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Band structures and the densities of states (DOS) of (a) -BNBL and (b) -BNBL at . Solid and dashed lines illustrate the results for , and , respectively. is normalized to the inverse lattice constant , the valleys are indicated by K and K′ with their coordinates . The centric insets in (a) and (b) show the arrangements of four atoms in the unit cells, where B and N label the boron and nitrogen atoms and indicates the interlayer coupling strength.
(a),(b) Band structures of the - and -BNBLNR for and ; the dispersion relations localized at two different edges are distinguished by the solid and dashed lines within the bulk gap, respectively, where indicate the positions of the chosen Fermi energies and A, B, C, D are four quantum edge states at . (c), (d) Evolution lines of Wannier function centers obtained in cases (a) and (b), respectively. (e), (f) Distributions of edge states A, B, C, D labeled in (a) and (b): the solid and dashed lines denote the spin-up and spin-down edge states, respectively.
(a) Phase diagram of -BNBLNR by changing and at , where the dispersion relations of bulk -BNBL on critical points and are indicated, denotes the cross point of two phase boundaries, and the band structures of three points A, B, C are shown in the right insets. (b) Quantized conductance as a function of energy in the different cases: (1) the bilayer zigzag-edged graphene nanoribbon at and ; (2) the -BNBLNR at and ; (3) the -BNBLNR at and ; (4) the -BNBLNR at .
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