# Tunable Mid-Infrared Graphene Plasmonic Cross-Shaped Resonator for Demultiplexing Application
## 2 Resonator and Analysis  
Two-dimensional top and side views of the proposed graphene plasmonic MIR resonator are shown in Figure 1a,b. The structure is composed of a cross-shaped resonator embedded between the horizontal input and vertical output graphene waveguides. The whole structure is placed on SiO\({}_{2}\)/Si. SiO\({}_{2}\) is used as a buffering layer and it has significant influence on the transmission spectrum of the proposed MIR resonator. Previous studies have reported that the thickness of SiO\({}_{2}\) cannot be lower than 50 nm due to the invariant transmission caused by the fabrication errors and faults [49; 50]. Two Au electrodes are used for applying the bias voltage to graphene in the proposed concept. The materials selection and the device design are made so that they are reasonable from the fabrication point of view. The Au film and SiO\({}_{2}\)/Si spacer can be deposited by using the conventional electron beam evaporation technique [51]. Then, the monolayer graphene can be coated on the top of the SiO\({}_{2}\) spacer after a chemical vapor deposition (CVD). Finally, the cross-shaped graphene MIR resonator with its input and output waveguides can be fabricated by using electron beam lithography and oxygen plasma etching [17]. The incident electromagnetic lightwave is launched to the resonator from its input port in MIR region and SPPs are excited along the input graphene waveguide. Two monitors are used at \(P_{in}\) and \(P_{out}\) to monitor the input and output powers. The input and output monitors are respectively 2D X-normal and 2D Y-normal. The transmission is calculated by \(T~{}=~{}\frac{P_{out}}{P_{in}}\). Only the incident wavelength, which satisfies the resonance condition of the cross-shaped resonator, can be coupled effectively to the resonator and pass through to the output port while the others are suppressed. Thus, the band pass filtering effect is obtained in MIR region. Structural and material parameters of the proposed graphene plasmonic resonator are given in Table 1. Numerical simulations are performed by use of three-dimensional finite-difference time-domain (3D-FDTD) method in Lumerical FDTD Software package under appropriate boundary conditions of perfectly matched layers (PML). The assumed mesh sizes of the proposed resonator in three different directionsof x, y, and z are given in Table 2. The optical conductivity of graphene \(\sigma_{G}\), is expressed by the Kubo formula [22]  
\[\sigma_{G}=\ \frac{2 ie^{2}k_{B}T}{\pi\hbar^{2}(\omega+i\tau^{-1})}ln\left[2 cosh \left(\frac{E_{f}}{2k_{B}T}\right)\right]+\frac{ie^{2}}{4\pi\hbar}ln\left[\frac {2E_{f}-\left(\omega+i\tau^{-1}\right)\hbar}{2E_{f}+\left(\omega+i\tau^{-1} \right)\hbar}\right] \tag{1}\]  
where \(e\), \(k_{B}\), \(T\), \(\hbar\), \(\omega\), \(E_{f}\), and \(\tau\) are respectively the electron charge, Boltzmann constant, temperature, reduced Planck constant, angular frequency, and Fermi energy level, and carrier relaxation time. \(\tau\) can be calculated by \(\tau=\frac{\mu E_{f}}{e\sigma_{f}^{2}}\), where \(\mu\) is the carrier mobility. The constant \(v_{f}=10^{6}\) m/s is the Fermi velocity of graphene. As depicted in Figure 1b, graphene should be biased by external gate voltage. \(E_{f}\) is related to the bias voltage \(V_{bias}\) as [52]  
\[E_{f}=\ \sqrt{\frac{\pi\varepsilon_{0}\varepsilon_{Fi02}V_{bias}}{e\varepsilon_{ Gi02}}} \tag{2}\]  
where \(\varepsilon_{0}\) is the permittivity of air with the value of 1. \(\varepsilon_{Fi02}\) and \(t_{SiO2}\) are respectively the relative permittivity and the thickness of the buffer layer. Moreover, the permittivity of graphene can be determined by  
\[\varepsilon_{G}\ =\ 1+\frac{i\sigma_{G}}{\varepsilon_{0}\omega t} \tag{3}\]  
where \(t\) is the thickness of graphene and is assumed to be 1 nm.