Band structure and edge states of star-like zigzag graphene nanoribbons

Connecting three zigzag graphene nanoribbons (ZGNRs) together through the sp³ hybrid bonds forms a star-like ZGNR (S-ZGNR). Its band structure shows that there are four edge states at k=0.5, in which the three electrons distribute at three outside edge sites, and the last electron is shared equally (50%) by two sites near the central site. The lowest conductance step in the valley is 2, two times higher than that of monolayer ZGNR (M-ZGNR). Furthermore, in one quasi-three-dimensional hexagonal lattice built, both of the Dirac points and the zero-energy states appear in the band structure along the z-axis for the fixed zero k-point in the x-y plane. In addition, it is an insulator in the x-y plane due to band gap 4 eV, however, for any k-point in the x-y plane the zero-energy states always exist at k_z=0.5.