# Effect of Nanodisks at Different Positions on the Fano Resonance of Graphene Heptamers
## 2 Simulated Methods and Models  
As shown in Figure 1a, the oligomer consisting of seven graphene nanodisks is placed on a calcium fluoride (CaF\({}_{2}\)) substrate with a refractive index of 1.4 and surrounded by air with a refractive index of 1. In the infrared spectral region, CaF\({}_{2}\) is transparent and this structure has no effects of substrate phonon [29]. Therefore, using CaF\({}_{2}\) as substrate can eliminate the coupling problem between the plasmon and phonon in the graphene plasmonic structure [30]. According to the requirements of the substrate, calcium fluoride also can be replaced by potassium bromide or other infrared transparent materials. In order to further understand the structural parameters, the projection of the oligomer on the XOY plane is given in Figure 1b. Six graphene nanodisks of the same size are evenly placed on a circle with a radius R = 160 nm and a nanodisk with a radius R\({}_{2}\) = 90 nm is in the center of this structure. The radius and chemical potential of the satellite nanodisks are 50 nm and 0.5 eV, respectively. The distance between center nanodisk and ring nanodisks d is kept at 10 nm. In this system, the power of incident light is set as \(3.6\times 10^{7}\) W. In order to avoid the reflected light fields, the perfectly matched layer (PML) is set around the nanostructure. In addition, the thicknesses of seven graphene nanodisks are meshed by at least five layers and the maximum element size in the graphene layer is set as 2 nm to ensure the accuracy of simulation. Simultaneously, the mesh size increases gradually from graphene nanodisks to the PML. Figure 1: (**a**) The incident light is polarized along the \(y\) axis. The graphene oligomer is surrounded by air with a refractive index \(n_{1}\)= 1 and placed on calcium fluoride substrate with a refractive index \(n_{2}\) = 1.4. (**b**) The top view of the schematic structure of the graphene oligomer. In this study, the complex surface conductivity of graphene \(\sigma_{g}\) is written as \(\sigma_{g}=\sigma_{intm}+\sigma_{inter}\), where \(\sigma_{intm}\) and \(\sigma_{inter}\) are intraband electron-photon scattering and interband electron-electron transition, respectively [31]. The intraband electron-photon scattering \(\sigma_{intm}\) is described by  
\[\sigma_{intm}=\frac{2e^{2}k_{B}T}{\pi\hbar^{2}}\cdot\frac{i}{\omega+i\tau^{-1 }}\bigg{[}\ln\bigg{(}2\cosh\bigg{(}\frac{\mu_{c}}{k_{B}T}\bigg{)}\bigg{)}\bigg{]}, \tag{1}\]  
and the interband electron-electron transition \(\sigma_{inter}\) is given by  
\[\sigma_{inter}=\frac{e^{2}}{4\hbar}\Bigg{[}\frac{\sinh(\frac{\hbar\omega}{k_{ B}T})}{\cosh\bigg{(}\frac{\mu_{c}}{k_{B}T}\bigg{)}+\cosh\bigg{(}\frac{\hbar \omega}{2k_{B}T}\bigg{)}}-\frac{i}{2\pi}\ln\frac{(\hbar\omega+2\mu_{c})^{2}}{ \left(\hbar\omega-2\mu_{c}\right)^{2}+\left(2k_{B}T\right)^{2}}\Bigg{]}, \tag{2}\]  
where \(\hbar\) is the reduced Planck constant, \(\omega\) is the radian frequency, \(T\) is the absolute temperature, \(\tau\) is the electro momentum relaxation time, and \(\mu_{c}\) is the chemical potential. In this article, \(T\) and \(\tau\) are set as 300 K and 0.5 ps, respectively. The complex surface conductivity of graphene is related to the complex permittivity. The complex permittivity is written as [32]  
\[\varepsilon=1+\frac{i\sigma_{g}\eta_{0}}{\kappa_{0}\Delta}, \tag{3}\]  
where \(\eta_{0}=377\)\(\Omega\) represents the impedance of the free space, and \(k_{0}\) stands for the wavenumber of the light in air. In our model, the thickness \(\Delta\) of a single carbon atom is set as 0.334 nm and the incident light is polarized along the \(y\) axis, which is shown in Figure 1a. In order to research the electromagnetic properties of the graphene heptamer, the extinction cross-section \(\sigma_{ext}\) is calculated by the commercial finite element method (FEM) software COMSOL Multi-Physics, RF module.