Accurate hyperfine tensors for solid state quantum applications: case of the NV center in diamond

Point defects in wide-bandgap semiconductors are emerging as leading candidates for quantum computing applications due to their high coherence and robustness, even at room temperature. Examples include the NV center in diamond, silicon vacancies in silicon carbide (SiC), and divacancy-related defects in SiC. The coherence of these spin qubits is often limited by spin-spin interactions with the surrounding nuclear spin bath. The hyperfine interaction, parameterized by the hyperfine tensor, governs this coupling. While accurate calculations of hyperfine tensors exist for nuclei close to the defect, the accuracy diminishes significantly for more distant nuclei. This study addresses the limitations of existing computational methods for accurately predicting hyperfine parameters for distant nuclear spins in the NV center in diamond, a critical component for developing high-precision quantum technologies and nano-NMR applications.

Previous theoretical work predominantly focused on hyperfine interactions with the nearest nuclear spins, with limited discussion on the accuracy of predictions for more distant spins. Studies have shown remarkable accuracy (~4.7% mean absolute relative error) for close nuclear spins using density functional theory (DFT) and wave-function-based methods. However, a supercell-size scaling test revealed that the accuracy sharply decreases for nuclear spins farther than 1-5 Å from the defect, likely due to periodic boundary conditions and finite-size effects. While using larger clusters can mitigate the periodic boundary condition issues, the computational cost increases drastically. Existing methods often struggle to handle long-range interactions accurately.

The authors first demonstrate the inaccuracies of hyperfine parameter calculations using the industry-standard VASP code. They then introduce a novel real-space integration method to address finite-size effects. This method combines a large support lattice that considers nuclear spins outside the supercell's boundaries with a real-space integration to avoid the periodicity limitations of standard methods. This calculation uses the full spin density on a fine grid, instead of the pseudo-spin density as in standard approaches. They conduct large-scale calculations for the NV center in diamond using various exchange-correlation functionals, including PBE and HSE06 with different mixing parameters (α). The ground state calculations are performed using 512-atom and 1728-atom supercells with optimized defect structures. The spin density is obtained from the converged 1728-atom supercell on a real-space grid with a 0.036 Å spacing. The hyperfine tensor elements are calculated using the spin density, the positions of the nuclear spins, and the gyromagnetic ratios of the electron and nucleus. The calculated hyperfine values are compared to various experimental datasets to assess accuracy. The positioning of nuclear spins for comparison with the experimental data involved finding the closest theoretical values to experimental ones within the error bars of measurements, utilizing the symmetry of the system.

The authors' improved real-space integration method significantly reduces the absolute relative error (ARE) of calculated hyperfine parameters. Using the HSE06 functional with a 0.2 mixing parameter, they achieve a mean absolute percentage error (MAPE) of 1.7% for nuclear spins located 6-30 Å from the NV center, a substantial improvement over the VASP-based calculations which showed over 100% ARE for distant nuclei. For different experimental datasets, the MAPEs range from 1.5% to 3.8%. The accuracy of the hyperfine values is highly dependent on the accurate computation of the Fermi contact term. Analysis reveals a correlation between the largest errors and the magnitude of the Fermi contact term, and a distance dependence on the mean absolute relative error. A comparison with calculations using the PBE functional highlights the superior performance of the HSE06 functional, particularly for nuclei close to the NV center. The point spin density approximation demonstrates a significantly higher MARE of 76%, emphasizing the need for accurate DFT spin density in the calculations. The authors make their high-accuracy hyperfine tensors (≈10⁶ lattice sites) and volumetric hyperfine data (<0.1 Å spatial resolution) publicly available online.

The significant improvement in accuracy achieved by the authors' real-space integration method resolves a critical limitation in simulating NV center-based quantum systems. The method effectively eliminates finite-size effects arising from periodic boundary conditions commonly encountered in standard first-principles calculations. The remaining errors are primarily attributed to inaccuracies in the Fermi contact term calculation. The observed correlation between error magnitude and the Fermi contact term suggests that further refinement of the functional and numerical techniques used to compute this term could further enhance the accuracy of the calculated hyperfine parameters. The availability of the comprehensive hyperfine data allows for more realistic and precise simulations of NV center-nuclear spin interactions, enabling advancements in quantum sensing, quantum computing, and nano-NMR applications.

This work presents a significant advancement in accurately computing hyperfine tensors for the NV center in diamond. The developed real-space integration method drastically improves accuracy, reducing the mean absolute relative error by two orders of magnitude compared to standard methods. The high-accuracy data provided will facilitate more precise simulations and pave the way for advancements in NV center-based quantum technologies. Future work could focus on improving the accuracy of the Fermi contact term calculations through enhanced numerical techniques or further functional optimization. Larger supercell calculations could also potentially reduce residual errors.

While the new method significantly improves accuracy, some residual errors remain, largely attributed to the challenges in accurately calculating the Fermi contact term. The study focuses on the NV center in diamond, and the generalizability of this method to other point defects requires further investigation. The positioning of the nuclear spins for comparing theoretical and experimental results relied on matching theoretical values to experimental ones within error margins, which introduces some uncertainty.