Introduction

Optically addressable spin defects in wide band-gap materials are promising solid-state qubit platforms that enable cutting-edge quantum applications such as quantum computation1, quantum sensing2, and quantum network3,4. Spin defects in diamond5, which show long spin coherence times and many other attractive features such as high-temperature quantum functionality6 and spin-to-photon interfaces7, is one of the leading qubit systems for quantum applications. Progresses made in the research on diamond have inspired several pioneering studies, wherein quantum spin defects have been developed in silicon carbide and nitrides, broadening the palette of spin qubits in materials8,9,10,11. Notably, the spin qubits in non-diamond hosts provide unique opportunities for building advanced platforms for quantum systems by taking advantage of the well-established material technologies developed for their hosts12,13. Recently, the search for qubit systems in two-dimensional (2D) van der Waals (vdW) materials has gained significant attention owing to their potential superiority in light extraction, heterostructuring, defect positioning, strain engineering, and nano-photonic integration14,15,16,17.

Among the 2D vdW materials, quantum spin defects in hexagonal boron nitride (h-BN) are gaining prominence for the development of optically active spin qubits. Owing to its wide bandgap of ~6 eV, h-BN hosts a variety of color centers from near infrared to ultraviolet15,16,18. Some of the color centers have been found to emit bright single photons even at room temperature, and these findings have sparked worldwide research efforts in this direction19,20,21,22,23,24,25. Subsequently, the search for optically addressable spin qubits in h-BN has become the focus of materials research, and several breakthroughs have been made recently. Notably, negatively charged boron vacancies (VB) were discovered as optically addressable spin qubits in h-BN26,27,28. Since the first report by Gottscholl et al.26, several important achievements have been made29, including the realization of coherent Rabi oscillations30, deterministic defect generations31,32,33, nano-scale quantum sensing34, and coupling to nano-photonic structures35. In addition, carbon-related defects36,37,38 and boron vacancy complexes have also been recently identified as optically accessible spin qubits in h-BN. Overall, significant advances have been made in the development of defect-based spin qubits in h-BN and their use in quantum applications. Nonetheless, further research is required to realize robust spin qubits in h-BN.

One of the most compelling issues for h-BN spin qubits is their short spin coherence time (T2) due to the dense nuclear spin bath in the h-BN lattice26,39. For VB, the Hahn-echo T2 time has been measured to be several microseconds30. Notably, several well-known schemes, such as isotopic purification40, dynamical decoupling (DD)41, and clock transitions (CT)42,43, are available to extend the spin coherence times in materials. However, in h-BN, a conventional isotopic purification is impossible because all the naturally occurring boron (10B and 11B) and nitrogen (14N and 15N) isotopes exhibit non-zero nuclear spins. In addition, the coherence protection schemes such as DD and CT are often limited by the intrinsic Hahn-echo T2 time. When the T2 time is extremely short like in h-BN, pulse requirements in the DD and CT schemes become considerably challenging11. Considering the fundamental role of the coherence time in determining the retention time of the quantum information, the short T2 time of the h-BN spin qubits is one of the most pressing problems that needs to be solved for advancing these platforms in h-BN.

In this study, by taking the VB spin as a representative spin qubit system in h-BN, we developed two unconventional methods to enhance the T2 time of spin qubits significantly in h-BN via isotopic enrichment and strain engineering. We combined density functional theory (DFT) and cluster correlation expansion (CCE) to compute the theoretical T2 time of VB in h-BN single layer and multiple layers to be 45.9 μs and 26.6 μs, respectively, in the presence of an intrinsic nuclear spin bath in natural h-BN. Subsequently, we showed that the T2 times were increased to 143.4 and 81.1 μs in the single- and multi-layer h-10BN, respectively, which is enriched with the 10B isotope. This result is somewhat counterintuitive as the 10B isotope has a larger nuclear spin (I = 3) than that of the 11B isotope (I = 3/2). We found that this T2 enhancement by a factor of three resulted from the smaller gyromagnetic ratio of 10B than that of 11B, despite the much larger nuclear spin of 10B. In addition, we demonstrated that the T2 time could be further increased to 207.2 μs and 161.9 μs for the single- and multi-layer h-10BN, respectively, by inducing inhomogeneous strain around the VB. We considered a Gaussian-shaped bubble around VB as a representative inhomogeneous strain and showed that the T2 increased in proportion to the Gaussian width and height, demonstrating the impact of the local inhomogeneous strain on the T2 time. We found that a spatial inhomogeneity in the nuclear spin quadrupole interaction effect, induced by the inhomogeneous strain, plays a crucial role in suppressing the nuclear spin flip-flop dynamics, thus enhancing the VB spin coherence. Our results pave the way for effective protection of the spin coherence in h-BN, and this method should be applicable to VB as well as to any potential spin qubits in h-BN44,45. Improved spin coherence of VB would also open possibilities of further coherence extension using DD or CT, thereby advancing spin qubits in h-BN as promising platforms for quantum information science and technology.

Results

Spin decoherence of VB− in natural h-BN

We considered a central spin model to compute the decoherence dynamics of a VB spin interacting with the nuclear spin bath in the h-BN lattice. Figure 1a shows a schematic of the spin model, in which a VB defect is created in the middle of a large h-BN supercell. The ground state of VB is a spin-triplet state, and the spin density is highly localized at the vacancy site (see inset of Fig. 1a). In our model, we assumed a localized S = 1 at the vacancy site and treated its +1 and 0 spin sublevels as qubit states. The nuclear spin bath, which strongly interacts with the VB spin via hyperfine interaction, is derived from the spin-bearing boron and nitrogen nuclei in h-BN. Notably, all the naturally occurring boron and nitrogen isotopes exhibit non-zero nuclear spins: 19.9% of 10B with I = 3, 80.1% of 11B with I = 3/2, 99.6% of 14N with I = 1, and 0.4% of 15N with I = ½. In our model, we randomly distributed the nuclear spins in the h-BN lattice according to their natural abundance. To compute the homogeneous dephasing time (T2) of the VB spin, we considered the Hahn-echo pulse sequence46 and used the CCE method to expand the many-body nuclear-spin correlation effects systematically on the VB spin decoherence47. Notably, the CCE method combined with DFT calculations enables the quantum many-body computation of T2 without the assumption of any adjustable theoretical parameter. Owing to its predictive power, the CCE method was successfully applied to a wide range of solid-state qubit systems, yielding excellent agreement with the experimental results11,48,49,50,51,52 (further details on the theoretical methods and the system’s spin Hamiltonian are provided in the Method and Supplementary Information sections; see Supplementary Note 1 and 2, and Supplementary Fig. 1).

Figure 1b presents the computed spin coherence of VB in the bulk of natural h-BN. Evidently, the spin coherence rapidly decays during the free evolution time, within tens of microseconds. By fitting the coherence with a stretched exponential function ($${{{\mathrm{exp}}}}\left( {t_{{{{\mathrm{free}}}}}/T_2} \right)^n$$), we computed the T2 time and stretching exponent (n) of the decay to be 26.64 μs and 2.64, respectively. In addition, we determined T2 and n of the VB spin in the single-layer h-BN to be 45.85 μs and 2.27, respectively, owing to the reduced number of nuclear spins surrounding VB (see Supplementary Fig. 2). Notably, the computed T2 time corresponds to the upper limit of T2 set by the intrinsic nuclear spin bath of h-BN. The experimentally measured T2 time can be smaller than the theoretical T230, if other decoherence sources such as other paramagnetic defects are present in h-BN.

To examine the nuclear bath dynamics generating the intracrystalline magnetic noise, we analyzed the impact of the spin Hamiltonian terms on the VB spin decoherence. Importantly, we find that the nuclear quadrupole interaction53 plays an important role in determining the bath dynamics in h-BN. Figure 1b shows the VB decoherence computed with a partial spin Hamiltonian, in which the quadrupole Hamiltonian terms are excluded from the model. We observe that the spin coherence decays much faster in the “without-the-quadrupole” model than in the “with-the-quadrupole” model. Without the quadrupole interaction effect in the model, the T2 time of VB is reduced to 17.92 μs and 34.98 μs for the multi- and single-layer h-BN, respectively39. Notably, the nuclear spin flip-flop transitions, driven by the magnetic dipolar coupling between the nuclear spins, are the dominant sources of intracrystalline magnetic noise in a nuclear spin bath48,52,54. Our results show that the nuclear quadrupole interaction in h-BN plays a significant role in suppressing such nuclear flip-flop transitions in the h-BN.

To understand the role of quadrupole interaction in the pairwise nuclear spin transitions, we show the energy levels of two 10B nuclear spins, defined as $$\left| {m_{I_1},m_{I_2}}\rangle \right.$$, where $$m_I$$ = 3, 2, 1, 0, −1, −2, −3, in Fig. 1c–e. Figure 1c shows the energy level splitting due to the Zeeman interaction (HZ) and hyperfine field imposed by the VB spin (HHyper). The Zeeman splitting yields spin manifolds, which are separated from each other in energy by 13.7 MHz in the presence of an external magnetic field of 3 T. Each manifold contains certain number of two-spin states such as $$\left\{ {\left| {{{{\mathrm{ - 3, - 1}}}}}\rangle \right.{{{\mathrm{,}}}}\left| {{{{\mathrm{ - 2, - 2}}}}}\rangle \right.{{{\mathrm{,}}}}\left| {{{{\mathrm{ - 1, - 3}}}}} \right.} \rangle\right\}$$, which have the same Zeeman energy, as shown in Fig. 1d. These states in a manifold are split by a small energy of the order of hundreds of Hz because of the hyperfine field (denoted as ΔHF in Fig. 1d). In the “without-the-quadrupole” decoherence model, the magnetic dipolar interaction drives the flip-flop transitions between the states, confined in each manifold, with a transition rate of hundreds of Hz (denoted as ΩFF in Fig. 1d). However, the transitions are strongly suppressed between the states belonging to different manifolds because of a large Zeeman energy mismatch. Remarkably, the energy levels in a manifold exhibit significant splitting in the presence of the quadrupole interaction, as shown in Fig. 1e. We found that the quadrupole-induced splitting (ΔHF+Q in Fig. 1e) ranges from tens of kHz to a few MHz, owing to the large nuclear quadrupole interaction in h-BN (see Table 1). Thus, the flip-flop transitions within a Zeeman manifold are significantly suppressed, compared to those evaluated using the “without-the-quadrupole” model, because of the large quadrupole-driven energy mismatch. Our analysis reveals the importance of the quadrupole interaction in determining the nuclear bath dynamics, which in turn governs the spin decoherence of the spin qubits in h-BN.

Isotopic enrichment of h-BN to increase T2

Figure 2 depicts the computed T2 of VB in h-BN as a function of the composition ratio of 10B and 14N in h-BN. Surprisingly, we find that the T2 time substantially increases as the ratio of 10B and 14N increases toward 100% in the lattice, despite their larger nuclear spins than those of 11B and 15N. When h-BN is 100% enriched with 10B and 14N (i.e., h-10B14N), the T2 time is increased to 143.39 μs and 81.11 μs in the single- and multi-layer h-10B14N, respectively. These values are three times larger than those in natural h-BN. In contrast, the T2 times in single- and multi-layer h-11BN are 45.85 μs and 26.64 μs, respectively, which are unanticipated, because a smaller nuclear spin (e.g., I = 3/2 of 11B vs. I = 3 of 10B) results in a smaller number of flip-flop transition channels in the nuclear spin–spin interaction as well as lower intracrystalline noise in h-11B15N than in h-10B14N. However, our results show that the opposite is true—the intracrystalline magnetic noise is significantly reduced in h-10B14N.

To elucidate the microscopic origin of the enhanced T2 in h-10B14N, we consider a hypothetical central spin model, wherein the gyromagnetic ratio (γB) and the nuclear spin number (IB) at the boron sites are variable. Figure 3 shows T2 as a function of γB and IB. Considering the actual IB and γB of 10B and 11B (2.875 $${{{\mathrm{rad}}}}\;{\mathrm{G}}^{{{{\mathrm{ - 1}}}}}\;{{{\mathrm{ms}}}}^{{{{\mathrm{ - 1}}}}}$$ and 8.585 rad G−1 ms−1 55, respectively), we consider the range of γB to be from 1 to 10 $${{{\mathrm{rad}}}}\;{\mathrm{G}}^{{{{\mathrm{ - 1}}}}}\;{{{\mathrm{ms}}}}^{ - 1}$$, and that of IB from ½ to 3 in Fig. 3. We find that for a given γB, the T2 time decreases as IB increases, indicating that the intracrystalline magnetic noise increases as the total number of nuclear spin flip-flop channels in the bath (proportional to (2IB)2) increases. In addition, T2 increases quadratically as γB decreases. For instance, for IB = 3/2, the T2 changes from 24 μs to 695 μs when γB changes from 10 to 1 $${{{\mathrm{rad}}}}\;{\mathrm{G}}^{{{{\mathrm{ - 1}}}}}\;{{{\mathrm{ms}}}}^{ - 1}$$. The γB-dependent change in T2 observed mainly because the magnetic dipolar coupling strength is proportional to (γB)2. Thus, decreasing γB reduces the flip-flop transition rate quadratically in the nuclear spin bath. In h-10BN, the total number of flip-flop transitions is increased by four times, but the flip-flop transition rate is substantially decreased by approximately nine times compared to those in h-11BN. In the case of 14N isotope, the same analysis yields four times larger number of flip-flop transition channels and approximately two times smaller flip-flop transition rates in h-B14N compared to those in h-B15N. Therefore, based on the result shown in Fig. 3, we can conclude that in h-BN, the spin coherence of VB is much more sensitive to the change in γB than that in IB, and the increased T2 in h-10B14N stems from the smaller γB of the 10B and 14N nuclear spins despite their large nuclear spin numbers.

Effects of inhomogeneous lattice strain on spin coherence

Next, we investigate the impact of inhomogeneous lattice strain on the T2 of VB spin in h-BN. We consider a curved h-BN to create an inhomogeneous strain around VB. Reportedly, various feasible methods have been developed to study the curvature-induced phenomena in h-BN by creating bubbles56,57, pillars33, folds58, and wrinkles20. As a representative model of the local curvature and inhomogeneous strain, we consider a Gaussian lattice deformation as illustrated in Fig. 4a. The lattice deformation can be described in terms of the full width at half maximum (FWHM) and height of the Gaussian function, and the VB defect is created on top of the Gaussian deformation. To cross-check the dependence of our results on the shape of lattice deformation, we also considered two other types of lattice deformations, which are Lorentzian-shape and pillar-shape deformations (see Supplementary Fig. 3). We employed DFT to compute the spin Hamiltonian parameters in the deformed h-BN lattice. The computed effects were subsequently included into the spin Hamiltonian to examine the effect of the inhomogeneous lattice strain on the spin coherence of VB.

Figure 4b shows the computed T2 time of the VB in natural h-BN (no isotopic modification) as a function of the FWHM and height of the Gaussian lattice deformation. Evidently, the T2 time increases significantly with the increasing function height and FWHM. At a Gaussian height of 3.5 Å, the T2 in the single- and multi-layer h-BN is 58.7 μs and 34.2 μs, respectively, as shown in Fig. 4b, c. These values are 12.9 μs and 7.5 μs greater than those in natural and flat h-BN. In Supplementary Fig. 4, we computed the VB spin decoherence in the presence of the Lorentzian-shape and pillar-shape lattice deformations. We found that the VB spin coherence is consistently enhanced in these situations: when the deformation height is 3.5 Å, the T2 time in the single-layer h-BN reaches 54.6 μs and 72.2 μs in the presence of the Lorentzian- and the pillar-shape deformations, respectively. This result shows that the T2 enhancement due to inhomogeneous strain is valid regardless of lattice deformation shapes.

To understand the curvature-induced enhancement of the T2 time, we analyze the impact of the lattice deformation on the VB spin decoherence in terms of two factors: modification of the nuclear spin–spin distances and spatial inhomogeneity induced in the nuclear spin quadrupole interaction in the deformed h-BN lattice. In Supplementary Fig. 6, we consider a model that does not include the nuclear spin quadrupole interaction, and compute T2 as a function of FWHM and function height same as before (Fig. 4). We find that the T2 time shows negligible changes, as shown in Supplementary Fig. 6, implying that the enhanced VB spin coherence, obtained using the Gaussian lattice deformation, is not derived from the mere modification of the nuclear spin–spin distances; instead, it is mediated by the modified nuclear quadrupole interaction produced by the inhomogeneous strain.

To visualize the role of nuclear quadrupole interaction in determining the strain-driven enhancement of T2, we compute the first-order quadrupole-interaction-induced energy shift (ζ) of a nuclear spin in h-BN. As described in Supplementary Note 3, the energy shift is calculated as: QC $$\times \;\zeta \; \times \left( {3m_I^2 - I\left( {I{{{\mathrm{ + 1}}}}} \right)} \right)$$, where QC is the coefficient of quadrupole interaction (6), and ζ = Vzz − (Vxx + Vyy)/2, where Vii (i = x, y, z) is the electric field gradient computed using DFT. Figure 5a–c shows ζ as a function of position with respect to the VB site, indicating that a difference of ζ between two different points, shown in Fig. 5a–c, is proportional to the difference between the quadrupole-induced energy shifts of the two nuclear spins at these two different points. As shown in Fig. 5a–c, the inhomogeneity of ζ becomes broader and larger as the Gaussian lattice deformation increases. In Supplementary Fig. 7, we also show the quadrupole coupling constant (Cq = eQVzz/ħ) and the asymmetry parameter (ηas = (Vxx − Vyy)/Vzz) in the presence of the lattice deformation. In terms of the 10B–10B two-spin energy levels presented in Fig. 1e, a larger difference of ζ between two 10B spins results in a larger level splitting in their two-spin energy levels.

Nuclear spin flip-flop transitions are well understood in terms of a pseudospin model48,52, which can model a flip-flop transition between any two 10B–10B nuclear spin states having the same Zeeman energy. The two states of a pseudospin are separated by an energy gap (Δ), and the transition between them is mediated by the magnetic dipolar interaction with a flip-flop transition rate (Ω) (see Supplementary Note 4). Figure 5d, e shows the histogram of (log(Ω/Δ)) for all the possible pseudospins in the Gaussian-deformed h-BN. Notably, a small log(Ω/Δ) indicates a high suppression in the flip-flop transition. Furthermore, a large Gaussian deformation around the VB defect produces a large number of pseudospins with small log(Ω/Δ) values appearing in the histogram. Our results demonstrate that the energy gap between numerous 10B–10B pseudospins in h-BN increases as the Gaussian lattice deformation becomes larger, which in turn produces a larger inhomogeneous strain around VB. This results in a considerable suppression of the nuclear flip-flop transitions in h-BN, leading to the enhancement of the T2 time of VB.

Maximizing the spin coherence time

The two methods suggested in this study can be combined to maximize the T2 time. Figure 6a, b presents the computed T2 as a function of the Gaussian height and FWHM for the VB in the Gaussian-deformed h-10B14N. Evidently, the maximum achievable T2 is 207.17 μs and 161.92 μs in the single- (Fig. 6c) and multi-layer h-BN (Fig. 6d), respectively. These values are approximately six times larger than that in natural and flat h-BN. Supplementary Fig. 5 shows the VB spin decoherence in the Lorentzian-deformed single-layer h-10B14N and in the presence of pillar-shape lattice deformation as well. We found that T2 time reaches 188.9 μs and 308.4 μs for the Lorentzian- and the pillar-shape deformation, respectively, in the single-layer h-10B14N. We remark that the combined effect of the smaller gyromagnetic ratio of 10B and the inhomogeneous quadrupole interaction induced by the inhomogeneous strain leads to the enhancement of T2 of the VB spin.

Discussion

In summary, we proposed two theoretically effective methods to extend the spin coherence time of the VB spin qubit in h-BN. The spin coherence time is otherwise limited to a few microseconds because of the strong intracrystalline magnetic noise caused by the dense nuclear spin bath in the h-BN lattice. To predict the T2 time of the spin qubit accurately, we performed first-principles calculations by combining the DFT and CCE theory. In a natural h-BN host, the theoretical upper limit of the VB T2 time is 45.85 μs and 26.64 μs in single- and multi-layer h-BN, respectively. Next, we demonstrated that the T2 time can be increased to 143.39 μs and 81.11 μs in single- and multi-layer h-10BN, respectively, in which all the boron atoms in the lattice are replaced by the 10B isotope. It is evident that such an isotopic enrichment technique has been already developed and applied to h-BN59,60. By analyzing the magnetic dipolar interaction between the boron nuclear spins, we showed that the smaller gyromagnetic ratio of 10B plays a key role in suppressing the nuclear flip-flop dynamics, despite its large nuclear spin compared to that of 11B. Then, we showed that the T2 time of VB can be further increased by introducing an inhomogeneous strain around the VB defect. Applying a Gaussian-type lattice deformation (as a representative model of the inhomogeneous strain), we identified that the inhomogeneous strain contributes to the suppression of the nuclear spin flip-flop dynamics in the bath by producing spatially varying nuclear quadrupole interactions in h-BN. We found that applying both the isotopic enrichment and inhomogeneous strain could increase the T2 time of VB by six times that in a pristine h-BN bulk, and consequently, the T2 can reach up to 207.17 μs and 161.92 μs for the single- and multi-layer h-BN, respectively.

Our study on VB not only provides a fundamental understanding of the decoherence of VB spins in h-BN but paves the way to engineer their T2 times. With an increased T2 time, the VB spin coherence could be further enhanced by combining other active quantum control schemes such as DD or CTs. In addition, the essential physics developed in this study can be applied to any localized spin in any 2D vdW material. Thus, the proposed approach provides a universal tool for engineering the coherence of potential spin qubits in 2D vdW materials. Combining these engineering schemes with the unique characteristics of 2D vdW materials would be a promising strategy for the development of robust low-dimensional quantum systems.

Methods

Quantum-bath approach for decoherence and CCE

According to the quantum-bath theory, the spin qubit decoherence occurs because of the entanglement between the qubit and its environment61. In our spin model, the VB spin serves as a qubit, and the main environmental degrees of freedom are the nuclear spins residing in the h-BN lattice. The nuclear spins are coupled to each other via magnetic dipolar coupling, and the nuclear spin bath interacts with the qubit through hyperfine interactions. The full spin Hamiltonian for the entire quantum many-body system is described in Supplementary Note 1. To compute the homogeneous dephasing time—T2—of the VB ensembles, the Hahn-echo pulse sequence, which features a π pulse between two free evolution time τ, is considered. The decoherence of the qubit is obtained by computing the off-diagonal elements of the reduced density matrix of the qubit after tracing out the bath degrees of freedom at the end of the $$2\tau$$ free evolution time, as: $${{{\mathcal{L}}}}\left( {2\tau } \right) = \frac{{{{{\mathrm{Tr}}}}\left[ {\rho \left( {2\tau } \right){{{\mathrm{S}}}}_ + } \right]}}{{{{{\mathrm{Tr}}}}\left[ {\rho \left( 0 \right){{{\mathrm{S}}}}_ + } \right]}}$$, where S+ is the electron spin raising operator, and ρ is the density operator of the entire system. To compute the coherence function, we employed the CCE method47,62, which systematically expands the coherence function. We find that the CCE expansion converges at the CCE-2 level of the theory, implying that the nuclear spin–spin pairwise correlation effect is the dominant source of decoherence. Extensive numerical convergence tests were performed in terms of the CCE order, bath size, and coupling radius, and the corresponding results are summarized in Supplementary Note 2 and Supplementary Fig. 1.

DFT calculations

We performed DFT calculations using plane-wave basis functions with an energy cutoff of 85 Ry, as implemented in the Quantum Espresso (QE) code63 and the GIPAW module64,65, to compute the nuclear quadrupole interaction and the hyperfine interaction in h-BN containing VB with a Gaussian lattice deformation. We used the Perdew–Burke–Ernzerhof exchange-correlation functional along with the projector-augmented wave (PAW) pseudopotentials66,67. To simulate an isolated VB defect in the h-BN, we used large supercells containing either single- or multi-layer h-BN. For the single-layer h-BN calculations, we built a 336-atom supercell starting from a 4-atom orthorhombic unit cell. The h-BN layer in the supercell was separated from its periodic images in the out-of-plane direction by a 10-Å-thick vacuum space. For the multi-layer h-BN, a 672-atom supercell was used.

To cross-check our results obtained with the orthorhombic supercell, we computed the spin Hamiltonian parameters and the VB spin decoherence by using a 450-atom hexagonal supercell for h-BN. We show the results in Supplementary Tables 1, 2, 3, and Supplementary Figs. 8, 9 in Supplementary Information. We found that the computed EFG and hyperfine parameters obtained with the hexagonal supercell are consistent with the orthorhombic supercell results except for a few atoms near the VB defect. In particular, the EFG parameters show a noticeable difference for a few N atoms near the VB due to a slight difference in the atomic relaxation depending on the supercell shape (see Supplementary Fig. 8 and Supplementary Table 2). We found, however, that this difference did not affect the main conclusion of this study. In Supplementary Fig. 9, we compare the VB spin decoherence in multi-layer h-BN, computed with the orthorhombic supercell and with the hexagonal supercell. We found that the two results show only a minor difference in T2: 25.43 μs and 26.64 μs for the hexagonal cell and the orthorhombic cell, respectively. We remark that the sub-μs variation found in the T2 time depending on the supercell shape does not affect the main conclusion of our study as our prediction on the T2 enhancement in isotopically enriched h-BN and in strain-engineered h-BN is on the order of tens of μs.