# Mode Coupling Properties of the Plasmonic Dimers Composed of Graphene Nanodisks
## 3 Results and Discussion  
When two nanodisks approach each other, the EM fields of the plasmon interact, and the collective modes appear. First, we study the collective resonant EM mode field distribution as a function of the gap between the two nanodisks, where \(\mu_{\text{c2}}=\mu_{\text{c3}}=0.5\) eV, \(\mu_{\text{c1}}=0.1\) eV and \(R=50\) nm. The z component of the typical electric field distribution of TM\({}_{2,1}\) modes of the dimer with a gap of 0 nm and 15 nm are illustrated in Figure 2a,b, respectively. According to the symmetry with respect to the X and Y directions, the four categories of the graphene plasmonic dimer modes are defined as \([\,|+1\textgreater_{\times}\,,|+1\textgreater_{\text{y}}\,]\), which is symmetric with respect to both the X and Y directions; \([\,|+1\textgreater_{\times}\,,|-1\textgreater_{\text{y}}\,]\), which is symmetric with respect to the X direction and anti-symmetric with respect to the Y direction; \([\,|-1\textgreater_{\times}\,,|+1\textgreater_{\text{y}}\,]\), which is anti-symmetric with respect to the X direction and symmetric with respect to the Y direction; and \([\,|-1\textgreater_{\times}\,,|-1\textgreater_{\text{y}}\,]\), which is anti-symmetric with respect to both the X and Y direction. More specifically, we follow Ref. [12] and regard the modes which are anti-symmetric with respect to the Y direction as the anti-bonding modes. Normally, the anti-bonding modes are located at the higher energy states. On the other hand, the modes that are symmetric with respect to the Y direction are regarded as the bonding modes, and are located at the lower energy states [12]. The frequency and the quality (Q) factor of the collective modes, as a function of the gap \(g\), are plotted in Figure 3a,b, respectively. It is shown in Figure 3a that the frequencies of the anti-bonding modes decrease with the increasing of the gap \(g\), while the counterparts of the bonding modes demonstrate the converse trend. The coupling strength between the WG modes of the nanodisks is indicated by the frequency splitting of the bonding or anti-bonding modes. When the gap \(g\) increases, the coupling strength between the plasmonic modes of the individual nanodisks degrades. Thus, the frequency splitting of the anti-bonding and the bonding modes decreases accordingly [12]. When \(g\) further increases to more than 50 nm, all the four mode categories combine together, the result of which is identical to the resonant mode of a single isolated nanodisk. This means that there is no interaction between the two nanodisks of the dimer. Figure 1: The 2D schematic of the graphene nanodisk dimer molecule, where \(R\) is the radius of the nanodisk; \(\mu_{\text{c2}}\) and \(\mu_{\text{c3}}\) are the chemical potentials of each nanodisk. It is also shown in Figure 3b that the Q factor of the collective plasmonic mode of the graphene dimer demonstrates an identical trend to that of the resonant frequency as a function of the gap \(g\). To understand this behavior, we plot the real part and imaginary part of the propagation constant \(\beta\) as a function of frequency in Figure 4, where the chemical potential is 0.5 eV. In general, the Q factor is given by \(Q_{\text{total}}^{-1}=Q_{\text{rad}}^{-1}+Q_{\text{abs}}^{-1}\)[33; 34], where \(Q_{\text{rad}}\) relates to the radiation loss and \(Q_{\text{abs}}\) relates to the absorption loss. The radiation loss is determined by the radius and material of the cavity, and \(Q_{\text{abs}}\) is calculated by  
\[Q_{\text{abs}}=\frac{\text{Re}(\beta)}{2\times|\text{Im}(\beta)|} \tag{5}\]  
where \(\text{Re}(\beta)\) and \(|\text{Im}(\beta)|\) are the real part and the absolute value of the imaginary part of the propagation constant \(\beta\). For a graphene sheet with a chemical potential of 0.5 eV, when the frequency of the plasmon increases from 80 THz to 83.5 THz, \(\text{Re}(\beta)\) and \(|\text{Im}(\beta)|\) increase together. However, the rising rate of \(\text{Re}(\beta)\) is even much faster than \(2|\text{Im}(\beta)|\).