# _electronics_
## 2 Waveguide Design  
The proposed hybrid hollow-core topological waveguide comprises silicon pillars sandwiched between two parallel gold plates. Such a hybrid structure can be realized by exploiting the multiwafer silicon-based fabrication process, which we have developed for fabricating a similar terahertz chip--a hybrid photonic crystal hollow-core waveguide [31]. During the microfabrication process, a 200 \(\upmu\)m gold layer was deposited on silicon wafers that was thicker than the skin depths (143-175 nm) of the gold over 0.2-0.3 THz. Throughout this work, the refractive index of high-resistivity silicon was fixed at 3.418, and the complex dielectric constant of gold is expressed using its Drude model, which is the same as our previous work [31]. We start the waveguide design from a honeycomb-latticed unit cell where six silicon pillars formed a hexagonal cluster. The parameters are defined in Figure 1a, where \(\Lambda\) is the lattice constant, \(d\) is the distance between the pillar and the honeycomb center, and \(r\) is the pillar radius. Keeping the \(C_{6}\) symmetry invariant, the six-pillar cluster is either expanded or compressed (see Figure 1a where the dashed circles represent the original position of six pillars) to open up a topological bandgap at the \(\mathbf{\Gamma}\) point in the Brillouin zone, as shown in Figure 1b. Such deformation led to a pseudospin degree of freedom in the electromagnetic system that is similar to the quantum spin Hall effect [32; 33]. To achieve the bandgap appearing at the target transmission band over 0.25-0.3 THz, \(\Lambda\) was determined to be 530 \(\upmu\)m. To further explore the possibility of improving the topological bandgap bandwidth, we investigated the influences of \(d\) and \(r\) on the relative bandwidth using finite element method models of the unit cell where period boundary conditionswere applied to simulate an infinite 2D unit cell array. For more details on finite element method-assisted topological band simulations, please see Ref. [34]. From the results shown in Figure 1c, the relative bandwidth of the topological bandgap was strongly governed by the cluster deformation (i.e., \(d/\Lambda\)) but less affected by the relative pillar dimension (i.e., \(r/\Lambda\)). We should note here that the maximum possible bandwidth was limited by both \(d\) and \(r\) since the cluster cannot be excessively expanded or compressed; meanwhile, the pillars should not be oversized in order to avoid adjacent pillars touching each other. Therefore, \(d\) and \(r\) values were, respectively, selected to be 0.375 (0.278) \(\Lambda\) and 0.111 \(\Lambda\) for a relative bandwidth of 16.1% (0.256-0.3 THz, i.e., the crossings between the dashed horizontal line and the red curve), which could also reduce the fabrication complexity. Next, one can assemble the expanded and compressed clusters to achieve an interface that propagates topological pseudospin states, as the red zigzag path shown in Figure 2. Considering a 3D waveguide model, the silicon pillar height, namely the distance between two parallel gold plates, was defined as \(h\) (see the inset of Figure 2a). To reveal the effects of structural parameters (including the cluster number on each side of the cladding \(Nc\) and the pillar height \(h\)) of the topological waveguide on transmission properties, we carry out numerical 3D supercell (shaded region in Figure 2a) simulations. Additionally, we use scattering boundary conditions in the \(xz\)-plane of the supercell (shown in Figure 2a) so as to evaluate the lateral confinement. Figure 2b presents the topological band diagram for a waveguide with \(Nc=11\) and \(h=\Lambda\). The dashed line in Figure 2b represents the light line. Two modes propagating along either \(+x\) (forward) or \(-x\) (backward) axes are supported in the topological bandgap range.