# Electrically Tunable Perfect Terahertz Absorber Using Embedded Combine Graphene Layer
## 2 Theoretical Background  
The conductivity surface plays a major role in creating the resonant peak through the absorption band for graphene-based absorbers. Graphene's conductivity surface is expressed by Kubo formula [33], where interband and intraband contributions are as follows:  
\[\sigma\Big{(}\omega,E_{f},\,\Gamma,T\Big{)}=\sigma_{int\,\,er}+\sigma_{int\,\,ra}, \tag{1}\]  
\[\sigma_{int\,\,ra}=\frac{2k_{B}Te^{2}}{\pi\hbar^{2}}\ln\left(2 cosh\frac{E_{f}} {2k_{B}T}\right)\frac{i}{(\omega+i\Gamma)}=\frac{\alpha}{-i\omega+\Gamma} \tag{2}\]  
\[\sigma_{int\,\,er}=\frac{e^{2}}{4\hbar}\Bigg{[}H\Big{(}\frac{\omega}{2}\Big{)} +i\frac{4\omega}{\pi}\int_{0}^{\infty}\frac{H(\Omega)-H\big{(}\frac{\omega}{2 }\big{)}}{\omega^{2}-4\Omega^{2}}d\Omega\Bigg{]}\,, \tag{3}\]  
and,  
\[H(\Omega)=\frac{sinh\Big{(}\frac{\hbar\Omega}{k_{B}T}\Big{)}}{\Big{[}cosh \Big{(}\frac{\hbar\Omega}{k_{B}T}\Big{)}+cosh\Big{(}\frac{E_{f}}{k_{B}T} \Big{)}\Big{]}} \tag{4}\]  
where \(\omega\) is the angular frequency, \(E_{f}\) describes graphene's Fermi energy, \(T\) is the temperature, \(K_{B}\) and \(e\) is Boltzmann constant and elementary charge respectively, and \(\hbar\) also depicts the reduced [PERSON]'s constant. It has been proved, when the Fermi energy becomes bigger than photon energy(\(\hbar\omega/2\)), the interband part of Equation (1) can be ignored in comparison with the amount of intraband part of the equation due to Pauli blocking [34]. Also, the conductivity of graphene can be described by selecting the intraband contribution of the Drude model, as presented in Equation (2). On this basis, the plasma frequency of graphene is obtained as below that \(t_{g}\) is graphene layer's thickness in the structure:  
\[\omega_{p}=\left[\frac{2e^{2}k_{B}T}{\pi\hbar^{2}\epsilon_{0}t_{g}}\ln\left(2 cosh\frac{E_{f}}{2k_{B}T}\right)\right]^{1/2}=\sqrt{\frac{\alpha}{\epsilon_{0}t_{g}}} \tag{5}\]  
On the other hand, the stimulation of graphene through external voltage bias and then changing Fermi level counts as one of the popular features for graphene. It has been established that if a DC bias voltage is applied to the structure positioned between two graphene layers, the Fermi energy (chemical potential) can be tuned. Therefore, the conductivity surface is adjustable. The approximated formula to estimate the relationship between bias voltage and Fermi energy level (chemical potential), is given as follows [35; 36]:  
\[\left|E_{f}\right|=\hbar v_{f}\sqrt{\frac{\pi\epsilon_{r}\epsilon_{0}V_{DC}}{ \epsilon t_{s}}} \tag{6}\]  
Here, \(\epsilon_{r}\) and \(\epsilon_{0}\) are the primitivity of spacer and vacuum, respectively. \(t_{s}\) also define spacer's thickness, and \(V_{DC}\) shows the external bias voltage. Furthermore, \(v_{f}\) is the Fermi velocity that is equivalent to 1.1 \(\times\) 10\({}^{6}\) (m/s). The Fermi level can be set over a broad range between \(-\)1 eV to 1 eV. As an example, a Fermi level is tunned between 0.2 eV to 0.6 eV by applying DC voltages between \(-\)240 V to 240 V, correspondingly. For these kinds of absorbers with the spacer between the graphene layer to avoid dielectric breakdown, silicon dioxide (SiO\({}_{2}\)) and Kapton are also suggested as appropriate insulators because of high dielectric breakdown and low voltage bias.