Square-root topological graphene nanoribbons modulated by electric field

Graphene nanoribbons (GNRs) have unique edge structures and can be designed to have rich topological behavior. Extensive research has been conducted on nontrivial boundary states in GNRs, showing great potential for applications in carbon-based quantum computing devices. To harness these boundary states, it is desirable to achieve spatial manipulation and to be able to control the corresponding fractional charges. In this study, through first-principles calculations and tight-binding modeling, we present a class of square-root topological GNRs that exhibit enriched nontrivial band structures when subjected to a transverse electric field. A boundary state can be created between regions of different directions of the electric field, which can be experimentally engineered to move, to add, or to delete such a state. As we illustrate with several examples, the fractional charge of the boundary state can be further tuned by strategic inclusion or exclusion of zigzag edge units. These results suggest the potential for realizing spatially controllable, arbitrary fractional charges with GNRs, thereby providing new opportunities and design principles for GNRs in quantum information science.