# Effect of Nanodisks at Different Positions on the Fano Resonance of Graphene Heptamers
## 3 Simulation Results and Discussions
### The Effect of the Chemical Potential of the Central Nanodisk on Fano Resonance
In a previous study [9], the phenomenon of Fano resonance was verified by the theory that it can be divided into two independent subgroup modes. Therefore, the lineshape of the Fano resonance containing peak B and peak C can be explained by a mode wherein the central nanodisk does not influence the collective behavior, and another mode wherein the central nanodisk couples strongly with the satellite nanodisks. In order to observe the coupling between the central nanodisk and the ring nanodisks, the extinction spectrum of the central nanodisk is calculated, as shown in Figure 2d. The amplitude of the plasmonic resonance peak of the central nanodisk is smaller than that of the hexamer. However, when a central nanodisk is added into the hexamer, the coupling between the nanodisks can produce two peaks and a dip, as presented in Figure 3b. For the structure with central and ring nanodisks, the central nanodisk is the key to producing Fano resonance without changing the symmetry [12; 36]. In this structure, the resonance of the nanodisks in the ring are in phase when the central nanodisk is not taken into account; the corresponding extinction spectrum is presented in Figure 2b. Therefore, six nanodisks with in-phase oscillation generate the superradiant bright mode. However, the subradiant dark mode is caused by the interference between the central nanodisk and the ring nanodisks when a central nanodisk is added into structure [26]. The Fano resonance occurs when two modes couple with each other. The plasmonic resonance peak B and the modulation depth of the Fano resonance are unclear when the chemical potential of the central nanodisk is 0.5 eV. In order to get a deeper modulation depth of the Fano resonance, we vary the chemical potential of the central nanodisk from 0.5 eV to 0.6 eV with steps of 0.05 eV. The distribution of the chemical potential in the graphene heptamer is presented in Figure 3a. The chemical potential of local graphene nanodisks can be turned by electrostatic and chemical doping in this process. For electrostatic doping, the local chemical potential can be manipulated via supplying the top gate voltage in an appropriation top gate configuration. For chemical doping, the required graphene nanodisks can be exposed to HNO\({}_{3}\) vapor, while other nanodisks should avoid contact with HNO\({}_{3}\) vapor. For variation of the chemical potential of the central nanodisk, the structure still has spatial symmetry along the y-axis, which does not destroy the symmetry of the heptamer structure. Figure 3b shows the extinction spectra of a series of graphene oligomers with different chemical potentials. As the chemical potential of the central nanodisk increases, the contrast between peak B and peak C is enhanced. According to the electric field (\(|\)E\(|\)) distribution from peak B to peak G (see Figure 3c), two subgroup modes remain unchanged. Nevertheless, the extinction spectra show blue shift and the modulation depth at Fano resonance is enhanced with the increase of the chemical potential. These phenomena are relative to the effective refractive index of graphene. The effective refractive index of graphene presents a tendency to decrease as the chemical potential increases [28]. Therefore, the ability of the central graphene nanodisk to confine the incident light is weakened. This results in a decrease of the interaction between surface plasmons on the central nanodisk and the satellite nanodisks, which explains the phenomenon of blue shift in the extinction spectra. The modulation depth of the Fano resonance becomes more pronounced because of the blue shift of the superradiant bright mode and the subradiant dark mode. Accordingly, if the chemical potential of the central nanodisk continues to increase in a certain range, the bright mode and dark mode overlap more closely, which makes the lineshape of the Fano resonance more symmetrical. In order to analyze the electromagnetic behavior at Fano resonance, the near-field electric field distribution of the Fano resonance is calculated.