Fig. 5.

Topological surface states with the existence of symmetry-preserving defects. a, b The simulated band structures of a ribbon-like supercell where the symmetry-preserving defect layers with $d_{\mathrm{intra}}=0.875a$ and d_intra= 0.125a are inserted, respectively. The structures of the supercells are depicted in the insets with the defect layers colored in red. The green and gray colors respectively denote the original topological ($d_{\mathrm{intra}}=0.85a$) and trivial ($d_{\mathrm{intra}}=0.15a$) SCs that have been reported in Fig. 2a. It is seen that in the bulk bandgap, the topological surface states emerge. The field distributions for these surface states at the high symmetry points of the surface Brillouin zone are presented in c and d, indicated by the colored dots (the defect layer is highlighted by the green blocks). We also label the mirror eigenvalues for the surface states in each field distribution. In addition to the topological surface states, extra defect modes are introduced (shown by the red curves in the band structures in a and b), whose field maps are presented in e and f.