# Mode Coupling Properties of the Plasmonic Dimers Composed of Graphene Nanodisks
## 3 Results and Discussion
Figure 5: The frequency splitting \(\Delta\)F of bonding and anti-bonding whispering-gallery (WG) modes of TM\({}_{\text{m},1}\) as a function of m from 2 to 8, where the distance \(g\) is 5 nm. to the TM\({}_{3,1}\) mode. When \(\mu_{\rm c3}\) reduces from 0.425 to 0.34 eV, the EM field in the right disk remains stable in the TM\({}_{3,1}\) mode. However, the counterpart in the left disk is first weakened then is enhanced again. Also, it switches from the TM\({}_{2,1}\) mode to the TM\({}_{1,1}\) mode, and the resonant frequency reduces linearly from 83 to 72 THz with \(\mu_{\rm c3}\). When \(\mu_{\rm c3}\) further decreases from 0.34 to 0.325 eV, another resonant frequency platform appears at 72 THz. The EM mode of the left disk remains in the TM\({}_{1,1}\) mode, and the field in the right disk is first weakened then enhanced again; the corresponding mode switches from TM\({}_{3,1}\) to TM\({}_{1,2}\). When \(\mu_{\rm c3}\) further reduces to less than 0.325 eV, the plasmonic field in the right disk remains in the TM\({}_{2,1}\) mode, but the field in the left disk disappears. Meanwhile, the resonant frequency decreases linearly with the reduction of \(\mu_{\rm c3}\). As for the \([\,|\,-1\)\(>_{x},\,|\,+1\)\(>_{y}]\) mode, when \(\mu_{\rm c3}\) reduces from 0.5 to 0.4 eV, the plasmonic mode in the right disk remains stable in the TM\({}_{2,1}\) mode, while the mode in the left disk is first weakened and then enhanced. Also, the plasmonic mode switches from the TM\({}_{2,1}\) mode to the TM\({}_{1,1}\) mode in the left disk. At the same time, the resonant frequency decreases linearly from 83 THz to 71 THz with the reduction of \(\mu_{\rm c3}\). Similar to the behavior of the \([\,|\,-1\)\(>_{x},\,|\,+1\)\(>_{y}]\) mode, a resonant frequency platform appears at around 71 THz when \(\mu_{\rm c3}\) decreases from 0.4 to 0.33 eV. In this region, the plasmonic mode of the left disk remains stable in the TM\({}_{1,1}\) mode, and the mode of the right disk is weakened first and then enhanced; also, the TM\({}_{2,1}\) mode is switched to the TM\({}_{3,1}\) mode. When \(\mu_{\rm c3}\) further reduces to less than 0.325 eV, the mode field in the left disk weakens gradually and then eventually disappears. However, the right disk remains in the TM\({}_{3,1}\) mode. Meanwhile, the resonant frequency decreases with the reduction of \(\mu_{\rm c3}\). It should be noted that the bonding/anti-bonding state of the dimmer may be changed during the mode evolution procedure [35]. Figure 6: The variation of the frequency and Q factor of the fundamental mode as a function of the chemical potential \(\mu_{\rm c3}\). (**a**) Resonance frequency; (**b**) Q factor. As for the modes which are symmetric with respect to the X axis, [\(\,|\,+1\text{$>$}_{\times},\,|\,+1\text{\[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]0}_{\text{y}}\)] and [\(\,|\,+1\text{\[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]1}_{\times},\,|\,-1\text{\[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]2}_{\text{y}}\)], there is also an avoid-crossing point at around 0.4 eV, which is shown in \[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]3 However, it is not as obvious as that of the asymmetric cases. This may be due to the large frequency splitting, which prevents the strong coupling effect between these two modes. Figure 8 plots the resonance frequency of the [\(\,|\,+1\text{\[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]4}_{\times},\,|\,+1\text{\[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]5}_{\text{y}}\)] mode as a function of \(\mu_{\text{c3}}\) in a range from 0.25 to 0.5 eV with a gap distance from 0 to 15 nm. The insets show the details in a range between 0.37 and 0.42 eV. Interestingly, with the increasing of the gap distance, the \"platform\" effect on the resonance frequency becomes more significant, accompanied by the red-shift of the peak position of the Q factor.