Theoretical study of edge states in BC2N nanoribbons with zigzag edges

In this paper, electronic properties of BC2N nanoribbons with zigzag edges are studied theoretically using a tight binding model and the first-principles calculations based on the density functional theories. The zigzag BC2N nanoribbons have the flat bands when the atoms are arranged as B-C-N-C along the zigzag lines. In this arrangement, the effect of charge transfer is averaged since B and N atoms are doped in same sublattice sites. This effect is important for not only the formation of flat bands but also for the validity of the tight binding model for such system.


Background
Graphene nanoribbons are finite-width graphene sheets, which are the one of the famous examples of nanocarbon materials [1,2]. The electronic properties of graphene nanoribbons strongly depend on the edge structures. Graphene nanoribbons with zigzag edges have the socalled flat bands at the Fermi level [1,2]. The states corresponding the flat bands are localized at the zigzag edges, i.e., the namely edge states [1,2]. In the honeycomb lattice, there are two inequivalent sites, A and B sublattices. For the formation of edge states, this sublattice structure plays decisive role [1,2].
Graphite-like BC 2 N sheet is one of the example of BCN, which was synthesized using chemical vapor depositions of boron trichloride, BCl 3 , and acetronitrile, CH 3 CN [15,16]. The stabilities and electronic properties of BC 2 N sheets were investigated by several authors [17][18][19]. The *Correspondence: KANEKO.Tomoaki@nims.go.jp 2 Computational Material Science Unit, NIMS, 1-2-1 Sengen, Tsukuba 305-0047, Japan Full list of author information is available at the end of the article electronic and magnetic properties of nanoribbons made with BC 2 N sheets were also investigated by several authors [20][21][22][23][24]. The magnetism in BC 2 N nanoribbons is predicted [20,21,23,24]. Xu et al. reported the presence of linear dispersion when atoms are arranged as C-B-N-C in the transverse direction [22].
Previously, the authors reported that the flat bands appear in zigzag BC 2 N nanoribbons where the atoms are arranged as B-C-N-C along the zigzag lines using a tight binding (TB) model [24]. The TB approximation is an efficient method to describe the electronic properties compared with the density functional theories (DFT). In the TB approximation, however, the effect of the charge transfer is absent, resulting in the failure of TB model for B-and N-doped nanocarbon system. The purpose of the paper was to investigate the effect of charge transfer in BC 2 N nanoribbons theoretically.
In this paper, we investigate the electronic properties of BC 2 N nanoribbons with zigzag edges using the TB model and the first-principles calculations based on DFT. The zigzag BC 2 N nanoribbons have the flat bands and edge states when atoms are arranged as B-C-N-C along the zigzag lines. The validity of TB approximation is discussed.

Methods
We shall consider four different structures of BC 2 N nanoribbons with zigzag edges, as shown in Figure 1 Let N be the number of zigzag lines of BC 2 N nanoribbons. The dashed rectangles represent the unit cell of BC 2 N nanoribbons. It should be noted that these nanoribbons were made of the same BC 2 N sheet indicated by the yellow-shaded dotted lines in Figure 1 which is the model-I introduced in [17]. The four different models are constructed by cutting the same BC 2 N sheet by changing the cutting positions. In these models, the atoms on the edges are different, as shown in Figure 1. It should be noted that the atoms are arranged as B-C-N-C along zigzag lines in models A and B while do not in models C and D.
We performed the first-principles calculations based on DFT using the local density approximation (LDA) and the projector augmented wave method implemented in VASP code. The cell size in the one-dimensional direction was measured by the lattice constant of BC 2 N sheet, a = 4.976 Å, and the ribbons were isolated by vacuum region with about 12 Å in thickness. The outermost atoms are terminated by single H atoms. The geometry was fully optimized when the maximum forces fell down below 10 −3 eV/Å. The cutoff energy of the plane wave basis set was chosen to be 400 eV, and the k-point sampling was chosen to be 12 in the one-dimensional direction. Although we found the finite spin polarization in BC 2 N nanoribbons, we restricted spin unpolarized calculations. The results of spin-polarized band structures will be reported in future publications elsewhere together with other models of BC 2 N nanoribbons.
The Hamiltonian of the system within TB model of π-electrons is given by where E i is an energy of π electron at the site i; c † i and c i are the creation and annihilation operators of electrons at the lattice site i, respectively; i, j stands for summation over the adjacent atoms; and t i,j is the hopping integral of π electrons from jth atom to ith atom. E i are the site energies, E B , E C , and E N , at the B, C, and N sites, respectively. Following the work of Yoshioka et al., we shall assume that the hopping integrals are constant regardless of the atoms, i.e., t i,j ≡ t, and E N = −E B and E C = 0 [25]. For the numerical calculations, we shall choose E B /t = 0.7, 1.0 and 1.3 [24,25].

Results and discussion
First, we shall discuss the stability of BC 2 N nanoribbons. Calculated formation energies of BC 2 N nanoribbons are summarized in Table 1. Here, the formation energy is defined as where E BC 2 N , E Gr , E BN , and E H 2 are total energies of BC 2 N nanoribbons, graphene, boron nitride sheet, and hydrogen molecules, respectively. The model C and D BC 2 N nanoribbons are stable compared with models A and B http://www.nanoscalereslett.com/content/8/1/341 due to the large number of C-C and B-N bonds. Previously, we considered the BCN nanoribbons where the outermost C atoms were replaced with B and N atoms. In these nanoribbons, H atoms tend to be adsorbed at B atoms [26]. For the model C and D BC 2 N nanoribbons, however, a termination of the outermost B atoms is not energetically favorable compared with a termination of the outermost N atoms. Similar behavior can be found for the zigzag and armchair BN nanoribbons [27]. The outermost B (N) atoms are connected with single N (B) atoms for the model C and D BC 2 N nanoribbons, while the outermost B and N atoms are connected with only C atoms for the previous models' nanoribbons. Such difference between atomic arrangement should lead different tendency on the enegetics. The calculated band structures of BC 2 N nanoribbons for N = 8 are summarized in Figure 2. The band structure of the model A nanoribbon within DFT shown in Figure 2a(image i) have nearly degenerate band around the Fermi level. In Figure 2a(images ii, iii, and iv), the band structures of the model A nanoribbons within TB model are shown. We observed that the flat bands and the degree of degeneracy depend on E B /t [24]. The band structure for E B /t = 0.7 has the doubly degenerate flat bands at E = 0, but the twofold degeneracy was lifted with increasing E B [24]. The band structure within DFT resembles to that within TB for E B /t = 1.3 shown in Figure 2a (image iv). The length of the flat bands increase with increasing of E B , since the shift of the Dirac point of BC 2 N sheet increases [24].
The band structures of the model B nanoribbons also show similar dependence. It should be emphasized that the atoms are arrange as B-C-N-C along the zigzag lines in the model A and B nanoribbons [24]. As reported before [24], we can expect that the bands around the Fermi level would degenerate with increasing of N.
In the model C nanoribbons, the band structure within DFT shows the flat bands around the Fermi level, but they are not degenerate. It should be noted that electronhole symmetry is broken in the model C nanoribbons and atoms are not arranged as B-C-N-C along the zigzag lines. On the other hand, the band structures within TB model do not have the flat bands at E = 0. While such prominent bands are not described well, we can see the correspondence between the result within DFT and that of TB model for E B /t = 1.3. Due to the relation E N = −E B , the positive energy states of the model C becomes negative in model D, vice versa. Therefore, we can find similar effect to model C in the band structures, i.e., the band structure within TB model of E B /t = 1.3 is similar to that of DFT except around the Fermi level.  We tried to describe the band structure of models C and D using TB model by introducing the extra site energies at the edges. In this study, we added E B /2 at the outermost N atoms for the model C nanoribbon and −E B /2 at the outermost B atoms for the model D nanoribbon, because such prescription found to show the relatively good performance. The results for E B /t = 1.3 are shown in Figure 2c we added E B /2 (−E B /2) for the outermost C atoms connected with B (N) atoms. As the results, we could produce the linear dispersion for these nanoribbons as indicated in the blue dashed curves in Figure 3b(image ii). It should be emphasized that all the improved cases have the edge character. Therefore, this prescription works well if the target band keeps the edge character. The prescription does not work for several BC 2 N nanoribbons. As an example, we shall consider the BC 2 N nanoribbon shown in Figure 4a, which was introduced in [20] as BB-CC model. Here, we shall call the nanoribbons as model F. The model F nanoribbon is made of the other BC 2 N sheet of the model II defined in [17]. Note that the size of unit cell of this nanoribbon is different from those discussed above and the atoms are not arranged as B-C-N-C along zigzag lines in the model F nanoribbons.
Calculated band structures are presented in Figure 4b. As shown in Figure 4b the other hand, the band structure within DFT has a tiny direct bandgap of 0.12 eV at the X point. The decrease of band gap was reported by Lu et al. [20]. It should be noted that we confirmed that the band structure was not improved even if we introduce the site energies at the outermost atoms. Therefore, the arrangement of B-C-N-C along zigzag lines plays a decisive role for the applicability of TB model for BC 2 N nanoribbons.
For the zigzag nanoribbons with unit cell being a single primitive cell, the energy at the X point, i.e., k = ±π, can be solved analytically. Since the matrix elements along the zigzag lines are proportional to −te ±ik/2 , the hopping along the zigzag lines vanishes at k = ±π ( Figure 5), and the nanoribbons are effectively disconnected as indicated by the shaded region in the right side of Figure 4. Let E a and E b be the site energies at a and b sites shown in Figure 4. In this case, the energies at k = ±π are given by Therefore, the energy bands concentrate on these values at k = ±π except edge sites, suggesting that the introduction of the edge site energy is not sufficient to improve the description of electronic properties of BC 2 N nanoribbons within TB model.
In the model F nanoribbons, the degeneracy at k = π within TB model is lifted in the band structure within DFT. The BC 2 N nanoribbons where atoms are arranged as C-B-N-C in the transverse direction do not have such degeneracies. These results indicate that the effect of charge transfer penetrates into interior of nanoribbons, resulting in a formation of transverse gradient of electrostatic potential. In the model C and D nanoribbons, on the other hand, the edge dominant states could not be described within TB calculations. For these nanoribbons, the direction of B-N bonds should play important role. In the BC 2 N sheet shown in Figure 1, the direction of BN dimers is arranged alternately. Then, the formation of transverse gradient of electrostatic potential in the nanoribbons is suppressed due to alternate arrangement of BN dimers in slant angle.
Previously, the authors reported that the arrangement of B-C-N-C along zigzag lines plays a decisive role for the formation of the edge states in zigzag BC 2 N nanoribbons [24]. This arrangement has other meaning. Within the TB approximation, effect of charge transfer is not described. On the other hand, B (N) atoms act as acceptors (donors) in graphene. Since B and N atoms occupy the same sublattice sites, the effect of charge transfer is canceled when the atoms are arranged as B-C-N-C along zigzag lines. Therefore, TB model is applicable for the zigzag BC 2 N nanoribbons when the atoms are arranged as B-C-N-C along zigzag lines.

Conclusions
The electronic properties of BC 2 N nanoribbons with zigzag edges have been studied theoretically using the tight binding model and the first-principles calculations. When atoms are arranged as B-C-N-C along the zigzag lines, the zigzag BC 2 N nanoribbons have the flat bands. Then, the tight binding model can become applicable for these systems. In this arrangement, the charge transfer is averaged effectively since B and N atoms are substituted in same sublattice sites, and such effect plays an important role for the formation of the edge states. For the tight binding model, the ratio of the site energies of B atom to the hopping integral is larger than unity. We tried to describe the band structure of BC 2 N nanoribbons where the atoms are not arranged as B-C-N-C along the zigzag lines using the tight binding model by introducing the extra site energies at the outermost atoms, but such method does not work for some BC 2 N nanoribbons. Therefore, study on the electronic properties of BC 2 N nanoribbons should be done within the first-principles calculations. http://www.nanoscalereslett.com/content/8/1/341 Author details 1 Nanosystem Research Institute, AIST, 1-1-1 Higashi, Tsukuba 305-8565, Japan. 2 Computational Material Science Unit, NIMS, 1-2-1 Sengen, Tsukuba 305-0047, Japan.