Local atomic-morphology-resolved edge states in twisted bilayer graphene nanoribbons.

We study the properties of edge states for a selected (10,1)[(4,3)] twisted bilayer graphene (TBG) nanoribbon with minimal edges but a majority of zigzag edges. By using the tight-binding and Green's function methods, we find a remarkable rule of a local electronic transfer for the edge states. As the energy away from the Fermi level, the transfer is in the order of convex AB-, concave AB-, concave AA- and convex AA-stacked regions of the ribbon curve edges. We illustrate that this rule comes from the difference in interlayer couplings among the four types of local geometries at edges. Further, an in-plane transverse electric field can rearrange the edge bands and enlarge the energy regimes, leading to the lowest energy states modified from AB-stacked edge states to AA-stacked ones. The realignment of the edge bands results from the interplay between the interlayer coupling and the potential difference induced by the transverse electric field, which results in different bonding and antibonding edge states, i.e. the edge bands. In contrast, the total energy regime of the edge bands remain nearly unchanged under a relative strong off-plane perpendicular electric field, and the typical AA-stacked edge states are still maintained even the rotational symmetry of two layers is broken. Until a sufficiently strong value, the TBG nanoribbon tends to behave as two noninteracting monolayer ribbons except for a band distortion in low-energy regime. The conductance spectra reflects the edge bands well. We also discussed the influence of edge defects in the TBG nanoribbon on transport properties. It is found that the contributed conductance of each type of edge states shows different degrees of suppression for a monatomic vacancy in the corresponding region of edges.