Introduction

Isolated optically active atomic defects in wide-band-gap materials serve as single-photon emitters (SPEs), which are key components for quantum information technologies1,2. Hexagonal boron nitride (hBN) is a layered van der Waals (vdW) material and a favorable host for SPEs due to its fabrication versatility and compatibility with lithographic processing3,4,5,6,7. The observed SPEs in hBN feature high brightness, room-temperature stability, sharp zero-phonon-line (ZPL) peaks around 2 eV, and short excited-state lifetimes8,9,10,11,12,13,14,15,16. Furthermore, coherent control of single electron spins has been recently demonstrated in hBN17, including systems operating at room temperature18,19, where electron spin initialization and readout rely on optical excitation and emission of defect spins, a technique known as optically detected magnetic resonance (ODMR).

One major challenge is the identification of the exact defect structures responsible for SPEs and single-spin ODMR centers, which is a prerequisite for realizing deterministic formation and control. The observed photoluminescence (PL) spectra exhibit varying ZPL energies and phonon sidebands (PSBs), and many show similar optical lineshapes9. These emissions may originate from various defects, but the similarities in optical lineshape imply the presence of common defect types in diverse crystalline environments20,21,22.

An S = 1/2 paramagnetic defect with strong hyperfine interaction involving two equivalent nitrogen nuclei has been observed by electron paramagnetic resonance (EPR)23, and we assigned this signal to the negatively charged ONVB defect—i.e., oxygen substituting nitrogen adjacent to a boron vacancy—based on excellent agreement between experimental and simulated EPR spectra24. Notably, the existence of the ONVB defect was confirmed by subsequent annular dark-field scanning transmission electron microscopy (ADF-STEM) measurements25. In addition, carbon and oxygen substitutions were simultaneously observed nearby using the same technique. This provides strong evidence that the extra charge on the ONVB defect giving rise to the EPR signal could originate from donor-like substitutions of boron by carbon (CB) or nitrogen by oxygen (ON)26. In other words, CB or ON may form donor-acceptor pairs (DAPs) with ONVB, described as \({{{{\rm{C}}}}}_{{{{\rm{B}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{\rm{V}}}}}_{{{{\rm{B}}}}}^{-}\) or \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) in the ground state, where the S = 1/2 spin state arises from spin density localized around the \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) component of the DAP. In this sense, the common defect type is the ONVB acceptor, while the variation in optical properties is governed by the type and position of the donor partner.

Here, we perform comprehensive theoretical calculations on the optical properties of DAPs with varying separation distances. We find that the donor (CB and ON) indeed donates an electron to the ONVB defect, rendering it negatively charged. The donor-acceptor distance significantly influences the electronic structure, offering a possible explanation for the ZPL variation observed in experiments. We show that the ON–ONVB DAP is photostable, with quantum yields comparable to those of the isolated negatively charged ONVB. In contrast, the CB–ONVB pair exhibits metastable dim states at certain separations, which act as non-radiative decay pathways. Spin-flipping within these dim states can mix doublet and quartet multiplets, leading to spin polarization in the S = 1/2 ground state of the negatively charged ONVB when external magnetic fields lift the Kramers degeneracy. As a consequence, ODMR spectrum24 may arise from the CB–ONVB defect pair in the S = 1/2 ground state when subjected to a constant magnetic field.

Results and discussion

Optical properties of carbon and oxygen DAPs

Our modeling relies on recent ADF-STEM results, where single and multiple carbon and oxygen atoms adjacent to the VB monovacancy—particularly substituting the innermost nitrogen atoms—were imaged25. However, monovacancy structures with multiple first-neighbor substitutions do not yield bright ZPL emission at 2 eV, which is frequently observed in experiments, as discussed in Supplementary Note 1. Instead, another interesting phenomenon was observed in ADF-STEM measurements: carbon and oxygen substitutional defects appear in vacancy-free regions. The presence of CB, CN, and ON defects can be readily confirmed, while the OB defect has a high formation energy and is therefore rarely observed25. A previous theoretical study indicated the donor characteristics of CB and ON defects26. We emphasize that the oxygen-related ONVB defect may exhibit 2 eV ZPL emission in its negative charge state24. Hence, we propose that some of the observed 2 eV SPEs originate from \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\), with the additional charge supplied by either CB or ON. In Supplementary Fig. 1, we illustrate that the single-electron levels of CB or ON lie higher in energy than the empty a1 state of ONVB, facilitating electron transfer from the donor to ONVB. Furthermore, we calculate the isolated defects in different charge states and find that the total energy of the \({{{{\rm{C}}}}}_{{{{\rm{B}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) configuration is lower than that of the neutral \({{{{{\rm{C}}}}}_{{{{\rm{B}}}}}}^{0}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\) pair, confirming the stability of the DAP. Notably, charge transfer within DAPs is typically distance-dependent: in wide-band-gap materials with localized donor and acceptor states, the probability of charge transfer rapidly decays as the donor-acceptor distance increases. Therefore, we focus on several DAP structures characterized by relatively short separation distances.

We begin by examining the properties of the CB–ONVB DAPs, as illustrated in Fig. 1. The atomic sites, or coordinates, of the donor species relative to the location of the acceptor are labeled in Fig. 1a. The first-neighbor site, which could form a direct bond to oxygen, was not considered due to a potential repulsive interaction between ON and CB12. The CB1 and CB2 sites are located near the oxygen atom, resulting in a significant influence of the donor on the DAP properties.

Fig. 1: CB–ONVB DAPs in hBN.
figure 1

a The numbers indicate the positions of CB at various distances in the hBN lattice (boron and nitrogen atoms are shown as gray and green spheres, respectively). In this panel, CB is placed at site 1 (brown sphere). b Distance-dependent ZPL energies corresponding to the intrinsically bright optical transition, calculated without including non-radiative processes. Source data are provided as a Source Data file. c Electronic structure of the defects in the ground state. Number 10 corresponds to the isolated \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) defect, shown for reference. The irreducible representations of defect levels under C2v symmetry are indicated. a1 and b1 denote levels with in-plane localized wavefunctions, while a2 and b2 correspond to out-of-plane extended orbitals. Filled and unfilled triangles represent occupied and unoccupied defect levels within the gap, respectively, and the direction of the triangle indicates spin-majority or spin-minority character.

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The ONVB defect possesses C2v symmetry, so we label the in-gap localized defect levels according to their irreducible representations, based on the symmetry operations of the wavefunctions. The corresponding localized wavefunctions are shown in Supplementary Fig. 1. In the isolated \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) defect, the optical transition occurs between the a1 and b1 levels, which are in-plane extended and localized on two equivalent nitrogen atoms along the C2v axis. However, strong interaction at the CB1 and CB2 positions alters the ordering of the occupied defect levels: the a1 level shifts downward, while the a2 or b2 levels shift upward. The presence of CB lowers the symmetry to C1h, merging the original a2 and b2 labels. Since the a2 (b2) orbitals are out-of-plane while b1 remains in-plane, the resulting transition dipole moment is oriented out-of-plane with very small magnitude (~0.12 Debye), making the optical transition weak. Thus, the bright emission still originates from the a1 → b1 transition. The large energy separation between these levels yields high ZPL energies: 3.7 and 3.9 eV for CB1 and CB2, respectively, located in the ultraviolet region.

As the donor–acceptor distance increases, the defect-level ordering reverts to that of the isolated \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\), and the wavefunction of CB recovers its D3h symmetry. Another observed trend is that the unoccupied levels of CB shift downward, while the b1 level shifts upward. In the CB5 configuration, the empty orbital from CB lies below the b1 orbital of ONVB. Nevertheless, the transition dipole moment from a1 to CB remains small (~0.16 Debye), so we focus on the bright emission localized on ONVB, where the ZPL energies gradually converge to 1.97 eV. We note that the charge stability of CB–ONVB DAPs is broader than that of the isolated ONVB defect, as the Coulombic attraction lowers the formation energy and shifts the charge transition levels toward the band edges, as discussed in Supplementary Note 2.

The above-discussed bright emission is a local excitation within \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\). In addition, the charge transfer process from \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) to \({{{{{\rm{C}}}}}_{{{{\rm{B}}}}}}^{+}\) may also result in luminescence, although it is intrinsically dim. The dim metastable state (MS) has two sub-states: an S = 1/2 doublet or an S = 3/2 quartet, depending on the relative spin orientations within the DAP. With Kohn-Sham density functional theory, we reliably calculate the total energy and geometry of the S = 3/2 state. Supplementary Fig. 4 shows the Kohn-Sham energy differences and excitation energies for which Kohn-Sham defect state pairs can be associated with intrinsically dim and bright optical transitions (without considering non-radiative processes or rates here).

Generally, the bright local excitation is independent of the distance between donor and acceptor, while the dim one shows a strong dependence. The bright emission is the first optical transition in CB3 and CB4; however, when CB resides at larger distances from ONVB, the dim emission appears at lower energy. This suggests the presence of a non-radiative decay path from the local excited state to the ground state. A simple model to capture the dim emission mechanism can be expressed as

$$E({R}_{{{{\rm{i}}}}})={E}_{{{{\rm{gap}}}}}-({E}_{{{{\rm{D}}}}}+{E}_{{{{\rm{A}}}}})+\frac{{e}^{2}}{\epsilon {R}_{i}}{{,}}\,$$
(1)

where ED and EA are the energy levels of the donor and the acceptor in the band gap, respectively. Here, we use the charge transition levels (CTLs) of CB (0∣ + 1) and ONVB(−1∣0) from a previous study26. ϵ = 6.93 is the in-plane dielectric constant of hBN27, and Ri is the separation distance between donor and acceptor. The last term represents Coulombic interaction, and the reciprocal function leads to a fast convergence of E(Ri) to the CTL difference (1.27 eV) for DAPs with large Ri, as shown in Supplementary Fig. 5. The S = 3/2 state converges properly and may serve as a reliable reference for predicting the energy of the S = 1/2 state, as their energy difference arises from spin-spin exchange interaction, which is typically less than 1 meV. As summarized in Supplementary Fig. 4, the total energy of the S = 3/2 state lies 1.68 to 2.36 eV above that of the ground state, depending on DAP separation. This means that the bright excited state (~2.0 eV) and the metastable states may exchange their energy ordering.

ON–ONVB DAPs are different from CB–ONVB DAPs (see Fig. 2). Even at short distances between ON and ONVB, the \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) character is still well preserved, likely due to the shallow donor nature of ON. The empty levels from ON lie very close to the conduction band minimum and therefore have negligible influence on the optical transition within ONVB. Except for the ON1 site, the ZPL energies are all below 2.23 eV. Figure 3a shows the evolution of the transition dipole moment for the bright excitation across the considered configurations, which is significantly stronger than that of the dim transition due to the charge transfer process, as shown in Supplementary Fig. 4. It rapidly converges to the value of the isolated \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) as the DAP separation slightly increases. The large energy difference between the charge transition levels of ON(0∣ + 1) and ONVB(−1∣0) results in E(Ri) exceeding 2.89 eV; for example, the S = 3/2 level of ON4 lies 3.59 eV above the ground state. Accordingly, no metastable states arise for the \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) pair, which makes the optically and spin-active \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) defect a photostable emitter, albeit with minor variations in ZPL wavelengths depending on the actual distance between the donor and the acceptor.

Fig. 2: ON–ONVB DAPs in hBN.
figure 2

a The numbers indicate the positions of ON at various distances in the hBN lattice (boron and nitrogen atoms are shown as gray and green spheres, respectively). In this panel, ON is placed at site 1 (red sphere). b Distance-dependent ZPL energies. Source data are provided as a Source Data file. c Electronic structure of the defects in the ground state. Position 10 corresponds to the isolated \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) defect, shown for reference. Filled and unfilled triangles represent occupied and unoccupied defect levels within the gap, respectively. The orientation of each triangle indicates spin-majority or spin-minority character.

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Fig. 3: Two types of emission from DAPs.
figure 3

a Transition dipole moment evolution across the considered defect configurations. The dashed line indicates the transition dipole moment of the isolated \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\). Coordinates labels are defined in Figs. 1 and 2. Source data are provided as a Source Data file. b Type 1 and (c) Type 2 mechanisms of optical transitions for DAPs in semiconductors. The two types differ whether charge transfer or direct recombination occurs: the excited electron can be trapped by A0 or D+, altering the charge state before relaxing back to the ground state.

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The fluorescence mechanisms in DAP

The fluorescence mechanisms within DAPs, as shown in Fig. 3b, c, can be classified into two types according to the relative energy levels of the donor and acceptor. If the donor level is lower than the acceptor level, then charge transfer cannot occur spontaneously. Recombination can be induced by illumination from D0 to A0 (neutral charge state) with photon energy hνa, creating an ionized donor D+ and acceptor A−. In other words, the electron from the occupied state of D0 is excited to the empty state of A0. The electron returns via radiative decay (charge recombination), and a photon is emitted with energy hνe. If the donor level is higher than the acceptor level, the electron from the donor may spontaneously transfer to the acceptor when they are at short distances. Subsequently, photoexcitation converts the charged DAP to its neutral state. The charge transfer from donor to acceptor is distance-dependent, and a recent study investigated this process based on the Marcus theory framework28. In contrast to the previous cases, photoexcitation here generally acts on orbitals localized on the acceptor, which is not included in Fig. 3, and may lead to bright emission from the excited state. This is particularly true for the \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) pair. The shallow donor character of ON places the dim \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}^{0}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\) states at significantly higher energies than the bright excited state.

In the \({{{{{\rm{C}}}}}_{{{{\rm{B}}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) DAP, the situation differs, as depicted in the optical loop of Fig. 4a. Optical excitation first drives the doublet ground state to the optically active doublet excited state (D+ + A−). The small energy difference between the optically active excited state and the intrinsically dim metastable states (D0 + A0) enables charge transfer from ONVB (A−) to CB (D+) via non-radiative decay. The doublet (Ds) and quartet (Qs) metastable levels are separated by the spin-spin exchange interaction J. The ground and optically active doublet excited states are connected to the Ds state by either weak optical transitions or internal conversion (IC), while transitions to the Qs state occur via intersystem crossing (ISC), mediated by spin-flip processes. Using the same orbitals in both Ds and Qs, it is widely accepted that IC is significantly faster than ISC29. Therefore, the Ds state becomes preferentially populated within the metastable manifold. At zero magnetic field, the Ds state provides both radiative and non-radiative decay pathways, shortening the optical lifetime of \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) in this DAP compared to that in the \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) DAP. We note that depending on the magnitude of the non-radiative rate (r1 rate, see also Supplementary Fig. 6), non-radiative decay to the metastable Ds may dominate, and dim optical transitions may subsequently occur from the Ds DAP state.

Fig. 4: The optical pumping loop of CB–ONVB.
figure 4

a Bright excitation (red arrow) from the ground state (GS) to the excited state (ES) is localized on \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\). Through internal conversion, the ES can relax to the doublet metastable state of \({{{{\rm{C}}}}}_{{{{\rm{B}}}}}^{0}\)--\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\), denoted as Ds. The Ds level is separated by an energy J > 0 from the quartet metastable state Qs of the same \({{{{\rm{C}}}}}_{{{{\rm{B}}}}}^{0}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\) configuration. The  ± 1/2 and  ± 3/2 spin sublevels within Qs are split by zero-field splitting. The black dot indicates spin mixing between Ds and Qs, while the black circle indicates the population magnitude in the ground state. b Spin sublevels in Ds and Qs under magnetic fields, shown for three values of J:  − 3000, 0, and  + 3000 MHz. Source data are provided as a Source Data file. c Population ratio between the \(| {{{{\rm{D}}}}}_{{{{\rm{s}}}}},+1/2\rangle\) and \(| {{{{\rm{D}}}}}_{{{{\rm{s}}}}},-1/2\rangle\) spin sublevels in the metastable state as a function of applied magnetic field.

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At non-zero magnetic fields, the Kramers doublets split in the \({{{{{\rm{C}}}}}_{{{{\rm{B}}}}}}^{+}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) DAP, where the presence of the Qs state plays an important role in spin-dependent non-radiative decay. For instance, the higher branch (mS = + 1/2) in Ds can mix with the lower branch (mS = − 3/2) in Qs. In other words, Ds acquires some Qs character, and vice versa. Therefore, the mS = − 1/2 spin sublevel localized on \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{-}\) in Ds can become more populated than the mS = + 1/2 sublevel through the non-radiative decay channel r3 (see Fig. 4). This spin-selective decay populates the mS = − 1/2 spin state in the electronic ground state preferentially over the mS = + 1/2 state, making the mS = − 1/2 level appear brighter. As a consequence, an ODMR signal can be observed for an S = 1/2 ground state. The key feature here is the quasi-degeneracy of the Ds and Qs states in the metastable \({{{{{\rm{C}}}}}_{{{{\rm{B}}}}}}^{0}\)–\({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\) configuration, which corresponds to a spin pair of S = 1/2 (see ref. 30) and S = 1 (see ref. 24) defects, respectively. This interaction can lead to spin polarization of the split S = 1/2 Kramers doublet ground state, as observed in ODMR.

Such effects in DAP systems have already been reported in wide-band-gap materials such as zinc sulfide31 and diamond32, but only phenomenological models have been proposed to explain the observations. Intersystem crossing (ISC) processes for spin pairs have mostly been discussed in the context of organic chromophores33,34,35,36,37, involving dipolar mixing (\({\hat{H}}_{{{{\rm{di}}}}}\)), spin-orbit coupling (\({\hat{H}}_{{{{\rm{soc}}}}}\)), and hyperfine coupling (\({\hat{H}}_{{{{\rm{hf}}}}}\)).

However, none of these terms can be easily determined in our system due to their dependence on both donor-acceptor distance and the Ds–Qs energy splitting. The Ds–Qs splitting scales with distance as R−6. In comparison, ISC mediated by hyperfine interaction decays much more slowly than that caused by dipolar or spin-orbit coupling.

For a specific distance Ri, the rate of spin-flip can be evaluated using Fermi’s golden rule:

$$\Gamma=\frac{2\pi }{\hslash }{\left| \langle \psi ({{{\rm{D}}}}_{{{\rm{s}}}})\right| {\hat{H}}_{{{\rm{di}}}}+{\hat{H}}_{{{\rm{soc}}}}+{\hat{H}}_{{{\rm{hf}}}}\left| \psi ({{{\rm{Q}}}}_{{{\rm{s}}}})\rangle \right| }^{2}L(J)\,{\mbox{,}}\,$$
(2)

where ψ(Ds) and ψ(Qs) are the doublet and quartet wavefunctions, respectively, and L( J) is the lineshape function associated with the phonon overlap between the two states. Using the strongest hyperfine coupling of 55 MHz arising from the two 14N nuclear spins of ONVB in the CB5 configuration, the estimated rate has an upper bound of 108 Hz, assuming that the doublet and quartet states are nearly degenerate in both energy and geometry. This upper bound may be reduced, for instance, through L( J), but it still represents a sufficiently fast process for efficient spin mixing.

The distance between the DAP constituents should be long enough to reduce the energy gap J between Ds and Qs, but not so long that the spin-mixing rate becomes negligible. Unfortunately, direct calculation of the Ds and Qs states and their splitting J is beyond the scope of the present study, as it requires multi-reference methods that are not feasible for the supercell sizes needed to model DAPs. Nevertheless, an effective spin Hamiltonian for the \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\)–\({{{{{\rm{C}}}}}_{{{{\rm{B}}}}}}^{0}\) system, composed of coupled spin pairs with S1 = 1 and S2 = 1/2 in a magnetic field B, may be written as

$${H}_{{{\rm{spin}}}}={S}_{1}{{\bf{D}}}{S}_{1}+\sum\limits_{i}{H}_{\,{\mbox{hf}}}^{(i)}+gB{S}_{1}+gB{S}_{2}+J{S}_{1}{S}_{2}{\mbox{,}}\,$$
(3)

where the electron gyromagnetic factor is g ≈ 2.0023 (due to the very weak spin-orbit interaction) for each constituent, J is the isotropic exchange interaction between the spin pairs, D is the zero-field splitting tensor of \({{{{\rm{O}}}}}_{{{{\rm{N}}}}}{{{{{\rm{V}}}}}_{{{{\rm{B}}}}}}^{0}\) [D = 3.8 GHz and E = 0.091 GHz], and we include only the largest hyperfine couplings from the two 14N (I = 1) nuclear spins. We numerically solve Eq. (3) as a function of magnetic field B (applied perpendicular to the hBN sheet) and spin exchange coupling J, using the Easyspin software package38.

Figure 4b shows the splitting of spin sublevels under magnetic field for various J values, which can exhibit different magnitudes and signs depending on the relative orientation of the coupled spins, as observed in our DFT calculations. Due to the zero-field splitting of the triplet sublevels of ONVB, the \(| {{{{{\rm{Q}}}}}_{{{{\rm{s}}}}}}, \pm {1/2} \rangle\) levels always lie lower in energy, and the dominant spin mixing occurs between \(| {{{{\rm{Q}}}}}_{{{{\rm{s}}}}}, \pm 3/2 \rangle\) and \(| {{{{\rm{D}}}}}_{{{{\rm{s}}}}}, \pm 1/2\rangle\).

The mixing primarily involves \(| {{{{\rm{D}}}}}_{{{{\rm{s}}}}},-1/2\rangle\) and \(| {{{{\rm{Q}}}}}_{{{{\rm{s}}}}},+3/2 \rangle\) when J < 0, while for J > 0, it occurs between \(| {{{{\rm{D}}}}}_{{{{\rm{s}}}}},+1/2 \rangle\) and \(| {{{{\rm{Q}}}}}_{{{{\rm{s}}}}},-3/2\rangle\). Differences in the coefficients of these spin sublevel eigenstates imply variations in spin population, which can lead to ODMR contrast that is tunable via an external magnetic field. More specifically, variation in mixing with the quartet states results in different non-radiative relaxation rates, giving rise to hyperpolarization and detectable ODMR contrast. For each finite J-coupling magnitude, we observe a strong dependence of the mixing coefficient ratios on the magnetic field, as illustrated in Fig. 4b. This difference becomes particularly pronounced near the avoided crossing, where the ratio reaches a maximum of approximately 1.4. Ultimately, such asymmetry in mixing is expected to translate into differences in internal conversion rates back to the ground state. Importantly, for J = 0—i.e., when the two defects are non-interacting—both doublet states mix equally with the quartet manifold at each magnetic field, resulting in no discernible difference in non-radiative decay.

To further substantiate the role of spin mixing in mediating the ODMR signal, we calculated the internal conversion rates to and from the metastable Ds state. We approach this process by evaluating transitions between the excited state and Ds for r1, and between Ds and the doublet ground state for r2 and r3, respectively. Due to the small transition dipole moment associated with the charge transfer process, as shown in Supplementary Fig. 4, the non-radiative decay channels are expected to dominate. At short donor-acceptor distances below 1 nm, the small energy difference between the optically active doublet excited state and the dim metastable state, combined with strong electron-phonon coupling, leads to a fast internal conversion rate on the order of 1010 MHz. Consequently, the system relaxes to the ground state without producing a detectable optical signal through the dim state. The orbital overlap integral—and the resulting electron-phonon coupling—decays with increasing distance. The R-dependent value of the overlap integrals can be approximated by the overlap of two Slater orbitals, which scales as e−R 39. Using two configurations, CB5 and CB7, we extrapolate the electron-phonon coupling and internal conversion rate to longer donor-acceptor separations that cannot be directly computed using DFT, as discussed in Supplementary Note 3. The estimated internal conversion rate r1 is approximately 6-8 MHz at an 18 Å separation. This yields competitive radiative and non-radiative pathways from the optically active excited state of the CB–ONVB DAP defects. Accordingly, the ZPL wavelength, quantum yield, and brightness of these DAP defects are expected to depend on the actual donor-acceptor distance. Notably, both the weak radiative and the non-radiative (r3) internal conversion pathways connect the metastable state to the doublet ground state (see Supplementary Note 3), thereby completing the optical spin-polarization loop in these DAP systems.

With around 2 nm separation between CB and ONVB, the present coupled spin pair model may account for the previously reported ODMR signals originating from ground-state S = 1/2 defects in hBN sheets and nanotubes12,40,41. We underline that if the metastable state is sufficiently long-lived to carry out spin operations (e.g., ST1 defect in diamond42), then this spin-pair model can be extended to ODMR centers in hBN where spin polarization is observed in the metastable state. In this case, the ground state may be a singlet, formed by an S = 1/2 acceptor and an S = 1/2 donor, whereas the metastable states comprise singlet and triplet spin pairs. We note recent works40,43,44 published during the preparation of our manuscript, which observe this phenomenon without providing microscopic models for the underlying processes. Indeed, other types of coupled spin pairs may exist in hBN beyond those discussed in detail in our study. In Supplementary Note 4, we examine additional DAP configurations, such as CB–VB and CB–CNVB. The CB–VB pair represents a prototype of an S = 1 ground-state system with triplet and singlet spin pairs in the metastable state, while CB–CNVB is an example of a system with an S = 1/2 ground state. The latter is similar to the CB–ONVB configuration but exhibits distinct magneto-optical characteristics. We also note that the previously studied CB–CN DAPs30 serve as prototypes of an S = 0 ground state, with triplet and singlet spin pairs emerging in the metastable state. These findings demonstrate that our basic DAP model offers a new pathway for defect spin identification in hBN.

Discussion on color center identification

Generally, the atomic structure of optical emitters in hBN is challenging to determine experimentally. Direct imaging through high-resolution transmission electron microscopy can provide geometric information about defects on the top layer of hBN45,46. However, the optical signals sometimes originate from defects located deeper within the crystal.

EPR is a powerful tool for identifying paramagnetic point defects (with one or more unpaired electrons), based on their zero-field splitting (ZFS) and hyperfine interactions. In hBN, the interpretation of EPR signals is challenging but feasible in principle23,47, because both boron and nitrogen have 100% natural abundance of non-zero nuclear spin isotopes—a factor that increases the complexity of extracting the appropriate spin Hamiltonian. Furthermore, the ZFS of high-spin defects provides an important parameter for theoretical benchmarking19,48. However, EPR cannot detect non-magnetic defects, and the observed signals may originate from multiple overlapping defect types.

Confocal microscopy and photon antibunching measurements can offer optical insights, such as ZPL energy, PSB, and fluorescence lifetime of individual defects. Still, distinct defects may produce similar optical features, leading to potential misassignments. By applying external fields—such as strain49 or electric fields13,50—the optical response of the emitters can be modulated, offering additional information for defect identification. In our model, we propose that the observed optical signals could originate from spin pairs rather than isolated single defects. Multiple optical transition pathways within such spin-pair systems can lead to spin polarization and ODMR contrast, providing another route for identifying quantum defects.

In summary, a comprehensive analysis of SPEs or qubits in hBN requires a multi-dimensional approach that integrates structural, magnetic, optical, and field-dependent characterizations. Our study shows that some observations can be interpreted as direct interaction between defect pairs which underscores the complexity of defect identification of emitters and qubits in wide-band-gap materials such as the 2D hBN.

Conclusion

In summary, we propose that DAP emission is responsible for the visible ZPL emission at around 2.0 eV. The variance in ZPL wavelengths and optical lifetimes reported in experiments can be explained by the type and spatial separation of the donor defect relative to a key acceptor defect, ONVB. Our simulations reveal the ODMR mechanism associated with spin pairs in certain DAP structures, which are activated via dim metastable states and manifest as ODMR signals from an S = 1/2 ground state under non-zero constant magnetic fields. Our model also explains the challenge of producing indistinguishable SPEs in hBN, as the spatial arrangement of defect pairs must be precisely engineered. DAPs are common in wide band gap semiconductors and can account for both optical emissions and ODMR signals in Kramers doublet spin systems. The general mechanisms proposed here offer insight not only into defect engineering in hBN for quantum information processing but also into broader optoelectronic applications involving defects in other semiconductors.

Methods

Density functional theory calculations

In this paper, we performed density functional theory (DFT) calculations using the Vienna Ab Initio Simulation Package (VASP) code51,52 with a plane-wave basis set. We applied a plane-wave cutoff energy of 450 eV, and the convergence test is provided in Supplementary Note 6. The valence electrons and the core regions were described using the projector augmented wave (PAW) potentials53,54.

A 8 × 8 two-layered supercell was used to avoid interactions between periodic images, and this size was sufficient to apply the Γ-point sampling scheme. The interlayer van der Waals interactions were described using the DFT-D3 method of Grimme55.

Using a mixing parameter of α = 0.32, the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE)56 reproduces the experimental optical band gap of approximately 6 eV (excluding electron-phonon renormalization effects). This functional was used for geometry optimization and electronic structure calculations. The convergence threshold for atomic forces was set to 0.01 eV/Å.

Electronic excited states were calculated using the ΔSCF method57. Band alignment in charge correction was performed based on the core level energy of a selected atom located far from the defect center, both in the defective and perfect supercells.

The zero-field splitting due to dipolar electron-spin interaction was calculated within the PAW formalism58, as implemented in VASP by Martijn Marsman. We applied spin decontamination by averaging values over two symmetric spin-0 configurations to arrive at the final results59. The hyperfine tensor was calculated with the Fermi-contact term included, considering only the major contribution from the two nitrogen atoms60.

For the electron-phonon coupling, we first generated configuration coordinate diagrams of the relevant states, then computed the electron-phonon matrix elements using wavefunction overlap and linear response theory. Non-radiative decay rates were calculated using Fermi’s golden rule, as implemented in the NONRAD code61.