Investigations of Vacancy Structures Related to Their Growth in h-BN Sheet

The atomic, electronic, and magnetic properties of vacancy structures with triangular shape related to their growth in single hexagonal boron nitride (h-BN) sheet are investigated using density functional theory calculations. We find that the optimized structures of triangular vacancies depend on the vacancy sizes with N-terminated zigzag edge. Then, vacancy structures obtained during the vacancy evolution in h-BN sheet are considered by removing a boron-nitrogen pair (BN pair) from edges of triangular vacancies. The magnetic properties of those vacancy structures are investigated by local density of states and spin densities. It is found that the stability of the optimized structures with a BN missing pair depends on the BN-pair missing position: the most stable structure is a BN-pair missing structure at the edge face region with the smallest magnetic moment.


Background
Hexagonal boron nitride (h-BN) sheet is a single-layered material similar to graphene, consisting of equal numbers of boron and nitrogen atoms and it has attractive physical properties in relation to the application of nanodevices. During its synthesis, single-layer h-BN sheet has various defects such as vacancies and grain boundaries [1,2]. These defects can change atomic and electronic structure of single-layer h-BN sheet and thus affect performance of h-BN-based devices.
Because h-BN sheet consists of two types of atoms, in contrast to graphene sheet, the edge structures of its clusters, nanoribbons, or nanoholes divide into two types: Nterminated and B-terminated. The most stable structure of the edge of cluster has N-terminated edge with zigzag structure [3,4]. In previous theoretical studies, the atomic and electronic structures of vacancy structures in singlelayer h-BN sheet depend on the type of termination atoms and their vacancy size [3][4][5][6][7][8][9][10][11]. That is, the calculated stability of triangle vacancy structures and the magnetic properties were found to depend on the type of terminated atoms and vacancy sizes of triangle vacancy due to lone electrons at edge atoms. The triangular vacancy structures were found in experiments for using a free-standing h-BN sheet [12][13][14][15]. Electron beam irradiation results in increasing size of vacancy structures that maintain triangular shape [12,13] regardless of the vacancy size.
Recently, we reported the study for growth of triangular vacancy of single-layer h-BN sheet [15]. It was observed in the experiment that atoms in h-BN sheet are ejected in the form of the bundles, not each atom, at the edge of vacancy structures. Furthermore, we briefly mentioned theoretical results to explain the growth of vacancy in h-BN sheet with triangular shape.
In this paper, we address the detailed study of the atomic structures of triangular vacancy of single-layer h-BN sheet. The locally stable structures of triangular vacancies are found to depend on the vacancy sizes with N-terminated zigzag edge. Then, by increasing the vacancy size, we investigate the stability of the optimized structures with a BN missing pair and their magnetic properties.

Computational methods
We have performed the density functional theory calculations using the Vienna ab initio simulation package (VASP) [16,17]. The plane-wave basis set with the energy cutoff of 400 eV is employed to describe electronic wave functions. The ions are represented by projector-augmented wave potentials [18,19] and generalized gradient approximation is employed to describe the exchange-correlation functional [20,21]. To take the weak van der Waals (vdW) interactions, we adopt Grimme's DFT-D2 vdW correction [22] based on a semi-empirical GGA-type theory.
The atomic positions of all structures are relaxed with residual forces smaller than 0.01 eV/Å. For the Brillouin-zone integration, we use only gamma point in the Monkhorst-Pack special k-points scheme. The lattice constant of our model is calculated to be 2.56 Å, which is in agreement with experimental value [23]. To study the difference in the reconstructed structures after BN-pair missing, we consider (9 × 9) and (15 × 15) supercell in our calculations.

Results and Discussion
Triangular Vacancy in h-BN Sheet First, we have considered several vacancy sizes of single h-BN sheet to study the size effect of vacancy structures. Because N-terminated vacancy structure of h-BN sheet is more stable structure than B-terminated one [3,4], we mainly focus on N-terminated triangular vacancy structures. To control vacancy sizes of h-BN sheet, we increase the number of ejected atoms in h-BN sheet maintaining the triangular shape. The B-terminated vacancy structures after relaxation result in small distortion in their vertex region with weakly binding between B atoms (not shown here) while the N-terminated structures shows a distinct change at the vertices of their triangular vacancy. Among different vacancy sizes of Nterminated triangular shape, we find two types of optimized (i.e., locally stable) structures. One is a symmetric structure (denoted as N-symm) in which no noticeable change of structure at vertex of triangular vacancy is found when compared with the pristine h-BN sheet, whereas the other is a distortion structure (denoted as NN-bond) that shows N-N bonds at all vertices of triangular hole vacancy in h-BN sheet.
In the cases of B monovacancy (V 1B ) in h-BN sheet, the optimized structure shows only one configuration that is the N-symm structure. Due to strong repulsive force between N atoms located at the vertex of triangular vacancy, the distance between N atoms increases (2.66 Å) compared to that of pristine h-BN sheet (2.48 Å) and B-N bond lengths at the edge of the triangular vacancy decrease.
When the size of triangular vacancy of h-BN sheet is increases to give V 3B+1N and V 6B+3N structures, where V mB+nN represents a triangular vacancy with m missing B atoms and n missing N atoms, the optimized structures can have both N-symm and NN-bond structures, as shown in Fig. 1. These results agree with the previous theoretical study for the vacancy structures [6].
The optimized structure of larger vacancy sizes than that of V 6B+3N structure represents only one configuration, i.e., an NN-bond structure. This large vacancy structure has the longer edge length of triangular shape than that of small vacancy structure, which means that the B-N bonds around the vacancy hole is less affected by the formation of N-N bond at vertex of vacancy hole in large vacancy structures (and thus the bond lengths between B and N atoms remain almost the same at the edge of vacancy structure). Calculated bond lengths between N atoms at the vertices of triangular vacancies and relative energies of two types of vacancy structures are given in Table 1. We find that the N-N bond lengths and relative energies depend on the size of vacancies. The difference in relative energies between N-symm and NN-bond structures decreases with increasing size of triangular vacancy structure. In contrast, the Bterminated vacancy structures turn out to be only one structure with a weak B-B bond at the vertex regardless of their sizes (see Table 1).
The calculated total magnetic moments of vacancy structures vary depending on the vacancy size, terminated atoms, and optimized structures (see Table 1). In the Nsymm structures, the value of the magnetic moment in units of μ B is equal to the number of nitrogen atoms located at the edge of triangular vacancy structures because these N atoms have dangling bonds after missing of atoms and breaking of B-N bonds in the h-BN sheet. However, total magnetic moments of N-N bond structures with various vacancy sizes are calculated to be different from those of N-symm structures due to formation of the N-N bonds (homopolar sigma bond) at vertices of triangular vacancy structures. The total magnetic moments for the V 3B+1N , V 6B+3N , and V 10B+6N structures with N-N bonds at vertex of vacancy are 0, 3, and 6 μ B, respectively. Figure 1b  size of vacancy hole structures was observed to be expanded through missing of B and N atoms at the edge of triangular vacancy structures in the experiment [14]. It was also reported that when the vacancies grow maintaining triangular shape in h-BN sheet, B and N atoms are ejected preferentially with pairs or bundles from the edge face of vacancy structures [15].
To study stability of vacancy structures depending on the missing position, we increase the supercell size of h-BN sheet up to 15 × 15 unit cell and obtain the larger vacancy size such as V 15B+10N and V 21B+15N . It is found that the optimized relaxations for those vacancies result in only one stable atomic configuration, i.e., the NNbond configuration. The N-N bond lengths at the vertices and total magnetic moments are shown in Table 1. We select a large N-terminated V 21B+15N triangular vacancy structure embedded in the supercell to consider more missing positions (Fig. 2a). As shown in the It is found that other structures (missing numbering 2-6) have one N dimer with pentagonal shape at the edge, shown in Fig. 2d-g, except Fig. 2c with N dimer located at the vertex. That is, the N atoms near the missing position have the dangling bond due to the missing and form the N dimer (see Fig. 2d-g). The presence of the N dimer in each structure influences its stability and magnetic properties. We calculate the relative energies and total magnetic moments of BN-pair missing structures obtained from V 21B+15N vacancy structure, which are listed in Table 2.
Based on the relative energies, we find that the stability of the BN-pair missing structures increases when the Table 1 The distances d 0 (Å) between N atoms (N-terminated vacancy) or B atoms (B-terminated vacancy) at vertices of vacancy structures, calculated total magnetic moments M (μ B ), and the relative energies E r (eV) for triangular vacancy configurations of h-BN sheet N-terminated vacancy   Table 2). Calculated total magnetic moments caused by terminated N atoms at the edge of optimized vacancy structures depend on the missing position. The magnetic moments of two corner missing structures are the same (M = 12μ B ). After the missing, the number of the terminated N atoms is 13 in the corner missing structures, which might give the magnetic moment M = 13μ B . However, the magnetic moment of an N atom in the distorted BN open rings vanishes due to the rearrangement of charge distribution as mentioned above.
The magnetic moments of other structures vary depending on the missing positions due to the presence of reconstructed BN open ring and/or N dimer located near the missing point. Figure 3 shows the spin densities of optimized structures obtained after BN-pair missing. From these spin densities, we know where the magnetic moments listed in Table 2 come from. For accurate analysis of differences between optimized structures depending on the missing positions, we select three configurations (corner-1, face-2, and face-3) among six BN-pair missing structures and calculate their electronic density of state (DOS). In the DOS plots, the defect states are located inside the band gap of the pristine h-BN sheet, as shown Fig. 4, where the valence band maximum and conduction band minimum of the pristine h-BN sheet are indicated by VBM and CBM, respectively. In the local DOS (LDOS) plot, the grey shaded regions and the red solid lines indicate LDOS of N atoms of the vacancy structures before and after BN-pair missing, respectively. Especially, the states of edge N atoms are concentrated around the Fermi level in the LDOS plot. As shown in the DOS and LDOS plots, the spin states of edge N atoms show the asymmetrical features. The corner-1 missing structure in Fig. 4a shows dangling bond states of N atoms in the range of −0.5 to 1.0 eV of the DOS and LDOS plots: noticeably, the dangling bond states localized only in the edge face region come mostly from spin-down states of LDOS (see spin density plots related to peak positions numbered 3 to 6 in LDOS plot). In the LDOS plots of two face-missing structures (Fig. 4b, c), not only the spin-down states but the spin-up states of edge N atoms also appear as the dangling bond states localized only in the edge face region near the Fermi level (−0.5~1.0 eV). That is, these spin-up and spindown plots are related to peak positions numbered 3 to 6 in the LDOS plot of Fig. 4b and those numbered 2 to 5 in the LDOS plot of Fig. 4c. On the other hand, all the BNpair missing structures have the energy band gap. The band gaps are about 0.35, 0.24, and 0.36 eV for the corner-1, face-2, and face-3 missing structures, respectively.

Conclusions
We have investigated the structural and electronic properties of the triangular vacancy structures of h-BN sheet  Fig. 3 The spin density (ρ spin up -ρ spin down ) distributions for the optimized structures of BN-pair missing. Yellow and light blue isosurfaces indicate the positive and negative values of spin densities, respectively using first-principles calculations. The optimized triangular vacancy structures were found to depend on their vacancy size. The most stable configuration of large vacancy structures has the N-N bond at each vertex of triangular vacancy, which determines its magnetic moments. When the missing of a BN pair occurs at the edge of triangular vacancy structure with a large hole size in the h-BN sheet, as observed in the experiment, the most stable structure is found to be a face missing structure with formation of N-N bonds. The magnetic moments and LDOS of the optimized structures depend on the missing positions of BN-pair at the edge of triangular vacancy.