# Mode Coupling Properties of the Plasmonic Dimers Composed of Graphene Nanodisks
## 1 Introduction  
Similar to electronic atom and molecule systems, photonic molecules (PhMs) and plasmonic molecules (PMs) are composed of interacting nanostructures in which the specific collective electromagnetic (EM) behaviors are demonstrated [1; 2]. These behaviors include Fano resonance [3; 4; 5], slow light [6; 7], EM-induced transparency [8; 9; 10] etc., which result from the coupling between the plasmons and the external disturbance. In the past decade, PhMs and PMs have found significant applications in broad areas including fundamental sciences, the processing of information and biochemics due to their excellent properties of high quality (Q) factors, flexible structure design, and high sensitivity to the variation of the environments [11]. Among the reported works, [PERSON] et al. [12] proposed and fabricated a hybrid photonic-plasmonic molecule composed of two coupled identical Ag-Al\({}_{2}\)O\({}_{3}\)-Si sandwich disks. This hybrid photonic-plasmonic molecule has sub-wavelength properties and provides a novel platform for the exploration of the light-matter interaction. [PERSON] et al. [13] reported the energy-level evolution of coupled non-identical microdisks (CNMs), which were characterized by their avoided crossing which is intrinsically associated with the exceptional point or the non-Hermitian degeneracy. [PERSON] [14] analyzed the wavelength of the Q factors of the whispering-gallery (WG) modes in the PhM micro-disks as a function of the parameter of the PhM numerically, and demonstrated the crossing and avoid-crossing characteristics. However, all the above-mentioned micro or nanodisk based cavities were made by a conventional novel metal such as Ag and Au, or by dielectric materials such as SiO\({}_{2}\). Thus, the huge ohmic loss would degrade the properties of the devices, for example, Q factors. Furthermore, frequency tunability is impossible as long as the geometric structures of the devices are determined. All the intrinsic shortcomings of the conventional plasmonic materials become an obstacle to practical application. Recently, graphene, which is composed of a single layer of carbon atoms arranged in a two-dimensional (2D) honeycomb lattice, has attracted great attention in the fields of electronics and photonics all over the world due to its excellent material properties [15; 16; 17]. More specifically, in the field of the plasmonics, graphene offers the advantages of high confinement capability, low damping loss and tunability of working frequency. Consequently, graphene is a reliable material to substitute conventional metal materials in the field of plasmonics. [PERSON] et al. investigated plasmon-phonon coupling in large-area graphene dot and antidot arrays [18]. [PERSON] and [PERSON] reported asymmetric and connected graphene dimers for a tunable plasmonic response [19]. [PERSON] et al. showed that an electrically small chemical potential interruption in a graphene layer may possess a quasi-static resonance that launches a mixed-type semilocalized plasmon wave [20]. They also designed a system that is extremely sensitive to the quantized spectrum of graphene Landau levels and predicted up to 90\({}^{\circ}\) of Faraday rotation with just a single sheet of graphene [21]. We have proposed and numerically explored the plasmonic dimers composed of coupled graphene nanodisks of different sizes, which are surrounded by another chemical potential on the same piece of infinite graphene monolayer [22]. The split modes with non-degenerate wavelengths appear in the proposed structure. Further, the coupling between fundamental and first order WG modes has been pointed out. In the current work, we further analyze the field properties of the collective mode including the resonant frequency; the Q factor in a graphene plasmonic dimer with different chemical potentials in each nanodisk; and we discuss the physical principles behind the EM behaviors of the dimers.