Decoherence of V${}_{\mathrm{B}}^{-}$ spin defects in monoisotopic hexagonal boron nitride

Abstract

Spin defects in hexagonal boron nitride (hBN) are promising quantum systems for the design of flexible two-dimensional quantum sensing platforms. Here we rely on hBN crystals isotopically enriched with either ¹⁰B or ¹¹B to investigate the isotope-dependent properties of a spin defect featuring a broadband photoluminescence signal in the near infrared. By analyzing the hyperfine structure of the spin defect while changing the boron isotope, we first confirm that it corresponds to the negatively charged boron-vacancy center (${\mathrm{V}}_{\mathrm{B}}^{-}$). We then show that its spin coherence properties are slightly improved in ¹⁰B-enriched samples. This is supported by numerical simulations employing cluster correlation expansion methods, which reveal the importance of the hyperfine Fermi contact term for calculating the coherence time of point defects in hBN. Using cross-relaxation spectroscopy, we finally identify dark electron spin impurities as an additional source of decoherence. This work provides new insights into the properties of ${\mathrm{V}}_{\mathrm{B}}^{-}$ spin defects, which are valuable for the future development of hBN-based quantum sensing foils.

Recently, coherent control of spin defects in hBN has been realized, enabling future applications in quantum sensing technologies. Here the authors perform a systematic study of isotope-dependent spin coherence properties of the negatively-charged boron-vacancy defect in monoisotopic hBN crystals.

Introduction

Optically active point defects in semiconductors are currently attracting intense research attention owing to prospective applications in the growing field of quantum technologies¹. These defects, that feature localized electronic states with energy levels deeply buried inside the bandgap of the host material, behave as artificial atoms with efficient optical transitions. Importantly, the electronic spin of the defect can offer a fertile quantum resource for the development of a wide variety of applications. Besides its use as a quantum bit for the realization of advanced quantum information protocols^2–4, the electronic spin state can also be exploited for quantum sensing purposes, by taking advantage of its extreme sensitivity to external perturbations such as magnetic and electric fields, pressure or temperature⁵. In this context, the nitrogen-vacancy (NV) defect in diamond is undoubtedly the most advanced solid-state quantum sensing platform, which has already found numerous applications in condensed matter physics and life sciences^6–8. However, the search for alternative material platforms that could expand the range of quantum sensing functionalities offered by diamond remains very active worldwide⁹. Along this research line, a particular attention is currently being paid to layered van der Waals crystals because such materials promise the design of flexible, atomically thin quantum sensing foils^10–13.

Among the broad variety of layered van der Waals crystals, hexagonal boron nitride (hBN) is an ideal material for hosting optically active spin defects owing to its large bandgap (~6 eV)¹⁴. During the last years, many papers have reported the isolation of individual defects in hBN, showing very bright optical emission and perfect photostability, even at ambient conditions^15–19. However, the detection of the electron spin state associated with these point defects has long remained a missing ingredient towards the development of quantum sensors. This gap has been recently filled with the demonstration of optically detected magnetic resonance on two different types of defects, which have been tentatively assigned to negatively charged boron vacancy centers^20,21 and carbon-related impurities^22–24. Identifying the microscopic structure of optically active spin defects remains however a highly difficult task given the large number of potential candidates, which can involve extrinsic substitutional or interstitial impurities as well as complex assemblies of vacancies and antisite defects. Density functional theory is a powerful tool to reduce the number of possible atomic structures through ab initio calculations of defect properties such as formation energies, charge state transition levels, photoluminescence (PL) spectra or electron spin densities^25–30. On the experimental side, valuable informations can be obtained by analysing the hyperfine structure of the spin defect^20,23, which results from its interaction with neighboring nuclear spin species.

In hBN, each lattice site is occupied by an atom with a non-zero nuclear spin, in contrast to other materials such as diamond or silicon carbide. Nitrogen naturally occurs as the ¹⁴N isotope with a nuclear spin I = 1 (99.6% natural abundance), while boron has two stable isotopes, ¹¹B with a nuclear spin I = 3/2 (80% natural abundance) and ¹⁰B with a nuclear spin I = 3 (20% natural abundance). To date, all the studies of spin defects in hBN were performed on crystals with the natural content of boron isotopes^20–24, making the interpretation of hyperfine spectra a difficult task³⁰. In this work, we circumvent this problem by employing hBN crystals containing a single boron isotope. Using such monoisotopic crystals, we focus our study on a spin defect featuring a broadband PL signal centered at a wavelength of 800 nm. The analysis of its hyperfine structure while changing the boron isotope confirms that this defect corresponds to the negatively charged boron-vacancy center (${\mathrm{V}}_{\mathrm{B}}^{-}$). We then investigate the spin coherence properties of ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers hosted in monoisotopic hBN crystals. We observe a slight increase of the spin coherence time T₂ in ¹⁰B-enriched samples, which is supported by numerical simulations using cluster correlation expansion methods. Using cross-relaxation spectroscopy, we finally identify dark electron spin impurities as an additional source of decoherence for the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center in hBN.

Results

Sample description

Monoisotopic, millimeter-sized h¹⁰BN and h¹¹BN crystals were synthesized through the metal flux growth method described in ref. ³¹, while using isotopically enriched boron powders with either ¹¹B (99.4%) or ¹⁰B (99.2%). The resulting crystals were irradiated with thermal neutrons to produce point defects. Two different processes can lead to the creation of defects via neutron irradiation: either by damages produced by neutron scattering through the crystal or via neutron absorption leading to nuclear transmutation. The latter process strongly depends on the isotopic content of the hBN crystal. Indeed, the thermal neutron capture cross-sections of ¹¹B ( ~0.005 barn) and ¹⁴N ( ~1.8 barn) are orders of magnitude smaller than that of ¹⁰B ( ~3890 barn)³². As a result, neutron irradiation of the h¹⁰BN crystal likely creates boron vacancy-related defects through nuclear transmutation and introduces interstitial ⁷Li atoms resulting from the fission of the ¹⁰B isotope^32,33. Conversely, the h¹¹BN crystal is almost transparent to neutrons and point defects are solely created by neutron scattering. To partially compensate for the isotope-dependent efficiency of defect creation by neutron irradiation, the h¹¹BN crystal was irradiated with a dose of ~2.6 × 10¹⁷ n cm⁻² and the h¹⁰BN crystal with a smaller dose of ~2.6 × 10¹⁶ n cm⁻².

Optical characterization

The monoisotopic content of our neutron-irradiated hBN crystals was first verified through Raman scattering spectroscopy. Figure 1a shows the Raman spectra recorded from the h¹⁰BN and h¹¹BN crystals around the energy of the intralayer E_2g phonon mode that corresponds to the in-plane vibration of boron and nitrogen atoms³⁴. The energy of this phonon mode is at 1358 cm⁻¹ for h¹¹BN and is shifted to 1394 cm⁻¹ for h¹⁰BN, as expected from the isotopic mass difference^35,36. The full-width at half maximum of the Raman line is around 4.5 cm⁻¹, attesting for the high crystalline quality of the hBN samples. This value is however slightly larger than that usually measured on similar crystals not subjected to neutron irradiation (~3 cm⁻¹)³⁵.

Fig. 1. Optical properties of neutron-irradiated monoisotopic hBN crystals.

a Raman scattering spectra recorded at room temperature on the h¹¹BN (red dots) and h¹⁰BN crystals (blue dots) after neutron irradiation. The solid lines are data fitting with Lorentzian functions. b Typical lateral PL raster scan of the h¹⁰BN crystals under green laser illumination with a power of 1 mW. c Axial confocal scan showing a high PL signal localized at the crystal surfaces. d PL spectra recorded from the h¹¹BN (red) and h¹⁰BN crystals (blue). The data have been normalized and vertically shifted for the sake of clarity.

The PL properties were then analyzed with a scanning confocal microscope operating at ambient conditions. A laser excitation at 532 nm was focused onto the sample with a high numerical aperture microscope objective (NA = 0.95) mounted on a three-axis piezoelectric scanner. The PL signal was collected by the same objective, focused in a 50-μm-diameter pinhole and finally directed either to a spectrometer or to a silicon avalanche photodiode operating in the single-photon counting regime. A lateral PL raster scan of the h¹⁰BN crystal is shown in Fig. 1b. It reveals a uniform PL signal, which is characterized by a broadband emission spectrum centered around λ ≈ 800 nm [Fig. 1d]. It was recently proposed to assign this emission line to negatively charged boron-vacancy centers (${\mathrm{V}}_{\mathrm{B}}^{-}$)^20,21,28. A similar PL signal was observed on the h¹¹BN crystal [Fig. 1d], though with an intensity decreased by roughly one order of magnitude despite a larger neutron irradiation dose. This confirms that point defects are efficiently created through nuclear transmutation in the h¹⁰BN crystal. Interestingly, a confocal PL scan recorded along the depth of the crystal shows that optically active defects are tightly localized at the sample surfaces [Fig. 1c]. Here the width of the hBN layer producing a PL signal is indeed limited by the longitudinal spatial resolution of the confocal microscope ( ~2 μm). This may be due either to the efficient migration of vacancy-related defects towards the surfaces or, more likely, to the stabilization of their charge state in an optically active configuration through a variation of the Fermi level close to the sample surface³⁷. Although a precise understanding of this effect is beyond the scope of the present work, this is an important information for the future integration of exfoliated hBN layers doped with optically active spin defects into van der Waals heterostructures.

Electron spin resonance spectroscopy

The point defect associated with the emission line at λ ≈ 800 nm exhibits very similar magneto-optical properties to those of the NV defect in diamond^20,21. It features a spin triplet ground state S = 1, that can be polarized through optical pumping, coherently manipulated with a microwave excitation, and readout by recording its spin-dependent PL signal. These properties enable the detection of electron spin resonances (ESR) by optical means^20,21. Optically detected ESR spectra were recorded by monitoring the PL signal while sweeping the frequency of a microwave field applied through a copper microwire directly spanned on the sample surface. When the microwave frequency crosses a transition between the ground state electron spin sublevels, the magnetic resonance is evidenced as a drop of the PL signal. At zero external magnetic field, the ESR frequencies are given by ν_± = D ± E, where D and E denote the longitudinal and transverse zero-field splitting parameters, respectively. A typical ESR spectrum recorded at zero field on the h¹⁰BN crystal is displayed in Fig. 2a. From this experiment, we infer D = 3.47 GHz and E = 61 ± 1 MHz, in good agreement with previous works^20,21.

Fig. 2. Analysis of the hyperfine structure.

a Optically detected ESR spectra recorded on the h¹⁰BN crystal at zero field (top panel) and for a magnetic field B = 14 mT applied along the c axis, i.e., perpendicular to the sample surface. b, c Hyperfine structure of the ESR line measured in b h¹¹BN and c h¹⁰BN crystals. The solid lines are data fitting with a sum of seven Gaussian functions. d, e Simulated hyperfine spectra of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ defect in hBN obtained via the procedure described in the main text while using Δ_s = 30 MHz. The inset shows the atomic structure of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center in hBN.

Analysis of the hyperfine structure

To investigate the hyperfine structure of the spin defect, a static magnetic field was applied along the c axis in order to shift the electron spin sublevels via the Zeeman effect and thus isolate a single ESR line in the spectrum [Fig. 2a]. The hyperfine structure recorded on the h¹¹BN crystal is displayed in Fig. 2b. Seven hyperfine lines can be resolved, with a characteristic splitting $\mathcal{A}~44$ MHz. This structure is analogous to that observed in hBN crystals with a natural content of boron isotopes, that corresponds to 80% of ¹¹B^20,21. The same experiment was performed for spin defects hosted in the monoisotopic h¹⁰BN crystal, leading to a very similar hyperfine spectrum [Fig. 2c]. This observation unambiguously indicates that boron atoms are not involved in the seven-line structure of the hyperfine spectrum, and are therefore not localized at the first neighboring lattice sites of the spin defect. Considering the intrinsic nuclear spin species in hBN, the hyperfine structure can only be explained by the interaction of a central electronic spin with three equivalent ¹⁴N nucleus (I = 1). This situation can be obtained either for a boron-vacancy center or for a substitutional impurity with zero nuclear spin localized at a boron site. Substitutional carbon C_B could be a plausible candidate, since carbon is a well-identified contaminant in hBN with a dominant spinless isotope (¹²C, 99%). However C_B defects exhibit a very different hyperfine structure, as discussed in ref. ³⁰. We thus conclude that the broad emission line centered at a wavelength of 800 nm in hBN corresponds to the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center, as proposed in recent experimental works^20,21 and supported by ab initio theoretical calculations²⁸.

The hyperfine interaction of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center with the six equivalent boron atoms placed in the second neighboring lattice sites cannot be resolved and leads to a broadening of the ESR lines. At first glance, it might be surprising to observe similar linewidths in h¹¹BN and h¹⁰BN crystals, because the ¹⁰B isotope has a higher nuclear spin (I = 3) than ¹¹B (I = 3/2). This is due to the difference in nuclear gyromagnetic ratio between the two isotopes, γ_n = 4.6 MHz/T for ¹⁰B and γ_n = 13.7 MHz/T for ¹¹B. Since the hyperfine coupling strength scales linearly with γ_n, it is reduced by a factor of three for ¹⁰B, which compensates for the impact of a higher nuclear spin in the broadening of the ESR line. To illustrate further this effect, simple simulations of the ESR spectra were performed. An exact calculation using a direct diagonalisation of the spin Hamiltonian cannot be done with conventional computer ressources given the large number of nuclear spins involved. We therefore rely on a simplified model for which we only consider the longitudinal component ${\mathcal{A}}_{zz}$ of the hyperfine tensor. Here z denotes the quantization axis of ${\mathrm{V}}_{\mathrm{B}}^{-}$, which corresponds to the c-axis of the hBN crystal²⁰. The hyperfine coupling constants were taken from the ab initio calculations reported in ref. ²⁸. The simulation includes (i) the three equivalent ¹⁴N first-neighbors with ${\mathcal{A}}_{zz}=48$ MHz, (ii) the six second-neighbors boron atoms with ${\mathcal{A}}_{zz}=-4.6$ MHz (resp. −1.5 MHz) for ¹¹B (resp. ¹⁰B), and (iii) six equivalent ¹⁴N third-neighbors with ${\mathcal{A}}_{zz}=4.5$ MHz. After calculating the position of the hyperfine lines, a convolution was applied with a Gaussian profile to take into account the inhomogeneous broadening of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ electron spin transitions. The hyperfine spectra were simulated for various full-width at half maximum Δ_s of the Gaussian function. In Fig. 2d, e, we display the results obtained with Δ_s = 30 MHz for which a fair agreement with experimental data is obtained. These simulations illustrate how the weaker hyperfine interaction with ¹⁰B leads to a similar width of the hyperfine lines despite a higher nuclear spin.

Spin coherence properties

We now study the spin coherence properties of an ensemble of ${\mathrm{V}}_{\mathrm{B}}^{-}$ defects hosted in monoisotopic hBN crystals. To this end, Rabi oscillations were first recorded by using the experimental sequence shown in Fig. 3a. After applying a first laser pulse to polarize the ${\mathrm{V}}_{\mathrm{B}}^{-}$ electron spins by optical pumping, coherent spin manipulation is performed by applying a resonant microwave pulse of varying duration, and the resulting spin state is finally readout optically by recording the spin-dependent PL signal. A typical Rabi oscillation of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ electron spins is shown in Fig. 3a. This experiment provides a calibration of the duration of the microwave pulses (π and π/2) employed in pulsed ESR measurements. We note that the decay of the Rabi oscillation cannot be used to directly infer the electron spin dephasing time $T_2^{\star }$. Indeed, Rabi measurements mostly probe populations of the quantum system. As a result, the decay of the Rabi oscillation is usually much longer than $T_2^{\star }$^38,39 (see Supplementary Fig. 1). The longitudinal spin relaxation time T₁ was measured by applying the usual experimental sequence sketched in Fig. 3b. Data fitting with an exponential decay leads to T₁ ~16 μs for the two monoisotopic hBN crystals. This value, which is consistent with that recently reported in the literature, is limited by the interaction of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ electron spin with lattice phonons²¹. A spin echo sequence was then applied to infer the spin coherence time T₂ of the ensemble of ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers [Fig. 3c]. By fitting the measured echo signal with an exponential decay, we obtain T₂ = 46 ± 3 ns for the h¹¹BN crystal and T₂ = 62 ± 3 ns for the h¹⁰BN crystal.

Fig. 3. Spin coherence properties.

a Optically detected Rabi oscillations of an ensemble of ${\mathrm{V}}_{\mathrm{B}}^{-}$ spin defects recorded by applying the experimental sequence shown in the top panel. The spin-dependent PL signal S is integrated at the beginning of the readout laser pulse and normalized by a reference signal S_ref, which corresponds to the steady-state PL signal measured at the end of the initialization laser pulse. b T₁ measurements in the two monoisotopic crystals. Here we plot the difference of the normalized PL signal S/S_ref measured with and without applying the resonant microwave π-pulse (see top panel). Solid lines are data fitting with an exponential decay, leading to T₁ ~ 16 μs in both samples. c Measurement of the spin coherence time T₂ using a spin echo pulse sequence. Solid lines are data fitting with an exponential decay. All experiments are performed with a static magnetic field B ~ 15 mT applied along the c axis. The duration of the laser pulses is 8 μs with a power of 1 mW and the integration time window of S and S_ref is fixed to 300 ns. The duration of the microwave π/2 pulse is set to 15 ns.

Theoretical analysis

The coherence time of spin defects in solids is commonly limited by the magnetic noise produced by a bath of fluctuating paramagnetic impurities, which includes electron spin impurities in the host material, dangling bonds at the sample surface and nuclear spins. To better understand the origin of the short spin coherence time observed for ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers in our monoisotopic hBN crystals, we first investigate the impact of the nuclear spin bath by performing numerical simulations with the generalized cluster correlation expansion method (gCCE)^40,41. Within this theoretical framework, one can account for decoherence effects of different levels of complexity. When considering a bath dominated by nuclear spin species, the most common sources of dephasing are (i) nuclear spin flip-flops driven by the hyperfine interaction with the central electron spin, (ii) nuclear spin flip-flops induced by nuclear spin-nuclear spin dipolar interaction, and (iii) many-body correlation effects^40,41. We find that all these effects up to three-nuclear spin correlation contribute to the decoherence of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center in hBN. We therefore rely on the third order approximation (gCCE3) in our simulations (see Supplementary Note 1 and Supplementary Figs. 2 and 3). In addition, we use the hyperfine coupling constants calculated in ref. ²⁸, which include the Fermi contact interaction. While this term is often neglected owing to the highly localized nature of the electron spin density associated with point defects in semiconductors⁴², it plays an important role in the decoherence of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center in hBN, as discussed below.

The simulated spin echo decay curves of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center in h¹¹BN and h¹⁰BN are shown in Fig. 4a, b. Both curves are fitted with a stretched exponential function, $\exp \left[-{\left(t/T_2\right)}^n\right]$, leading to T₂ = 92 ns and T₂ = 115 ns for h¹¹BN and h¹⁰BN, respectively, with an exponent n ~1.35. The dependence of the coherence time with the isotopic content is depicted in Fig. 4c. It reveals a linear increase with the ¹⁰B abundance. This effect results from the reduced nuclear gyromagnetic ratio of ¹⁰B, that weakens the hyperfine interaction and the boron nuclear spin flip-flop rate, both of which has a positive impact on the coherence time of the central spin. The convergence of the simulation with respect to the size of the nuclear spin bath provides important additional informations on the decoherence processes. Indeed, our results indicate that the coherence time is mostly limited by the fluctuations of nuclear spins that are not farther than 0.8 nm from the central spin (see Supplementary Fig. 4). We thus conclude that decoherence of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ electron spin in a nuclear spin bath is governed by highly localized interactions. As a result, nuclear spin flip-flops driven by the hyperfine interaction, which includes both the Fermi contact and the dipolar terms, play a critical role in the decoherence process. If the contact interaction and the first order coherence contribution are neglected in the simulation, we obtain a spin coherence time up to 23 μs, a value in line with earlier theoretical works⁴². The two orders of magnitude difference in the calculated coherence times, however, clearly falsifies the validity of these approximations. In fact, by taking the Fermi contact term into account, the ${\mathcal{A}}_{zz}$ hyperfine term of the first and second neighbor shell atoms of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center increase by a factor of ~28 and 2.7, respectively. Since the phase shift induced by the nuclear spin flip-flops is determined by the ${\mathcal{A}}_{zz}$ hyperfine terms, neglecting the Fermi contact has a drastic impact on the spin coherence function of all CCE orders. Furthermore, the first order CCE coherence function (C1) can solely be neglected when the Zeeman splitting of the nuclear spins exceeds the strength of the hyperfine coupling. Since the latter is in the 10 MHz range for neighboring nuclear spins, the required magnetic field to suppress the first order coherence function falls far beyond reach, in the 10–100 Tesla region. These results strikingly illustrates that the Fermi contact term and the first order coherence function cannot be neglected in the calculation of the coherence properties of spin defects in hBN.

Fig. 4. Nuclear spin bath induced decoherence.

a, b Simulated spin echo decay curve and corresponding stretched exponential fit for the ${\mathrm{V}}_{\mathrm{B}}^{-}$ defect in a h¹¹BN and b h¹⁰BN. c Calculated spin coherence time T₂ as a function of the abundance of ¹⁰B isotope. The T₂ time is enhanced by the reduced hyperfine and nuclear spin-nuclear spin interaction strengths in ¹⁰B-enriched samples. The black dashed line is a linear fit.

Cross-relaxation spectroscopy

Even though our experimental results reproduce the slight increase of the spin coherence time in h¹⁰BN, the measured T₂ values are about 50% smaller than the theoretical predictions [Fig. 3c]. This discrepancy could be explained by an additional source of decoherence in the experiments, such as electron spin impurities in the environment. Several paramagnetic centers with electronic spin S = 1/2 have been identified in neutron-irradiated hBN crystals through electron paramagnetic resonance (EPR) measurements^43–45. Some of these defects were recently assigned to N_BV_N and C_NV_B centers⁴⁵. To probe the presence of such S = 1/2 paramagnetic impurities in the vicinity of ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers, we measured the longitudinal spin relaxation time T₁ as a function of a static magnetic field applied along the c axis. When the ESR transition of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center is brought in resonance with a target spin, cross-relaxation is induced by dipole-dipole interaction, which leads to a reduction of T₁ [Fig. 5a]^46–48. Considering a paramagnetic center with S = 1/2, the resonance condition is obtained for a magnetic field B_c = D/2γ_e, where γ_e = 28 GHz/T is the electron spin gyromagnetic ratio. For the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center, it corresponds to B_c ~62 mT. The experimental results are displayed in Fig. 5b. As anticipated, a drop of the longitudinal spin relaxation time is observed around 62 mT, which reveals the existence of dark electron spin impurities with S = 1/2 in the close vicinity of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ defects. The reduced value of the T₂ time in our experiments compared to theoretical predictions is attributed to fluctuations of these paramagnetic centers.

Fig. 5. Cross-relaxation spectroscopy.

a Energy levels of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ spin triplet ground state (top panel) and of a paramagnetic impurity with S = 1/2 (bottom panel) as a function of a static magnetic field B. The two spin systems are brought in resonance for B = B_c ~ 62 mT. b T₁ time of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ spin defect in the h¹⁰BN crystal as a function of a magnetic field applied along the c-axis. The field amplitude is inferred by recording the Zeeman shift of the ESR frequency of the ${\mathrm{V}}_{\mathrm{B}}^{-}$ defect. The solid line is data fitting with a Gaussian function. The error bars correspond to the uncertainty of fitting the T₁ decay curve with an exponential function.

We note that the few studies found in the literature on the spin coherence properties of ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers in hBN provide very different results. In ref. ⁴⁹, a coherence time around T₂ ~80 ns was measured in a neutron-irradiated hBN samples with a natural content of boron isotopes. This value is analogous to that obtained in the present work and is supported by our theoretical predictions. On the other hand, refs. ^21,50 have reported the observation of a few μs-long coherence time in similar hBN samples. In these studies, the contrast of the echo signal was however very weak and the experimental data at short time time scale were not shown. Such long T₂ values, which are not captured by theoretical simulations using CCE methods, could be alternatively attributed to nuclear spin coherence.

Discussion

To conclude, we have illustrated how hBN crystals isotopically enriched with either ¹⁰B or ¹¹B can be used to identify the structure of spin defects through the analysis of their hyperfine structure. Beyond the ${\mathrm{V}}_{\mathrm{B}}^{-}$ center studied in this work, these methods could be applied to other spin impurities in hBN, such as carbon-related defects whose microscopic structure remains unidentified^22–24,30. An in-depth study of the spin coherence properties of ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers indicates a slight increase of T₂ time in ¹⁰B-enriched crystals, which is supported by numerical simulations using cluster correlation expansion methods. These simulations highlight the importance of the spin density distribution, the corresponding Fermi-contact hyperfine fields, and first order correlation effect for determining the coherence time of solid-state spin qubits interacting with a dense nuclear spin bath. This finding, which generally applies to any defect hosted in a solid with a high nuclear spin density, constrains the validity of prediction of spin coherence times using hypothetical defects⁵¹. Using cross-relaxation spectroscopy, we finally identified dark electron spin impurities as an additional source of decoherence in hBN. Although such paramagnetic impurities have a detrimental effect of the spin coherence time, they could also be turned into a useful resource for quantum sensing applications⁵². This work provides key insights into the spin relaxation dynamics of ${\mathrm{V}}_{\mathrm{B}}^{-}$ centers in hBN, which is a promising spin defect for the future development of flexible, atomically thin quantum sensing foils.

Author contributions

V.J. conceived and coordinated the project. A.H., R.T., N.M., A.Du., F.F., A.Dr., T.M., B.G, G.C., and V.J. conducted and analyzed the experiments. J.L. and J.H.E. performed the crystal growth. V.I. and A.G. performed the numerical simulations. V.I. and V.J. wrote the manuscript. All the authors discussed the data and commented the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

Data availability

The datasets generated as part of the current study are available from the corresponding authors upon reasonable request.

Code availability

The codes used for the analysis included in the current study are available from the corresponding authors upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-022-31743-0.