Low-dimensional condensed matter physics, particularly in transition metal dichalcogenides and van der Waals heterostructures, demands high spatial resolution sensing. Point defect qubits in semiconductors, like NV centers in diamond and silicon vacancies in SiC, offer capabilities in this area. However, their bulk nature and surface chemistry dependence limit their applicability in low-dimensional structures. Layered van der Waals semiconductors, such as hBN, provide a solution due to their chemical stability and thickness controllability. hBN's wide bandgap accommodates numerous optically active electronic states of defects and impurities, leading to the observation of various single photon emitters and ODMR centers. The negatively charged boron vacancy (VB) center, while showing potential, suffers from a low photoluminescence emission rate. Other, brighter ODMR centers exhibiting narrow lines have been observed, often linked to carbon contamination. This study focuses on theoretically investigating the neutral charge state of nitrogen-centered (C4N) and boron-centered (C4B) symmetric carbon tetramer structures in hBN to determine their suitability as spin qubits for sensing. The low formation energy of these tetramers, explained by Baird aromaticity, makes them particularly attractive.

Existing literature extensively covers point defect qubits in 3D semiconductors, highlighting their capabilities in high-resolution sensing. However, the challenge lies in integrating these sensors into low-dimensional structures due to limitations associated with surface chemistry and bulk nature. Research on van der Waals semiconductors, especially hBN, offers a promising alternative due to their surface stability and thickness tunability. Several studies have demonstrated single photon emitters and ODMR centers in hBN, with some identifying the negatively charged boron vacancy center (VB). While the VB center exhibits potential, its low photoluminescence rate hinders applications. Other bright emitters with narrow ODMR lines have been reported, often linked to carbon contamination. Several theoretical works have studied carbon impurities and related defects in hBN, predicting defects with optical properties resembling experimental observations and some with high-spin ground states. This study expands upon these works by investigating the specific characteristics of symmetric carbon tetramers and their potential for quantum sensing.

This research employs a combined approach utilizing both periodic supercell models and molecular models to analyze symmetric C4 defects (C4N and C4B) in hBN. Periodic models leverage Kohn-Sham density functional theory (DFT) calculations with the VASP package, employing a plane-wave basis set (450 eV), PAW core potentials, and the HSE06 hybrid exchange-correlation functional with 0.32 exact exchange fraction (HSE(0.32)). The D3 correction accounts for van der Waals interactions. Supercells of 162 atoms (monolayer) and 768 atoms (bulk) were used. ZPL energies are calculated from energy differences between ground and excited states using spin-conserved constrained DFT methods. Molecular models use the ORCA program system employing CASSCF-NEVPT2 and TD-DFT methods to analyze the many-body electronic structure, low-energy excitation spectra, spin-orbit coupling, radiative lifetimes, and Huang-Rhys factors. Hyperfine coupling parameters are determined using periodic hybrid-DFT in a bulk model. The Golden Rule rate equation was applied for determining excited-state dynamics including decay rates, Huang-Rhys factors, photon emission energies, and intensities. Calculations consider the effects of strain and isotopic variation on spin properties and spectral characteristics.

The study reveals that both C4N and C4B defects exhibit triplet ground states. C4N possesses an optically allowed triplet excited state (³E') at 2.32 eV above the ground state (³A₂), while C4B has a similar structure but with an additional dark triplet excited state (³A'') below ³E'. For C4N, spin-selective decay is facilitated by out-of-plane distortions induced by strain. The calculated photoluminescence spectrum of C4N shows excellent agreement with experimental spectra of unidentified carbon-related qubits. Spin-orbit coupling is weak in the undistorted C4N structure but increases significantly under strain (10-30 GHz range), enabling spin-dependent non-radiative decay. For C4B, strong spin-orbit coupling (48.81 GHz) between the ³A₂(ms=±1) and ¹E' states leads to spin-selective decay. Electron spin resonance (ESR) simulations show that both C4N and C4B exhibit narrow linewidths (8-31 MHz), attributable to spin density localization on carbon atoms, suggesting potential for high sensitivity. Analysis of chemical stability reveals that complex formation with boron and carbon interstitials is energetically favorable except for C4N with carbon interstitials, which demonstrates a high energy barrier suggesting stability up to ~630 K. Annealing temperatures for restoring isolated C4N and C4B defects are estimated between 690 and 1190 K.

The findings demonstrate that C4N and C4B defects show characteristics ideal for spin qubits in hBN for sensing. The narrow ESR linewidths in C4N could significantly improve sensing capabilities. The strain sensitivity of C4N allows for tuning the spin contrast, further enhancing the functionality of these qubits. The potential for high-temperature and low-magnetic nuclear hyperpolarization further enhances the value of C4 defects for boosting the sensitivity of conventional NMR and MRI applications. The predicted ZPL energy and phonon sideband of C4N correlate well with experimental data from unidentified carbon-related qubits, suggesting a possible identification. The discrepancy in the zero-field splitting parameter (D) between theoretical predictions and some experimental findings suggests that these defects may not be associated with all the previously observed ODMR centers. Overall, the study highlights the significant potential of symmetric carbon tetramers as spin qubits in hBN-based sensing.

This study provides a comprehensive theoretical analysis of neutral symmetric carbon tetramers (C4N and C4B) in hBN, establishing their potential as spin qubits for sensing applications. The predicted optical and spin properties, including narrow ESR linewidths and strain-dependent behavior, suggest superior sensitivity. The findings encourage further experimental investigation to confirm the theoretical predictions and explore the full potential of these defects for advancing low-dimensional quantum sensing. Future research could focus on exploring the effects of various defects and their interactions and developing effective fabrication methods for these defects in hBN.

The study relies on theoretical predictions. Experimental verification is needed to confirm the findings. The accuracy of the calculations is limited by the approximations inherent in the computational methods used. The chemical stability analysis considers only a limited set of possible defect complexes, and other complex formations might be relevant. The study focuses on the neutral charge state, while further investigations into charged states could provide valuable insights. The impact of environmental factors, such as the substrate material and defects concentration, might influence the results.