With the first-principle method, we studied the effects of the type and position of defects on the defect formation energy, electronic band structure, and electron mobility of the 4-layer hexagonal system silicon carbon (4H–SiC). The vacancy defect formation energy is smaller than the interstitial defect formation energy. The C vacancy defect formation energy is the smallest, while the Si interstitial defect formation energy is the largest. The defect formation energy is little affected by the defect position. The electronic band structure shows semi-metallic property due to the vacancy defect and the interstitial defect, and it shows a smaller bandgap due to the antisite defect. The electronic band structure changes little while the defect position changes. The electron mobility is reduced in varying degrees according to different defect types. The electron mobility changes little while the defect position changes.

Silicon carbide (SiC) is outstanding in radiation conditions because of its wide bandgap, high breakdown voltage, high radiation resistance, and high saturated electron mobility.1–3 Among the commercially available polytypes SiC, the 4-layer hexagonal system SiC (4H–SiC) is more promising for nuclear application.4,5

4H–SiC has been studied and utilized in nuclear systems for decades.6 In nuclear reactors, 4H–SiC deals with the fabrication of the inner walls of the toroidal chamber facing the plasma, extracting the heat generated by the fusion reactions and transferring that heat to the coolant.7–10 In radiation detectors, the grown 4H–SiC epilayers are controllable and reproducible, and diodes realized on them have shown superior detection properties.11 However, due to radiation defects, some degradations of its radiation responses was observed in the above applications, resulting in shortened service life and reduced accuracy.12 In serious cases, the radiation defects lead to the failure of applications and then affect the normal operation of the application system.13 

Thus, studying the effects of defects on the 4H–SiC performance degradation is meaningful for reducing degradation, ensuring the normal and effective operation, and finding possible solutions for radiation response improvements. Up until now, many valuable studies have been devoted to the effects of defects on 4H–SiC. Li et al. studied the threshold displacement energy of Si and C toward different directions and the defect generation process caused by Si in 4H–SiC.14 Iskandarova et al. studied the effects of vacancy and antisite defects on carrier mobility degradation in 4H–SiC.15 Vali et al. studied the effects of defects caused by thermal neutron radiation on structural and electrical properties of n-type 4H–SiC.16 Lebedev et al. studied the effects of radiation defects on the radiation resistance of heterostructures 3C–SiC/4H–SiC.17 Given all of that, the effects of defects on 4H–SiC deserves more research. At present, the effects of defects on the defect formation energy, electronic band structure, and electron mobility need more investigation.

The purpose of this study was to investigate the effects of the type and position of defects on the defect formation energy, electronic band structure, and electron mobility in 4H–SiC. The presented results provide a fundamental insight into the microscopic mechanism of the effects of defects on 4H–SiC and advance the understanding of the radiation responses of 4H–SiC.

The research method of this study is the first-principle theory, which has been the most important method for obtaining the electronic properties of realistic systems.18 All calculations were performed based on the first-principle theory using the OpenMX3.8 code.19 The exchange-correlation functional uses the Perdew–Burke–Ernzerhof (PBE) formulation of the generalized gradient approximation (GGA).20 The ambient temperature is 0 K. The cut-off energy used to calculate the matrix and solve the Poisson equation is 300 ry. The maximum number of iterations is set to 700 to ensure the computational integrity of structures with defects. The electron density hybrid method (or density matrix) to generate the input electron density (Hamiltonian) in the Self-Consistent Field (SCF) calculation step is the residual minimization method in the direct inversion iterative subspace with Kerker metric (RMM-DIISK), which is helpful for fast SCF convergence calculation.21 The input electron density (Hamiltonian) at the next SCF step is estimated based on the output electron densities (Hamiltonian) in the several previous SCF steps. The first 12 SCF steps in the estimation use the Kerker hybrid method.22 The absolute deviation between the eigenvalue energy at the current and previous SCF steps in the convergence criterion (Hartree) for the SCF calculation is 1.0 × 10−6 (Hartree). The basis sets are fully relativistic pseudopotentials (VPS) and pseudoatomic orbitals (PAOs).23,24 The PAO basis functions are two s-state, two p-state, and one d-state radial functions (s2p2d1) for C and Si atoms. The periodic boundary condition is used to expand the calculation dimension. The 2 × 1 × 1 supercell is used for calculation. As shown in Fig. 1, the gray ones are C atoms, and the yellow ones are Si atoms. The lattice constants a, b, and c are 3.078, 3.078, and 10.045 Å, respectively. The array rule of 4H–SiC is shown in Fig. 2, and the 4H–SiC crystal is a hexagonal crystal composed of four layers of Si and C atoms. Layers from bottom to top are defined as layer 1 to layer 4. The defect position is expressed by layer 1 to layer 4. The defect types are vacancy, interstitial, and antisite.25,26

FIG. 1.

(a) Unit cell of 4H–SiC and (b) supercell of 4H–SiC.

FIG. 1.

(a) Unit cell of 4H–SiC and (b) supercell of 4H–SiC.

Close modal
FIG. 2.

The array rule of 4H–SiC.

FIG. 2.

The array rule of 4H–SiC.

Close modal

The defect formation energy determines the defect concentration of some important processes.27 The defect formation energy is defined as the energy needed to form a stable structure determining the difficulty of producing defects, which is calculated by19,28,29

Ef=EdefEpri+iΔniμi,
(1)

where Edef is the total energy of defective 4H–SiC, Epri is the total energy of pristine 4H–SiC, △ni is the change in the number of species i (i is C or Si), and μi is the chemical potential of species i, which is calculated by28,30

μi=Eibulkni,μi=Eibulkniμi,
(2)

where Eibulk is the total energy of bulk i (i is C or Si), Eijbulk is the total energy of bulk ij (ij is SiC), and n is the number of species i or species ij. All the total energies are calculated by OpenMX3.8.

When i is C, μi = −5.84 Ha and μj = −4.16 Ha; this condition is called the C-rich condition. When a is Si, μi = −4.14 Ha and μj = −5.86 Ha; this condition is called the Si-rich condition.

The defect formation energies under the C-rich condition are different from those under the Si-rich condition. As shown in Fig. 3, the C vacancy defect formation energies and the C antisite defect formation energies decrease from the C-rich condition to the Si-rich condition. The defect formation energies of the Si antisite, Si vacancy, and C interstitial defects increase from the C-rich condition to the Si-rich condition. Under the C-rich condition, the C antisite defect formation energy is greater than the Si vacancy defect formation energy. Under the Si-rich condition, the C antisite defect formation energy is greater than the Si vacancy defect formation energy.

FIG. 3.

The changes of the defect formation energies from the C-rich condition to the Si-rich condition.

FIG. 3.

The changes of the defect formation energies from the C-rich condition to the Si-rich condition.

Close modal

The defect formation energies of different defect types and layers are calculated. As shown in Fig. 4, different defect types lead to different defect formation energies. The defect types and layers have different effects on defect formation energies. The C vacancy defect is the most easily formed defect. The average C vacancy defect formation energy is 3.31 eV. The Si interstitial defect is the most challenging defect to form. The average Si interstitial defect formation energy is 463.61 eV. The Si interstitial defect formation energy is much greater than other defect formation energies, so it is not shown in the figure. C is more accessible to break away from the lattice than Si because the atomic mass of Si is greater than that of C. Vacancy defects are more accessible to be formed than interstitial defects because an atom needs more energy to enter the interstitial between atoms after breaking away from the lattice. Compared with the defect types, the effects of defect layers on the defect formation energy is very small. In the same layer, the defect formation energy remains unchanged with the change in the interstitial defect position. The defect formation energy of a defect in layer 1 is approximately equal to that in layer 2. The defect formation energy of a defect in layer 3 is roughly equivalent to that in layer 4. The defect type has a more significant effect on the defect formation energy than the defect type.

FIG. 4.

The defect formation energies of 4H–SiC with defects under (a) C-rich condition and (b) Si-rich condition.

FIG. 4.

The defect formation energies of 4H–SiC with defects under (a) C-rich condition and (b) Si-rich condition.

Close modal

Electronic band structure is an important characteristic parameter of semiconductors.30–32 The electronic band structure of pristine 4H–SiC is calculated. As shown in Fig. 5, the Fermi energy is 0 eV. The minimum energy of the conduction band (CBM) is 0.85 eV at the Γ point, and the maximum energy of the valence band (VBM) is −1.67 eV at the M point. The bandgap is 2.52 eV, which is 0.75 eV less than the experimental value of 3.27 eV.33 The bandgap simulated is underestimated because the correlation between excited electrons is underestimated by the GGA-PBE approximate method. The bandgap simulated is generally smaller than the experimental value.34 

FIG. 5.

The electronic band structure of pristine 4H–SiC.

FIG. 5.

The electronic band structure of pristine 4H–SiC.

Close modal

The electronic band structures of 4H–SiC with one vacancy or interstitial defect are calculated. As shown in Figs. 6 and 7, 4H–SiC with one vacancy or interstitial defect in layer 1 has the same electronic band structure as that in layer 2. 4H–SiC with one vacancy or interstitial defect in layer 3 has the same electronic band structure as that in layer 4. In 4H–SiC with one vacancy defect, each of the four atoms nearest to the vacancy defect has an unpaired electron; the electron forms an unsaturated covalent bond. In 4H–SiC with one interstitial defect, the interstitial atom has four electrons that do not form covalent bonds; these electrons become free electrons when getting certain energy. As a result of the vacancy or interstitial defect, impurity energy bands appear, leading to the semi-metallic properties.

FIG. 6.

The electron band structures of 4H–SiC with (a) one C vacancy defect in layer 1 or 2, (b) one C vacancy defect in layer 3 or 4, (c) one C interstitial defect in layer 1 or 2, and (d) one C vacancy defect in layer 3 or 4.

FIG. 6.

The electron band structures of 4H–SiC with (a) one C vacancy defect in layer 1 or 2, (b) one C vacancy defect in layer 3 or 4, (c) one C interstitial defect in layer 1 or 2, and (d) one C vacancy defect in layer 3 or 4.

Close modal
FIG. 7.

The electron band structure of 4H–SiC with (a) one Si vacancy defect in layer 1 and layer 2, (b) one Si vacancy defect in layer 3 and layer 4, (c) one Si interstitial defect in layer 1 and 2, and (d) one Si vacancy defect in layer 3 and 4.

FIG. 7.

The electron band structure of 4H–SiC with (a) one Si vacancy defect in layer 1 and layer 2, (b) one Si vacancy defect in layer 3 and layer 4, (c) one Si interstitial defect in layer 1 and 2, and (d) one Si vacancy defect in layer 3 and 4.

Close modal

The C defect types and layers have different effects on electronic band structures. C vacancy and interstitial defects lead to impurity energy bands. The impurity energy bands caused by the C vacancy defects cross the Fermi level and overlap with the conduction band. The impurity energy bands caused by the C interstitial defects cross the Fermi level and overlap with the valence band.

The Si defect types and layers have different effects on electronic band structures. Si vacancy and interstitial defects lead to impurity energy bands. The impurity energy bands caused by the Si vacancy defects cross the Fermi level and overlap with the valence band. The impurity energy bands caused by the Si interstitial defects cross the Fermi level and overlap with the conduction band.

The electronic band structures of 4H–SiC with one antisite defect are calculated. As shown in Fig. 8, there is no impurity energy band caused by antisite defects because C has the same number of valence electrons as Si. However, the atomic masses of C and Si are different, resulting in unequal coulomb forces, which make different electronic band structures. However, the bandgap is reduced by antisite effects.

FIG. 8.

The electron band structure of 4H–SiC with (a) one C antisite defect in layer 1 and layer 2, (b) one C antisite defect in layers 3 and 4, (c) one Si antisite defect in layer 1 and layer 2, and (d) one Si antisite defect in layers 3 and 4.

FIG. 8.

The electron band structure of 4H–SiC with (a) one C antisite defect in layer 1 and layer 2, (b) one C antisite defect in layers 3 and 4, (c) one Si antisite defect in layer 1 and layer 2, and (d) one Si antisite defect in layers 3 and 4.

Close modal

The bandgaps of 4H–SiC with one antisite defect are calculated. In Fig. 9, the orange histograms are CBM, the green histograms are VBM, and the line is bandgap. The CBM of 4H–SiC with one antisite defect is higher than that of pristine 4H–SiC. The VBM of 4H–SiC with one antisite defect is lower than that of pristine 4H–SiC. The bandgaps of 4H–SiC with one antisite defect are from large to small: Si antisite defect in layer 3, Si antisite defect in layer 1, C antisite defect in layer 3, and C antisite defect in layer 1.

FIG. 9.

The CBM, VBM, and bandgaps of 4H–SiC with one antisite defect. Pristine is the pristine 4H–SiC, Xan/m (X = C or Si, n or m = 1, 2, 3, or 4) is the X antisite defect in layer n or m.

FIG. 9.

The CBM, VBM, and bandgaps of 4H–SiC with one antisite defect. Pristine is the pristine 4H–SiC, Xan/m (X = C or Si, n or m = 1, 2, 3, or 4) is the X antisite defect in layer n or m.

Close modal

Electron mobility is a fundamental parameter that must be accurately known for device design and circuit simulation.35 Electrons are scattered due to defects. Electron mobility is a physical quantity that describes the frequency of scattering events. The direction of the electron mobilities is from point Γ to point A. The calculation formula of electron mobility is provided by36 

μ=eτm*,
(3)

where μ is the electron mobility, and e is the electron charge, which is 1.602 × 10−19 C. τ is the relaxation time. m* is the effective mass of carrier, which is calculated by36 

m*=2d2E(k)dk21,
(4)

where is the reduced Planck constant, k is the wave vector, and E(k) is the energy corresponding to k. τ and E(k) are calculated by OpenMX3.8.

The effective mass of pristine 4H–SiC is 0.12 me, and me is the electron rest mass, which is 9.109 × 10−31 kg. The effective masses of 4H–SiC with one defect are summarized in Table I. The effective mass of 4H–SiC with one defect in layer 1 is equal to that in layer 2. The effective mass of 4H–SiC with one defect in layer 3 is equal to that in layer 4. The effective mass of 4H–SiC with one defect is greater than that of pristine 4H–SiC except for 4H–SiC with one Si antisite defect in layer 1 or 2.

TABLE I.

The effective mass (me) at the point Γ (unit: me).

Defect typeLayer 1Layer 2Layer 3Layer 4
C vacancy 0.18 0.18 0.25 0.25 
C interstitial 0.19 0.19 0.23 0.23 
C antisite 0.15 0.15 0.19 0.19 
Si vacancy 0.17 0.17 0.15 0.15 
Si interstitial 0.30 0.31 0.31 0.31 
Si antisite 0.12 0.12 0.17 0.17 
Defect typeLayer 1Layer 2Layer 3Layer 4
C vacancy 0.18 0.18 0.25 0.25 
C interstitial 0.19 0.19 0.23 0.23 
C antisite 0.15 0.15 0.19 0.19 
Si vacancy 0.17 0.17 0.15 0.15 
Si interstitial 0.30 0.31 0.31 0.31 
Si antisite 0.12 0.12 0.17 0.17 

The electron mobility of pristine 4H–SiC is 0.29 m2/Vs, and the electron mobilities of 4H–SiC with one defect are summarized in Fig. 10. The electron mobility of 4H–SiC with one defect in layer 1 is equal to that in layer 2. The electron mobility of 4H–SiC with one defect in layer 3 is equal to that in layer 4. Compared with the pristine 4H–SiC, the electron mobility of 4H–SiC with one defect decreases except for the electron mobility of 4H–SiC with one Si antisite defect in layer 1 or 2. The Si antisite defect in layer 1 or 2 makes the largest electron mobility, average 0.29 m2/Vs, which is approximately equal to that of pristine 4H–SiC. 4H–SiC with one Si interstitial defect shows the smallest electron mobility. Therefore, the electron mobility of 4H–SiC is affected by defects depending on the type and layer.

FIG. 10.

The electron mobility (unit: m2/vs along the z-direction of 4H–SiC with (a) one C defect and (b) one Si defect.

FIG. 10.

The electron mobility (unit: m2/vs along the z-direction of 4H–SiC with (a) one C defect and (b) one Si defect.

Close modal

The first-principle theory calculations were carried out to investigate the effects of the type and position of a defect on the defect formation energy, electronic band structure, and electron mobility in 4H–SiC.

The defect formation energy affected by defect types: Under the C-rich condition, the defect formation energies of 4H–SiC with one defect are from large to small: Si interstitial defects, C interstitial defects, Si vacancy defects, C antisite defects, Si antisite defects, and C vacancy defects; Under the Si-rich condition, the defect formation energies of a defect are from large to small: Si interstitial defects, C interstitial defects, C antisite defects, Si vacancy defects, Si antisite defects, and C vacancy defects. The effect of the defect location on the defect formation energy is small.

The electronic band structures show that vacancy and interstitial defects lead the bandgap to zero. The antisite defects lead the bandgap to a smaller value. The bandgaps of 4H–SiC with one antisite defect are from large to small: Si antisite defect in layer 3, Si antisite defect in layer 1, C antisite defect in layer 3, and C antisite defect in layer 1.

Electron mobility is reduced in varying degrees according to different defect types. The electron mobility of one C defect in layer 1 or 2 is larger than that in layer 3 or 4. The electron mobility of one Si vacancy defect in layer 1 or 2 is smaller than that in layer 3 or 4. The electron mobility of one Si interstitial defect in layer 1 or 2 is larger than that in layer 3 or 4. The electron mobility of one Si antisite defect in layer 1 or 2 is larger than that in layer 3 or 4.

This work was supported by the Hunan Province Key Laboratory for Ultra-Fast Micro/Nano Technology and Advanced Laser Manufacture (Grant No. 2018TP1041) and by the Molecular Dynamics Simulation of Irradiation Damage of Doped 4H–SiC Project (No. CX20200943).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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