# Design of Tunable Broadband Graphene-Based Metasurface with Amplitude-Phase Modulation
## 2 Theoretical Analysis  
### Graphene Conductivity  
Graphene is a new type of high-performance material that was isolated from graphite in 2004 by [PERSON] and [PERSON] at the University of Manchester [18]. It is the first single-atomic layer 2D material to be prepared and processed in the laboratory. Graphene materials have a bipolar electric field effect [19]. Therefore, when an electric field is applied to the graphene material or chemical doping is performed, the carriers in the graphenecrystal can be converted between electrons and holes [20]. As the voltage of the applied electric field can be seamlessly adjusted, the carrier concentration in the corresponding graphene can also be continuously regulated. This relationship can be expressed through Equation (1) [21; 22]. \[n=\left(n_{0}^{2}+\alpha^{2}\Big{|}\Delta V|^{2}\right)^{1/2} \tag{1}\]  
where \(n\) is the carrier concentration in graphene, \(n_{0}\) is the residual carrier concentration, and \(\alpha\) is the capacitance of the control voltage gate, which is generally related to the electrode configuration. \(\big{|}\Delta V\big{|}=\big{|}V_{CNP}-V_{S}\big{|}\), \(V_{S}\) is the voltage of the applied electric field; \(\big{|}V_{CNP}\big{|}\) is the compensation voltage when the concentration of carriers is 0. The relationship between the Fermi level and carrier concentration of graphene at room temperature \(T\) = 300 K (\(k_{B}T\sim\) 0.026 eV) is shown in Equation (2), where the Fermi velocity \(v_{f}=10^{6}\) m \(\cdot\) s\({}^{-1}\). If it belongs to the structure of the parallel plate capacitor, the concentration of carrier aggregation caused by the capacitance effect is \(n=\epsilon_{0}\epsilon_{d}V_{S}/ed\). \[\Big{|}E_{f}\Big{|}=\hbar v_{f}\Big{(}\pi n\Big{)}^{1/2} \tag{2}\]  
The relationship between relaxation time and Fermi level is shown in Equation (3). When other conditions have been determined, the relaxation time \((\tau)\) is proportional to the Fermi level \((E_{f})\). When the Fermi level is higher, the relaxation time is longer. If the graphene carrier mobility \(\mu=10^{4}\)cm\({}^{2}\)V\({}^{-1}\)s\({}^{-1}\), the relaxation time can be calculated to be \(0.1\times 10^{-12}\) s at the Fermi level \(E_{f}=0.1\) eV. \[\tau=\mu E_{f}/ev_{f}^{2} \tag{3}\]  
The effect of temperature on graphene carrier mobility is small, and the effect of impurity scattering on graphene carrier mobility is relatively large. Therefore, the graphene carrier mobility can be improved by increasing the purity of graphene at room temperature. [PERSON], a Japanese scientist, proposed that the [PERSON] formula [23] can be applied to modeling the transport characteristics of graphene. As shown in Equations (4)-(6), the conductivity of graphene is induced by its interband and intraband transitions. In the visible and near-infrared ranges, electrical tuning of the device is not possible, as graphene exhibits universal conductance (\(\sigma=q^{2}/4\hbar\)) dominated by interband transitions. However, from the mid-infrared to microwave bands, the conductivity of graphene can be dynamically controlled by electrostatic gating. Based on the Pauli Blocking effect [24], when the Fermi level \(E_{f}\)? \(\hbar\omega=2\), the interband transition of carriers is forbidden, and intraband transitions have a significant influence on the dielectric properties of graphene layer. In this work, Graphene is used in the design of W-band metasurfaces, where its conductivity can be approximated by Equation (5). \[\sigma=\sigma_{intra}+\sigma_{inter} \tag{4}\]  
\[\sigma_{intra}=j\frac{q^{2}k_{B}T}{\pi\hbar(\hbar\omega+jI_{c})}\left[\frac{ E_{f}}{k_{B}T}+2\text{ln}\Big{(}e^{-E_{f}/k_{B}T}+1\Big{)}\right] \tag{5}\]  
\[\sigma_{inter}=j\frac{q^{2}}{4\pi\hbar}ln\left[\frac{2\Big{  
states. Based on Equation (5), the sheet impedance of graphene is inversely proportional to both carrier mobility and the Fermi level. Therefore, by applying a voltage to graphene, the concentration of carriers in graphene is controlled and the Fermi level of graphene will be changed.