# Topological Corner Modes in Graphene-Coated Nanowire Waveguide
## 2 Triangle Lattice  
We start by studying the corner modes in the triangle structure, which is illustrated in Figure 1. The nanowire waveguide is placed in a triangle lattice and the SPP propagation direction is the \(z\) axis. Each unit cell carries three nanowires and the diameter of the nanowire is denoted as \(2a\). Each nanowire is coated by monolayer graphene, which can be realized by the CVD (chemical vapor deposition) growth. The structure has two different spatial spacings, which are denoted by \(d_{1}\) and \(d_{2}\), respectively. The relative permittivity of the background dielectric is represented by \(\varepsilon_{\text{b}}\), which is assumed to be air with \(\varepsilon_{\text{b}}=1\) in this study. The nanowire is made up of SiO\({}_{2}\) with the permittivity being \(\varepsilon_{\text{d}}=3.9\) in the infrared frequency. As a 2D material, the optical material property of graphene depends on surface conductivity \(\sigma\). In contrast to bulk metal, the graphene surface conductivity can be adjusted as desired. The surface conductivity \(\sigma\)(\(T\), \(\omega\), \(\mu\), \(\tau\)) is figured out according to the Kubo formula [6], including inter- and intra-band transitions. The electronic band structures of graphene will be affected when graphene is rolled [52; 53]. The geometric potential is introduced and the chemical potential of the curved graphene monolayer can be expressed by \(\mu_{\rm c}\)' = \(\mu_{\rm c}-\hbar\
u_{\rm F}/\rm a\), where \(\mu_{\rm c}\) is the chemical potential of monolayer graphene sheet, \(\
u_{\rm F}\approx\) c/300 is the Fermi velocity, and \(a\) is the curvature radius. The Kubo formula can be still used with a modified chemical potential. The chemical potential will decrease when the bending is considered. The main conclusions of our manuscript are still valid except that the effective refractive index of SPPs becomes larger. In this work, we calculate the effective mode index and modal distributions of supermodes using the finite element method [6]. The current that applies in graphene is j = \(\sigma\)E with E denoting the electric field vector. Some simulating parameters are chosen as follows and remain unchanged, T = 300 K, \(\mu\) = 0.15 eV, \(\tau\) = 0.5 ps, and \(\lambda\) = 12 \(\upmu\)m, \(a\) = 40 nm. The parameters are tuned in advance such that each graphene nanowire is a single mode waveguide of the SPPs. The effective mode index of the SPP mode in isolated nanowire is almost \(n_{\rm eff}\approx\) 20. The coupled mode theory can be used to perform the semi-empirical simulations. The effective refractive index of a single nanowire can be analytically calculated. The coupling strength can be extracted from a double-nanowire coupler. Then, the effective refractive index of the nanowire array can be calculated by the coupled mode theory. Here, we mainly show the results obtained by a full wave simulation. Topological bound modes depend on the bulk topology, which appears at the boundary of a structure with a nontrivial bulk topology. The structure shown in Figure 1 is similar to the extended Su-Schrieffer-Heeger (SSH) model in 2D arrays, whose topology is completely governed by the relative value of the coupling strength between the different nanowires [42]. The system topology of the bulk mode is trivial when the spacing \(d_{1}\) between the different waveguides in the same cell is smaller than the spacing \(d_{2}\) between the different cells. In contrast, the bulk topology is nontrivial as \(d_{1}>d_{2}\). Figure 2 plots the effective mode index of all eigenmodes for trivial and nontrivial structures. The total number of nanowire waveguides is \(N\) = 45, corresponding to five layers of a triangle lattice. The three corners of the triangle lattice are identical. In the above two figures in Figure 2, we plot the complex effective mode index for all supermodes as \(d_{1}<d_{2}\). The spatial spacing is \(d_{1}\) = 400 nm and \(d_{2}\) = 320 nm.