# Tuning of Graphene-Based Optical Devices Operating in the Near-Infrared
## 2 Graphene Optical Properties
### Graphene Conductivity  
The graphene surface conductivity (\(\sigma_{G}\)) is described by the high-frequency expression obtained from the Kubo model [57]  
\[\sigma_{G}=\frac{ie^{2}}{\pi h^{2}(\omega-i2\Gamma)}\int_{0}^{\infty}\xi\left( \frac{\delta f_{d}(\xi)}{d\xi}-\frac{\delta f_{d}(-\xi)}{d\xi}\right)d\xi- \frac{ie^{2}(\omega-i2\Gamma)}{\pi h^{2}}\int_{0}^{\infty}\frac{f_{d}(-\xi)-f _{d}(\xi)}{(\omega-i2\Gamma)^{2}-4\left(\frac{\xi}{h}\right)^{2}}d\xi \tag{1}\]  
where \(e\) is the electron charge, \(\omega\) is radian frequency, \(h=h/2\pi\) is the reduced Planck constant, \(\Gamma\) is a phenomenological scattering rate that is assumed to be independent of energy \(\xi\) (\(\tau=1/\Gamma\)-relaxation time, can also be used) and \(f_{d}(\xi)=\left(e^{(\xi-\mu_{c})/k_{B}T}+1\right)^{-1}\) is the Fermi-Dirac distribution. As follows from [57], and given that \(\sigma=\sigma t+i\sigma^{\prime\prime}\) (where \(\sigma^{\prime}\) is \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\Gamma\) and \(\tau\) & phenomenological scattering rate and \(\tau=1/\Gamma\), relaxation time \\ \(f_{d}(\xi)\) & \(f_{d}(\xi)=\left(e^{(\xi-\mu_{c})/k_{B}T}+1\right)^{-1}\), Fermi-Dirac distribution (\(\xi\)–energy) \\ \(k_{B}\) & [PERSON]’s constant \\ \(T\) & temperature \\ \(\mu_{c}\) & chemical potential \\ \(\mu\) & charge carrier mobilities \\ \(v_{F}\) & Fermi velocity \\ \(t_{G}\) and \(t_{I}\) & graphene and insulator thicknesses \\ \(L\) & graphene length \\ \(d\) & spacing between minima or free spectral range \\ \(D_{G}\) & distance between graphene electrodes \\ \(\varphi\) & phase shift \\ \(V_{G}\) and \(V_{Dirac}\) & applied voltage and flat-band voltage corresponding to the \\  & charge-neutral Dirac point \\ \(C_{ox}\) & \(C_{ox}=\varepsilon\epsilon_{0}(W_{m}l/t_{I})\), oxide (insulator) capacitance per unit area \\ GE & Graphene Electrode \\ GfW & Graphene-Insulator-Waveguide \\ GIG & Graphene-Insulator-Graphene \\ EGWC & Electrolyte/ion-gel-Graphene-Waveguide+PhC Cavity \\ GEin & Graphene-Insulator-Graphene inside waveguide \\ M-ZM & Mach–Zehnder Graphene Modulator \\ CNM & Crystal Nanocavity-based Modulator \\ MRR & Micro-ring Resonator \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._the intraband conductivity \(\sigma_{intra}\) and \(\sigma^{\prime\prime}\) is the interband conductivity \(\sigma_{inter}\)), this expression can be reduced to  
\[\sigma_{G}=\frac{ie^{2}k_{B}T}{\pi h^{2}(\omega+i2\Gamma)}\left[\frac{\mu_{c}}{ k_{B}T}+2 ln\left(e^{\frac{-\mu_{c}}{k_{B}T}}+1\right)\right]+\frac{ie^{2}}{4\pi h }ln\left[\frac{2|\mu_{c}|-(\omega+i2\Gamma)h}{2|\mu_{c}|+(\omega+i2\Gamma)h}\right] \tag{2}\]  
where \(k_{B}\) is Boltzmann's constant, \(T\) is the temperature and \(\mu_{c}\) is a chemical potential. Based on the surface conductivity, the permittivity can be obtained as [5]  
\[\varepsilon_{G}=1+i\frac{\sigma_{G}}{\omega t_{G}\varepsilon_{0}} \tag{3}\]  
where \(t_{G}\) is the monolayer graphene thickness (\(t_{G}\cong 0.34\) nm) and \(\varepsilon_{0}\) is the vacuum permittivity. The refractive index \(n_{G}\) and the extinction coefficient \(k_{G}\) can be retrieved by means of the following equations  
\[n_{G}=\frac{\sqrt{\sqrt{\varepsilon_{G_{\_}}{re^{2}}+\varepsilon_{G_{\_}}{ im^{2}}}+\varepsilon_{G_{\_}}{re}}}{2}\quad k_{G}=\frac{\sqrt{\sqrt{ \varepsilon_{G_{\_}}{re^{2}}+\varepsilon_{G_{\_}}{im^{2}}}-\varepsilon_{G_{\_}} {re}}}{2} \tag{4}\]  
The real part of the permittivity \(\varepsilon_{G_{\_}}{re}\) is the phase term (essentially independent on the intraband scattering rate), while the imaginary part \(\varepsilon_{G_{\_}}{im}\) is the absorption term (may vary by orders of magnitude depending on the relaxation time, i.e., on the material quality) [21].