Geometry of graphene ribbons separated by a spacing of l 0 perpendicular to the direction of the ribbon edge. The ribbons have finite width in the y direction and are assumed to be infinite along the x direction. The solid rectangles shows the supercell with length a Z (a A ) and b Z (b A ) in the x and y direction for ZGNRs (AGNRs), respectively. Here, the GNRs, which are single-layer and coplanar, are laterally parallel to each other. There is an IRD of a Z /2 and a A /2 along the x direction between two adjacent ribbons in the assembled structures (b) and (d) in comparison with that in structures (a) and (c), respectively.
(Color online) (a) The difference between the inter-ribbon distances before and after optimization and (b) the total energy of 8-(D)ZGNRs with the different initial inter-ribbon distances l 0. The total energy of the system with l 0 = 15 Å is set as the reference point for energy. Atomic structure and charge density distribution of 8-ZGNRs (c, e) and 8-DZGNRs (d, f) systems with different l 0: 3 Å (c, d) and 10 Å (e, f). Here, Angstrom is used as the unit of length, the charge density is drawn from the graphene plane, and a common scale is adopted. The symbol a0 denotes the Bohr radius, and the black balls represent C atoms, similarly hereinafter.
(Color online) Spin density distribution for (a) the 8-ZGNRs (b) the 8-DZGNRs with l 0 = 6 Å. The dark (green) and light (gray) isosurfaces in the images represent the spin-up and spin-down spin densities, respectively.
(Color online) Spin-up (solid curves) and spin-down (dotted curves) energy-band structure of 8-ZGNRs with initial inter-ribbon distance l 0 = 3 Å (a), 4 Å (b), 6 Å (c), 8 Å (d), 9 Å (e), 10 Å (f) and 8-DZGNRs with l 0 = 3 Å (g), 4 Å (h), 6 Å (i), 8 Å (j), 9 Å (k), 10 Å (l), respectively. The insets are magnified plots of the less energy regions in plots (a), (c), and (e)-(f). Dashed lines represent the Fermi energy and E F = 0. The up- and down-spin edges states U Z and D Z are located below and above the Fermi level, respectively.
(Color online) Energy-band structure of 14-AGNRs with l 0 = 3 Å (a), 4 Å (b), 5 Å (c), 6 Å (d), and 14-DAGNRs with l 0 = 3 Å (e), 4 Å (f), 5 Å (g), 6 Å (h), respectively. Except for AGNRs with l 0 = 3 Å [spin-up (solid curves) and spin-down (dashed curves)], the up-and down-spin states are degenerate for all the other systems mentioned here. Dashed lines represent Fermi energy and E F = 0. Edges states S A 1 and S A 2 are located above and below the Fermi level, respectively.
The band gap of 14-AGNRs (14-DAGNRs) with the different inter-ribbon distance l 0. The result marked with a superscript "d" means that it is "direct" band gap; otherwise, "indirect" one.
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