## 3 Results and Discussion  
### General Depiction of the Mode of Graphene Plasmonic Trimers  
The typical induced surface charge density distributions of graphene plasmonic trimers are depicted in the first row of Figure 2. The direction of the dipole of each \"atom\" is from negative charge to positive charge, and the corresponding dipole distribution of the whole plasmonic molecule is shown in the second row. The A\({}_{1}\)' mode in the first column of the first row belongs to the equilateral  
Figure 1: (**a**) Three-dimensional geometry of the proposed structure: (**b**) vertical view, (**c**) left view. Here, the radius (\(R\)), height (\(\Delta\)), and chemical potential (\(\mu_{c}\)) of graphene are 50 nm, 0.335 nm, and 0.5 eV, respectively. \(\theta\) is the vertex angle of trimeric plasmonic molecules (PMs), shown in the inset of (**b**), and g = 1 nm is the distance between the individual plasmonic “atoms.” Outside the graphene is the free space, where the length (\(L\)), width (\(W\)), and height (\(H\)) of the computation window are 450 nm, 450 nm, and 300 nm, respectively. triangle (\(\mathbf{D_{3h}}\) symmetry), which is the highest-symmetry case of a trimeric PM. Here, the direction of the base of the triangle is defined as longitudinal, which in Figure 2 is the \(x\) axis. The vertical direction is defined as perpendicular to the base, which in Figure 2 is the \(y\) axis. For a succinct expression to describe the distribution of dipole moments, we define the zero net dipole moment along one axis as symmetrical with this axis and the nonzero net dipole moment as asymmetrical. Note that the total moment of the dipole of \(\mathbf{D_{3h}}\) symmetry is zero due to the symmetry, as shown in the corresponding diagram in the second row of the first column. When the vertex angle of the plasmonic trimer increases, \(\mathbf{D_{3h}}\) symmetry reduces to \(\mathbf{C_{2v}}\) symmetry. The total dipole of the whole trimer in the longitudinal direction remains zero, while the counterpart in the vertical direction becomes nonzero, as shown in the second row of Figure 2b. This nonzero momentum suggests that the whole PM is optically active with the polarized light incidents. When the vertex angle becomes \(180^{\circ}\), the PM turns out to be a linear chain with a symmetrical group of \(\mathbf{D_{\text{o}h}}\), where the electromagnetic field demonstrates a \(\prod_{\text{u}}\) mode, shown in Figure 2c. In this paper, only the fundamental mode of each plasmonic atom is considered. So we are only concerned about the point dipole located in the center of each plasmonic atom induced by the surface charge. The properties of the quadrupole induced by the surface charge are beyond the scope of this paper. Note that the incident angle of the light does not have a significant effect on either the near field or the far field of the plasmonic molecule; only the in-plane mode field distribution and the scattering spectra with respect to the normal incident electromagnetic field are considered here. Furthermore, the electromagnetic properties of the plasmonic molecule system are computed by solving the classic [PERSON]'s equations and the quantum effects, such as electron tunneling between the plasmonic atoms, are ignored, and even the distance between the nanodisks is in nanometer scale. The trimer plasmonic molecules have six mode categories [2] due to the \(\mathbf{D_{3h}}\) symmetry, shown in Figure 3. Similar to the molecule orbit, which is composed of linear combinations of each atom, the collective modes of the trimer are made of linear combinations of each plasmonic mode. These six mode categories are divided into two groups, one corresponding to the higher energy of antibonding modes, and the other corresponding to low-energy bonding modes, which is consistent with the plasmon hybridization calculation [1]. In the antibonding group, there are doubly degenerated modes related to the irreducible representation of E' symmetry.