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 (e.g. inversion) with only termination patterns commensurate with inversion-or mirror-symmetric unit cells. In this work, we prove that a symmetry-protected Z 2 topological classification exists for any type of termination. In these cases the Berry phase summed up over all occupied bands turns out to be π-quantized in the presence of the chiral symmetry. However, it does not always provide the correct corresponding Z 2 as one would have expected.We show that only the origin-independent part of the Berry phase gives the correct bulk-boundary correspondence by its π-quantized values. The resulting Z 2 invariant depends on the choice of the 1D unit cell (defined by the nanoribbon termination) and is shown to be connected to the symmetry eigenvalues of the wave functions at the center and boundary of the Brillouin zone. Using the cove-edged GNRs as examples, we demonstrate the existence of localized states at the end of some 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.
The integration of quantum materials like topological insulators (TIs) with magnetic insulators (MIs) has important technological implications for spintronics and quantum computing. Here, we report excellent crystallinity of c-axis oriented epitaxial TI films of Bi2Se3 grown on MI films, a rare earth iron garnet (ReIG), such as thulium iron garnet (Tm3Fe5O12, TmIG), by molecular beam epitaxy using Se-buffered low-temperature growth technique. Strained-TmIG films with robust perpendicular magnetic anisotropy were deposited by off-axis sputtering. We demonstrated a streaky reflection high-energy electron diffraction pattern starting from the very first quintuple layer of Bi2Se3, indicating the high-quality interface between TmIG and Bi2Se3, a prerequisite for studying interfacial exchange coupling effects. The strong interfacial exchange interaction was manifested by the observation of an anomalous Hall effect in the Bi2Se3/TmIG bilayer and a shift of the ferromagnetic resonance field of TmIG induced by Bi2Se3. We have reproducibly grown high-quality Bi2Se3/ReIG and interfaces using this TI growth method, which may be applied to grow other types of van der Waals hetero-structures.
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