## 3 Optical Conductivity
### RPA Model  
In the Drude model, the nonlocal effect is ignored. To describe the organized oscillation of electrons due to the long-range nature of Coulomb force among them, the Drude model is insufficient when the oscillation is characterized by a large wave number \(q\). To account for the nonlocality, the optical conductivity of graphene as a function of both \(\omega\) and \(q\) can be derived within the random phase  
Figure 1: Scattering rate of carriers in graphene as a function of (**a**) the carrier energy; and (**b**) the Fermi energy. Curves from top to bottom: the total scattering rate \(\tau^{-1}=\tau_{\mathrm{ac}}^{-1}+\tau_{\mathrm{imp}}^{-1}+\tau_{\mathrm{sp}}^ {-1}\) (blue curves), \(\tau_{\mathrm{imp}}^{-1}+\tau_{\mathrm{sp}}^{-1}\) (red curves), and \(\tau_{\mathrm{imp}}^{-1}\) (black curves). Curves plotted in (**c**) are scattering rates plotted in (**b**) but normalized to the total scattering rate. approximation (RPA). In the RPA, the out-of-phase response of electrons to the external field is assumed to average out to zero due to the random location of the electrons in a large quantity [39]. The polarizability of graphene within the RPA is given as [40]  
\[\Pi\big{(}q,\omega\big{)}=-\frac{4}{A}\sum_{\mathrm{k_{S}}^{\prime}}\frac{f\big{(} sE_{\mathrm{k}}\big{)}-f\big{(}s^{\prime}E_{\mathrm{k}^{\prime}}\big{)}}{h \omega+sE_{\mathrm{k}}-s^{\prime}E_{\mathrm{k}^{\prime}}+\mathrm{i}\delta} \frac{1+ss^{\prime}\cos\theta}{2} \tag{8}\]  
where \(A\) is the area of graphene, \(s\) and \(s^{\prime}\) denote the band indices having the values of 1 for the conduction band or \(-1\) for the valance band, \(\delta\) is an infinitesimal number, \(\mathbf{k}^{\prime}=\mathbf{k}+\mathbf{q}\), \(\theta\) is the scattering angle between \(\mathbf{k}\) and \(\mathbf{k}^{\prime}\), and \(E_{\mathrm{k}}=\hbar v_{\mathrm{F}}\left|\mathbf{k}\right|.\) The analytical expression for Equation (8) has previously been obtained in the limit of \(\mu>>k_{\mathrm{B}}T\)[40, 41] and in the collisionless limit of \(\tau^{-1}\to 0\). To account for the finite scattering rate, we replace \(\omega\) with \(\omega_{\gamma}=\omega+i\tau^{-1}\big{(}E_{\mathrm{F}}\big{)}\) in Equation (8) and follow the same procedure as the one in References [40, 41] assuming \(\mu>>k_{\mathrm{B}}T\); the result is  
\[\Pi\big{(}q,\omega\big{)}=\frac{2k_{\mathrm{F}}}{\pi v_{\mathrm{F}}\hbar}- \frac{q^{2}}{4\pi\hbar\sqrt{\omega_{\gamma}^{2}-v_{\mathrm{F}}^{2}q^{2}}} \Bigg{[}G\bigg{(}\frac{2k_{\mathrm{F}}v_{\mathrm{F}}+\omega_{\gamma}}{v_{ \mathrm{F}}q}\bigg{)}+\mathrm{sgn}\big{(}\omega-2k_{\mathrm{F}}v_{\mathrm{F} }\big{)}G\bigg{(}\frac{2k_{\mathrm{F}}v_{\mathrm{F}}-\omega_{\gamma}}{v_{ \mathrm{F}}q}\bigg{)}\Bigg{]} \tag{9}\]  
where \(G(x)=x\sqrt{x^{2}-1}-\ln\Big{(}x+\sqrt{x^{2}-1}\Big{)}\); \(\mathrm{sgn}\big{(}x\big{)}=1\) if \(x\geq 0\) and \(-1\) otherwise. Equation (9) is identical to Equation (7) of Reference [42] in a slightly different form. Apparently, by setting \(\tau^{-1}\big{(}E_{\mathrm{F}}\big{)}\to 0\) for \(\omega_{\gamma}\to\omega\), Equation (9) reduces to the collisionless results in References [40, 41]. To comply with local electron conservation, Equation (9) needs to be modified to [43, 44]  
\[\Pi_{\gamma}\big{(}q,\omega\big{)}=\frac{\omega_{\gamma}}{\omega\Pi\big{(}q, \omega\big{)}^{-1}+\mathrm{i}\tau^{-1}\big{(}E_{\mathrm{F}}\big{)}\Pi\big{(}q,0\big{)}^{-1}} \tag{10}\]  
where \(\Pi\big{(}q,\omega\big{)}\) is given by Equation (9) and \(\Pi\big{(}q,0\big{)}\) is obtained by setting both \(\tau^{-1}\big{(}E_{\mathrm{F}}\big{)}\) and \(\omega\) to zero in Equation (9). Equation (10) is called the RPA relaxation time (RPA-RT) approximation, which properly accounts for the finite scattering rate and the conservation of local electrons [44].