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Lateral in-plane coupling between graphene nanoribbons: A density functional study
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Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

(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.

Image of FIG. 4.
FIG. 4.

(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.

Image of FIG. 5.
FIG. 5.

(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.


Generic image for table
Table I.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Lateral in-plane coupling between graphene nanoribbons: A density functional study