Graphene is a two-dimensional (2D) material with a honeycomb sheet consisting of sp²-bonded carbon atoms.^[1–4] Since its birth, graphene has received much attention of researchers due to its excellent properties such as outstanding mechanical properties^[5–9] and electrical properties,^[10–17] unique optical properties,^[18–22] high thermal conductivity,^[23] etc. These unique properties make graphene an ideal candidate for a number of applications in semiconductor, charge storages, sensors, and field-emission devices^[24–26] based on graphene.

To promote the application of graphene in the field of nanoelectronics in the future, the ability to finely control the electronic and transport properties of this material is required. In recent years, many different methods have been used to improve the electronic properties of graphene, such as edge shape, chemical doping, and geometrical deformation. Ritter and Lyding^[27] used the tunneling spectroscopy to find that graphene nanoribbons (GNRs) with a higher fraction of zigzag edges exhibit a smaller energy gap than a predominantly armchair-edge ribbon of similar width. Many studies have shown that doping atoms and defects can improve the electronic structure of graphene and open the energy gap between the valence band and the conduction band, such as B, N, P, via vacancies, and Stone–Wales defects.^[28] Pereira and Neto^[29] investigated the electronic structures of armchair-edge graphene nanoribbons under a small uniaxial strain by tight-binding calculation and the first-principles calculation. It was found that a small asymmetrical strain introduces a band gap for metallic armchair-edge graphene nanoribbons (AGNRs) and modifies the band gaps for semiconducting AGNRs near the Fermi level. However, the in-depth understanding of the charge distribution, one of the basic issues affecting the graphene’s electronic property has not been described up to date.

It was reported that the charges of edge carbon atom accumulate and the charge density significantly increases near the strip edges.^[30] The edge effects have been investigated for graphene nanoribbons with minimum lateral dimension in a range of 1.0–4.5 nm by using the first-principles calculations.^[11] The distribution of net electric charges and capacitance in finite and multilayer graphene were studied by using both a constitutive atomic charge–dipole interaction model and an approximate analytical solution to Laplace’s equation.^[31] However, at present, the influence of graphene morphology (e.g., round, rectangle, etc.) on charge distribution, especially the effect on the charge distribution of carbon atoms on different edge shapes has not been detailed. An investigation on charge distribution in graphene is also important for understanding the charge impurities or chemical doping in graphene and promoting its application. At present, in most of the study of graphene, especially the electronic structure, the first-principles calculation method has been used. In Ref. [32], the first-principles calculation was used as a convenient tool to calculate and analyze the electronic structures of some crystalline materials due to the electronic structure that is the most important characteristic of a solid material. Zhou et al.^[33] have studied hydrogen storage on graphene with Li atoms using the first-principles calculations. They found that hydrogen storage capacity can reach 16 wt.% by adjusting Li coverage on graphene to (3 × 3) on both sides. Ding et al.^[34] have studied the structural and electronic properties of monolayer porous graphene (C), BN, and BC2N sheets by using the first-principles calculations. They found that all the porous C, BN, and BC2N sheets with one-hydrogen passivation exhibit direct-band-gap semiconducting behaviors and the porous C, BN, and BC2N sheets have semiconducting behaviors with practical band engineering by different hydrogen passivations. However, the efficiency of the first-principles calculation is too low.

In this paper, we use a semi-empirical molecular orbital program MOPAC devised by Stewart^[35] for our calculation. Firstly, we check the validity of the program with a linear-scaling density functional theory software package ONETEP (order-N electronic total energy package).^[36] Then we construct some graphene models with different morphologies and calculate their charge distributions by using MOPAC. Finally, we report the local charge distribution in graphenes.

MOPAC is a general-purpose, semi-empirical molecular orbital program for the study of chemical reactions involving molecules, ions, and linear polymers. This model uses self-consistent field (SCF) method to optimize the molecule structure and electron density based on the minimum energy optimization. The electrostatic repulsion and exchange stabilization are key factors in calculation process. This SCF method uses a restricted basis set of one s orbital and three p orbitals (p_x, p_y, and p_z) per atom and ignores their overlap integrals in the secular equation. The main quantities calculated by MOPAC are atomic charge (Q), molecular orbital (P), molecular geometry (G), and heat of formation (H). The results of MOPAC SCF calculations can be compared with experimental results or used in subsequent calculations. In the SCF calculations of MOPAC, molecular orbitals, charges, bond orders, and valences^[37,38] were calculated from quantum mechanics. Before the calculation, the geometry must be modified from the input geometry in order to be able to compare with experimental measurements and calculations.

We verify the validity of MOPAC by ONETEP. In Fig. 1, in order to test the validity of MOPAC, we use ONETEP,^[36] a linear-scaling density functional theory software package, along with MOPAC to calculate the charge distribution in spiro-OMeTAD (C₈₁N₄O₈H₆₈). The spiro-OMeTAD is a hole transportation material for the perovskite solar cells. The charges in this 161-atom molecule from MOPAC and those from ONETEP are shown in Fig. 1. It is found that eight O atoms have the same negative charges (shown in Fig. 1), while four N atoms have also identical negative charges (shown in Fig. 1). Figure 1 also indicates that sixty eight H atoms have slightly different positive charges, which can be linearly fitted (shown in Fig. 1, in blue line, R² = 0.99); figure 1 also indicates that twenty four C atoms have positive charges, on the contrary, fifty seven C atoms have negative charges. The charges for C atom (red line in Fig. 1) and H atom (blue line) are linearly fitted. This calculation suggests that the MOPAC is feasible to calculate the charge distribution of graphene sheet. Compared with the precise first-principles calculation, the semi-classical quantum MOPAC can be greatly time effective.

Fig. 1.

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	Fig. 1. (color online) Comparison between charges from MOPAC and ONETEP for spiro-OMeTAD (C81N4O8H68).

The charge distributions in different morphology graphenes and atomic chains are investigated by semi-quantum calculations. Our main findings are as follows. (i) The charge transfer at edge is much larger than in the internal part. The charge distribution has the same symmetry as that of the carbon atom distribution. (ii) The charge of double-bonded atom at zigzag edge has a strong dependence on bond angle, while the charge distribution of double-bonded atoms at the armchair edge has a strong dependence on the area of the triangle. (iii) The charge of triple-bonded atom at the edge is affected by the standard deviation of three bond lengths around it and also affected by the position (armchair edge or zigzag edge) of atom simultaneously.