# Mode Coupling Properties of the Plasmonic Dimers Composed of Graphene Nanodisks
## 3 Results and Discussion
Consequently, the Q factor increases as the increasing of frequency, and finally results in a similar frequency trend as a function of given g, which are shown in Figure 3a,b. Next, we investigate the frequency splitting \(\Delta F\) of the corresponding anti-bonding and bonding modes as a function of the azimuthal mode number m of the plasmonic WG mode TM\({}_{\text{m,1}}\) of the graphene dimers. Figure 5 shows the frequency splitting \(\Delta F\) of bonding and anti-bonding of WG modes as a function of m, where the distance \(g\) is 5 nm. When the azimuthal number increases from 2 to 8, the frequency \(\Delta F\) of the bonding mode decreases from 0.567 to 0.002 THz, while the counterpart of the anti-bonding mode decreases from 0.275 to 0.0002 THz. When the azimuthal number increases, the wavelength reduces. However, the EM field confinement capacity of the graphene dimer becomes stronger as the resonant wavelength decreases. So the coupling between the plasmonic modes of the  
Figure 3: (a) Frequency and (b) Q factor of the collective modes TM\({}_{\text{2,1}}\) as a function of g of the four categories of symmetry, where \(\mu_{\text{c2}}=\mu_{\text{c3}}=0.5\) eV, \(\mu_{\text{c1}}=0.1\) eV and \(R=50\) nm. Figure 2: The electric field z component distribution of the collective mode TM\({}_{\text{2,1}}\). (a) \(g=0\) nm and (b) \(g=15\) nm. nanodisks becomes weaker, which further results in the small frequency splitting for both the bonding and anti-bonding mode, as shown in Figure 5.  
Then, we study the frequency and the Q factor of the plasmonic mode TM\({}_{2,1}\) of the graphene plasmonic dimer as a function of the chemical potential of the nanodisk at the right side. Figure 6a,b plots the resonant frequency and the corresponding Q factor as a function of \(\mu_{\text{c3}}\) for the four mode categories. In the range from 0.25 to 0.5 eV, one can see that the frequencies of the two modes which are anti-symmetric with respect to the X direction, i.e., \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,+1\text{-}\text{>}_{\text{y}}]\) and \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,-1\text{-}\text{>}_{\text{y}}]\) modes, are similar. There is a notch in the Q factor of the \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,-1\text{-}\text{>}_{\text{y}}]\) mode at \(\mu_{\text{c3}}=0.33\) eV, while there is a peak in the counterpart of the \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,+1\text{-}\text{>}_{\text{y}}]\) mode at the same point. This is a typical avoid-crossing point [11; 14]. However, unlike the photonic dimer composed of the coupled dielectric micro-disks, the Q factors of the \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,+1\text{-}\text{>}_{\text{y}}]\) and \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,-1\text{-}\text{>}_{\text{y}}]\) modes demonstrate a general climbing trend as the \(\mu_{\text{c3}}\) increases. This is due to the fact that the mismatching between the plasmons inside and outside the nanodisks becomes stronger when \(\mu_{\text{c3}}\) becomes higher, which also prevents the plasmons from escaping from the nanodisk and eventually results in a general increasing of the Q factor. In order to analyze the avoid-crossing property, we study the EM mode evolution as a function of \(\mu_{\text{c3}}\) and the corresponding diagrams are shown in Figure 7. For the \([\,|\,-1\text{-}\text{>}_{\text{x}}\,,|\,-1\text{-}\text{>}_{\text{y}}]\) mode, when \(\mu_{\text{c3}}\) reduces from 0.5 to 0.425 eV, the resonant frequency remains stable at a platform of 82.5 THz. Also, the EM field of the left nanodisk does not change significantly and remains in the TM\({}_{2,1}\) mode. However, the EM field of the right nanodisk is first weakened, then is enhanced again and switched from the TM\({}_{2,1}\) mode  
Figure 4: The real part and the absolute value of the imaginary part of the propagation constant as a function of frequency, where the chemical potential of the graphene is 0.5 eV, respectively.