As a promising material for quantum technology, silicon carbide (SiC) has attracted great interest in materials science. Carbon vacancy is a dominant defect in 4HSiC. Thus, understanding the properties of this defect is critical to its application, and the atomic and electronic structures of the defects needs to be identified. In this study, density functional theory was used to characterize the carbon vacancy defects in hexagonal (h) and cubic (k) lattice sites. The zerophonon line energies, hyperfine tensors, and formation energies of carbon vacancies with different charge states (2^{−}, ^{−}, 0,^{+} and 2^{+}) in different supercells (72, 128, 400 and 576 atoms) were calculated using standard Perdew–Burke–Ernzerhof and Heyd–Scuseria–Ernzerhof methods. Results show that the zerophonon line energies of carbon vacancy defects are much lower than those of divacancy defects, indicating that the former is more likely to reach the excited state than the latter. The hyperfine tensors of V_{C}^{+}(h) and V_{C}^{+}(k) were calculated. Comparison of the calculated hyperfine tensor with the experimental results indicates the existence of carbon vacancies in SiC lattice. The calculation of formation energy shows that the most stable carbon vacancy defects in the material are V_{C}^{2+}(k), V_{C}^{+}(k), V_{C}(k), V_{C}^{−}(k) and V_{C}^{2−}(k) as the electronic chemical potential increases.
HIGHLIGHTS

The properties of carbon vacancy defects under different supercells of 4HSiC are compared.

The energy bands of 4HSiC under different correlation functionals are studied.

The effect of carbon vacancy defects on the properties of 4HSiC is explored.
1. Introduction
As a wide band gap semiconductor, silicon carbide (SiC) plays an important role in the power electronics industry owing to its advantages of high electronic breakdown field, high thermal conductivity, and chemical stability.^{1–3} SiC has different polytypes, of which 3C, 4H, and 6H are the most common ones. Moreover, 4HSiC is suitable for manufacturing highpower electronic devices because of its large forbidden band width (3.26 eV), good thermal conductivity, and relatively small anisotropy.^{4–6}
So far, research on the growth of 4HSiC has made remarkable achievements.^{7–9} Carbon vacancy (V_{C}) is a significant defect with strong and harmful effects on the carrier lifetime of 4HSiC.^{10,11} Previous studies used electron paramagnetic resonance to show V_{C}^{+} defects in Ptype 4HSiC annealed at high temperatures and analyzed the influence of external environmental changes on the defects, but some basic electrical properties of V_{C}^{+} defects remain to be characterized.^{12} Point defects in SiC were theoretically studied employing pseudopotential calculations without full ionic relaxations and using only a small supercell.^{13} Recent studies have performed accurate pseudopotential calculations with full ionic relaxations employing a large supercell.^{14–17} Nevertheless, the present understanding of point defects in SiC remains incomplete.
Therefore, the present study used the firstprinciples method to characterize the geometric structure and electrical properties of the carbon vacancy in 4HSiC supercell. The defects at cubic and hexagonal lattice sites were considered separately in our calculations.
The outline of the article is as follows. Section 2 describes the calculation details and model. Section 3 presents the results for the carbon vacancy defects: zerophonon line energy, hyperfine tensor, and formation energy curves. Finally, Section 4 discusses the results of the properties of carbon vacancy defects in SiC.
2. Calculation details andmodel
2.1. Details of calculation
Density functional theory (DFT) calculations were carried out using the 5.4.4 version of Vienna Ab Initio Simulation Package (VASP).^{18,19} The projector augmented wave (PAW) method was employed to describe the core electrons.^{18,20,21} Spinpolarized electronic structure calculations were carried out using the Perdew–Burke–Ernzerhof (PBE)^{22} and Heyd–Scuseria–Ernzerh of (HSE06)^{23} exchangecorrelation functionals. For SiC, the HSE06 functional provides accurate results while considering the contribution of the spin polarization of the core electrons to the Fermicontact term.^{23}
Planewave basis with kinetic energy up to 420 eV was used to describe the Kohn–Sham electronic states. For defect modeling, the unit cell must be optimized. The optimized lattice constants and band gaps for the 4H–SiC unit cell were obtained using a Monkhorst–Pack grid of 9 × 9 × 5 with a planewave cutoff energy of 420 eV.^{24} In this calculation, the electronic and ionic convergence thresholds were set as 1 × 10^{−5} eV and 1 × 10^{−4} eV, respectively.
2.2. Model
The basic structural unit of 4HSiC is SiC tetrahedron, wurtzite structure and belongs to the hexagonal crystal system. Its stacking method is ABCB type. The theoretical model of 4HSiC is shown in Fig. 1. The large atoms in the figure are silicon, and the small atoms are carbon. The crystal of this structure is an ABtype covalent bond crystal, and each silicon atom is surrounded by four carbon atoms bonded by oriented strong tetrahedral sp3 bonds. Each 4HSiC unit cell contains four silicon atoms and four carbon atoms. The space group of 4HSiC is P6_3 mc, and the lattice constants are a = b = 3.073 Å and c = 10.053 Å. For defect modeling, large 72, 128, 400, and 576 atom supercells were applied for 4HSiC. Fig. 2 shows the structure of a carbon vacancy defect containing 576 atomic supercells.
3. Results and discussion
The basic structure and energy band of the 4HSiC unit cell were calculated to ensure the accuracy of the 4HSiC model. Only under the premise of obtaining sufficiently good results can the electrical properties of the 4HSiC supercell structure containing carbon vacancy defects be calculated.
3.1. 4HSiCunit cell calculation results
3.1.1. Structure of crystal lattice
The structure, including the atomic position and lattice parameters, of the 4HSiC unit cell is optimized using the PBE exchangecorrelation functional. Fig. 3 shows the optimization results of the lattice constants of the 4HSiC unit cell structure. Fig. 3(a) is the optimization of lattice constants a and b, and Fig. 3(b) is the optimization of lattice constant c. Finally, the lattice constants of the 4HSiC unit cell are a = b = 3.094 Å and c = 10.125 Å. The results are the same as those calculated by Davidsson et al.^{25} Compared with the experimental value, calculated lattice constants differ by about 0.7%, which agrees well with the experimental value. This result indicates that the PAW pseudopotential and PBE exchangecorrelation functional used in the simulation calculation can describe the structure of the 4HSiC well.
3.1.2. Energy band structures
On the basis of the previously optimized 4HSiC unit cell structure, the electrical properties of the 4HSiC unit cell can be calculated nonselfconsistently. The choice of exchangecorrelation functional in the calculation may lead to a large difference in the energy band calculation results. Two exchangecorrelation functionals (PBE and HSE06) are used in the calculation of the 4HSiC unit cell to obtain accurate calculation results. The energy band results under different exchangecorrelation functionals were compared, and the result closer to the true value was selected for the subsequent nonselfconsistent calculation of the supercell.
Given that 4HSiC has a hexagonal crystal structure, the high symmetric Kpoint path of the Brillouin zone used in the calculation is Γ → M → K → Γ → A → L → H → A, where the Γ point is Brillouin in the center of the district. The path of highsymmetry points is shown in Fig. 4, and the K point used is 9 × 9 × 5. Fig. 5 shows the calculation results of the two exchangecorrelation functionals.
Figure 5(a) shows the energy band structure calculated using the PBE exchangecorrelation functional method. The 4HSiC band gap width is 2.63 eV, which is different from the experimental result of 3.26 eV but is similar to the calculated result of Cheng et al.^{26} This discrepancy can be ascribed to the fact that the PBE functional underestimates the correlation between excited state electrons, which explains the low calculation result of the energy band. The energy band result using the HSE06 functional obtained is shown in Fig. 5(b). In addition, the band gap of the 4HSiC unit cell is 3.18 eV, which is close to the experimental result of 3.26 eV with a difference of about 2.4%. The result is similar to the 3.19 eV calculated by Yan et al.^{27} The valence band maximum of the 4HSiC energy band diagram is at Γ point, and the conduction band minimum (CBM) is at M point, which is an indirect band gap semiconductor. The comparison shows that the HSE06 functional is more accurate than the PBE functional in describing the energy band structure of 4HSiC.
3.2. Electrical properties of carbon vacancy defects in 4HSiC supercells
On the basis of the previously optimized unit cell structure, 4 supercell models of 3 × 3 × 1, 4 × 4 × 1, 5 × 5 × 2 and 6 × 6 × 2 were established, respectively, that is, the supercells contain 72, 128, 400 and 576 atoms, respectively. In actual calculations, the calculation cost of the HSE06 functional is excessively large, and the calculation time is excessively long. Therefore, the PBE functional was used to simulate the supercell structure. It has little effect on comparing the results of different super cells. The lattice constant, Fermi level, zerophonon line energy, formation energy, and hyperfine tensor of the carbon vacancy defects were calculated, and the structural and electrical properties of the carbon vacancy defects at two different lattice positions were systematically studied.
3.2.1. Lattice constant
The supercell structure needs to be optimized. Table 1 shows the corresponding lattice structure parameters of each supercell.
Atoms .  72 .  128 .  400 .  576 .  

Lattice constants .  a (Å) .  c (Å) .  a (Å) .  c (Å) .  a (Å) .  c (Å) .  a (Å) .  c (Å) . 
Supercell  3.094  10.125  3.094  10.125  3.094  10.125  3.094  10.125 
V_{C}(h)  3.086  10.139  3.088  10.129  3.089  10.120  3.090  10.119 
V_{C}(k)  3.087  10.126  3.088  10.126  3.089  10.120  3.090  10.119 
Atoms .  72 .  128 .  400 .  576 .  

Lattice constants .  a (Å) .  c (Å) .  a (Å) .  c (Å) .  a (Å) .  c (Å) .  a (Å) .  c (Å) . 
Supercell  3.094  10.125  3.094  10.125  3.094  10.125  3.094  10.125 
V_{C}(h)  3.086  10.139  3.088  10.129  3.089  10.120  3.090  10.119 
V_{C}(k)  3.087  10.126  3.088  10.126  3.089  10.120  3.090  10.119 
For the same supercell, the lattice constant of the supercell containing carbon vacancy defects is smaller than that of the supercell without defects. This result can be ascribed to the fact that the supercell containing carbon vacancy defects loses a carbon atom compared with the perfect supercell. As a result, the force between atoms adjacent to the carbon atom weakens, and the lattice constant becomes smaller. The carbon vacancy defects at two different lattice positions have almost the same effect on the lattice constant of the material because they both lose a carbon atom, and the interaction forces of carbon atoms at different lattice positions are almost the same. As the supercell increases (72 atoms to 576 atoms), the lattice constant a of the defect gradually increases, and the lattice constant c gradually decreases because the defect concentration decreases with the increase in supercells.
3.2.2. Fermi level
The charged defects may have a certain impact on the Fermi level. Thus, the Fermi level of the carbon vacancy defects with a positive charge on the two lattice positions was calculated. A static selfconsistent calculation was performed on the basis of the optimized supercell to obtain its Fermi level, as shown in Table 2.
Atoms .  72 .  128 .  400 .  576 . 

Supercell  0  0  0  0 
V_{C}(h)  1.415  1.401  1.509  1.530 
V_{C}(k)  1.399  1.384  1.414  1.431 
V_{C}^{+}(h)  0.878  0.921  0.879  0.832 
V_{C}^{+}(k)  0.957  0.941  0.875  0.919 
Atoms .  72 .  128 .  400 .  576 . 

Supercell  0  0  0  0 
V_{C}(h)  1.415  1.401  1.509  1.530 
V_{C}(k)  1.399  1.384  1.414  1.431 
V_{C}^{+}(h)  0.878  0.921  0.879  0.832 
V_{C}^{+}(k)  0.957  0.941  0.875  0.919 
For the same supercell, the Fermi levels of V_{C}(h), V_{C}(k), $ V c + $(h), and $ V c + $(k) all move to a higher energy because the supercell with defects is smaller than the perfect supercell, and the number of electrons carried by carbon atoms is less than the number of electrons carried by silicon atoms. Thus, the number of valence electrons per unit volume is increased, andthe Fermi level of the supercell with defects increases.
3.2.3. Zerophonon line energy
The zerophonon line energy is the difference between the energy of the ground state and the energy of the excited state. Therefore, understanding the groundstate and excitedstate electronic structures of the carbon vacancy defects is necessary, as shown in Fig. 6. In this section, the zerophonon line energy of the doublevacancy defect was also calculated to understand the carbon vacancy defect. The calculation result is shown in Fig. 7.
Figure 7 shows a comparison of the zerophonon line energy of the defects under different supercells. Results show that the zerophonon line of the defect gradually converges as the supercell increases. In the case of a supercell with 72 atoms, the defect concentration is large, which makes the results unreliable. When the supercell contains 576 atoms, the zerophonon line energy of the doublevacancy defect is 1.046 eV, which is similar to the calculation results of others.^{25,28} The zerophonon line energy of the V_{C}(h) defect is 0.216 eV, and the zerophonon line energy of the V_{C}(k) defect is 0.414 eV. As seen in Fig. 7, the calculated values of ZPL energy using 4 × 4 × 1 supercell (128 atoms) are very close to those calculated values which use 5 × 5 × 2 and 6 × 6 × 2 supercells, therefore, 4 × 4 × 1 supercell is large enough for the calculations of ZPL energy of carbon vacancy and double vacancy. The zerophonon line energy of the carbon vacancy defects is much lower than that of the doublevacancy defects. This result indicates that the singlevacancy defects more easily reach the excited state than the doublevacancy defects.
3.2.4. Formation energy
Defect formation energy is an important concept that directly reflects the difficulty of formation of defects and the degree of stability of defects in the material. It can also reflect the concentration of defects in the material to a certain extent. Therefore, calculating defect formation energy is important for the study of carbon vacancy defects in 4HSiC. This section mainly studies the formation energies of carbon vacancy defects with different charges based on the structure of these two lattice positions. The formation energy of charged defects in this article is calculated as follows^{29}:
where E_{def}(q) represents the total energy of supercells containing carbon vacancy defects, E_{0} represents the total energy of supercells without defects, n_{i} represents the number of atoms added or removed in the supercell (for carbon vacancy defects, it means losing an atom), u_{i} represents the chemical potential of adding or removing atoms, that is, the chemical potential of carbon atoms, E_{VBM} represents the VBM corresponding to the supercell containing the defect, and E_{F} represents the energy corresponding to the Fermi level (i.e., the electronic chemical potential), and it varies from zero to the forbidden band width.
However, when calculating the defect formation energy, false interactions of charged defects occur because the periodic boundary conditions are used to construct the supercell model. Such interactions affect the total energy of the supercell. For such a situation, the Freysoldt, Neugebauer, and Van de Walle (FNV) scheme is used to make corrections to obtain realistic results.^{29} The key idea of the FNV scheme is to introduce the real charge distribution to simulate the actual distribution of the charge carried by the defect.
A supercell with 128 atoms is used to calculate the formation energy of defects with different charges and lattice positions in 4HSiC. The calculation parameters are the same as the previous calculations. The calculation results are shown in Fig. 8.
The left image in Fig. 8 shows the variation in the formation energy corresponding to the 10 types of carbon vacancy defects with the electronic chemical potential, and the right image is an enlarged view of the box in the left image. Calculation results show that the formation energy of V_{C}(k) defects in 4HSiC is 4.857 eV, and the formation energy of V_{C}(h) defects is 4.917 eV. The results are similar to those calculated by Kobayashi et al.^{30} The formation energy does not change with the change in electronic chemical potential. That is, the formation energy of neutral chargestate defects remains unchanged regardless of whether the 4HSiC material is Ntype or Ptype. The charged defects change with the crystal environment and with the increase in the electronic chemical potential. In the transition of 4HSiC from Ptype to high purity to Ntype, the most stable defects in 4HSiC are $ V c 2 + $(k), $ V c + $(k), V_{C}(k), $ V c \u2212 $(k), and $ V c 2 \u2212 $(k). Cheng et al.^{31} also showed that the concentration of carbon vacancy defects is the highest in 4HSiC. In Ptype 4HSiC, the formation energy of positively charged carbon vacancy defects is the lowest, indicating that most of the carbon vacancy defects are positively charged. In Ntype 4HSiC, the formation energy of negatively charged carbon vacancy defects is the lowest. This result shows that most of the carbon vacancy defects are negatively charged, whereas the defects in highpurity silicon carbide (SiC) are generally not charged. Among the carbon vacancies of the same charge, the formation energy of the carbon vacancy defects in the hexagonal lattice position is always greater than that in the cubic lattice position. This result indicates that the carbon vacancy defects in the cubic lattice position form more easily and have a higher concentration than those in the hexagonal lattice position. It is consistent with the conclusion of Capan et al.^{32}
3.2.5. Hyperfine tensor
Hyperfine coupling (also called hyperfine interaction) refers to the interaction between unpaired electron spin and nuclear spin. If magnetic nuclei exist around the unpaired electrons, a single EPR line will split into many narrower lines under the hyperfine interaction. These lines are the hyperfine structure of the spectrum. In the previous work,^{12} the hyperfine tensors of the two types of carbon vacancy defects were obtained by electron paramagnetic resonance. In this section, the hyperfine tensors for these two types of carbon vacancy defects were calculated. The results are shown in Tables 3 and 4. These tables only record the hyperfine tensor value of the carbon vacancy defects and the four nearest silicon atoms, which is sufficient for identifying the defect structure. Among them, x, y and z are the hyperfine tensor values in three directions, and the bolded parts in these tables are the results of Ref. 33. The supercell selected in Ref. 33 contains 128 atoms.
V_{C}^{+}(h) .  Atoms .  Hyperfine tensor (MHz) .  

x .  y .  z .  
A(Si_{1})  72  199.9  199.9  290.0 
128  313.9  313.9  461.0  
400  307.9  308.0  455.1  
576  307.2  307.2  454.5  
Ref. 33  275  275  400  
A(Si_{2–4})  72  125.1  124.3  156.1 
128  21.8  19.9  36.2  
400  21.0  18.9  35.5  
576  21.1  19.1  35.34  
Ref. 33  22  20  43 
V_{C}^{+}(k) .  Atoms .  Hyperfine tensor (MHz) .  

x .  y .  z .  
A(Si_{1})  72  154.2  154.2  207.8 
128  292.3  292.3  409.0  
400  336.5  336.5  476.9  
576  341.5  341.5  484.9  
Ref. 33  122  114  197  
A(Si_{2–4})  72  157.5  157.4  200.3 
128  68.7  68.0  88.3  
400  35.8  33.6  48.6  
576  33.2  30.7  45.4  
Ref. 33  93  78  155 
As listed in Table 3, for the carbon vacancy at the hexagonal lattice position $ V c + $(h), the hyperfine tensor results converge gradually with the increase in supercell and are close to the experimental results in the previous work.^{12} The interaction between the nucleus and the electron cloud can be well described by using the PAW method and the PBE functional. In addition, $ V c + $(h) has C_{3v} symmetry. Table 3 shows that the hyperfine tensor results corresponding to the supercell with only 72 atoms have a large deviation compared to results for larger cells because the supercell of 72 atoms is too small, which leads to the high concentration of defects in the material. As a result, the hyperfine tensor results are not accurate. However, Table 4 displays that for the carbon vacancy in the cubic lattice $ V c + $(k), the exchangecorrelation functional PBE may not describe the hyperfine tensors of carbon vacancy defects in SiC. In addition, the pseudopotential in the VASP software selected in this article is not the same as the references. With the increase of supercell atoms, the hyperfine tensors of $ V c + $(h) defects tend to be $ V c + $(k) defects, and the hyperfine tensor values of $ V c + $(k) defects in the VASP system do not reach a steady state, leading to the transformation of defects.
4. Conclusions
The energy levels and electrical properties of carbon vacancy defects in 4HSiC were calculated by firstprinciples VASP software. Aiming at the carbon vacancy defects in SiC, this study investigated the effect of defect concentration on the experiment by changing the size of the supercell. This article plays a guiding role in the preparation of highpurity SiC materials. Given the existence of two lattice positions in 4HSiC, two types of carbon vacancy defects appear in different lattice positions: $ V c + $(h) and $ V c + $(k). Considering the influence of defect concentration on the calculation results, the lattice constants and Fermi levels of supercells with different sizes (72, 128, 400 and 576 atoms) are calculated. The results show that the lattice constants of supercells with carbon vacancy defects are smaller than those of perfect supercells because supercells with defects lose a carbon atom relative to the perfect supercells, which weakens the interaction force between the atoms adjacent to the carbon atom, thus decreasing the lattice constant. For the Fermi level, with the increase in the number of atoms in the supercell, the Fermi level of the perfect lattice moves to the lower energy band. For the same supercell, the Fermi levels of V_{C}(h), V_{C}(k), $ V c + $(h), and $ V c + $(k) all move to the higher energy. Then, the zerophonon lines of two kinds of defect structures were calculated. The results show that for the twocarbon vacancy defects, their zerophonon line energy is considerably far lower than the doublevacancy zerophonon line energy, that is, the carbon vacancy defects more easily reach the excited state than the doublevacancy defects. Then, the formation energies of 10 types of defect structures in 4HSiC were calculated. The formation energy of V_{C}(k) defects in 4HSiC is 4.857 eV, and the formation energy of V_{C}(h) defects is 4.917 eV. The results also show that with the increase in the electronic chemical potential, the most stable defect in 4HSiC is $ V c 2 + $(k), $ V c + $(k), V_{C}(k), $ V c \u2212 $(k) and $ V c 2 \u2212 $(k). In the same charge state, the formation energy of carbon vacancy defects in the hexagonal lattice is always greater than that in the cubic lattice, which indicates that the carbon vacancy defects in the cubic lattice form more easily and have higher concentration than those in the hexagonal lattice.
Declaration of Competing Interest
None.
Acknowledgments
The study is supported by the National Natural Science Foundation of China (No. 51575389, 51761135106), the National Key Research and Development Program of China (No. 2016YFB1102203), the State Key Laboratory of Precision Measuring Technology and Instruments (Pilt1705), and the “111” Project by the State Administration of Foreign Experts Affairs and the Ministry of Education of China (No. B07014). Computational research performed at the University of Helsinki was supported by the EU Project M4F (Project ID: 755039). CSCIT Center for Science, Finland, is acknowledged for providing the computational resources. The authors thank Dr. Ilja Makkonen and Dr. Joel Davidsson for the helpful discussion. The authors also thank for the valuable discussions with Mr. Fei Ren from Lanzhou University and Prof. Dejun Wang from Dalian University of Technology.
References
Xiuhong Wang, State Key Laboratory of Precision Measuring Technology and Instruments, School of Precision Instruments and Optoelectronics Engineering, Tianjin University. Ms.Wang is studying for Master degree. Her research interests include silicon carbide material and deep level defect.
Junlei Zhao, PhD, Department of Physics, University of Helsinki; Department of Electrical and Electronic Engineering, Southern University of Science and Technology. His research interests include: multiscale computational modeling of nanomaterials, wide bandgap semiconductor and highentropy alloy, etc.
Zongwei Xu, Associate Professor, State Key Laboratory of Precision Measuring Technology and Instruments, School of Precision Instrument and OptoElectronics Engineering, Tianjin University. His research interests include: defect engineering in wide band gap semiconductor, micro/nanofabrication using focused ion beam, Raman and fluorescence spectrum, etc.
Flyura Djurabekova, Professor in Materials in Extreme Environments, Department of Physics, University of Helsinki. Her group studies the materials for accelerator technology within the Accelerator Technology project at HIP by means of different computational methods.
Mathias Rommel, FriedrichAlexanderUniversität ErlangenNürnberg (FAU) and Fraunhofer Institute for Integrated Systems and Device Technology IISB, Germany. His research interests include: focused ion beam (FIB), nanoimprint lithography, electrical scanning probe microscopy, and deep level transient spectroscopy (DLTS).
Ying Song is working in the area of manufacturing and spectral characterization of silicon carbide color centers, including the preparation of silicon carbide color centers by ionimplantation, threedimensional Raman and photoluminescence spectral characterization, and model of spectral depth profiling.
Fengzhou Fang, Professor, State Key Laboratory of Precision Measuring Technology and Instruments, School of Precision Instrument and OptoElectronics Engineering, Tianjin University. His research interests are in the areas of micro/nano manufacturing, optical freeform manufacturing, biomedical manufacturing, and ultraprecision machining and metrology.