# Recent Progress on Graphene-Functionalized Metasurfaces for Tunable Phase and Polarization Control
## 2 Material Properties of Graphene  
The optical response of graphene is governed fundamentally by its surface conductivity. Since the intuitive optical characterization is usually done through the permittivity or the mode index, we summarize in this section the conductivity, the effective permittivity, and the mode index of the surface plasmon (SP) wave supported by graphene, so as to perceive the transformation of optical response with Fermi level and in different frequency bands. The surface conductivity of graphene can be modeled by the well-known Kubo formula derived with the random phase approximation (RPA) theory [36; 37; 38]. \[\sigma=\frac{-ie^{2}}{\pi\hbar^{2}(\omega+i/\tau)}\int_{0}^{\infty}\epsilon \left[\frac{\partial f_{d}(\epsilon)}{\partial\epsilon}-\frac{\partial f_{d}(- \epsilon)}{\partial\epsilon}\right]d\epsilon+\frac{ie^{2}(\omega+i/\tau)}{\pi \hbar^{2}}\int_{0}^{\infty}\frac{f_{d}(\epsilon)-f_{d}(-\epsilon)}{(\omega+i/ \tau)^{2}-4(\epsilon/\hbar)^{2}}d\epsilon \tag{1}\]  
where \(\epsilon\) is the electron charge, \(\hbar\) is the reduced Planck constant, and \(\omega\) is the angular frequency. \(f_{d}(\epsilon)\) is the Fermi-Dirac distribution as \(f_{d}(\epsilon)=(e^{(\epsilon-E_{F})/k_{B}T}+1)^{-1}\). \(k_{B}\) is the Boltzmann constant. \(T\) is the temperature. \(E_{F}\) is the Fermi level. \(\tau\) is the carrier relaxation time defined by \(\tau=\mu E_{F}/(ev_{f}^{2})\). \(v_{f}=10^{6}\) m/s is the Fermi velocity, and \(\mu\) is the carrier mobility fluctuating in a large range determined by the fabrication process. \(\mu\) is around 0.1 m\({}^{2}\)/Vs in the chemical vapor deposition (CVD) graphene [39], 1 m\({}^{2}\)/Vs in the mechanically-exfoliated graphene [40], and even 100 m\({}^{2}\)/Vs in the suspend grapheneat low temperature [41]. The carrier density \(n\) is related to the Fermi level by \(n=E_{F}^{2}/(\pi\hbar^{2}v_{f}^{2})\). The dynamic modulation of \(n\) with \(E_{F}\) makes graphene either metallic, semiconductive, or dielectric. The first term in Equation (1) is the intraband contribution, which can be explicitly expressed as [38]:  
\[\sigma^{intra}=\frac{2k_{B}Te^{2}}{\pi\hbar^{2}}\text{ln}(2\text{cosh}\frac{E _{F}}{2k_{B}T})\frac{i}{\omega+i/\tau} \tag{2}\]  
When \(E_{F}\gg k_{B}T\), Equation (2) is simplified as the Drude model \(\sigma^{intra}=\frac{i\epsilon^{2}E_{F}}{\pi\hbar^{2}(\omega+i/\tau)}\). The second term in Equation (1) originates from the interband contribution and can be approximated, if \(E_{F}\gg k_{B}T\), as [38]:  
\[\sigma^{inter}=\frac{ie^{2}}{4\pi\hbar}\text{ln}\left(\frac{2E_{F}-(\omega+i/ \tau)}{2E_{F}+(\omega+i/\tau)}\right) \tag{3}\]  
If the thickness \(d\) is considered, the effective permittivity of graphene relative to the vacuum permittivity is defined as:  
\[\epsilon_{r}=e_{av}+\frac{i\sigma}{\epsilon_{0}\omega d} \tag{4}\]  
where \(e_{0}\) is the vacuum permittivity and \(\epsilon_{av}\) is the average relative permittivity of the surrounding environment. The mode index of the SP wave supported in the graphene monolayer is derived from the dispersion equation as [42; 43]:  
\[n_{sp}=\sqrt{\epsilon_{av}-(\frac{2\epsilon_{av}}{\sigma\eta})^{2}} \tag{5}\]  
with \(\eta\) being the wave impedance. In Figure 1, the three parameters above are plotted at different frequencies associated with different types of light-matter interaction. The parameters are specified in the caption. In the sub-THz region (Figure 1a), the real part of the conductivity \(\sigma^{{}^{\prime}}\), corresponding to the imaginary part of the permittivity \(\epsilon_{r}^{{}^{\prime}}\), is dominant, indicating graphene as a lossy film. The loss is weakly dependent on the frequency, but is very sensitive to the variation of \(E_{F}\). Electric gating of the graphene-loaded metallic metasurface in this frequency band is mainly embodied in the amplitude modulation of the local resonances together with a limited range of phase variation due to the transition between underdamped and overdamped phases.