Topological Properties of Gapped Graphene Nanoribbons with Spatial Symmetries

To date, almost all of the discussions on topological insulators (TIs) have focused on two- and three-dimensional systems. One-dimensional (1D) TIs manifested in real materials, in which localized spin states may exist at the end or near the junctions, have largely been unexplored. Previous studies have considered the system of gapped graphene nanoribbons (GNRs) possessing spatial symmetries with terminations commensurate with inversion- or mirror-symmetric unit cells. In this work, we prove that a symmetry-protected Z₂ topological classification exists for any type of termination. Instead of the Berry phase, only the origin-independent part of it gives the correct bulk-boundary correspondence by the π-quantized values. The resulting Z₂ invariant depends on the 1D unit cell and is connected to the symmetry eigenvalues at the center and boundary of the Brillouin zone. Using cove-edged GNRs, we demonstrate the existence of localized states at the end of GNR segments and at the junction between two GNRs based on a topological analysis. The current results are expected to shed light on the design of electronic devices based on GNRs as well as the understanding of the topological features in 1D systems.