# Topological Corner Modes in Graphene-Coated Nanowire Waveguide
## 3 Rhombic Lattice  
The topological corner modes are determined by the bulk property, which emerge in the structure with a nontrivial bulk topology. Moreover, the corner modes also depend on the corner shape [49]. Some corners are not able to allow for zero modes even if the nanowire is of a nontrivial topology. Such phenomenon is promptly revealed in the rhombic lattice. Figure 4a shows the structure of the graphene-covered nanowires placed in a rhombic lattice. In Figure 4b,c, we plot the real and imaginary part of the effective mode index. The whole number of nanowire waveguides is \(N=40\). The spectrum of the effective mode index possesses a band gap. One can see the gap only supports a single corner mode, denoted as a red dot. The effective mode index of the corner mode remains unchanged compared with that in a triangle lattice, which is figured out by \(n_{\text{eff}}=19.5+0.25i\). Along with the corner modes, the system also supports six topological edge modes. They are also located at the band gap, denoted by the green dots. Just like the situation in the triangle lattice, the real effective mode index corner mode is larger than that of the edge modes, while \(\text{Im}(n_{\text{eff}})\) of the corner mode is smaller than the edge mode. Therefore, the corner mode in the rhombic lattice also has a smaller modal wavelength and longer propagation distance than that of the edge modes. The discussed triangular lattice has only one type of corner and can support three degenerate corner modes. In contrast, the rhombic lattice has three different types of corners, the lower acute-angled corner, the up acute-angled corner, and two other obtuse-angled corners. The lower acute-angled corner is similar to that in the triangle lattice with a nontrivial topology, whi  
Figure 4: The topological corner mode in a rhombic array. (**a**) Geometry of a rhombic graphene-coated cylinder waveguide. (**b,c**) demonstrate the real and imaginary mode index for \(d_{1}=400\) nm and \(d_{2}=320\) nm. Figure 3: The typical modal profiles (**IE**) for the different types of modes. (**a**) Corner mode, (**b**) edge mode, and (**c**) bulk mode. corner is similar to that in the triangle lattice with a trivial topology. Therefore, one may expect the field will be confined at the lower acute-angled corner. Figure 5a,b exhibits the field distributions (\(|\textbf{E}|\)) of one topological edge mode and the topological corner mode, respectively. The corresponding mode indexes are \(n=15\) and \(16\). In Figure 5a, the fields of the edge mode are located at the left and lower edges of the triangle. In Figure 5b, the energy is mainly confined at the lower acute-angled corner and exponentially decreases away from the corners. There is no energy confined at the two obtuse-angled corners nor the upper trivial acute-angled corner. From the field distribution, one can also see the field of the edge mode is confined in several waveguides while the field of the corner modes mostly is concentrated in the single waveguide at the corner. Therefore, the corner mode has a smaller mode volume than the edge modes. Here, we try to explain the influences of some parameters on the corner mode. The effective refractive index of the graphene-coated nanowire will increase with the decrease in the nanowire radius. The corner mode will persist with different \(n_{\text{eff}}\) when the nanowire radius is altered. When the total number of nanowires increases, the corner modes will sustain as well. However, the corner modes may disappear if the number of nanowires is too small. This is because the corner mode is the result of the bulk property. The porosity is mainly determined by the spatial spacing. When the spacing is too large, the band gap will be decreased. The field confinement of the corner modes will become weaker. For small spacing, the nanowires are strongly coupled. The three corner modes will not degenerate and have a different effective refractive index.