Abstract
Despite the recognition of twodimensional (2D) systems as emerging and scalable host materials of singlephoton emitters or spin qubits, the uncontrolled, and undetermined chemical nature of these quantum defects has been a roadblock to further development. Leveraging the design of extrinsic defects can circumvent these persistent issues and provide an ultimate solution. Here, we established a complete theoretical framework to accurately and systematically design quantum defects in widebandgap 2D systems. With this approach, essential static and dynamical properties are equally considered for spin qubit discovery. In particular, manybody interactions such as defect–exciton couplings are vital for describing excited state properties of defects in ultrathin 2D systems. Meanwhile, nonradiative processes such as phononassisted decay and intersystem crossing rates require careful evaluation, which competes together with radiative processes. From a thorough screening of defects based on firstprinciples calculations, we identify promising singlephoton emitters such as Si_{VV} and spin qubits such as Ti_{VV} and Mo_{VV} in hexagonal boron nitride. This work provided a complete firstprinciples theoretical framework for defect design in 2D materials.
Introduction
Optically addressable defectbased qubits offer a distinct advantage in their ability to operate with high fidelity under room temperature conditions^{1,2}. Despite the tremendous progress made in years of research, systems that exist today remain inadequate for realworld applications. The identification of stable singlephoton emitters (SPEs) in 2D materials has opened up a new playground for novel quantum phenomena and quantum technology applications, with improved scalability in device fabrication and leverage in doping spatial control, qubit entanglement, and qubit tuning^{3,4}. In particular, hexagonal boron nitride (hBN) has demonstrated that it can host stable defectbased SPEs^{5,6,7,8} and spin triplet defects^{9,10}. However, persistent challenges must be resolved before 2D quantum defects can become the most promising quantum information platform. These challenges include the undetermined chemical nature of existing SPEs^{7,11}, difficulties in the controlled generation of desired spin defects, and scarcity of reliable theoretical methods which can accurately predict critical physical parameters for defects in 2D materials due to their complex manybody interactions.
To circumvent these challenges, the design of promising spin defects by highintegrity theoretical methods is urgently needed. Introducing extrinsic defects can be unambiguously produced and controlled, which fundamentally solves the current issues of the undetermined chemical nature of existing SPEs in 2D systems. As highlighted by refs. ^{2,12}, promising spin qubit candidates should satisfy several essential criteria: deep defect levels, stable high spin states, large zerofield splitting (ZFS), efficient radiative recombination, high intersystem crossing (ISC) rates, and long spin coherence and relaxation time. Using these criteria for theoretical screening can effectively identify promising candidates but requires theoretical development of firstprinciples methods, significantly beyond the static and meanfield level. For example, accurate defect charge transition levels in 2D materials necessitates careful treatment of defect charge corrections for removal of spurious charge interactions^{13,14,15} and electron correlations for nonneutral excitation, e.g. from GW approximations^{15,16} or Koopmanscompliant hybrid functionals^{17,18,19,20}. Optical excitation and exciton radiative lifetime must account for defect–exciton interactions, e.g. by solving the Bethe–Salpeter equation (BSE), due to large excitonbinding energies in 2D systems^{21,22}. Spinphonon relaxation time calls for a general theoretical approach to treat complex symmetry and state degeneracy of defective systems, along the line of recent development based on abinitio density matrix approach^{23}. Spin coherence time due to the nuclei spin and electron spin coupling can be accurately predicted for defects in solids by combining firstprinciples and spin Hamiltonian approaches^{24,25}. In the end, nonradiative processes, such as phononassisted nonradiative recombination, have been recently computed with firstprinciples electron–phonon couplings for defects in hBN^{26}, and resulted in less competitive rates than corresponding radiative processes. However, the spin–orbitinduced ISC as the key process for pure spin state initialization during qubit operation has not been investigated for spin defects in 2D materials from firstprinciples indepth.
This work has developed a complete theoretical framework which enables the design of spin defects based on the critical physical parameters mentioned above and highlighted in Fig. 1a. We employed stateoftheart firstprinciples methods, focusing on manybody interaction such as defect–exciton couplings and dynamical processes through radiative and nonradiative recombinations. We developed a methodology to compute nonradiative ISC rates with an explicit overlap of phonon wavefunctions beyond current implementations in the Huang–Rhys approximation^{27}. We showcase the discovery of transition metal complexes such as Ti and Mo with a vacancy (Ti_{VV} and Mo_{VV}) to be spin triplet defects in hBN, and the discovery of Si_{VV} to be a bright SPE in hBN. We predict Ti_{VV} and Mo_{VV} are stable triplet defects in hBN (which is rare considering the only known such defect is \({\,\text{V}}_{\text{B}\,}^{}\)^{28}) with large ZFS and spinselective decay, which will set 2D quantum defects at a competitive stage with NV center in diamond for quantum technology applications.
Results
In the development of spin qubits in 3D systems (e.g. diamond, SiC, and AlN), defects beyond sp dangling bonds from N or C have been explored. In particular, large metal ions plus anion vacancy in AlN and SiC were found to have potential as qubits due to triplet ground states and large ZFS^{29}. Similar defects may be explored in 2D materials^{30}, such as the systems shown in Fig. 1b–d. This opens up the possibility of overcoming the current limitations of the uncontrolled and undetermined chemical nature of 2D defects, and unsatisfactory spindependent properties of existing defects. In the following, we will start the computational screening of spin defects with static properties of the ground state (spin state, defect formation energy, and ZFS) and the excited state (optical spectra), then we will discuss dynamical properties including radiative and nonradiative (phononassisted spin conserving and spinflip) processes, as the flow chart shown in Fig. 1a. We will summarize the complete defect discovery procedure and discuss the outlook at the end.
Screening triplet spin defects in hBN
To identify stable qubits in hBN, we start by screening neutral dopantvacancy defects for a triplet ground state based on total energy calculations of different spin states at both semilocal Perdew–Burke–Ernzerhof (PBE) and hybrid functional levels. We considered the dopant substitution at a divacancy site in hBN (Fig. 1b) for four different elemental groups. The results of this procedure are summarized in Supplementary Table 1 and Note 1. With additional supercell tests in Supplementary Table 2, our screening process finally yielded that only Mo_{VV} and Ti_{VV} have a stable triplet ground state. We further confirmed the thermodynamic charge stability of these defect candidates via calculations of defect formation energy and charge transition levels. As shown in Supplementary Fig. 1, both Ti_{VV} and Mo_{VV} defects have a stable neutral (q = 0) region for a large range of Fermi levels (ε_{F}), from 2.2 to 5.6 eV for Mo_{VV} and from 2.9 to 6.1 eV for Ti_{VV}. These neutral states will be stable in intrinsic hBN systems or with weak ptype or ntype doping (see Supplementary Note 2).
With a confirmed triplet ground state, we next computed the two defects’ ZFS. A large ZFS is necessary to isolate the m_{s} = ± 1 and m_{s} = 0 levels even at zero magnetic field allowing for controllable preparation of the spin qubit. Here we computed the contribution of spin–spin interaction to ZFS by implementing the planewavebased method developed by Rayson et al. (see the “Methods” section for details of implementation and benchmark on NV center in diamond)^{31}. Meanwhile, the spin–orbit contribution to ZFS was computed with the ORCA code. We find that both defects have sizable ZFS including both spin–spin and spin–orbit contributions (axial D parameter) of 19.4 GHz for Ti_{VV} and 5.5 GHz for Mo_{VV}, highlighting the potential for the basis of a spin qubit with optically detected magnetic resonance (ODMR) (see Supplementary Note 3 and Fig. 2). They are notably larger than previously reported values for ZFS of other known spin defects in solids^{29}, although at a reasonable range considering large ZFS values (up to 1000 GHz) in transitionmetal complex molecules^{32}.
Screening SPE defects in hBN
To identify SPEs in hBN, we considered a separate screening process of these dopantvacancy defects, targeting those with desirable optical properties. Namely, an SPE efficiently emits a single photon at a time at room temperature. Physically this corresponds to identifying defects that have a single bright intradefect transition with a high quantum efficiency (i.e. much faster radiative rates than nonradiative ones), for example current SPEs in hBN have radiative lifetimes ~1–10 ns and quantum efficiency over 50%^{33,34}.
Using these criteria we screened the defects by computing their optical transitions and radiative lifetime at random phase approximation (RPA) (see Supplementary Note 4, Fig. 3, and Table 3). This offers a costefficient firstpass to identify defects with bright transition and short radiative lifetime as potential candidates for SPEs. From this procedure, we found that C_{VV}(T), Si_{VV}(S), Si_{VV}(T), S_{VV}(S), Ge_{VV}(S), and \({{\rm{Sn}}}_{{\rm{VV}}}\)(S) could be promising SPE defects ((T) denotes triplet; (S) denotes singlet), with a bright intradefect transition and radiative lifetimes on the order of 10 ns, at the same order of magnitude of the SPEs’ lifetime observed experimentally^{34}. Among these, Si_{VV}(S) has the shortest radiative lifetime, and in addition, Si has recently been experimentally detected in hBN with samples grown in chemical vapor deposition (the ground state of Si_{VV} is also singlet)^{35}. Hence we will focus on Si_{VV} as an SPE candidate in the following sections as we compute optical and electronic properties at higher level of theory from manybody perturbation theory including accurate electron correlation and electron–hole interactions. Note that C_{VV} (commonly denoted C_{B}V_{N}) has also been suggested to be an SPE source in hBN^{36}.
Singleparticle levels, optical spectra, and radiative lifetime
The singleparticle energy levels of Ti_{VV}, Mo_{VV}, and Si_{VV} are shown in Fig. 2. These levels are computed by manybody perturbation theory (G_{0}W_{0}) for accurate electron correlation, with hybrid functional (PBE0(α), α = 0.41 based on the Koopmans’ condition^{17}) as the starting point to address selfinteraction errors for 3d transition metal defects^{37,38}. For example, we find that both the wavefunction distribution and ordering of defect states can differ between PBE and PBE0(α) (see Supplementary Figs. 4–6). The convergence test of G_{0}W_{0} can been found in Supplementary Fig. 7, Note 5, and Table 4. Importantly, the single particle levels in Fig. 2 show there are welllocalized occupied and unoccupied defect states in the hBN bandgap, which yield the potential for intradefect transitions.
Obtaining reliable optical properties of these twodimensional materials necessitates solving the BSE to include excitonic effects due to their strong defect–exciton coupling, which is not included in RPA calculations (see comparison in Supplementary Fig. 8 and Table 5)^{39,40,41,42}. The BSE optical spectra are shown for each defect in Fig. 3a–c (the related convergence tests can be found in Supplementary Figs. 9 and 10). In each case, we find an allowed intradefect optical transition (corresponding to the lowest energy peak as labeled in Fig. 3a–c, and red arrows in Fig. 2). From the optical spectra we can compute their radiative lifetimes as detailed in the “Methods” section on “Radiative recombination”. We find the transition metal defects’ radiative lifetimes (tabulated in Table 1) are long, exceeding μs. Therefore, they are not good candidates for SPE. In addition, while they still are potential spin qubits with optically allowed intradefect transitions, optical readout of these defects will be difficult. Referring to Table 1 and the expression of radiative lifetime in Eq. (9) we can see this is due to their low excitation energies (E_{0}, in the infrared region) and small dipole moment strength (\({\mu }_{\mathrm {eh}}^{2}\)). The latter is related to the tight localization of the excitonic wavefunction for Ti_{VV} and Mo_{VV} (shown in Fig. 3d–f), as strong localization of the defectbound exciton leads to weaker oscillator strength^{43}.
On the other hand, the optical properties of the Si_{VV} defect are quite promising for SPEs, as Fig. 3c shows it has a very bright optical transition in the ultraviolet region. As a consequence, we find that the radiative lifetime (Table 1) for Si_{VV} is 22.8 ns at G_{0}W_{0} + BSE@PBE0(α). We note that although the lifetime of Si_{VV} at the level of BSE is similar to that obtained at RPA (13.7 ns), the optical properties of 2D defects at RPA are still unreliable, due to the lack of excitonic effects. For example, the excitation energy (E_{0}) can deviate by ~1 eV and oscillator strengths (\({\mu }_{{\mathrm {eh}}}^{2}\)) can deviate by an order of magnitude (more details can be found in Supplementary Table 5). Above all, the radiative lifetime of Si_{VV} is comparable to experimentally observed SPE defects in hBN^{34}, showing that Si_{VV} is a strong SPE defect candidate in hBN.
Multiplet structure and excitedstate dynamics
Finally, we discuss the excitedstate dynamics of the spin qubit candidates Ti_{VV} and Mo_{VV} defects in hBN, where the possibility of ISC is crucial. This can allow for polarization of the system to a particular spin state by optical pumping, required for realistic spin qubit operation.
An overview of the multiplet structure and excitedstate dynamics is given in Fig. 4 for the Ti_{VV} and Mo_{VV} defects. For both defects, the system will begin from a spinconserved optical excitation from the triplet ground state to the triplet excited state, where next the excited state relaxation and recombination can go through several pathways. The excited state can directly return to the ground state via a radiative (red lines) or nonradiative process (dashed dark blue lines). For the Ti_{VV} defect shown in Fig. 4a, we find the system may relax to another excited state with lower symmetry through a pseudoJahn–Teller distortion (PJT; solid dark blue lines), and ultimately recombine back to the ground state nonradiatively. Most importantly, a third pathway is to nonradiatively relax to an intermediate singlet state through a spin–flip ISC and then again recombine back to the ground state (dashed lightblue lines). This ISC pathway is critical for the preparation of a pure spin state, similar to the NV center in diamond. Below, we will discuss our results for the lifetime of each radiative or nonradiative process, in order to determine the most competitive pathway under the operation condition.
Direct radiative and nonradiative recombination
First, we will consider the direct ground state recombination processes. Figure 5 shows the configuration diagram of the Ti_{VV} and Mo_{VV} defects. The zerophonon line (ZPL) for direct recombination can be accurately computed by subtracting its vertical excitation energy computed at BSE (0.56 eV for Ti_{VV} and 1.08 eV for Mo_{VV}) by its relaxation energy in the excited state (i.e. Franck–Condon shift^{44}, ΔE_{FC} in Fig. 5). This yields ZPLs of 0.53 and 0.91 eV for Ti_{VV} and Mo_{VV}, respectively. Although this method accurately includes both manybody effects and Franck–Condon shifts, it is difficult to evaluate ZPLs for the triplet to singletstate transition currently. Therefore, we compared it with the ZPLs computed by the constrained occupation DFT (CDFT) method at PBE. This yields ZPLs of 0.49 and 0.92 eV for Ti_{VV} and Mo_{VV}, respectively, which are in great agreement with the ones obtained from BSE excitation energies subtracting ΔE_{FC} above. Lastly, the radiative lifetimes for these transitions are presented in Table 1 as discussed in the earlier section, which shows Ti_{VV} and Mo_{VV} have radiative lifetimes of 195 and 33 μs, respectively (red lines in Fig. 4).
In terms of nonradiative properties, the small Huang–Rhys (S_{f}) for the \(_{1}^{3}A^{\prime\prime} \rangle\) to \(_{0}^{3}A^{\prime\prime} \rangle\) the transition of the Ti_{VV} defect (0.91) implies extremely small electron–phonon coupling and potentially an even slower nonradiative process. On the other hand, S_{f} for the \(_{1}^{3}A\rangle\) to \(_{0}^{3}A\rangle\) the transition of the Mo_{VV} defect is sizable (22.05) and may indicate a possible nonradiative decay. Following the formalism presented in ref. ^{26}, we computed the nonradiative lifetime of the ground state direct recombination (T = 10 K is chosen to compare with the measurement at cryogenic temperatures^{45}). Consistent with their Huang–Rhys factors, the nonradiative lifetime of Ti_{VV} is found to be 10 s, while the nonradiative lifetime of the Mo_{VV} defect is found to be 0.02 μs. The former lifetime is indicative of a forbidden transition; however, the Ti_{VV} defect also possesses a PJT effect in the triplet excited state (red curve in Fig. 5a). Due to the PJT effect, the excited state (C_{S}, \(_{1}^{3}A^{\prime\prime}\rangle\)) can relax to lower symmetry (C_{1}, \(_{1}^{3}A\rangle\)) with a nonradiative lifetime of 394 ps (solid dark blue line in Fig. 4a, additional details see Supplementary Note 9 and Fig. 11). Afterward, nonradiative decay from \(_{1}^{3}A\rangle\) to the ground state (\(_{0}^{3}A^{\prime\prime} \rangle\)) (dashed dark blue line in Fig. 4a) exhibits a lifetime of 0.044 ps due to a large Huang–Rhys factor (14.95).
Spin–orbit coupling (SOC) and nonradiative ISC rate
Lastly, we considered the possibility of an ISC between the triplet excited state and the singlet ground state for each defect, which is critical for spin qubit application. In order for a triplet to singlet transition to occur, a spinflip process must take place. For ISC, typically SOC can entangle triplet and singlet states yielding the possibility for a spinflip transition. To validate our methods for computing SOC (see the “Methods” section), we first computed the SOC strengths for the NV center in diamond. We obtained SOC values of 4.0 GHz for the axial λ_{z} and 45 GHz for nonaxial λ_{⊥} in fair agreement with previously computed values and experimentally measured values^{27,46}. We then computed the SOC strength for the Ti_{VV} defect (λ_{z} = 149 GHz, λ_{⊥} = 312 GHz) and the Mo_{VV} defect (λ_{z} = 16 GHz, λ_{⊥} = 257 GHz). The value of λ_{⊥} in particular leads to the potential for a spinselective pathway for both defects, analogous to NV center in diamond.
To compute the ISC rate, we developed an approach which is a derivative of the nonradiative recombination formalism presented in Eq. (11):
Compared with previous formalism^{27}, this method allows different values for initial state vibrational frequency (ω_{i}) and final state one (ω_{f}) through explicit calculations of phonon wavefunction overlap. Again to validate our methods we first computed the ISC rate for NV center in diamond. Using the experimental value for λ_{⊥} we obtain an ISC rate for NV center in diamond of 2.3 MHz which is in excellent agreement with the experimental value of 8 and 16 MHz^{45}. In final, we obtain an ISC time of 83 ps for Ti_{VV} and 2.7 μs for Mo_{VV} as shown in Table 2 and light blue lines in Fig. 4.
The results of all the nonradiative pathways for the two spin defects are summarized in Table 2 and are displayed in Fig. 4 along with the radiative pathway. We begin by summarizing the results for Ti_{VV} first and then discuss Mo_{VV} below. In short, for Ti_{VV} the spin conserved optical excitation from the triplet ground state \(_{0}^{3}A^{\prime\prime} \rangle\) to the triplet excited state \(_{1}^{3}A^{\prime\prime} \rangle\) cannot directly recombine nonradiatively due to a weak electron–phonon coupling between these states. In contrast, a nonradiative decay is possible via its PJT state (\(_{1}^{3}A\rangle\)) with a lifetime of 394 ps. Finally, the process of ISC from the triplet excited state \(_{1}^{3}A^{\prime\prime}\rangle\) to the singlet state (\(_{0}^{1}A^{\prime} \rangle\)) is an order of magnitude faster (i.e. 83 ps) and is inturn a dominant relaxation pathway. Therefore the Ti_{VV} defect in hBN is predicted to have an expedient spin purification process due to a fast ISC with a rate of 12 GHz. We note that while the defect has a low optical quantum yield and is predicted to not be a good SPE candidate, it is still noteworthy, as to date the only discovered triplet defect in hBN is the negatively charged boron vacancy, which also does not exhibit SPE and has similarly low quantum efficiency^{9}. Meanwhile, the leveraged control of an extrinsic dopant can offer advantages in spatial and chemical nature of defects.
For the Mo_{VV} defect, its direct nonradiative recombination lifetime from the triplet excited state \(_{1}^{3}A\rangle\) to the ground state \(_{0}^{3}A\rangle\) is 0.02 μs. While the comparison with its radiative lifetime (33 μs) is improved compared to the Ti_{VV} defect, it still is predicted to have low quantum efficiency. However, again the ISC between \(_{1}^{3}A\rangle\) and \(_{0}^{1}A\rangle\) is competitive with a lifetime of 2.7 μs. This rate (around MHz) is similar to diamond and implies a feasible ISC. Owing to its more ideal ZPL position (~1eV) and improved quantum efficiency, optical control of the Mo_{VV} defect is seen as more likely and may be further improved by other methods such as coupling to optical cavities^{47,48} and applying strain^{5,26}.
Discussion
In summary, we proposed a general theoretical framework for identifying and designing optically addressable spin defects for the future development of quantum emitter and quantum qubit systems. We started by searching for defects with triplet ground state by DFT total energy calculations which allow for rapid identification of possible candidates. Here we found that the Ti_{VV} and Mo_{VV} defects in hBN have a neutral triplet ground state. We then computed ZFS of secondary spin quantum sublevels and found they are sizable for both defects, larger than that of NV center in diamond, enabling possible control of these levels for qubit operation. In addition, we screened for potential SPEs in hBN based on allowed intradefect transitions and radiative lifetimes, leading to the discovery of Si_{VV}. Next, the electronic structure and optical spectra of each defect were computed from manybody perturbation theory. Specifically, the Si_{VV} defect is shown to possess an exciton radiative lifetime similar to experimentally observed SPEs in hBN and is a potential SPE candidate. Finally, we analyzed all possible radiative and nonradiative dynamical processes with firstprinciples rate calculations. In particular, we identified a dominant spinselective decay pathway via ISC at the Ti_{VV} defect which gives a key advantage for initial pure spin state preparation and qubit operation. Meanwhile, for the Mo_{VV} defect, we found that it has the benefit of improved quantum efficiency for more realistic optical control.
This work emphasizes that the theoretical discovery of spin defects requires careful treatment of manybody interactions and various radiative and nonradiative dynamical processes such as ISC. We demonstrate the high potential of extrinsic spin defects in 2D host materials as qubits for quantum information science. Future work will involve further examination of spin coherence time and its dominant decoherence mechanism, as well as other spectroscopic fingerprints from firstprinciples calculations to facilitate experimental validation of these defects.
Methods
Firstprinciples calculations
In this study, we used the open source planewave code Quantum ESPRESSO^{49} to perform calculations on all structural relaxations and total energies with optimized normconserving Vanderbilt (ONCV) pseudopotentials^{50} and a wavefunction cutoff of 50 Ry. A supercell size of 6 × 6 or higher was used in our calculations with a 3 × 3 × 1 kpoint mesh. Charged cell total energies were corrected to remove spurious charge interactions by employing the techniques developed in refs. ^{15,51,52} and implemented in the JDFTx code^{53}. The total energies, charged defect formation energies and geometry were evaluated at the PBE level^{54}. Singlepoint calculations with kpoint meshes of 2 × 2 × 1 and 3 × 3 × 1 were performed using hybrid exchangecorrelation functional PBE0(α), where the mixing parameter α = 0.41 was determined by the generalized Koopmans’ condition as discussed in refs. ^{17,20}. Moreover, we used the YAMBO code^{55} to perform manybody perturbation theory with the GW approximation to compute the quasiparticle correction using PBE0(α) eigenvalues and wavefunctions as the starting point. The RPA and BSE calculations were further solved on top of the GW approximation for the electron–hole interaction to investigate the optical properties of the defects, including absorption spectra and radiative lifetime.
Thermodynamic charge transition levels and defect formation energy
The defect formation energy (FE_{q}) was computed for the Ti_{VV} and Mo_{VV} defects following:
where E_{q} is the total energy of the defect system with charge q, E_{pst} is the total energy of the pristine system, μ_{i} and ΔN_{i} are the chemical potential and change in the number of atomic species i, and ε_{F} is the Fermi energy. A charged defect correction Δ_{q} was computed for charged cell calculations by employing the techniques developed in refs. ^{15,51}. The chemical potential references are computed as \({\mu }_{{\mathrm {Ti}}}={E}_{{\mathrm {Ti}}}^{{\mathrm {bulk}}}\) (total energy of bulk Ti), \({\mu }_{{\mathrm {Mo}}}={E}_{{\mathrm {Mo}}}^{{\mathrm {bulk}}}\) (total energy of bulk Mo), \({\mu }_{{\mathrm {BN}}}={E}_{{\mathrm {BN}}}^{{\mathrm {ML}}}\) (total energy of monolayer hBN). Meanwhile the corresponding charge transition levels of defects can be obtained from the value of ε_{F} where the stable charge state transitions from q to \(q^{\prime}\).
Zerofield splitting
The firstorder ZFS due to spin–spin interactions was computed for the dipole–dipole interactions of the electron spin:
Here, μ_{0} is the magnetic permeability of vacuum, g_{e} is the electron gyromagnetic ratio, \({\hbar}\) is the Planck’s constant, s_{1}, s_{2} is the spin of first and second electron, respectively, and r is the displacement vector between these two electron. The spatial and spin dependence can be separated by introducing the effective total spin S = ∑_{i}s_{i}. This yields a Hamiltonian of the form \({H}_{{\mathrm {ss}}}={{\bf{S}}}^{{\mathrm {T}}}\hat{{\bf{D}}}{\bf{S}}\), which introduces the traceless ZFS tensor \(\hat{{\bf{D}}}\). It is common to consider the axial and rhombic ZFS parameters D and E which can be acquired from the \(\hat{{\bf{D}}}\) tensor:
Following the formalism of Rayson et al. ^{31}, the ZFS tensor \(\hat{{\bf{D}}}\) can be computed with periodic boundary conditions as
Here the summation on pairs of i, j runs over all occupied spinup and spindown states, with χ_{ij} taking the value +1 for parallel spin and −1 for antiparallel spin, and Ψ_{ij}(r_{1}, r_{2}) is a twoparticle Slater determinant constructed from the Kohn–Sham wavefunctions of the ith and jth states. This procedure was implemented as a postprocessing code interfaced with Quantum ESPRESSO. To verify our implementation is accurate, we computed the ZFS of the NV center in diamond which has a wellestablished result. Using ONCV pseudopotentials, we obtained a ZFS of 3.0 GHz for NV center, in perfect agreement with previous reported results^{29}. For heavy elements such as transition metals, spin–orbit (SO) coupling can have substantial contribution to ZFS. Here, we also computed the SO contribution of the ZFS as implemented in the ORCA code^{56,57} (additional details can be found in Supplementary Note 10, Fig. 12, and Table 6).
Radiative recombination
In order to quantitatively study radiative processes, we computed the radiative rate Γ_{R} from Fermi’s Golden Rule and considered the excitonic effects by solving BSE^{58}:
Here, the radiative recombination rate is computed between the ground state G and the twoparticle excited state S(Q_{ex}), \({1}_{{q}_{L},\lambda }\) and 0 denote the presence and absence of a photon, H^{R} is the electron–photon coupling (electromagnetic) Hamiltonian, E(Q_{ex}) is the exciton energy, and c is the speed of light. The summation indices in Eq. (8) run over all possible wavevector (q_{L}) and polarization (λ) of the photon. Following the approach described in ref. ^{58}, the radiative rate (inverse of radiative lifetime τ_{R}) in SI unit at zero temperature can be computed for isolated defect–defect transitions as
where e is the charge of an electron, ϵ_{0} is vacuum permittivity, E_{0} is the exciton energy at Q_{ex} = 0, n_{D} is the reflective index of the host material and \({\mu }_{{\mathrm {eh}}}^{2}\) is the modulus square of exciton dipole moment with length^{2} unit. Note that Eq. (9) considers defect–defect transitions in the dilute limit; therefore the lifetime formula for zerodimensional systems embedded in a host material is used^{8,59} (also considering n_{D} is unity in isolated 2D systems at the longwavelength limit). We did not consider the radiative lifetime of Ti_{VV} defect at a finite temperature because the first and second excitation energy separation is much larger than kT. Therefore a thermal average of the first and higher excited states is not necessary and the first excited state radiative lifetime is nearly the same at 10 K as zero temperature.
Phononassisted nonradiative recombination
In this work, we compute the phononassisted nonradiative recombination rate via a Fermi’s golden rule approach:
Here, Γ_{NR} is the nonradiative recombination rate between electron state i in phonon state n and electron state f in phonon state m, p_{in} is the thermal probability distribution of the initial state \(\left{\mathrm {in}}\right\rangle\), H^{e−ph} is the electron–phonon coupling Hamiltonian, g is the degeneracy factor and E_{in} is the energy of vibronic state \(\left{\mathrm {in}}\right\rangle\). Within the static coupling and onedimensional (1D) effective phonon approximations, the nonradiative recombination can be reduced to:
Here, the static coupling approximation naturally separates the nonradiative recombination rate into phonon and electronic terms, X_{if} and W_{if}, respectively. The 1D phonon approximation introduces a generalized coordinate Q, with effective frequency ω_{i} and ω_{f}. The phonon overlap in Eq. (12) can be computed using the quantum harmonic oscillator wavefunctions with Q−Q_{a} from the configuration diagram (Fig. 5). Meanwhile the electronic overlap in Eq. (13) is computed by finite difference using the Kohn–Sham orbitals from DFT at the Γ point. The nonradiative lifetime τ_{NR} is given by taking the inverse of the rate Γ_{NR}. Supercell convergence of phononassisted nonradiative lifetime is shown in Supplementary Note 11 and Table 7. We validated the 1D effective phonon approximation by comparing the Huang–Rhys factor with the full phonon calculations in Supplementary Table 8.
SOC constant
SOC can entangle triplet and singlet states yielding the possibility for a spin–flip transition. The SOC operator is given to zeroorder by^{60}
where c is the speed of light, m_{e} is the mass of an electron, p and S are the momentum and spin of electron i and V is the nuclear potential energy. The spin–orbit interaction can be rewritten in terms of the angular momentum L and the SOC strength λ as^{60}
where λ_{⊥} and λ_{z} denote the nonaxial and axial SOC strength, respectively. The SOC strength was computed for the Ti_{VV} and Mo_{VV} defect in hBN using the ORCA code by TDDFT^{56,61}. More computational details can be found in Supplementary Note 10.
Data availability
The data that support the findings of this study and the code for the firstprinciples methods proposed in this study are available from the corresponding author (Yuan Ping) upon reasonable request.
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Acknowledgements
We acknowledge Susumu Takahashi for helpful discussions. This work is supported by the National Science Foundation under grant nos. DMR1760260, DMR1956015, and DMR1747426. Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DEAC5207NA27344. T.J.S. acknowledges the LLNL Graduate Research Scholar Program and funding support from LLNL LDRD 20SI004. This research used resources of the Scientific Data and Computing center, a component of the Computational Science Initiative, at Brookhaven National Laboratory under Contract No. DESC0012704, the lux supercomputer at UC Santa Cruz, funded by NSF MRI grant AST 1828315, the National Energy Research Scientific Computing Center (NERSC) a U.S. Department of Energy Office of Science User Facility operated under Contract No. DEAC0205CH11231, the Extreme Science and Engineering Discovery Environment (XSEDE) which is supported by National Science Foundation Grant No. ACI1548562^{62}.
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Y.P. established the theoretical models and supervised the project, T.J.S. and K.L. performed the calculations and data analysis, Y.P. and J.X. discussed the results, and all authors participated in the writing of this paper. T.J.S. and K.L. contributed equally to this work.
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Smart, T.J., Li, K., Xu, J. et al. Intersystem crossing and exciton–defect coupling of spin defects in hexagonal boron nitride. npj Comput Mater 7, 59 (2021). https://doi.org/10.1038/s41524021005255
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DOI: https://doi.org/10.1038/s41524021005255
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