  Using first principles calculations, we studied a new class of graphdiyne
nanoribbons (GDYNR) with open hexagonal rings on the edges.To avoid the effects
from dangling bond, hydrogen or oxygen atoms were absorbed on the edges. There
are two kinds of GDYNR depending on the edge structures, armchair and zigzag.
The electronic structures show that all of them are semiconductors. The band
gap can be tuned by the width of GDYNR. As the width of nanoribbons increases,
the energy gap decreases firstly and then increases, and reaches a minimum gap
for both kinds. To understand the intriguing phenomenon, we constructed a
tight-binding model for GDYNR and found that the existence of the minimum of
the energy gap is due to the competition between the interaction within the two
edges and the coupling in between. Furthermore, topological unprotected edge
states are found in the band structure of a semi-infinite system by calculating
surface Green's function. If GDYNR could be synthesized in experiments, it
would be useful for the nanodevices in the future.
