#### Abstract

We investigate defect properties in hexagonal boron nitride (hBN) which is attracting much attention as a single photon emitter. Using first-principles calculations, we find that nitrogen-vacancy defect has a lower energy structure in symmetry in 1− charge state than the previously known symmetry structure. Noting that carbon has one more valence electron than boron species, our finding naturally points to the correspondence between and defects with one charge state difference between them, which is indeed confirmed by the similarity of atomic symmetries, density of states, and excitation energies. Since is considered as a promising candidate for the source of single photon emission, our study suggests as another important candidate worth attention, with its simpler form without the incorporation of foreign elements into the host material.

#### 1. Introduction

The conventional importance and interest of point defects in solids lie in their impacts on the electrical and optical properties of the material hosts, in which the behavior of a single specific defect has little meaning apart from others. Within more recently emerging quantum technology, on the contrary, the individual defect has significance as a building block of quantum computation [1, 2], quantum cryptography [3, 4], and quantum communication [5]. Recently, single photon emission (SPE) has been discovered [6–9] in hexagonal boron nitride (hBN), characterized by a strong zero-phonon line (ZPL) in the optical spectrum thanks to weak coupling to phonon modes [10]. With desirable properties such as a wide band gap and thermal and chemical stability [11, 12], hBN has become one of the most promising materials for quantum applications.

Besides ZPL at around 2 eV energy, other physical properties relevant to identify the source of SPE include the Huang–Rhys factor related to the electron-phonon coupling strength [10], spin states [13], and the polarization of the optical transition [10, 14]. Depending on their unique evaluation of these properties, different studies have reached different conclusions: a nitrogen vacancy [15], boron dangling bonds [16], a complex defect consisting of next to a carbon substitutional replacing a boron atom [9, 17–19], adjacent to a nitrogen antisite [20, 21], and a boron vacancy next to a carbon substitutional replacing nitrogen [22], have been suggested to be promising candidates.

While most of the previous theoretical studies focus on the accurate estimation of excited state properties adopting some advanced methods beyond standard density-functional theory (DFT), the basic ground state energetics seems to have received less attention. For example, it is known that has symmetry with only in-plane atomic displacements, either in the spin singlet [18, 19] or triplet [17] state. A recent study has demonstrated that this structure is actually unstable against the symmetry-lowering relaxation with out-of-plane atomic displacements, consistent with the Stark shift of SPE via an out-of-plane electric field [23]. However, there has not been enough discussions on its physical contents and impact on the validity of previously suggested candidate defects.

In the present study, we revisit the ground state atomic and electronic configurations of some important defect structures using first-principles DFT calculations. After symmetry with out-of-plane relaxation is confirmed as the ground state atomic configuration of the defect, we investigate the possible analogy between and with one more electron. Surprisingly, it is found that 1− charged undergoes the same symmetry lowering relaxation to as neutral . Moreover, neutral and 1+ charged also have similar atomic and electronic configurations as expected. Based on the correspondence between the two defect structures, we suggest as another possible candidate for SPE whose physical properties need to be thoroughly investigated.

Our first-principles calculations are based on DFT with the PAW potential [24, 25] as implemented in the VASP code [26, 27]. The generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [28] is adopted for the exchange-correlation functional, facilitating the easier comparison with previous calculational works that also used PBE. Electronic wave functions are expanded with plane waves up to a cutoff energy of 500 eV. A supercell consisting of primitive units of hBN is used to ensure minimal spurious interactions among a defect structure and its periodic images. We adopt the experimental lattice constants and internal coordinates are fully relaxed. The *K*-point sampling is performed on a Monkhorst–Pack grid [29].

#### 2. Result and Discussion

First, we investigate the stable atomic structure of , which has been suggested as one of the promising candidates as the source of SPE. As shown in Figure 1(a), the symmetry structure with out-of-plane displacements (0.6 Å for the carbon atom) is found to be the most stable for the neutral charge state in accordance with recent work [23], with no sign of stable spin-polarized solution (i.e., spin-triplet state). When an electron is removed to become 1+ charged state , it still retains symmetry, but with much reduced out-of-plane displacements (0.2 Å for the carbon atom), as shown in Figure 1(b), which is closer to symmetry than the neutral state. While carbon is a foreign species in the hBN host and its presence naturally breaks the local symmetry, there is also a possibility of spontaneous symmetry breaking in native defects consisting of only host species, boron and nitrogen. Replacing the carbon atom with boron turns to , and we find that the spontaneous symmetry breaking indeed takes place with one of three equivalent boron atoms in a defect displaced out-of-plane direction in symmetry, even in the case of starting the atomic relaxation from the symmetry (i.e., three equivalent boron atoms around missing nitrogen). Since carbon has one more valence electron than boron, we can expect similar properties between and , with the former in one more negatively charged state. In fact, larger out-of-plane atomic displacement of 0.4 Å is found for , while it has smaller 0.1 Å displacement in the neutral state , as shown in Figures 1(c) and 1(d), respectively.

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The correspondence between and defects and the symmetry breaking atomic relaxation can be understood by their electronic structures. Figure 2 shows the density of states (DOS) of the defects in different charge states. For , only one electron occupies localized defect states, shown as small peaks inside the band gap energy region. The presence of an unpaired electron results in a local magnetic moment of 1 and our spin-polarized calculation shows that only a spinup state is occupied below the Fermi energy while spindown states are above by the exchange splitting, resulting in a spin “doublet” state. The orbital character of the occupied defect level, as represented by a peak around −0.6 eV, is shown in Figure 3(c) as HOMO (highest occupied molecular orbital). The two boron atoms around the missing nitrogen atom form a bond, with the wave function amplitude connecting them, while the carbon atom forms an antibond with them, with vanishing wave function amplitude between the carbon and the two boron atoms. This wave function character can be readily seen in supplemental Figure 5 in [17]. This defect state becomes more stable by decreasing the distance between the two boron atoms in the bonding state while increasing the distance between the carbon and the two boron atoms which are in an antibonding state, which can be facilitated by the out-of-plane relaxation of the carbon atom as the driving force of the symmetry reduction from to . Meanwhile, in the neutral state with two electrons, both spinup and down states are occupied by an electron each, as shown in Figure 2(a), exhibiting a spin “singlet” state. The energy gain by the out-of-plane relaxation of the carbon atom is enhanced by the presence of two electrons compared with the case of only one electron in the 1+ charge state, explaining the enhanced symmetry breaking with the larger out-of-plane displacement of the carbon atom in the neutral state compared with the 1+ charge state as depicted in Figures 1(a) and 1(b). The electronic structure of defect has a close analogy with that of as expected, displayed in Figures 2(c) and 2(d). has one electron as does, and hence exhibits the spin doublet configuration. In the case of , which has two electrons, same with , the spin singlet configuration is obtained with the enhanced symmetry breaking out-of-plane atomic displacement, as shown in Figures 1(c) and 1(d). Previously, only in symmetry has been known, and we find that it is 0.14 eV more stable in the symmetry structure for the 1− charged case.

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On the other hand, the defect states are overall shifted to higher energies and closer to the bulk conduction states in than in , and this leads to the difference of charge transition levels (CTLs), as depicted in Figure 3(a). For , 1+/0 CTL, which corresponds the position of below (above) which 1+ charge (neutral) state is stabilized, is estimated to be 1.57 eV above the valence band minimum (VBM). Meanwhile, the corresponding 0/1− CTL in is found to be higher, 2.90 eV above VBM. Compared with our GGA band gap of 4.32 eV, is stable unless the system is p-doped, whereas is stabilized in slightly n-doped samples which can be realized, for example, in oxygen contaminated samples [16]. Noting that has been considered as a promising candidate for SPE, now we estimate its optical transition energies by differences between ground and excited states, along with its analogous . There are four different configurations of a defect-contained system for which total energies need to be calculated, as schematically depicted in Figure 3(b): the electronic ground state with the relaxed atomic structure (*A*), the electronically excited state with the atomic structure fixed to that for *A*(*B*), the electronically excited state with the atomic structure fully relaxed (*C*), and the electronic ground state with the atomic structure fixed to that for *C*(*D*). The electronically excited state is obtained by holding an electron to the excited Kohn–Sham (i.e., one-electron) state just above the highest occupied defect state while performing the usual self-consistent-field (SCF) ground state calculation. This scheme is often referred to as [30] and was used in many previous calculations [9, 16–18]. Without appropriate band gap correction using techniques such as hybrid functional or GW, this method inevitably tends to underestimate transition energies.

The energy difference between configurations A and B, , corresponds to the absorption energy and is the emission energy, both of which do not involve the phonon. , meanwhile, is a ZPL where the electronic transition can take place between different atomic configurations via the overlap of the lowest phonon modes. The ZPL of is calculated to be 1.01 eV, as shown in the table in Figure 3(b), which must be underestimated due to the band gap error typical for standard DFT calculations. On the other hand, for is found to be 1.36 eV, 0.35 eV larger than that of . Since previous theoretical studies [9, 17–19] report that the error-corrected ZPL of lies within the experimentally observed ZPL range between 1.6 and 2.2 eV [7], the ZPL of is also expected to fall in within that energy range assuming that the difference of calculated ZPL between defects does not vary much after the band gap correction. Notably, atomic relaxation energies associated with the electronic excitation (i.e., energy differences between *B* and *C* and *A* and *D*) are smaller in , implying smaller electron-phonon coupling and Huang–Rhys factor. In the meantime, the lower symmetry of defects results in more allowed polarization directions for the optical transition. Figure 3(c) shows the squared amplitude of the highest occupied and lowest unoccupied molecular orbital (HOMO and LUMO, respectively) wave functions in . HOMO exhibits bonding character while LUMO originates from orbital of the carbon atom. In this case, optical transition is allowed for both and linear polarization (i.e., ) because both HOMO and LUMO are even functions along the direction. With planar atomic relaxation only, as in symmetry, the -like state would be even while the -like state is odd in the direction, so that only -polarized transition would be allowed.

Our work finds a new ground state for one of the most basic native defects, , which has a lower symmetry than the previously known symmetry. We also find a close correspondence between and defects with similar electronic structures and ZPLs. Many former studies, with advanced theoretical methods correcting typical DFT errors, found the properties of compatible with experiments. Therefore, our result naturally suggests that is another promising candidate for SPE. The fact that electron irradiation [7] or ion implantation [14] increases the formation probability of emitters further supports the plausibility of vacancy defects in general. Moreover, out-of-plane relaxation in is also consistent with the Stark effect observed for out-of-plane electric fields [23]. As our calculation results have limitations related to intrinsic DFT errors, further theoretical studies beyond standard DFT are necessary to thoroughly investigate their physical properties. Our result also calls attention to the possibility for symmetry-broken ground states of other defect structures in hBN.

#### 3. Conclusions

In conclusion, we suggest the close analogy between and defects with one charge state difference. Both defect structures with two electrons have symmetry with the spin singlet state and out-of-plane atomic displacements to gain in energy by increasing the distance between atoms in an anti-bonding state, in contrast to previous reports with in-plane displacements only and/or the spin triplet state. ZPL within standard DFT is calculated to be 1.01 and 1.36 eV for and , respectively. Based on the possibility of as a promising candidate for SPE in hBN as suggested by previous theoretical works, our result points to the necessity for the attention to the analogous defect and for the further investigation with more accurate theoretical schemes overcoming standard DFT errors in the excitation energy evaluation.

#### Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2016R1C1B1014715).