Abstract
The chemical and structural nature of defects responsible for quantum emission in hexagonal boron nitride (hBN) remain unknown. Optically detected magnetic resonance (ODMR) measured from these defects was reported in two recent papers. In one case, the ODMR was tentatively attributed to the negatively charged boron vacancy, \(V_{\mathrm{B}}^ \). Here we show how the key optical and magnetic properties vary with location within the bulk and along edges of hBN sheets for this and the negatively charged nitrogen vacancy, \(V_{\mathrm{N}}^ \). Sign changes of the axial zerofield interaction parameter D are predicted, as well interchange of singlet and triplet ground states. Based on the latest experimental information, we assign the observed ODMR signal to bulk \(V_{\mathrm{B}}^ \). The other observed ODMR has some features reminiscent of our calculations for \(V_{\mathrm{N}}^ \) edge defects.
Introduction
The discovery^{1,2,3,4} in 2016 of singlephoton emission (SPE) from defects in hexagonal boron nitride (hBN) has inspired significant experimental and theoretical research in an effort to understand the chemical origin of this emission. Excitement stems from the possibility that such SPE could be harnessed by nanophotonic^{5,6,7,8} industries to deliver applications such as quantum key distribution. Their exploitation, however, requires understanding of the chemical nature of the colour centres in hBN and the ability to control and tune their structural arrangements and spectroscopic properties. Whilst various colour centres have been proposed and examined computationally as potential sources of SPEs in hBN^{9,10,11,12,13,14,15}, none have been firmly identified^{13}. In parallel to this work, defect centres in hBN have also been demonstrated to produce strong paramagnetic effects, and for these chemical assignments have indeed been developed^{13}.
Recent advances, reported in two preliminary publications^{16,17}, depict the observation of roomtemperature optically detected magnetic resonance (ODMR) from hBN, a phenomenon involving both measurement of groundstate magnetic properties and photoluminescence (PL) on the same defect. This not only makes it much easier to identify the chemical nature of a colour centre, but also suggests possible future applications in quantum information and other technologies.
In one preliminary ODMR report, by Gottscholl et al.^{16}, based on the observed magnetic properties, the defect was tentatively assigned to the \(V_{\mathrm{B}}^ \) defect, a negatively charged defect^{9,11}, in which one boron atom is missing from the hBN lattice. For this, a zerofield splitting (ZFS) parameter D of −3.5 GHz was reported for a triplet defect ground state, with a triplet ground state having previously been predicted for \(V_{\mathrm{B}}^ \)^{9,11}. Nevertheless, our initial calculations for a bulk \(V_{\mathrm{B}}^ \) defect, reported herein, predicted a value of 3.32 GHz, in stark contrast to the reported value of −3.5 GHz. This inspired the remainder of the work presented herein, where we consider how the properties of defects may vary as the defect location is varied from bulk to some layer edge. This yielded another of our principle results, that the sign of D can change according to defect location. A relevant feature of the experiments is that the defects observed were located throughout the sample but were not native to the hBN, instead produced following exposure to beams of neutrons, Li, or Ga atoms. Hence the bulk regions of the hBN may sustain considerable damage, making the detailed atomic structure difficult to discern. This makes bulk versus edge differences all the more relevant; in native hBN, defects located at both bulk^{18,19} and edge^{20,21} locations have been observed, as well as at structural defects in the basal plane of the hBN^{22}. Resolution of key issues appears imminent, however, as in the published version of the Gottscholl et al.^{23}, the sign of D is changed to be positive. Another computational study confirms the large positive computed value of the zerofield splitting for \(V_{\mathrm{B}}^ \) defects^{16}.
In the other preliminary ODMR report by Chejanovsky et al.^{17} the ZFS parameter D was too small to measure (magnitude < 4 MHz) and, the hyperfine coupling constant (hfc) was determined to be around 10 MHz with an angular dependence consistent with an unpaired electron in a π orbital. In the report, the observed ODMR was tentatively associated with a substitutional carbon impurity or other low mass atom.
The most commonly applied computational approach to predicting the optical and magnetic properties of defects is densityfunctional theory (DFT)^{13}. The prediction of optical spectroscopic properties, such as the zero phonon line (ZPL) energy, by DFT is very difficult as most defect states are openshell in nature, displaying multireference characteristics, whereas DFT is intrinsically a singlereference approach^{24}. Timedependent DFT (TDDFT) can be reliable for defect excited states, provided that it is performed based on a wellrepresented reference state, as indeed is the case for the triplet manifold of \(V_{\mathrm{B}}^ \)^{13}. Nevertheless, DFT is typically quite accurate when it comes to the prediction of groundstate magnetic properties^{13}, for example using the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06)^{25,26}. Indeed, DFT predictions of the hfc and ZFS properties of the nitrogen vacancy (NV) centre in diamond^{27,28} were important for the identification of its chemical structure. Further, previously observed paramagnetic signals from hBN defects have also been assigned by DFT as the neutral nitrogen (V_{N}) and boron (V_{B}) vacancies^{29,30}. We therefore expect that appropriate combinations of DFT and TDDFT calculations will be reliable for the identification of defects associated with the observed ODMR.
Calculations pertaining to the Gottscholl et al.^{16} ODMR observations have become available recently in the literature^{31}. This also uses DFT to predict magnetic properties of bulk \(V_{\mathrm{B}}^ \) defects, a critical novel conclusion being that straininduced distortions away from full threefold symmetry can interpret the observed offaxis ZFS parameter of E = 50 MHz. They evaluated excitedstate energies using an ab initio multiconfigurational approach, empirically modified to use DFT orbitals and solved using densitymatrix renormalisationgroup (DMRG) technologies.
The current works explores other possible explanations for the initially reported ZFS value reported by Gottscholl et al.^{16}, as well as to develop understanding of bulk versus edge locality. We have also considered properties of the \(V_{\mathrm{N}}^ \) defect which like \(V_{\mathrm{B}}^ \), has native 3fold local pointgroup symmetry, D_{3h}. Our results suggest that the ODMR could instead be associated with edgelocated \(V_{\mathrm{N}}^ \) defects. As alternative defects, we also consider a defect involving an internal boron dangling bond^{14} and the \(C_{\mathrm{N}}^ +\) and \(C_{\mathrm{N}}^ \) defects in which either a positively or negatively charged carbon replaces a nitrogen, respectively. We employ the CAMB3LYP density functional^{32,33,34} for this purpose, this calibrated, rangecorrected hybrid density functional being an example of an entrylevel method for the prediction of the spectroscopy of defects in molecular solids^{24}. State energies reported by Ivády et al.^{31} are of the same order of magnitude as the present results, but some differ in basic nature to key ones identified herein. Firstprinciplesbased approaches analogous to theirs are available that may prove useful for modelling defect spectroscopy^{35,36,37,38,39}, but are not applied herein.
Results
Zerofield splitting parameters
To find possible interpretations of reported ODMR, we investigated many possible edge locations for \(V_{\mathrm{B}}^ \), presenting also analogous results obtained for \(V_{\mathrm{N}}^ \) and, in addition, some results for also a proposed^{14} boron danglingbond site as well as for \(C_{\mathrm{N}}^ +\) and \(C_{\mathrm{N}}^ \). Initial investigations were performed examining calculated spin densities only, as these are straightforward to calculate reliably. Periodic sheets in either 1 or 2 dimensions are used in these investigations, the 1dimensional structures presenting infinite straight hBN edges. Results are given in detail in Supplementary Figs. 1–8, with Table 1 presenting a summary of some critical features of the calculated triplet ground states: defect natures, innerring geometry and location, and calculated value for the ZFS splitting parameter D.
A primary result is the demonstration that the ZFS parameter D can be very sensitive to the location of defects within the hBN layer. For \(V_{\mathrm{B}}^ \), calculated values change from 3.3 GHz (bulk) to 3.2 GHz (Hpassivated nitrogen edge) to −2.5 GHz (Hpassivated armchair edge) to −3.5 GHz (Hpassivated boron edge). Note that the related hyperfine coupling constants may not show analogous behaviour, with calculated values at the centre of hBN sheet being A = [49.1, 54.5, 102.5] MHz, while for the defect located at the edge they are A = [47.2, 49.39, 89.5] MHz, both in good agreement with measured xyplane values^{16,23} of 47 MHz. Hence all observable properties need to be individually considered. Concerning D, for \(V_{\mathrm{N}}^ \), the calculated values are also widely varying, ranging from −0.97 to 1.3 GHz. These results stress the need for detailed understanding of geometrical defect structure as a basis for the interpretation of ODMR, especially pertinent for experiments in which the defects of interest are produced following irradiation with particle beams^{16,23}.
Concerning the recently observed ODMR, we select for further investigation from amongst the systems listed in Table 1: \(V_{\mathrm{B}}^ \) in bulk, for which the calculated D agrees with the current experimental intepretations^{23}, \(V_{\mathrm{B}}^ \) at the postulated Hpassivated nitrogen edge, for which the calculated D value agrees with the original experimental interpretation^{16}, and the analogous structures for \(V_{\mathrm{N}}^ \), both for reasons of exploiting the analogy to understand basic principles as well as for the possible interpretation of the ODMR experiments that reported D < 0.004 GHz as the calculated bulk value of −0.046 GHz may be in agreement with this to within the accuracy of the calculations^{40}. Supporting this, the calculated values of the hfc tensor of A = [10.0, 10.0, 29.5] MHz for bulk and A = [11.11, 11.08, 30.01] MHz for Hpassivated boron edge indicate good agreement with the observed value of 10 MHz.
Origin of the edge effects for \({\boldsymbol{V}}_{\mathrm{B}}^ \) and \({\boldsymbol{V}}_{\mathrm{N}}^ \)
Figure 1a plots the calculated ZFS parameter D as a function of defect location for \(V_{\mathrm{B}}^ \). The onedimensional strip geometric model used for this work is shown in Fig. 1b, with the calculated spin distributions of the triplet ground states superimposed. The corresponding data and geometric model for \(V_{\mathrm{N}}^ \) are shown in Fig. 2a, b. In these figures, the defect is located adjacent to the edge, but the model supports moving the defect in towards the centre of the strip, emulating bulk defects. In both cases, the ZFS quickly approaches the bulk value as the defect moves away from the edge, but for \(V_{\mathrm{N}}^ \) the change is small and regular, whereas for \(V_{\mathrm{B}}^ \) an abrupt change reverses the sign of D upon moving just one site location away from the edge.
To understand the very different calculated qualitative properties, we note that lattice strains are already known to influence the spectral properties of some SPEs^{41}. We compute D as a function of hydrostatic pressure, for both defects \(V_{\mathrm{B}}^ \) and \(V_{\mathrm{N}}^ \) in bulk, as shown in Fig. 3. This hydrostatic pressure produces isotropic compressive and tensile strains around the defect centres, thus preserving the D_{3h} point group symmetry, with D found to vary linearly with strain. In our calculations, the strain dependent ZFS yields a slope of 7.9 MHz/GPa (for positive strain) for \(V_{\mathrm{B}}^ \). This number, in principle, can be compared to the experimentally observed variation in ZFS with temperature, which has been attributed to thermal expansion of the lattice, yielding a slope of −0.4 MHz/K, equivalent to a pressure coefficient of 2.9 MHz/GPa^{16}. These results do not entirely agree, unlike good agreement found between analogous calculations and observed properties for the NV centre in diamond^{28}. If the defect were located at a step edge, then thermal expansion would lead to slippage in the direction orthogonal to the edge and hence the sensitivity of the ZFS would be reduced; alternatively, the location of the defect within the bulk sufficiently close to other defects caused perhaps by exposure to atom beams could allow for strain mitigation without affecting other defect properties.
One may distinguish two contributions to the variation of ZFS as a function of pressure: purely geometrical changes around the defect centre and the variation of the defect’s spin density. The former may be described using the compressedorbital model^{28}, according to which the ZFS becomes scaled by a geometrical factor (d/d_{0}) determined by the atomic relaxation induced by pressure, in the proximity of the defect; here d and d_{0} are the nearestneighbour distances under pressure and at equilibrium, respectively. As shown in Fig. 3, the compressedorbital model describes well the calculated pressure dependence of ZFS for both \(V_{\mathrm{B}}^ \) and \(V_{\mathrm{N}}^ \), showing a negligible contribution arising from spin density changes. This is consistent with the results for NV centre in diamond, where the change in ZFS is attributed predominantly to geometrical changes^{28,42}.
We now discuss the microscopic origin of the calculated change in D from the bulk values of 3.32 GHz in \(V_{\mathrm{B}}^ \) and −46 MHz in \(V_{\mathrm{N}}^ \) to the edgevalues of −3.5 GHz and −30 MHz, respectively (Table 1). As the edge is approached, the nearestneighbour bond lengths inside the defect change (see Supplementary Fig. 1) from three equivalent 2.59 Å separations to one at 2.59 Å and two at 2.81 Å for \(V_{\mathrm{B}}^ \); for \(V_{\mathrm{N}}^ \), these distances are 2.04, and 2.30 Å or 2.04 Å, respectively, embodying the lowering of local pointgroup symmetry from D_{3h} to C_{2v} as the edge is approached. To demonstrate that geometrical changes to the innerring of defect atoms are responsible for the changes in ZFS parameters for \(V_{\mathrm{B}}^ \), we consider (Table 1 and Supplementary Fig. 3) an artificially constructed structure in which the innerring atoms are frozen at their geometry as calculated at the boron edge, reimbedding them inside the 2D bulk material. No significant change is calculated as a result of this reimbedding, indicating that geometry alone dominates its large edge effect. For \(V_{\mathrm{N}}^ \) for which the edge effect is small, analogous calculations indicate that the nature of the reimbedding also becomes relevant (Table 1 and Supplementary Fig. 4).
The observed^{16,23} nonzero value of E for \(V_{\mathrm{B}}^ \) is nonzero may also be attributed to the edge effect. Our calculated value of E for the defect at the edge is 80 MHz, which on an absolute scale is in a reasonable agreement with the experimental value of 50 MHz^{16,24}. Alternatively, residual strain influencing bulk defects could also account for the observed nonzero value of E^{31}.
Photoluminescence and electronic states of \({\boldsymbol{V}}_{\mathrm{B}}^ \)
For \(V_{\mathrm{B}}^ \), we focus on electronic states within the triplet manifold as previous calculations have suggested^{9,11} that it has a triplet ground state, a property that is most likely required to explain the observed magnetic ODMR properties. All energetics calculations are performed using CAMB3LYP on 3ring model compounds described in detail in Supplementary Table 1 with TDDFT used to depict excited states. The triplet ground state in the bulk is predicted to be \((1)^3A_2^\prime\), in accordance with recent calculations^{31} and expectations, with the defect displaying D_{3h} local point group symmetry^{9,10,11,12}. Such a ground state is consistent with the basic observations of Gottscholl et al.^{16,23}. Note that we use standard grouptheoretical axis conventions throughout^{43}. When the defect is located at an Hterminated boron edge, the pointgroup symmetry is lowered to C_{2v}, with the ground state then described as (1)^{3}B_{2}.
Table 2 lists critical calculated properties of the triplet manifold for defects located in the bulk and at the considered edge. These include the adiabatic transition energies ΔE_{0}, likely to be a good approximation for observed ZPL energies ΔE_{00}; vertical excitation energies \({\mathrm{{\Delta} }}E_{\mathrm{v}}^{\mathrm{A}}\), and the absorption and emission reorganisation energies λ^{A} and λ^{E}, respectively, that depict spectral bandwidths^{13}.
State symmetry for bulk defects is expressed in terms of both D_{3h} to C_{2v} symmetry^{43}, with spontaneous symmetry lowering mandated by Jahn–Teller distortion for doubly degenerate states; all edge defects intrinsically have C_{2v} symmetry; \(V_{\mathrm{N}}^ \) in the centre (bulk) is also unstable owing to warping of the plane to C_{2} symmetry. The CAMB3LYP TDDFT calculations are expected to be very reliable for the situation presented by \(V_{\mathrm{B}}^ \) but typically underestimate isolatedlayer transition energies by 0.3 ± 0.2 eV in situations relevant to \(V_{\mathrm{N}}^ \) at the edge, increasing to 0.4 ± 0.3 eV for \(V_{\mathrm{N}}^ \) at the centre^{13}. No corrections are made for the embedding of the defect layer inside bulk hBN, but cluster calculations show small change as the ring size is increased from 1 to 2 and ca. 0.03 eV in expansion from 2 to 3. The (1)^{3}B_{1}, (1)^{3}A_{2} states of the \(V_{\mathrm{B}}^ \) defect located in the centre and edge, respectively, presumably form a transitionstate on the (possibly distorted) Jahn–Teller potentialenergy surface. In addition the latter state along with the (1)^{3}A_{2} state at the centre and (2)^{3}A_{2} states are centre and edge are Franck–Condon allowed transitions with the ground state. Similarly for the \(V_{\mathrm{N}}^ \) defect, the (1)^{1}A_{2}, (1)^{3}B_{1}, (1)^{3}B_{2} states for centre location are possible transition states, while the (1)^{1}B_{1}, (3)^{1}A_{1}, (1)^{1}B_{2} states for the centre location and (2)^{3}B_{1}, (3)^{3}B_{1} states for the edge location are Franck–Condon allowed. The adsorption vertical oscillator strength for the (1)^{1}A_{2} state of \(V_{\mathrm{N}}^ \) located at the boron zigzag edge is 0.028. When \(\lambda ^{\mathrm{A}} + \lambda ^{\mathrm{E}}\) > \(E_{\mathrm{v}}^{\mathrm{A}}\), an energetically accessible conical intersection with the ground state is present, making an ultrafast relaxation pathway for initial excitation back to the ground state as well as for relaxation to a potentially stable defect isomer. This is the case for the (1)^{3}A_{2} and (1)^{3}B_{1} states of the \(V_{\mathrm{N}}^ \) defect located at the centre. The lowest vertical excitations from these two triplet states are of B_{1} type, at 2.19 eV for (1)^{3}A_{2} and 1.91 eV for (1)^{3}B_{1}.
Details of the dominant orbital excitations contributing to the example system of bulk \(V_{\mathrm{B}}^ \) at vertical excitation are presented in SI Section S3; this includes also a description of spincontamination effects, and embodies small contributions from hBN valenceband excitations to the excited states, listing all five excitedstate components of up to 3.5 eV in energy.
In the centre of an hBN layer, its first triplet excited state is predicted to be (1)^{3}E″, a state that undergoes a Jahn–Teller distortion to produce the (1)^{3}A_{2} state in C_{2v} symmetry at an adiabatic transition energy of 1.71 eV. This quantity is a good approximation for the ZPL energy and is in excellent agreement with observed^{16} photoluminescence properties. The transition is formally forbidden in D_{3h} symmetry, which may explain the very weak observed emission as the implied long radiative lifetimes could permit other processes to operate. The transition becomes allowed outofplane polarised once the Jahn–Teller distortion occurs. Also, (1)^{3}E″ appears to be very close in energy to \((1)^3A_1^{\prime\prime}\), a state that inherently supports allowed emission to the ground state (calculated transition moment 0.18 D). Of note too, \((1)^3A_1^{\prime\prime}\) is predicted to undergo an avoided crossing with (1)^{3}A_{2} as the symmetry is lowered to C_{2v}.
A critical feature is that the calculated reorganisation energy in emission λ^{E} of 0.29 eV is at least double the largest values that could be consistent with the observed spectral bandwidth^{16,23} if simple approaches that ignore the Jahn–Teller effect are used to model the spectra. The observed spectra are noted as arriving from many emitters rather than singlephoton sources, and it is possible that the observed bandwidths reflect inhomogeneity rather than intrinsic emitter spectral width. Improved experimental data is required so that the extent of the discrepancy between observation and calculation can be assessed. Possibilities that need to be investigated in future work include that the observed emission is significantly narrowed from naïve expectations owing to either the Jahn–Teller effect or the impact of acoustic phonons, and that the emission could arise from \((1)^3A_1^{\prime\prime}\), or else from within the singlet manifold^{31} following intersystem crossing.
Table 2 also shows changes to the calculated state properties for \(V_{\mathrm{B}}^ \) located at a Hterminated boron edge. Mostly, the changes are small, yet the impact could be profound as, e.g., the (1)^{3}A_{2} and (1)^{3}B_{1} structures reverse in relative energy, dramatically changing the nature of the (now pseudo) Jahn–Teller distortion. Most significantly, as expected, the calculated emission reorganisation energies λ^{E} increase significantly, becoming 0.53 eV for the lowestenergy triplet photoemission, depicting spectra very much broader than observed. Hence it is unlikely that the observed ODMR involves the triplet manifold of \(V_{\mathrm{B}}^ \) located at this edge.
Photoluminescence and the electronic states of \({\boldsymbol{V}}_{\mathrm{N}}^ \)
The \(V_{\mathrm{N}}^ \) defect in the bulk has been predicted to have a closedshell singlet \((1)^1A_1^\prime\) ground state^{10,11,12,30}, and our calculations support this prediction. ODMR can be observed for systems with singlet ground states that allow for intersystem crossings (ISC) from initially excited singlet states to triplet states^{44}, but this requires also significant buildup of population on the triplet state and hence unusual circumstances. The calculated photoluminescence properties of the singlet ground state are listed in Table 2 and indicate that it is highly unlikely that this could lead to any of the observed ODMR signals: these signals occur following excitation by light with under 2 eV energy per photon, whereas the lowestenergy singlet excited state is predicted to be located vertically at 3 eV. Further, the lowest excited state is predicted to undergo a huge Jahn–Teller distortion with a reorganisation energy of 1.19 eV, two orders of magnitude larger than observed spectral widths. It also manifests outofplane buckling and a conical intersection with the ground state that would facilitate extremely fast nonradiative energy loss.
Of note, however, is that our calculations predict that the lowestenergy triplet states are just 0.3–0.5 eV higher in energy than \((1)^1A_1^\prime\), numbers of the order of likely errors^{13} in the computational approach and hence the ground state may indeed be triplet in character. The lowestenergy transitions within the triplet manifold are predicted to be outofplane polarised at 2.19 eV for (1)^{3}A_{2} (Jahn–Teller transition states) and 1.91 eV for (1)^{3}B_{1} (Jahn–Teller local minima), consistent with observed PL properties.
Table 2 also shows that moving the defect to an edge has a profound effect on the properties of the ground and excited states of \(V_{\mathrm{N}}^ \), making a firm prediction of a (1)^{3}B_{1} ground state. For the edgelocated defect used in the calculations (see Table 2), the excitation energies within the triplet manifold now appear too large to account for the ODMR results, as also do the reorganisation energies. Nevertheless, the calculations raise the possibility that some other edgerelated \(V_{\mathrm{N}}^ \) defect could have the observed PL properties as they are established as being strongly site dependent. They show that relocating defects from the bulk to edges can have profound effects on electronic properties, analogous to the profound effects on magnetic properties predicted for \(V_{\mathrm{B}}^ \).
Discussion
Our calculations show that the optical and magnetic properties of defects may change considerably as a function of the location of the defect in regard to step edges in hBN. For \(V_{\mathrm{B}}^ \), the calculated ZFS parameter D changed sign on going from edge to centre, with only small changes to the PL, whereas for \(V_{\mathrm{N}}^ \), D only changed slightly for the triplet state but the nature of the ground state changed from triplet to singlet. Defect location and defect properties are strongly correlated.
The calculations presented here for ground state spin properties and the PL spectra provide strong evidence that the recently observed ODMR/EPR spectra for hBN by Gottscholl et al.^{16,23} arises from the \(V_{\mathrm{B}}^ \) defect. Our calculated values for the magnitude of the ZFS parameter D and hfc tensor A for bulklocated defects are in excellent agreement with those finally reported by Gottscholl et al.^{23}. Nevertheless, ignoring possible spectral changes owing to the Jahn–Teller effect, the calculated reorganisation energies for emission within the triplet manifold appear to be too large to explain the narrow observed photoemission linewidths obtained for an ensemble of emitters, and further experiments and calculations are required to understand native spectral line widths, the significance of the Jahn–Teller effect, the role of acoustic phonons, and alternate emission possibilities.
Also, the calculations suggest that the ODMR observations of Chejanovsky et al.^{17} could possibly be assigned to \(V_{\mathrm{N}}^ \) defects located near step edges. At the edge, the calculations strongly suggest a triplet ground state, with a singlet ground state slightly preferred in the centre with PL properties similar to those observed. The ZFS parameter D varies between −46 and −30 MHz for location in the bulk and at the edge respectively and hfc tensor of A = [10.0, 10.0, 29.5] MHz. These results are in a reasonable agreement with the experimental observations, whereas one considered simple defect involving carbon, as was originally suggested as a possible centre, was found to be inconsistent. The relationship of these results to more widely obtained results for other SPE’s^{13} needs to be established.
Methods
The computational parameters and theoretical details are presented in detail in Supplementary Methods^{25,26,45,46,47,48}. Briefly, periodic DFT calculations are performed by the Vienna Ab Initio Simulation Package (VASP)^{45,46} using the nonlocal Heyd–Scuseria–Ernzerhof hybrid functional (HSE06)^{25,26} in replicated images representing hBN monolayers, strips, and multilayer slabs. For the strips, we assume that all hBN edges are hydrogen terminated, with no access of hydrogens to partially fill defect vacancy sites, even if the defect is located adjacent to an edge. Magnetic properties are evaluated using standard methods^{28,30,49,50}. Excitedstate properties are evaluated on a 3ring model compound (see Supplementary Data) representing the defects, evaluating transition energies, reorganisation energies, and oscillator strengths by TDDFT using Gaussian16^{51}. The CAMB3LYP density functional^{32,33,34} is used with the 631G* basis set^{52}, this being a calibrated entrylevel method for the evaluation of defect spectroscopic properties^{24}.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by resources provided by the National Computational Infrastructure (NCI), Pawsey Supercomputing Centre, Australia and funding from the Australian Research Council (DP150103317 and DP160101301). K.S.T. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 773122, LIMA).
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A.S. conceived and proposed the main idea of this work. A.S. and J.R.R. contributed to all aspects of this work, with A.S. contributing calculations of the spinstate properties and J.R.R. the electronic state calculations. M.J.F. contributed to editing of the manuscript and analysis of results particularly with regard to understanding the previous literature and the relevance of the current work. K.S.T. contributed to discussions regarding the design and interpretation of the nano ribbons. All the Authors contributed to the analysis of results and writing of manuscript.
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Sajid, A., Thygesen, K.S., Reimers, J.R. et al. Edge effects on optically detected magnetic resonance of vacancy defects in hexagonal boron nitride. Commun Phys 3, 153 (2020). https://doi.org/10.1038/s4200502000416z
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