent DMs. That is, a higher value of \(\gamma^{\prime}\) leads to a higher scattering probability and reduces the number of coherent DMs. Hence, the energy of an EMF that originated from an ensemble of DMs for a lower \(\gamma^{\prime}\), with the same momentum, is larger than that for a higher \(\gamma^{\prime}\). In Figures 6 and 7, we plot the 3D TM SP spectrum and the \((\omega^{\prime},\gamma^{\prime})\) contour plot of SB and TB, respectively. As can be seen, for a fixed \(\gamma^{\prime}\), increasing the SP frequency leads to momentum increases as well. The bars in these figures indicate the allowed values in a region where SPs exist.