# Effect of Nanodisks at Different Positions on the Fano Resonance of Graphene Heptamers
## 3 Simulation Results and Discussions
### The Effect of a Single Nanodisk in the Ring on Collective Behavior  
In this section, we investigate the effect of a single nanodisk in the ring on the collective behavior of the heptamer. From Figure 4, it can be clearly seen that the contributions of the leftmost nanodisk and the rightmost nanodisk are negligible. Therefore, it is more meaningful to research the influence of the two top nanodisks and the two bottom nanodisks. Due to the symmetry of the heptamer and the polarization of the incident light, these four nanodisks have identical effects on the collective behavior. Therefore, the upper left nanodisk is selected as the focus in this study. The chemical potential of the upper left nanodisk is labeled as \(\mu_{c3}\), as presented in Figure 5a. The other five nanodisks in the ring and the central nanodisk are set as 0.5 eV and 0.6 eV, respectively. \(\mu_{c3}\) was varied from 0.5 eV to 0.8 eV and the corresponding spectra are shown in Figure 5c. It is clear that a new plasmonic resonance peak, marked as H, appears in the spectrum when the chemical potential of the upper left nanodisk differs from the other ring nanodisks. This phenomenon arises from the destruction of the symmetry [16]. It is also remarkable that the Fano resonance disappears with the increase of \(\mu_{c3}\). When \(\mu_{c3}\) increases, the effective refractive index of the upper left nanodisk decreases, which makes the interaction between surface plasmons weaker. Consequently, this leads to the net dipole moment in ring nanodisks no longer matching the dipole moment of the central nanodisk. Hence, the Fano dip disappears gradually in the extinction spectra. When \(\mu_{c3}\) is 0.8 eV, the Fano dip is barely observed in the spectrum. The radius distribution and the extinction spectra are shown in Figure 5b,d, respectively. The radius of the target nanodisk, labeled as \(R_{3}\), decreases gradually, while other nanodisks retain the original parameters. When \(R_{3}\) is reduced to 48 nm, it can be found that the extinction spectrum has a larger blue shift, whereas when \(R_{3}\) is reduced to 45 nm, the blue shift of the spectrum is smaller. When the radius of the upper left nanodisk decreases, the coupling of surface plasmons between the upper left nanodisk and the other three nanodisks gradually weakens, which explains the blue shift of the spectral position at Fano resonance. However, when the radius decreases slowly, the distance between the nanodisks becomes larger, which weakens the behavior of the upper left nanodisk participating in the whole process. Therefore, the blue shift of extinction spectrum decreases gradually with the radius decreasing. At the same time, the dipole moment of the upper left nanodisk decreases with the decreasing of the radius. This further enhances the mismatching between the net dipole moment of the ring and the center nanodisk. As a result, the modulation depth of the Fano resonance decreases.  
Figure 4: The E distribution at Fano resonance when the chemical potential of central nanodisk is 0.6 eV.