Mono-vacancy
Before investigating the geometric and electronic structures of monolayer h-BN with mono-vacancy, it is necessary to study pristine h-BN in order to distinguish geometric and electronic differences between pristine and defective h-BN. Thus, the geometric and electronic structures of pristine lattice structure were calculated. Figure 2 shows a relaxed structure, isodensity plot, and electronic band structure of pristine h-BN. As shown in Fig. 2a, the bonding length between B and N atoms was calculated to be 1.45 Å from the relaxed structure. As marked with a triangle in the figure, the distances between B atoms and N atoms are all of the same length (2.50 Å). The isodensity plot in Fig. 2b shows that when a vacancy does not exist in monolayer h-BN, electron density is evenly distributed throughout the supercell. Figure 2c displays the electronic band structure of pristine h-BN which was calculated along the Γ-K-M-Γ path in the Brillouin zone. The band structure reveals that pristine h-BN has an indirect band gap of 4.3 eV in the electronic structure, which is consistent with previous calculations [25, 28].
Different from graphene, h-BN can have two kinds of defective structures in the case of mono-vacancy (1VB and 1VN), depending on a vacancy element. Thus, both different configurations should be separately considered to study the effect of mono-vacancy on the geometric and electronic band structures. Figure 3a and b display relaxed supercell structures with B and N vacancy, respectively. To evaluate the degree of distortion, the distance between atoms around the vacancy site was measured in each case. For this, the atoms around mono-vacancy were connected by lines which lead to a triangle. When 1VB is created in the structure as seen in Fig. 3a, all distances between the atoms are measured to be 2.61 Å which is greater than 2.50 Å measured from pristine h-BN. The relaxed structure of the defected h-BN and the measured distance are consistent with the results reported by previous works [21, 24]. The elongated distance indicates that the presence of mono-vacancy distorted a geometric structure of monolayer h-BN. And, the bonding lengths between the atoms at the edge site around B vacancy are changed to 1.41 and 1.44 Å. This also means that a geometric distortion occurred by B-vacancy in comparison with pristine lattice structure. Any planar distortion perpendicular to the monolayer did not happen after relaxation. In the case of 1VN, a similar shape of deformation to B-vacancy appears in the relaxed structure, and the distance between B atoms is measured to be 2.31 Å. This relaxed structure and the distance are also consistent with the previous works [21, 24]. The bonding lengths of atoms existing at the edge of mono-vacancy ranges from 1.44 to 1.45 Å, indicating that the presence of N-vacancy causes a geometric distortion in the supercell. Also, there was no planar deformation to the vertical axis of h-BN lattice structure. Interestingly, the distance between B atoms in Fig. 3b is shorter than the length between N atoms in Fig. 3a. This reason can be found in the isodensity plot displayed in Fig. 3c and d. When B atom is missing in the supercell, the repulsion between N atoms around the vacancy breaks out as shown in Fig. 3c. Meanwhile, B atoms around N-vacancy attract one another, resulting in a shortened distance of 2.31 Å and an elongated bonding length of 1.45 Å. Therefore, the shortened distance can be attributed to the attraction between B atoms around N-vacancy. This different situation gives rise to different electronic structures as shown in Fig. 3e and f. In the case of 1VB, a gap in the band structure has a similar value to that of pristine monolayer h-BN. However, the presence of 1VB shifts the Fermi level downward valence band. In the case of 1VN, a new energy state is created along the Fermi level as shown in Fig. 3f, which might help electron jump to conduction band.
Di-vacancy
Different from an odd-number vacancy, di-vacancy in h-BN has only one defective configuration which includes all elements. B and N atoms positioned at A and B site in Fig. 1 were removed to model di-vacancy in the supercell. Figure 4a displays a relaxed structure of monolayer h-BN with 2VBN. To examine the degree of geometric distortion, the distance between atoms at the edge site around the vacancies was measured as marked with lines shown in Fig. 4a. As a result, the distance between B atoms around 2VBN was measured to be 1.99 Å while the distance between N atoms at the edge was calculated to be 2.33 Å. These values are less than the distance calculated in pristine lattice structure. In the case of the distance between B and N atoms around 2VBN, the length was calculated to be 3.07 Å in all cases as seen in Fig. 4a. It can be understood that this is due to the repulsion by the localized electrons around N atoms and the attraction between B atoms, resulting in an elongated distance of 3.07 Å greater than 2.50 Å. No planar distortion was observed toward the direction perpendicular to the monolayer. Also, the di-vacancy influences on the bonding length between atoms at the edge around the vacancies, where the bonding length is distributed from 1.40 to 1.5 Å. Consequently, this leads to a distortion in the geometric structure, and changes the electron density distribution as presented in Fig. 4b. The distribution also shows that a weak covalent bond is formed between B atoms due to the relatively high electron density. This apparently explains why the distance between B atoms was shortened to 1.99 Å in monolayer h-BN with 2VBN. The structural change and the presence of di-vacancy lead to an alteration in the electronic structure. As displayed in Fig. 4c, two new energy states, which do not show in the band structure of pristine h-BN, appear between the valence and conduction band. The new energy states can contribute to electron jumps to conduction band.
Tri-vacancy
Similar to mono-vacancy, monolayer h-BN with tri-vacancy is present in the form of two defective configurations such as 3V2BN and 3V2NB. Figure 5a shows the relaxed hexagonal lattice structure of monolayer h-BN with 3V2BN. To investigate the degree of distortion by the vacancies, the atoms at the edge around the vacancies were connected by lines. As seen in the figure, the lines form a pentagon and have different lengths of 2.43, 2.62, and 3.15 Å. The different distances indicate that the structure was deformed by the presence of 3V2BN in comparison with the pristine structure. This is also supported that the bonding lengths between atoms at the edge sites range from 1.34 to 1.47 Å, which are deviated from the value of 1.45 Å calculated from pristine h-BN. Figure 5b displays a deformed structure of monolayer h-BN by 3V2NB where the atoms at the edge sites are also connected by lines. The sides of pentagon have different values of 2.06, 2.52, and 2.98 Å, and the bonding lengths of atoms at the edge are in the range between 1.38 and 1.51 Å. As seen in mono- and di-vacancy, the distance between B atoms around the vacancies becomes shorter than 2.50 Å corresponding to the distance between B atoms in pristine h-BN. The isodensity plot shown in Fig. 5c and d can explain the reason why the distance between B atoms became short by the vacancies. In the case of 3V2BN, N atoms around the vacancies in Fig. 5c have much higher electron density than B atoms while almost few electrons exist around B atoms at the edge sites in Fig. 5c. However, in the case of 3V2NB, B atoms at the edge site have more electron density and form a weak covalent bond by attraction between two B atoms as seen in Fig. 5d. Consequently, these different electron densities in the lattice structure lead to different electronic structures. As displayed in Fig. 5e and f, new energy states are generated between valence and conduction band in both cases of 3V2BN and 3V2NB. However, in the case of 3V2BN, the new states are evenly distributed in the gap defined by valence and conduction band, and the Fermi level exists between the states. Meanwhile, in the case of 3V2NB, the new states are relatively unevenly positioned, and the Fermi level exists away from the new states, compared with the electronic structure created by 3V2BN. Nevertheless, it can be expected that the new energy states can help electrons jump from valence to conduction band, converting monolayer h-BN to an electrically conductive material.
Eform of the vacancies
It has been revealed that defective h-BN structures have different geometric deformations and electronic structures even under the same vacancy density. This means that each structure has unique Eform, depending on the configuration and the number of a vacancy defect. Therefore, calculating Eform of each defective structure can give information about which vacancy is an energetically preferable structure. Figure 6 exhibits a plot of Eform with respect to the number of a vacancy. In the case of mono-vacancy, Eform of 1VB has much lower value than that of 1VN, meaning that B-vacancy is an energetically preferable defect rather than N-vacancy. The Eform of 2VBN has a positive value of 8.9 eV, suggesting that di-vacancy is not preferable to be energetically formed. A similar trend seen in mono-vacancy is found in the case of tri-vacancy. While Eform of 3V2BN has a negative value of −75.2 eV, 3V2NB has a positive value of 98.1 eV. This means that tri-vacancy is energetically preferable to be present in the configuration of 3V2BN rather than of 3V2NB. From this result, it can be said that when h-BN has the same or greater number of B-vacancy than that of N-vacancy out of the total number of vacancies, the defective h-BN structure is energetically preferable to be formed. Therefore, it is predicted that multi-vacancy created in the monolayer h-BN will have more B-vacancies than N-vacancies.