## Introduction

Solid-state spin defects are an emerging platform with applications in quantum information science, sensing, and metrology1. The negatively charged nitrogen-vacancy centers (NV-) in diamond and neutral divacancy defects (VV) in silicon carbide (SiC) represent promising examples of defect complexes for use as spin qubits2,3,4. Divacancies in SiC—a widely used material in the semiconductor industry—are particularly attractive, as they offer all-optical spin initialization and readout capabilities5,6, nuclear spin control7, and a near-infrared high-fidelity spin-photon interface8, in addition to long coherence times9.

The optical and electronic properties of spin defects in SiC, including the silicon vacancy (VSi)10,11, NV centers12,13, and carbon antisite-vacancy complexes (CSiVC)14, have been characterized using a variety of techniques. These include density-functional theory (DFT) calculations, electron paramagnetic resonance spectroscopy (EPR), ﻿deep-level transient spectroscopy (DLTS), and photoluminescence spectroscopy (PL)15,16,17,18,19,20,21. Little is known, however, of how to control the selective formation and spatial localization of defect complexes. Addressing this challenge is critical for their integration with optical and electric devices and nanostructures18.

Divacancies and other spin defects in SiC are generally generated by ion implantation or electron irradiation, followed by thermal annealing22,23,24,25,26. The vacancy-to-VV conversion efficiency, however, is low—a few percent or less22,27. VV localization is difficult to control, as samples need to be thermally annealed at high temperatures (above ~700 oC) to generate the defect mobility necessary to create divacancies8. Furthermore, the spatial placement of VV appears to depend on the initial distribution of several intrinsic and extrinsic defect species (e.g., vacancies, interstitials, substitutional defects, and dopants) and the Fermi level of the system27.

The dynamics of vacancy-defect complexes in semiconductors, particularly SiC, present considerable challenges for theory and computation as well. In particular, the activation energies for vacancy migration in SiC can be on the order of several electronvolts19, thereby limiting the applicability of first-principles molecular dynamics (FPMD) simulations. As a result, prior DFT calculations in SiC have focused mainly on computing defect-formation energies and migration barriers for monovacancies at T = 0 K28,29,30,31,32,33. Recent kinetic Monte Carlo simulations considered the dynamics of vacancies in SiC34, but they did so by invoking a priori mechanisms for defect mobilization.

Here, we study how spin defects hosted in vacancy-defect complexes are formed in cubic, 3C-SiC, by coupling FPMD simulations with a neural-network-based enhanced sampling technique35,36 to efficiently probe phase space, and we compute the free energies and stabilities of several defects. In some cases, our FPMD results are augmented by classical MD simulations conducted with larger system sizes and over longer timescales. Additionally, we predict previously unidentified spin defects consisting of antisites and double vacancies. We focused on the 3 C polytype, given the increasing number of low-cost synthesis strategies developed for high-purity 3C-SiC, e.g., heteroepitaxial growth on silicon substrates37,38. Furthermore, 3C-SiC contains a single configuration for each vacancy defect (VSi, VC, and VV), and it is thus simpler than other polytypes (e.g., 4H-SiC) that can host multiple configurations. We find that the formation of divacancies is initiated by the migration and association of monovacancies, which occur in the same temperature regime as the crystallographic reorientation of VV. Our results reveal that a divacancy is a thermodynamically stable state, while VSi represents a kinetically trapped state that readily transforms into an intermediate carbon antisite-vacancy (CSiVC) defect. Our simulations also show that divacancy formation can be maximized by choosing initial conditions corresponding to a large fraction of VSi in the sample to mobilize monovacancies prior to the destabilization of VSi.

## Results

### Divacancy dynamics at high temperatures

The simulation of defect migration in covalently bonded materials, where activation energies may be as large as several electron volts, requires the collection of statistics for ~10–100 ns. Therefore, capturing the migration of vacancies in SiC using straightforward FPMD is at present prohibitively demanding from a computational standpoint. Hence, before conducting FPMD simulations coupled to enhanced sampling techniques, we explored the dynamics of vacancy defects in SiC using an empirical force field39, based on a widely used Tersoff-type bond-order formalism40,41. The force field was selected for its ability to yield temperature-dependent bulk densities and a decomposition temperature in agreement with experiments42,43 (see Supplementary Fig. 1). We found that classical MD using the empirical force field and FPMD simulations yields qualitatively similar energetics for vacancy migration processes (see Supplementary Note 1 for a detailed comparison between the two methods).

To probe the dynamics of VV and its formation from single-vacancy defects (see Fig. 1), which are only present in low concentrations, we carried out 10-ns-long classical MD simulations of a 4096-atom supercell of 3C-SiC containing a pair of VSi and VC (see “Methods”). Analyses of the MD trajectories reveal one or more of the following phenomena at temperatures between 1000 K and 1500 K: (1) monovacancy migration (Supplementary Movie 1), (2) VV formation from the pairing of two monovacancies (Fig. 2a–d, Supplementary Movie 3), (3) VV orientational changes (Fig. 2e–h, Supplementary Movie 4), and (4) conversion of VSi into a carbon vacancy–antisite complex, CSiVC (Fig. 2i, j, Supplementary Movie 2). Below 1000 K, we do not observe any vacancy diffusion, even after continuing our simulations for 100 ns; above 1500 K, divacancies dissociate into VSi and VC, and CSiVC also dissociates into CSi and VC (Fig. 2j–l). These results suggest that the formation of a VV is a temperature-activated process. VV generation occurs in the same temperature regime as the single-vacancy diffusion and, surprisingly, by the destabilization of silicon vacancies via the conversion of VSi into CSiVC.

Our results between ~1000 K and 1500 K are consistent with recent measurements performed on irradiated samples of 4H-SiC44. These annealing experiments reported that with increasing temperature above ~600 K, there is a strong correlation between decreasing deep-level transient-spectroscopy (DLTS) intensities of VSi and CSiVC and increasing DLTS and PL intensities of VC and VV. The experiments, therefore, are also consistent with the formation of VV and the formation of VC from VSi and CSiVC.

### Mechanism of VV formation and reorientation processes

Snapshots from our MD simulations above 1000 K are shown in Fig. 2. We provide a 3D representation of the free volume at defect sites (see Methods) to investigate changes to the local structure proximal to each defect. As shown in Fig. 2a, b, the size and shape of the free volume surrounding VC and VSi are different. In particular, VC sites have tetrahedrally shaped voids, while VSi sites have void volumes consisting of two overlapping tetrahedra centered at the VSi site. The void volume of the VSi site is larger than that of the VC site, indicating that silicon and carbon atoms surrounding VSi are more likely to be mobile than the atoms surrounding VC. When an atom in-between VC and VSi fills in a vacancy site, the two monovacancies join together to form a VV.

Our MD simulations show that the thermal annealing of SiC samples may also lead to a reorientation of the VV without dissociating the original VV, i.e., it leads to a change in the direction of the VSi–VC axis within the laboratory frame. A recent photoluminescence confocal microscopy study of NV- in diamond reported a crystallographic reorientation process, where the NV-defect symmetry axis in a thermally annealed sample was rotated from its starting configuration (before annealing) in the original reference frame45. Our results suggest that thermal annealing at temperatures between ~1000 K and 1500 K may be a route toward aligning the orientation of VVs, which would be desirable for sensing applications46,47.

### Free-energy landscape for divacancy formation and interconversions

To quantify the energetics of defect-interconversion processes and identify possible transition pathways, we computed the free energy or the potential of mean force (PMF) for VV formation at 1500 K and compared the PMF with free-energy profiles of other defect transformation processes that occur at the same temperature. In order to do so, we turned to DFT calculations, and we computed the PMFs by combining a neural-network-based enhanced sampling method with FPMD36, and we used the Cartesian coordinates of the mobile carbon or silicon atom as order parameters. We performed simulations at a Fermi level known to favor the formation of neutral divacancies (see Methods). As expected, the computed 2D-PMFs for defect transformations in 3C-SiC reveal multiple minima and maxima that depend on the specific location of the defects in the crystal structure (Fig. 3a–c).

In the free-energy profile shown in Fig. 3a and Supplementary Fig. 6a, we identify a transition state (T) separating the free-energy basins of VV and that of the two unassociated vacancies (VSi + VC). We find that the free energy of the VV is ~1.5 eV lower than that of the (VSi + VC), thus confirming that divacancies are stable species at 1500 K, consistent with high-temperature annealing experiments of as-grown 4H-SiC samples25,48. Additionally, the free-energy difference between the VV and the transition state is comparable to that of the monovacancy migration barrier (~3.0 eV for the VSi diffusion and ~3.9 eV for VC diffusion, see Supplementary Fig. 5) as both processes involve a next-nearest-neighbor atom substituting a vacancy site. The predicted VC migration barrier obtained from FPMD (~3.9 eV) agrees with the barrier measured by DTLS on annealed SiC samples (~3.7–4.2 eV)19.

Regarding the VV reorientation process, our simulations indicate that the diffusion free-energy barriers of C atoms atoms (Fig. 3d, e) are lower than that of the Si atom (Supplementary Fig. 6b) by ~0.4 eV. The PMF for the former process features two energetically equivalent local minima (M1 and M2) along the free-energy profile (Fig. 3d). By comparing the free-energy profiles for VV formation and reorientation processes (Fig. 3d, e), we find that the barrier for VV dissociation (VV → VSi + VC) is ~0.5 eV higher than that required to change its orientation. This result suggests that the reorientation of divacancies can occur at a temperature lower than the temperature at which divacancies dissociate.

First-principles free-energy calculations reveal that the height of the barrier for VSi conversion into CSiVC (~1.3 eV) is similar to that of the barrier for VV formation (~1.5 eV) (see Fig. 3d, f). Moreover, the activation energy for the formation of CSiVC from VSi is lower than that of the migration of VSi (~3.0 eV barrier; see Supplementary Fig. 5b). Thus, CSiVC formation is more likely to occur than silicon migration. Interestingly, the free energy of CSiVC is lower than that of VSi by ~1.3 eV at 1500 K.

### Optimal temperatures for divacancy stability in as-grown and irradiated SiC

To understand the behavior of divacancies at elevated temperatures, we first analyze the effect of annealing temperatures on the population of VVs in our classical MD simulations. As shown in Fig. 4a, the VV population decreases with increasing temperature above 1700 K, which is consistent with annealing experiments that measured the population of VV of as-grown SiC samples using electron paramagnetic resonance spectroscopy (EPR)48. We find that the kinetics of VV decay is faster in the presence of a higher-vacancy concentration in the sample (see Supplementary Fig. 8). Hence, it is likely that the VV decay observed experimentally is slower than the one found here, since the VV concentrations of the experimental samples are much lower than those considered in our simulations (see Methods). By fitting an Arrhenius equation to the experimental data, i.e., $$y\propto {{\exp }}\left(-\triangle E/{k}_{B}T\right)$$ (see Fig. 4a), we estimate the energy difference between a bound and a dissociated VV state to be ~2.1 eV. This value is on a similar order of magnitude as the free-energy difference between VV and VSi + VC obtained from FPMD (~1.5 eV).

After collecting classical MD trajectories for 10 ns starting with stable VVs, we find that multiple vacancy defects are formed as divacancies dissociate (Fig. 4b), leading to the presence of VC, VSi, and CSiVC. The dominant species detected at high temperatures are carbon vacancies. Other defects (VSi and CSiVC) are present in smaller fractions (less than ~10% after 10 ns), as they can convert to VC via the following process: VSi → CSiVC → CSi + VC. In order to validate our classical MD results, we calculated energy differences using DFT, and then we computed defect populations via kinetic Monte Carlo (KMC) simulations (see Supplementary Note 2). Similar to classical MD simulations, our KMC model shows that, at high temperatures, carbon vacancies are the dominant species (Fig. 4b). However, the KMC model predicts that the temperature required to dissociate CSiVC is higher than that found in classical MD, because the VC migration barrier is higher in FPMD (~3.9 eV) than in classical MD (~2.5 eV). The results from MD and KMC simulations of double vacancies and the computed free-energy landscape of VV formation demonstrate that temperature facilitates both the creation and dissociation of VV. We expect the optimal temperature regime to anneal defects and generate VV to be between ~1000 K and ~1500 K or 1250 $$\pm$$ 250 K. This finding is consistent with recent annealing experiments carried out for irradiated or ion-implanted samples of 4H-SiC6,44, in which PL data of VV also suggest that the concentration of VV defects is maximized at ~1200 K (Fig. 4c). More specifically, by fitting a Gaussian process model to experimental data, we estimate the optimal annealing temperature in experimentally irradiated samples to be ~1193 K.

### Compositional dependence in divacancy formation

Our simulations and free-energy calculations show that the kinetics of the VV formation is controlled by the diffusion of monovacancies, while the maximum number of divacancies is limited by the stability of VSi. These results suggest that the annealing temperature and relative concentration of VSi to VC affect the mono-to-divacancy conversion efficiency. Experimentally, the composition of point defects in the sample can be changed by growing either carbon-rich or silicon-rich samples49 or by controlling the Fermi level21. Thus, it is of interest to compute the mono-to-divacancy conversion efficiency at varying ratios of VSi to VC (see Methods and Supplementary Note 2).

In the KMC simulations carried out in the temperature range where the conversion CSiVC → CSi + VC occurs, we find that higher concentrations of VSi lead to an increase of the probability of VV formation, even in the absence of VC at t = 0. These results are in agreement with those of classical MD simulations at 1500 K (see Fig. 5a) and are intriguing, as one would expect no stable VV when the system has predominantly one type of vacancy (VC or VSi). The results obtained at lower temperature are more intuitive as the KMC simulations show that the maximum number of divacancies is formed when there is an equal concentration of VSi and VC in the sample. These findings reveal that at high temperatures, the optimal condition to form VV would be to start with a large initial fraction of VSi to mobilize monovacancy migration and CSiVC dissociation, so as to convert some VSi into VC and then generate additional VC, as shown schematically in Fig. 5b. A low initial concentration of VSi reduces the VV-conversion efficiency (see Fig. 5c) because there are no energetically favored pathways to form VSi from VC.

### MD simulations identify potential spin defects

The generation of VSi and CSiVC species during MD simulations of VV dissociation, validated by DFT-based KMC simulations (Fig. 4b), pointed out species that, in appropriate charged states, are interesting spin defects10,14. Prior studies have detected and characterized spin defects based on impurities and antisite defects using EPR measurements and DFT calculations at T = 0 K50,51. Remarkably, we also identified potential spin defects from MD simulations of divacancies at 2000 K and above through the process: VV → VCCSiVC → CSiVC + VC (see Supplementary Movie 5). Importantly, simulations show that these defects can be stabilized to room temperature (see Supplementary Note 3). Multiple structures of this vacancy complex are possible, depending on the relative orientation between CSiVC and VC, as well as the physical proximity between the two defects.

The results of hybrid DFT calculations52 of the electronic structure of the neutral antisite-vacancy complexes shown in Fig. 6, indicate that VCCSiVC (Fig. 6b) and one of the structures for [CSiVC + VC], [CSiVC + VC]n=3 (Fig. 6c), consist of defect levels with localized electron densities. The lowest electronic configuration of these defect levels is a triplet, as the NV- in diamond and VV0 in SiC. However, we note that in the antisite-vacancy complexes, the two carbon vacancies are separated by a carbon antisite and, therefore, their ground-state spin densities are more spatially delocalized than those of the VV state.

In the case of the VCCSiVC state, we find unoccupied defect levels within the bandgap close to the conduction band, similar to neutral VV spin defect in 3C-SiC53 (see also Fig. 6a). These results suggest that [CSiVC + VC]n=3 and, especially, VCCSiVC are possible spin defects. ﻿ Since VCCSiVC and [CSiVC + VC] are created when VV dissociates (T > ~1700 K), it is unlikely that these spin defects are simultaneously present with VVs in post-annealed samples. However, if present, high concentrations of these spin defects and/or isolated CSi species with localized electronic spins54,55 may unfavorably decrease the VV coherence times56.

Notably, our simulations did not incorporate any prior information on the formation mechanism of the spin defects found here. This highlights the potential of an atomistic approach to identify not only novel spin defects when combined with electronic-structure calculations but also microscopic mechanisms of their formation.

## Discussion

Our simulations of the relative stability of monovacancies and divacancies in 3C-SiC and their dynamics have revealed that, to generate a high concentration of VV in the material, a large ratio of silicon to carbon-vacancy concentration is needed. Several of the defect-transformation mechanisms observed in our simulations are consistent with and help explain annealing experiments, both in as-grown samples and irradiated/ion-implanted samples. ﻿

Divacancies are found to undergo crystallographic reorientation without dissociating. These findings indicate that the optimal annealing temperature for reorientation, whose control is of interest for sensing applications, is lower than that required to generate the defects.

Beyond having delineated the formation mechanisms of VV, VSi, and CSiVC spin defects, which arise from the dissociation of VV into CSiVC and another VC, we have also discovered plausible new spin defects and characterized their electronic structure using hybrid density-functional theory. Such defects provide new opportunities for quantum technology applications.

## Methods

### Classical MD simulations for divacancy dynamics

Bulk 3C-SiC was modeled as a cubic supercell containing 4096 atoms with 3D periodic boundary conditions. Energy minimization and MD simulations were carried out using LAMMPS57 and an environment-dependent interatomic potential (EDIP) force field39, chosen after benchmarking several empirical potentials for SiC (Supplementary Fig. 1). We equilibrated a 3C-SiC structure at the target temperature T and pressure of 1 atm in the NPT ensemble for 1 ns with a 1.0-fs time step. The system was further equilibrated for an additional 1 ns in the NVT ensemble. A Nose–Hoover thermostat and barostat were used to control temperature and pressure, respectively.

After the 2-ns long equilibration phase of the bulk crystal, three types of NVT-production runs were conducted with supercells containing one of the following: (1) a pair of VC and VSi and (2) a divacancy, and (3) 40 monovacancies with varying composition of monovacancies. In the first case, 20 independent 10-ns-long trajectories were sampled at each T between 500 K and 2500 K, every 500 K. In the second case, 100 independent 10-ns-long trajectories starting with a single divacancy were sampled at each temperature between 1500 K and 2500 K, every 100 K. In the last case, a high-vacancy concentration (40/4096 = 1%) was used to increase the probability of forming VVs from monovacancies within the simulation time of 100 ns. At each composition (ratio of VSi to VC), 25 independent 100-ns-long trajectories were collected at 1500 K. The VV-conversion efficiency at time t was calculated by dividing the number of divacancies at time t by half of the number of monovacancies at t = 0. Overall, we performed over 2200 MD simulations of 10–100 ns each.

### Void-volume analysis for vacancy defects

To identify and track the position of vacancy defects, we constructed Voronoi cells around each atom in a bulk crystal using VORO++ 58. The location of the vacancy defect was defined by the centroid position of an empty Voronoi cell after aligning the atoms in the bulk crystal to those in the MD snapshot. We computed the 3D void volume at the vacancy-defect site by calculating the distribution of voids on a discretized 3D grid with a spacing of 0.15 Å × 0.15 Å × 0.15 Å. The excluded volume of each atom was approximated as a sphere with a radius R (1.6 Å for carbon and 2.0 Å for silicon), where the ratio of the C and Si radii was the same as the ratio of their respective van der Waals radii.

### Determination of defect-energy levels using DFT calculations

Electronic-structure calculations were carried out using DFT as implemented in the Quantum Espresso59 and Qbox code60,61. Defects were modeled using 4 × 4 × 4 or 512-atom supercells of 3C-SiC. The interaction between core and valence electrons was described using Optimized Norm-Conserving Vanderbilt pseudopotentials from the SG15 library62 and we used a plane-wave basis with a kinetic energy cutoff of 55 Ry; the Brillouin zone was sampled with the Γ-point. For each defect calculation, a smearing with a width of 0.001 Ry was used in ground-state electronic-structure calculation, using the Marzari–Vanderbilt procedure63; geometries were optimized at the generalized gradient approximation (GGA) level of DFT using the ﻿Perdew–Burke–Ernzerhof (PBE) functional64 in Quantum Espresso. The defect-level energy diagrams were obtained using ﻿the dielectric-dependent hybrid (DDH) functional﻿ with a self-consistent Hartree–Fock mixing parameter of α = 0.15 for SiC52 at the ground-state triplet configuration using the recursive bisection method65 as implemented in the Qbox code.

### Free-energy calculations by coupling FPMD simulations with enhanced sampling

The free-energy landscape66,67,68,69 for defect migration and formation was computed using the combined force-frequency method or CFF-FPMD36 as implemented in the software package SSAGES70 coupled to the FPMD software Qbox. ﻿Unbiased FPMD and enhanced sampling simulations were performed in the NVT ensemble using the Bussi−Donadio−Parrinello thermostat71 with a time step of 0.967 fs at T = 1500 K. To evaluate energy and forces, ﻿DFT calculations were carried out using the PBE functional using plane waves with a 40-Ry kinetic energy cutoff and the Γ-point to sample the Brillouin zone. We used the same pseudopotential as in the defect energy-level calculations. For computational efficiency, the system was modeled with a 216-atom cell (see Supplementary Note 4 for the finite-size effect). Defect dynamical properties were simulated for the ground-state triplet configuration. The total charge of the supercell was neutral; however, charge rearrangements (i.e., charges near individual vacancy sites) during vacancy migration are allowed to occur in FPMD simulations. The x- and y-coordinates of a mobile atom displacing a vacancy site were used as the collective variables (CV), e.g., $${\xi }_{i,x}={X}_{{C}_{i}}$$ and $${\xi }_{i,y}={Y}_{{C}_{i}}$$ for the VC migration. During CFF-FPMD simulations, running average of forces and frequencies was recorded on a 2D grid with a grid spacing of 0.21 Å along $${\xi }_{i,x}$$ and $${\xi }_{i,y}$$. Bayesian-regularized artificial neural networks (BRANN) with two hidden layers (8 and 6 nodes in the first and the second layers, respectively) were used. For each free-energy calculation, up to 63 walkers/replicas were employed, and simulations were carried out up to 30 ps per walker. To compute the mean free-energy path (MFEP), we fitted the 2D-PMF to a BRANN to enable a smooth interpolation of free energies from a discretized 2D-CV grid36. The nudge elastic-band method72 with a spring constant of 5 eV/Å was applied to the BRANN-fitted PMF to find the MFEP between the initial and final states.