Edge states of zigzag bilayer graphite nanoribbons

Electronic structures of the zigzag bilayer graphite nanoribbons(Z-BGNR) with various ribbon width $N$ are studied within the tight binding approximation. Neglecting the small inter-layer hopping parameter $\gamma_4$, there exist two fixed Fermi points $\pm k^*$ independent of the ribbon width with the peculiar energy dispersion near $k^*$ as $\varepsilon (k) \sim \pm (k-k^*)^N$. By investigating the edge states of the Z-BGNR, we notice that the trigonal warping of the bilayer graphene sheets are reflected on in the edge state structure of the Z-BGNR. With the inclusion of $\gamma_4$, the above two Fermi points are not fixed, but drift toward the vicinity of the Dirac point with the increase of the width $N$ as shown by the finite scaling method and the peculiar dispersions change to the parabolic ones.