# Influence of Impurity Scattering on Surface Plasmons in Graphene in the Lindhard Approximation
## 4 Quality Factor  
To efficiently transport high-speed information and energy, devices must, to the highest extent possible, have the least number of obstacles or defects, e.g., impurities, so that the energy dissipation is the lowest. Devices with the best functionality are usually distinguished by the quality factor (QF) defined as the ratio of the stored to the dissipated energy. The higher the QF, the higher the device functionality and performance are [37]. The QF is defined by  
\[Q=\frac{U_{stor}}{U_{dis}}=\frac{U_{0}-U_{dis}}{U_{dis}}=\frac{U_{0}}{U_{dis}}-1, \tag{14}\]  
Figure 8: Spectrum of critical impurity versus SB and TB SP frequency.  
Figure 6: 3D bar and \((\gamma^{\prime},\omega^{\prime})\) contour plots of TM SB SPs.  
Figure 7: 3D bar and \((\gamma^{\prime},\omega^{\prime})\) contour plots of TM TB SPs.  
where \(U_{stor}\), \(U_{dis}\) and \(U_{0}\) refer, respectively, to the stored, dissipated, and total energy densities. \(U_{0}\) is the energy density of the incoming field, \(U_{0}=\epsilon_{0}|E_{I}|^{2}/2\). The dissipated energy of the SP, \(U_{dis}\), is  
\[U_{dis}=\frac{1}{2}\epsilon_{0}|E_{dis}|^{2}=\frac{1}{2}\epsilon_{0}|E_{I}|^{2 }\Big{|}\mathfrak{Im}\,t(q^{\prime}_{sp},\omega^{\prime}_{sp})\Big{|}^{2}, \tag{15}\]  
where \(\mathfrak{Im}\,t(q^{\prime}_{sp},\omega^{\prime}_{sp})\) is the imaginary part of the transmission amplitude at the plasmon momentum and frequency. For SB and TB transitions, the results are  
\[Q_{SB}=\Big{[}\eta\,\mathfrak{Im}\,C(\omega^{\prime}_{sp},\gamma^{\prime})/2 q^{\prime}_{sp}\Big{]}^{2}-1, \tag{16}\]  
\[Q_{TB}=\Big{[}\eta D(\omega^{\prime}_{sp},\gamma^{\prime})/2q^{\prime}_{sp} \Big{]}^{2}-1, \tag{17}\]  
with  
\[D(\omega^{\prime}_{sp},\gamma^{\prime})=\mathfrak{Im}\,C(\omega^{\prime}_{ sp},\gamma^{\prime})+(\pi q^{\prime 2}_{sp}/4\omega^{\prime})\,\Theta(\omega^{ \prime}-2), \tag{18}\]  
and \(\eta=4\kappa c/(v_{F}n_{1}n_{2})\). For low frequencies, by combining Equations (10) and (16) in the SB case, or Equations (11) and (17) in the TB case, we obtain  
\[Q\approx\alpha^{2}\lambda_{sp}^{2}/\gamma^{2}; \tag{19}\]  
this result depends only on the fine-structure constant, the SP wavelength, and the impurity strength.  
In Figure 9a,b, we show, respectively, the SB and TB QF of TM SPs versus their frequencies \(\omega\) for several values of \(\gamma^{\prime}\). As seen, the QF decreases strongly with increasing frequency. In addition, for a fixed \(\omega\), the QF decreases, in both cases, upon increasing \(\gamma^{\prime}\), which is expected since higher values of \(\gamma^{\prime}\) imply an increase in dissipation. According to Equations (10) and (11), for low frequencies \(\omega\), the SP wavelength is proportional to \(\omega^{-2}\), see also Figure 9, and in both cases the QF decreases as \(\omega^{-4}\). As the dissipation is stronger in the TB case than in the SB one, cf. Equations (4) and (5), we expect that the QF will be higher in the SB case; indeed, this is made clear in Figure 9. Correspondingly, the energy density stored in the SB case is higher than in the TB one.