^{1}, Xianqi Dai

^{1,a)}, Yawei Dai

^{1}, Bao Zhao

^{1}and Maohai Xie

^{2}

### Abstract

Properties brought about by lateral in-plane coupling between graphenenanoribbons (GNRs) are investigated using the first-principle total energy calculations. It is found that, when two GNRs approach each other, the lateral coupling between the two brings about edge state splitting. Between zigzag-edged graphenenanoribbons (ZGNRs), the coupling mainly results from Coulomb and spin-spin interaction, while for armchair-edged graphenenanoribbons (AGNRs), it is from Coulomb interaction only. It is further found that the maximum inter-ribbon distance for effective coupling depends on the type of ribbons, which is ∼10 Å for ZGNRs, but ∼6 Å for AGNRs. Also, displacements of the GNRs along the ribbon direction are found to affect the electronic properties of the coupled GNRs. The results may be important for the microminiaturization of future nanoelectronic and spintronic devices based on graphene.

This research work has been supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11047026 and Henan Science and Technology Innovation Talent Support Program (2008HASTIT030).

I. INTRODUCTION

II. MODELS AND METHODS

III. RESULTS AND DISCUSSIONS

A. Effects of inter-ribbon LIPC on ZGNRs

1. Atomic structure and charge density

2. Spin density

3. Energy band structure

B. Effects of inter-ribbon LIPC on AGNRs

1. Atomic structure and charge density

2. Energy band structure

IV. CONCLUSIONS

### Key Topics

- Graphene
- 29.0
- Band structure
- 10.0
- Nanomaterials
- 10.0
- Band gap
- 7.0
- Fermi levels
- 6.0

## Figures

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.

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 a_{0} denotes the Bohr radius, and the black balls represent C atoms, similarly hereinafter.

(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 a_{0} 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 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) 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.

(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.

## Tables

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.

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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