# Topological Corner Modes in Graphene-Coated Nanowire Waveguide
## 2 Triangle Lattice
The adjacent waveguides are evanescently coupled and the coupling strength is exponentially decreased with the increase in the spatial spacing. The relative coupling strength between adjacent waveguides in the same cell exceeds that between the different cells as \(d_{1}<d_{2}\). As a result, the bulk topology of the nanowires should be trivial and its edges and corners will not allow for topological modes. The results show that there is a bandgap in the spectra of the complex effective mode index. However, no modes are allowed in the bandgap. In the lower two figures in Figure 2, we reverse the relative values of \(d_{1}\) and \(d_{2}\) and the spatial spacing is \(d_{1}\) = 320 nm and \(d_{2}\) = 400 nm. The spatial spacing in the same cell  
Figure 1: Geometry of the proposed triangular nanowire waveguide. now exceeds that in the different cells, that is, \(d_{1}>d_{2}\). The intra-layer coupling should be weaker than the inter-layer coupling. The bulk topological property of the nanowire now comes to be nontrivial and is able to sustain topological modes at its boundary. The complex spectra of the effective mode index still possess a bandgap. Moreover, there are three degenerate corner modes denoted as red dots, which are with almost the same effective mode index as \(n_{\text{eff}}=19.5+0.25i\). The effective wavelength of SPPs is inversely proportional to \(\text{Re}(n_{\text{eff}})\). This indicates the oscillating wavelength of the corner states is approximately 1/20 of the incident wavelength. The subwavelength feature originates from the confinement of graphene plasmonics. Along with the corner modes, nanowires also support nine topological edge modes. Their eigenvalues, that is, the effective mode index, are also located at the bandgap, plotted by the green dots in Figure 2c,d. Results show that \(\text{Re}(n_{\text{eff}})\) of the corner modes is larger than that of the edge modes, indicating the corner modes would experience a smaller modal wavelength than the edge modes. In addition, the imaginary effective mode index of the three corner modes is smaller that of the edge modes, reflecting that the corner modes experience smaller propagation loss and that the propagation distance of the corner modes is longer than that of the edge modes. Although both the edge and corner modes are located in the bandgap, they can be well distinguished from their eigenmodes. Figure 3a-c illustrates the typical profiles of the eigenmodes (\(|\textbf{E}|\)) of the topological corner mode, topological edge modes, and bulk modes, respectively. The corresponding mode indexes are chosen as \(n=18\), 17, and 45. In Figure 3a, the field is mainly restricted to the three corners of the triangle and rapidly declines away from the corners. The waveguides at the three corners have the strongest field, the nearest adjacent two waveguides have a vanished field, and the next two waveguides have small fields. This alternating field distribution is also a typical characteristic of the topological mode and guarantees the mode is robust against some disorders. In Figure 3b, the energy is located at the three edges of the triangle and there is almost no energy present in the bulk waveguides nor the waveguides at the three corners. From the field distribution, one can see that the energy of the corner modes is mainly confined at the three waveguides at the corner and field, and the edge modes are extended into many waveguides that are located along the edge. Therefore, the corner modes have a smaller mode volume than the edge modes. In Figure 3c, we plot one case of a bulk mode. The fields almost expand into the entire structure. Figure 2: The effective mode index in the triangle nanowires. In (**a**,**b**), the spatial spacing in the nanowires is \(d_{1}=400\) nm and \(d_{2}=320\) nm. In (**c**,**d**), the spacing is reversed and \(d_{1}=320\) nm and \(d_{2}=400\) nm. The green dots represent the edge mode while the red dots denote the corner mode.