Carrier lifetime killer in 4H-SiC: carrier capture path via carbon vacancies

Abstract

Carbon vacancies are thermally stable and are the most commonly observed native point defects in 4H-SiC, the key wide-bandgap semiconductor in power electronics. They are also identified as the physical original of Z_1/2 and EH_6/7 deep levels which are important carrier lifetime killers. However, the microscopic recombination process and detailed carrier capture path around carbon vacancies remain unclear. Leveraging upon first principles calculations, this work comprehensively investigates the carrier capture path and corresponding capture coefficients of carbon vacancies in 4H-SiC which are consistent with experimental observations. The findings also reveal the metastable spin-triplet carbon vacancies as key transition states in completing the non-radiative carrier capture path, especially at the donor levels. These metastable carbon vacancies can be formed either during the materials growth or through spin-selective carrier capture. This finding helps address the discrepancy in the association of EH_6/7 and Z_1/2 levels in experimental observation and provides deeper insights into the nature of carrier recombination in 4H-SiC.

---

Tianqi Deng

Tianqi Deng received his BSc from the University of Science and Technology of China in 2011 and PhD from Nanyang Technological University in 2016. Currently, he is a Principal Investigator at the ZJU-Hangzhou Global Scientific and Technological Innovation Center, Zhejiang University. His research interests include developing computational tools for interactions among electrons, phonons, and defects from first principles calculations, and the corresponding transport properties in semiconductors for power electronic and thermoelectric applications.

---

1 Introduction

With the rising demand for reliable power electronics such as converters and inverters for electric vehicles, silicon carbide (SiC) is becoming a key material in advanced power semiconductor devices.¹ This is due to its superior physical and electrical properties including but not limited to the wide bandgap, high breakdown field, high temperature and radiation resistance.^2,3 Particularly, 4H-SiC (Fig. 1) is one of the most widely used polytypes in industrial processes and scientific research because of the higher bandgap and electron mobility compared with the other polytypes.^4,5 As with other crystals, devices based on SiC may experience a serious decline in performance and reliability suffering from defects.^6,7 For example, bipolar devices for power electronics require a long carrier lifetime⁸ which would be degraded by high density of deep levels induced by point defects inside the bandgap.^9,10 Therefore, an insight into the nature of the defects present in 4H-SiC is fundamental to the performance and reliability and has become a major area of interest.

Fig. 1 Schematic structure of 4H-SiC. Blue and yellow balls represent Si and C atoms, respectively. Conventional unit cell of 4H-SiC is outlined in the dotted box. The cubic sublattice sites are labeled with ‘k’, and hexagonal sublattice sites are labeled with ‘h’.

Carbon vacancies (V_C) represent the primary thermally stable point defects which generate deep levels in the bandgap. Through cross-studies employing deep level transient spectroscopy (DLTS), electron paramagnetic resonance (EPR), and theoretical calculations,^11–14 a preliminary correlation was established between carbon vacancies and a set of deep level centers labelled as Z_1/2 and EH_6/7. These deep levels are identified as the major limitations to the minority-carrier lifetime and low field-effect mobility in 4H-SiC devices.^15,16 For acceptor-like levels (Z_1/2 centers), a negative-U property was discovered through both DLTS and density functional theory (DFT) calculations. As a consequence, a two-electron emission cascade in V_C(−−/0) has been considered as the primary source of the Z_1/2 centers, and the Z₁ and Z₂ signals are assigned to two different sublattice sites.^17–19 Capan et al. proposed a two-step transition path for electrons, the majority-carrier in n-type 4H-SiC, from neutral to doubly negatively charged and calculated the associated capture barriers. However, an accurate capture cross-section value for each transition level remains unclear.²⁰

Different from the Z_1/2 centers, the origin of EH_6/7 is still controversial due to significant signal overlap and challenges in characterization, even with techniques specifically designed for EH_6/7 centers such as Laplace-DLTS or low resolution DLTS.^21,22 Some suggest that EH_6/7 and the Z_1/2 centers originate from the same defect center V_C but with different charge states.^23,24 However, data from multiple studies indicate inconsistencies in the magnitude ratio between EH_6/7 and Z_1/2.^24–26 Booker et al. suggested that the stronger peak of EH_6/7 in DLTS results from a combination of EH₇(0/+), EH₇(+/++), and EH₆(+/++) transitions, while the smaller peak arises solely from EH₆(0/+).²⁷ Another possible assignment was also proposed that EH_6/7 should be associated with a defect complex involving V_C.^21,26 Up to now, researchers have not treated the carrier capture path of EH_6/7 centers in much detail and the understanding of the capture cross-section is still limited due to the lack of comprehensive and microscopic theoretical investigations.

To clarify the carrier capture behaviour in 4H-SiC, the origin of Z_1/2 and EH_6/7 centers is revisited in this study by examining the carrier recombination path identified through theoretical simulation, providing microscopic insights into the experimental findings. We identify that the ground state carbon vacancies contribute to the Z_1/2 deep level centers. But for the EH_6/7 deep level centers, it may not only be contributed by the ground state carbon vacancies, but with some metastable spin-triplet states. This work set out to gain deeper insights into the nature of carrier recombination for 4H-SiC with the objective of improving carrier lifetimes through defect engineering and enhancing the performance of high-power devices based on 4H-SiC.²⁸

2 Methods

2.1 First-principles calculations

The calculations were made using the Vienna ab initio simulation package (VASP),^29–32 employing the projector-augmented-wave formalism (PAW) method³³ to circumvent explicit treatment of core electrons. A plane-wave energy cutoff of 520 eV was utilized for Kohn–Sham pseudo-wave-functions. The many-electron interactions was assessed using the hybrid density functional of Heyd–Scuseria–Ernzerhof (HSE06),^34,35 which combines semi-local and exact exchange interactions at short ranges, while handling long-range interactions within the simpler generalized gradient approximation as proposed by Perdew–Burke–Ernzerhof (PBE).³⁶ Compared to plain PBE calculations, HSE06 offers the key advantage of predicting a Kohn–Sham band gap of 3.12 eV for 4H-SiC, aligning closer to the experimental value of 3.23 eV.³⁷ It also mitigates the over-delocalization of defect states in PBE calculations^38,39 that could affect the carbon vacancy calculations.

We utilized 6 × 6 × 2 hexagonal supercells with 576 atoms from which a carbon atom was removed to create a V_C defect. The lattice parameters of 4H-SiC were fixed to a = 3.0798 Å and c = 10.0820 Å, consistent with experimental values.⁴⁰ All defect structures were optimized at the PBE level, employing a conjugate-gradient method until the forces acting on the atoms were below 0.01 eV Å⁻¹. Subsequently, self-consistent calculations were carried out at the HSE06 level for the relaxed supercell structure with a numerical accuracy of 10⁻⁶ eV using a single Γ point in the Brillouin zone.

2.2 Defect formation energy

For the calculation of formation energies, we follow the usual procedure introduced by Freysoldt et al.³⁹ for finite-size effect correction. The formation energy ΔH_f of a carbon vacancy V_C with charge state q in 4H-SiC can be defined as follows:

ΔH_f(V_C, q) = E_d(V_C, q) − E(bulk) + (μ_C + E_C) + q(ε_VBM + E_F) + E_corr(q) (1)

where E_d(V_C,q) is the total energy of the supercell containing the defect V_C with charge state q, E(bulk) is the total energy for the perfect crystal using an equivalent supercell, μ_C is the chemical potential of carbon element, E_C is the DFT energy of carbon element obtained from graphite, ε_VBM is the valence band edge, E_F is the Fermi energy, and E_corr(q) is the correction of finite volume in charge state q.

The chemical potential of the carbon element is determined by examining the stability of 4H-SiC with respect to μ_C and μ_Si where no competing phases are found. In this study, μ_C is constrained within the range of −0.59 eV < μ_C < 0 eV, where the upper and lower bounds correspond to C-rich and C-poor 4H-SiC crystals, respectively. In the following calculations, the chemical potential under C-poor conditions is utilized, which is the experimental condition in the sublimation growth.⁴

Moreover, for the correction of finite volume, the method generalized by Kumagai, and Oba (eFNV method)⁴¹ is employed in this study, after a comparison with the method introduced by Freysoldt, Neugebauer, and Van de Walle (FNV method),⁴² as listed in the Supplementary information Part I (ESI†).

A significant application of the eqn (1) is to identify the thermodynamic transition level ε(q₁/q₂) defined as the Fermi level position where the formation energies of charge states q₁ and q₂ are equal:

where ΔH_f(q, E_F = 0) is the formation energy of defect in the charge state q when the Fermi level is at the VBM.

2.3 Non-radiative capture coefficients

The first-principles technique has significantly advanced in recent years to determine the non-radiative carrier capture coefficients of defects in semiconductors^43–45 based upon the Shockley–Read–Hall recombination.⁴⁶ This approach has been employed in various studies, including the investigation of MAPbI₃ to successfully elucidate the role of hydrogen vacancies,⁴⁷ and the examination of inorganic halides to identify factors influencing the luminous efficiency of dopant centers,⁴⁸ among others.^49–51 To assess the non-radiative capture coefficients in this work, the multi-phonon emission methodology⁴⁴ is employed in this work. The evaluation of electron–phonon coupling matrix elements is performed within the linear-coupling approximation,⁵² utilizing the PAW in the VASP. Moreover, the Sommerfeld parameter is also taken into account to describe the enhancement or suppression of a free-carrier in the vicinity of the charged defect.⁵³

3 Results and discussion

3.1 Identifying ground-state structure of carbon vacancies

Constructing the defect and identifying its ground-state structure are foundational steps in first-principles calculations, yet they can be challenging tasks. Initially, a vacancy structure usually arises from the removal of an atom at a specific lattice site while leaving the other atoms in the original positions. Through a gradient-based optimization algorithm, the relaxed configuration is determined, typically locating into a local minimum, which may represent the ground minimum in many scenarios but not necessarily in 4H-SiC. This is why there have been several proposed “ground-state structures” that disagree with each other.^54–56 What plays a key role in identifying ground structure of carbon vacancies in 4H-SiC is symmetry-breaking reconstructions induced by the crystal field and Jahn–Teller effects.⁵⁷ Classical energy-lowering optimization typically does not actively break symmetry and is trapped by a local minimum, which can result in discrepancies in electron configuration and higher energy.

Several approaches have been devised to navigate the ground-state structure of defects. Arrigoni and Madsen purposed a machine learning based method for searching low-energy configurations by exploring the potential energy surface.⁵⁸ Recently, Mosquera-Lois et al. described a more pragmatic approach to exploring the configurational space and identifying ground-state structures.⁵⁹ Building upon this work and considering the fundamental characteristic of 4H-SiC, our study employs a further simplified approach, consisting of two steps: (1) random perturbation is imposed on all the atoms in a supercell to break the symmetry (as shown in the upper panel of Fig. 2), guided by crystal field and Jahn–Teller effects; (2) Different spin-configurations are evaluated and the most stable one is adopted for each charge state and applied in the structure optimization by adjusting the electron magnetic moment.

Fig. 2 Structure navigation method for V_C in 4H-SiC with random perturbation. The top half depicts the schematic structure of 4H-SiC before and after bond perturbation, while the bottom half illustrates four atomic configurations identified as the ground structure of V_C. Silicon atoms are denoted in blue, carbon atoms in yellow, and V_C highlighted. Grey lines indicate the formation of reconstructed bonds between silicon and its neighbors. The [0001] axis is parallel to the plane of the paper. Three silicon atoms forming a triangle parallel to the base plane (BP), with the central silicon atom parallel to the [0001] axis, below which the removed carbon atom would be located.

The detailed optimization results of V_C(h) and V_C(k) for various charge states obtained from the perturbed structure are presented in Supplementary Information Part III (ESI†). From this data, four different atomic configurations for the ground structure can be summarized, as can be seen from the bottom half of Fig. 2, depending on the sub-lattice site and charge state. The notable difference between these configurations is that only configuration A exhibits C_3v symmetry, while all other configurations only process C_s symmetry.

The configurations depicted in Fig. 2 are closely related to the occupation of the electron orbitals, where high symmetry configuration like V²⁺_C(h, A) and V²⁺_C(k, A) typically arise from the energy level degeneracy. However, as electron energy levels are filled, degenerate levels separate, leading to low symmetry configurations due to crystal field and Jahn–Teller effects, such as V⁺_C(k, B), V²⁻_C(h, D), and V²⁻_C(k, D), accompanied by an increasing trend in the breaking of original bonds and the formation of reconstructed bonds. The extent of level separation also varies with electron structures, resulting in different configurations for V⁻_C(h, C) and V⁻_C(k, D). The structural stability is further confirmed by ab initio molecular dynamics where the atoms only vibrate around the stable structure without vacancy migration or transformation, as detailed in the ESI.†

Overall, these results suggest that this method we propose performs well in 4H-SiC and exhibits good reproducibility. These results are consistent with those obtained by Coutinho et al.⁵⁷ and Huang et al.⁶⁰ The coordination environment of 4H-SiC is relatively simple, thus obviating the need to search the entire potential energy surface. However, owing to the crystal field and Jahn–Teller effects, as well as the similar valence electronic arrangement of silicon and carbon elements, high-spin occupation are prone to occur during the optimization process, necessitating the spin constraints. This method enhances the chance of efficiently obtaining the ground structure of V_C, with minimal increase in computation time. In the following sections, some key items involved in the calculation of the defect formation energy and non-radiative capture coefficient will be discussed based on these ground structures.

3.2 Formation energy of carbon vacancy

The formation energy of V_C in 4H-SiC, calculated using eqn (1), and thermodynamic transition levels, calculated using eqn (2), are depicted in Fig. 3. Specifically, the formation energy of neutral V_C(h) and V_C(k) in C-poor conditions is 4.434 eV and 4.309 eV, respectively, while in C-rich conditions it is 5.024 eV and 4.899 eV. These values align with the formation enthalpy of 5.0 eV deduced for V_C form 4H-SiC.²⁵ The formation energies correspond to an intrinsic V_C concentration of 8.77 × 10¹¹ cm⁻³ to 9.89 × 10¹³ cm⁻³ at the chemical potential of carbon-rich conditions and the typical growth temperature range from 2000 K to 2300 K, which also agrees well with experimental estimation.⁴

Fig. 3 (a) Formation energy of V_C(h) (red lines) and V_C(k) (blue lines) at different charge states (−2 ≤ q ≤ +2) as a function of the Fermi level under C-poor conditions. (b) Schematic illustration of thermodynamic transition levels of V_C(h) and V_C(k) using the eFNV methods, alongside the Z_1/2 and EH_6/7 deep levels measured by DLTS.

It is apparent from the Fig. 3 that the negative-U behaviour is only observed for the acceptor-like levels of V_C(k) (U = −31 meV), indicating that V⁻_C(k) is always metastable regardless of the position of the Fermi energy. This finding is consistent with the requirement of optical excitation in order to observe negatively charged vacancies by EPR, contrasting with the behaviour of donor-like levels. For the donor-like levels, a positive-U is identified for V_C(h) (U = 164 meV), as well as for V_C(k) (U = 43 meV). Although negative-U behaviour may be observed for donor-like levels, the possible interference of finite volume corrections cannot be ruled out. Following the addition of suitable finite volume corrections, a positive-U is found in the donor-like levels here.

More importantly, when comparing both acceptor-like and donor-like levels of V_C with Z_1/2 (E_C – 0.64 eV) and EH_6/7 (E_C – 1.54 eV) levels,⁷ there is a small gap, which serves as the crucial support for assigning Z_1/2 and EH_6/7 signals to V_C. Furthermore, due to the higher amplitude of Z₂ signal compared to Z₁ and the lower formation energy of V_C(k) compared to V_C(h), the Z₁ and Z₂ signals can be assigned to V_C(k)(−−/0) and V_C(h)(−−/−)(−/0), respectively. However, the correspondence between EH_6/7 and donor-like levels of V_C still lacks key evidence for conclusive assignment and it will be argued by carrier capture coefficients in the section that follows.

3.3 Recombination paths for carrier at the carbon vacancy

We first evaluate the recombination of electron and hole at acceptor-like levels of V_C in 4H-SiC. The capture cross-sections (converted from calculated non-radiative capture coefficients) at 300 K are illustrated in Fig. 4, while the overall non-radiative capture coefficients as a function of temperature as well as the 1D configuration coordinate diagrams relevant for the recombination are shown in Supplementary information Part II (ESI†). It is noteworthy that the two-step transitions for V_C(k) was both calculated at ε(−−/0) rather than ε(−−/−) or ε(−/0) because of the negative-U.

Fig. 4 Schematic illustration of recombination paths for the electron and hole at acceptor-like levels of V_C in 4H-SiC. The capture cross sections were calculated at 300 K.

First of all, it is apparent from Fig. 4 that the transition V⁰_C(k) + e⁻ → V⁻_C(k) dominates the electron recombination for the acceptor-like level. The capture cross-sections for electrons at Z_1/2 levels are approximately 10⁻¹³ to 10⁻¹² cm² as measured by DLTS,^4,20,27 which is similar to the results obtained in this study. Then, as observed in Fig. 4, the differences in capture cross-sections at acceptor-like levels of V_C between V_C(k) and V_C(h) are evident. The processes of the electron capture are both non-radiative, while notably, the capture cross-section for V_C(h)(−/0) is much smaller than that for V_C(k)(−/0). This discrepancy arises because the capture barrier for V_C(k)(−/0) transition is much smaller than that for V_C(h)(−/0) as shown in Fig. S5 (ESI†), and the thermal activation toward the crossing point becomes much easier. However, it is challenging to observe this difference by DLTS. On the other hand, these findings support the view that the influence of the capture barrier ΔE_b on the non-radiative capture cross section is much greater than the ground structure difference ΔQ in potential energy curves. This suggests that for 4H-SiC electron–phonon coupling is the main limiting factors of capture cross section rather than vibrational relaxation.

For the hole recombination at the acceptor-like levels, the transition V⁻_C(h) + h⁺ → V⁰_C(h) is the key factor. And the calculated capture cross-sections for hole are an order of magnitude larger than that for electron, which is in line with the previous study.⁶¹ Particularly, for the transition V²⁻_C + h + → V⁻_C, there is a rather unexpected result that no crossing point is found between the two potential energy surfaces of ground-state structure. As a consequence, there may be at least one extra state involved in the non-radiative capture process, if it could occur, with the most likely candidate being a metastable state of carbon vacancies. As noted by B. Dou et al.⁶² the metastable state center plays an important role in constructing an efficient carrier capture channel. And for 4H-SiC, due to the significant distortion induced by the crystal field and Jahn–Teller effects, numerous metastable states are found around the ground structure and some of them have been reported previously.^20,57 This conjecture is confirmed by the following recombination paths for the donor-like levels of V_C.

The recombination of electron and hole at donor-like levels of V_C have also been calculated using the same method. The 1D configuration coordinate diagrams relevant to the recombination, as well as the non-radiative capture cross sections as a function of temperature, are also illustrated in Supplementary information Part II (ESI†). Generally, due to the lack of a crossing point or the presence of a large capture barrier, the possibility of radiative capture for donor-like levels is significantly increased compared to acceptor-like levels. For example, for the transition V⁰_C(k, B) + h⁺ → V⁺_C(k, B), as illustrated in Fig. 5(a), the radiative capture would produce an optic radiation of around 1.2 eV, or a large capture barrier ΔE_b,p of around 1.33 eV would lead to a non-radiative capture cross-section of 1.95 × 10⁻²³ cm² which is consistent with data obtained by P. Dong et al.⁶¹ Obviously, the cross-section is too low for hole capture to occur.

Fig. 5 Calculated 1D cc diagram for (a) V⁺_C(k, B) ⇔ V⁰_C(k, B) charge-states transition, (b) V⁺_C(k, B) ⇔ V⁰_C(k, A) charge-state transition, and (c) intersystem transition between V⁰_C(k, B) and V⁰_C(k, A) in 4H-SiC. Symbols: calculated values; solid lines: parabolic fit and a fourth-order polynomial fit only for V⁰_C(k, A).

However, this behaviour is different when the metastable spin-triplet V⁰_C(k, A) is taken into account. As shown in Fig. 5(b), the capture barrier ΔE_b,p for transition V⁰_C(k, A) + h⁺ → V⁺_C(k, B) is only 0.37 eV and the non-radiative capture cross-section is around 7.68 × 10⁻¹⁷ cm² (Fig. S14, ESI†), which is consistent with the experimental value.^21,27 Meanwhile, the capture barrier ΔE_b,n for the electron non-radiative capture of V⁺_C(k, B) + e⁻ → V⁰_C(k, B) remains comparable to that of V⁺_C(k, B) + e⁻ → V⁰_C(k, A) and both barriers are around 0.24 eV. These results demonstrate that the metastable spin-triplet V⁰_C(k, A) can also be generated through the V⁺_C + e⁻ → V⁰_C process with probability comparable to that of the ground-state spin-singlet V⁰_C(k, B) on the recombination paths for the donor-like levels of V_C. Comparing the metastable triplet V⁰_C(k, A) with singlet V⁰_C(k, B), the energy difference ΔE is only 0.33 eV, as shown in Fig. 5(c). Considering the relatively weak spin–orbit coupling of Si and C, the probability of spontaneous intersystem crossing, for example the transition from spin triplet to singlet state, is expected to be low. Therefore, the generated spin-triplet V⁰_C(k, A) could possibly survive for a long enough time to capture the hole and contribute to the carrier recombination. On the other hand, the electron non-radiative capture cross-section for triplet V⁰_C(k, A) from V⁺_C(k, B) is also two orders of magnitude larger than that for singlet V⁰_C(k, B), as shown in Fig. S14 (ESI†). Therefore, the metastable triplet is expected to play an important role in the nonradiative carrier recombination at donor-like levels. Such spin-selective carrier recombination may also be useful for spintronics and quantum applications.⁶³

In summary, the electron configuration for the carbon vacancy involved in the donor-like level recombination can differ from that of the acceptor-like level. Such a triplet V⁰_C(k, A) can easily form in a lower concentration compared to the ground state V⁰_C(k, B) due to the crystal field and Jahn–Teller effects. And, the ground state V⁰_C(k, B) participates in the recombination for accepter level, while for donor-like level, it can be triplet V⁰_C(k, A). Consequently, this can partially result in the inconsistencies in the magnitude ratio between EH_6/7 and Z_1/2.

4 Conclusions

This study aims to determine the carrier recombination path for the acceptor-like and donor-like levels of carbon vacancy to determine the origin of Z_1/2 and EH_6/7 centers in 4H-SiC. First of all, in order to get the non-radiative capture cross-section accurately, the ground structure and formation energy of carbon vacancy are determined. A simple and efficient method is adopted in this work to navigate the ground structure of carbon vacancy with scarcely increased computing time. Then, the carrier recombination path for the acceptor-like and donor-like levels was discussed respectively. For the acceptor-like level, the electron would be captured through non-radiative transitions. V_C(k) is a fairly efficient non-radiative capture center compared to V_C(h), while for the hole, the main non-radiative capture center is V_C(h). Next, for the donor-like level of carbon vacancy, it is proved to be unable to independently accomplish the carrier recombination path through non-radiative processes. Further investigations reveal that the carrier recombination of EH_6/7 centers is not only contributed by the ground state carbon vacancies, but also metastable spin-triplet states. It is different from the Z_1/2 centers which are almost contributed by the ground state carbon vacancies. This finding could explain the inconsistencies in the magnitude ratio between EH_6/7 and Z_1/2 because the states of the carbon vacancies involved in this two recombination are not the same. The recombination paths identified in this work and their corresponding capture coefficients are expected to assist the experimental identification and understanding of carbon vacancies in 4H-SiC and inspire further exploration of defect engineering.

Author contributions

Xuanyu Jiang: data curation (lead), formal analysis (lead), investigation (lead), visualization (lead), and writing – original draft (lead). Yuanchao Huang: formal analysis (supporting), and validation (supporting). Rong Wang: conceptualization (supporting), investigation (supporting), and validation (supporting). Xiaodong Pi: conceptualization (lead), funding acquisition (lead), resources (lead), and writing – review & editing (lead). Deren Yang: conceptualization (lead), resources (lead), and supervision (lead). Tianqi Deng: conceptualization (lead), funding acquisition (lead), resources (lead), and supervision (lead).

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We gratefully acknowledge funding support from the National Natural Science Foundation of China (Grant No. 62204218), the “Pioneer” and “Leading Goose” R&D Program of Zhejiang Province (Grant No. 2022C01021 and 2023C01010), and the Leading Innovative and Entrepreneur Team Introduction Program of Hangzhou (TD2022012). The schematic of the crystal structure was drawn using VESTA.⁶⁴ The VASPKIT,⁶⁵ doped,⁶⁶ and NONRAD⁵² programs were used in this study for post-processing of the VASP calculated data.

Notes and references

1. G. Liu, B. R. Tuttle and S. Dhar, Appl. Phys. Rev., 2015, 2, 021307.
2. T. Kimoto, Jpn. J. Appl. Phys., 2015, 54, 040103.
3. X. She, A. Q. Huang, O. Lucia and B. Ozpineci, IEEE Trans. Ind. Electron., 2017, 64, 8193–8205.
4. T. Kimoto and J. A. Cooper, Fundamentals of silicon carbide technology: growth, characterization, devices and applications, John Wiley & Sons, 2014.
5. W. J. Schaffer, G. H. Negley, K. G. Irvine and J. W. Palmour, MRS Proc., 1994, 339, 595.
6. N. Iwamoto and B. G. Svensson, in Semiconductors and Semimetals, Elsevier, 2015, vol. 91, pp. 369–407.
7. H. F. Xiong, X. S. Lu, X. Gao, Y. C. Yan, S. Liu, L. H. Song, D. R. Yang and X. D. Pi, J. Semicond., 2024, 45, 072502.
8. M. Fanton, D. Snyder, B. Weiland, R. Cavalero, A. Polyakov, M. Skowronski and H. Chung, J. Cryst. Grow., 2006, 287, 359–362.
9. P. B. Klein, B. V. Shanabrook, S. W. Huh, A. Y. Polyakov, M. Skowronski, J. J. Sumakeris and M. J. O’Loughlin, Appl. Phys. Lett., 2006, 88, 052110.
10. F. La Via, M. Camarda and A. La Magna, J. Appl. Phys., 2014, 1, 031301.
11. T. Kimoto, A. Itoh, H. Matsunami, S. Sridhara, L. L. Clemen, R. P. Devaty, W. J. Choyke, T. Dalibor, C. Peppermüller and G. Pensl, Appl. Phys. Lett., 1995, 67, 2833–2835.
12. C. Hemmingsson, N. T. Son, O. Kordina, J. P. Bergman, E. Janzén, J. L. Lindström, S. Savage and N. Nordell, J. Appl. Phys., 1997, 81, 6155–6159.
13. T. Hiyoshi and T. Kimoto, Appl. Phys. Express, 2009, 2, 041101.
14. T. Hornos, A. Gali and B. G. Svensson, Mater. Sci. Forum, 2011, 679–680, 261–264.
15. U. Grossner, G. Alfieri and R. Nipoti, Mater. Sci. Forum, 2015, 821–823, 381–386.
16. P. Klein, J. Appl. Phys., 2008, 103, 033702.
17. C. G. Hemmingsson, N. T. Son, A. Ellison, J. Zhang and E. Janzén, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 58, R10119.
18. N. T. Son, X. T. Trinh, L. S. Løvlie, B. G. Svensson, K. Kawahara, J. Suda, T. Kimoto, T. Umeda, J. Isoya, T. Makino, T. Ohshima and E. Janzén, Phys. Rev. Lett., 2012, 109, 187603.
19. C. G. Hemmingsson, N. T. Son, A. Ellison, J. Zhang and E. Janzén, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 7768.
20. I. Capan, T. Brodar, J. Coutinho, T. Ohshima, V. P. Markevich and A. R. Peaker, J. Appl. Phys., 2018, 124, 245701.
21. G. Alfieri and T. Kimoto, Appl. Phys. Lett., 2013, 102, 152108.
22. B. Zippelius, A. Glas, H. B. Weber, G. Pensl, T. Kimoto and M. Krieger, Mater. Sci. Forum, 2012, 717–720, 251–254.
23. L. Storasta, J. P. Bergman, E. Janzén, A. Henry and J. Lu, J. Appl. Phys., 2004, 96, 4909–4915.
24. K. Danno and T. Kimoto, J. Appl. Phys., 2006, 100, 113728.
25. H. M. Ayedh, V. Bobal, R. Nipoti, A. Hallén and B. G. Svensson, J. Appl. Phys., 2014, 115, 012005.
26. S. A. Reshanov, G. Pensl, K. Danno, T. Kimoto, S. Hishiki, T. Ohshima, H. Itoh, F. Yan, R. P. Devaty and W. J. Choyke, J. Appl. Phys., 2007, 102, 113702.
27. I. D. Booker, E. Janzén, N. T. Son, J. Hassan, P. Stenberg and E. Ö. Sveinbjörnsson, J. Appl. Phys., 2016, 119, 235703.
28. R. Wang, Y. Huang, D. Yang and X. Pi, Appl. Phys. Lett., 2023, 122, 180501.
29. G. Kresse and J. Hafner, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 47, 558–561.
30. G. Kresse and J. Hafner, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 14251–14269.
31. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
32. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
33. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
34. J. Heyd, G. E. Scuseria and M. Ernzerhof, J. Chem. Phys., 2003, 118, 8207–8215.
35. A. V. Krukau, O. A. Vydrov, A. F. Izmaylov and G. E. Scuseria, J. Chem. Phys., 2006, 125, 224106.
36. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
37. C. Persson and U. Lindefelt, J. Appl. Phys., 1997, 82, 5496–5508.
38. K. Alberi, M. B. Nardelli, A. Zakutayev, L. Mitas, S. Curtarolo, A. Jain, M. Fornari, N. Marzari, I. Takeuchi, M. L. Green, M. Kanatzidis, M. F. Toney, S. Butenko, B. Meredig, S. Lany, U. Kattner, A. Davydov, E. S. Toberer, V. Stevanovic, A. Walsh, N.-G. Park, A. Aspuru-Guzik, D. P. Tabor, J. Nelson, J. Murphy, A. Setlur, J. Gregoire, H. Li, R. Xiao, A. Ludwig, L. W. Martin, A. M. Rappe, S.-H. Wei and J. Perkins, J. Phys. D: Appl. Phys., 2019, 52, 013001.
39. C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti and C. G. Van de Walle, Rev. Mod. Phys., 2014, 86, 53.
40. M. E. Levinshtein, S. L. Rumyantsev and M. S. Shur, Properties of Advanced Semiconductor Materials: GaN, AIN, InN, BN, SiC, SiGe, John Wiley & Sons, 2001.
41. Y. Kumagai and F. Oba, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 195205.
42. C. Freysoldt, J. Neugebauer and C. G. Van de Walle, Phys. Rev. Lett., 2009, 102, 016402.
43. L. Shi and L.-W. Wang, Phys. Rev. Lett., 2012, 109, 245501.
44. A. Alkauskas, Q. Yan and C. G. Van de Walle, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 075202.
45. L. Shi, K. Xu and L.-W. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 205315.
46. V. Abakumov, V. I. Perel and I. Yassievich, Nonradiative recombination in semiconductors, Elsevier, 1991.
47. X. Zhang, J.-X. Shen, M. E. Turiansky and C. G. Van de Walle, Nat. Mater., 2021, 20, 971–976.
48. R. Hao and C.-K. Duan, Inorg. Chem., 2024, 63, 3152–3164.
49. D. Han, J. Wang, L. Agosta, Z. Zang, B. Zhao, L. Kong, H. Lu, I. Mosquera-Lois, V. Carnevali, J. Dong, J. Zhou, H. Ji, L. Pfeifer, S. M. Zakeeruddin, Y. Yang, B. Wu, U. Rothlisberger, X. Yang, M. Grätzel and N. Wang, Nature, 2023, 622, 493–498.
50. Y. Y. Illarionov, T. Knobloch, M. Jech, M. Lanza, D. Akinwande, M. I. Vexler, T. Mueller, M. C. Lemme, G. Fiori, F. Schwierz and T. Grasser, Nat. Commun., 2020, 11, 3385.
51. J. S. Park, S. Kim, Z. Xie and A. Walsh, Nat. Rev. Mater., 2018, 3, 194–210.
52. M. E. Turiansky, A. Alkauskas, M. Engel, G. Kresse, D. Wickramaratne, J.-X. Shen, C. E. Dreyer and C. G. Van de Walle, Comput. Phys. Commun., 2021, 267, 108056.
53. M. E. Turiansky, A. Alkauskas and C. G. Van de Walle, J. Phys.: Condens. Matter, 2024, 36, 195902.
54. L. Torpo, M. Marlo, T. E. M. Staab and R. M. Nieminen, J. Phys.: Condens. Matter, 2001, 13, 6203.
55. T. Kobayashi, K. Harada, Y. Kumagai, F. Oba and Y.-I. Matsushita, J. Appl. Phys., 2019, 125, 125701.
56. X. Wang, J. Zhao, Z. Xu, F. Djurabekova, M. Rommel, Y. Song and F. Fang, Nanotechnol. Precis. Eng., 2020, 3, 211–217.
57. J. Coutinho, V. J. B. Torres, K. Demmouche and S. Öberg, Phys. Rev. B, 2017, 96, 174105.
58. M. Arrigoni and G. K. H. Madsen, npj Comput. Mater., 2021, 7, 71.
59. I. Mosquera-Lois, S. R. Kavanagh, A. Walsh and D. O. Scanlon, npj Comput. Mater., 2023, 9, 25.
60. Y. Huang, X. Jiang, T. Deng, D. Yang and X. Pi, J. Appl. Phys., 2024, 135, 135701.
61. P. Dong, X. Yan, L. Zhang, S. Jin, F. Dai, Y. Zhang, Y. Cui, X. Yu and B. Huang, IEEE Trans. Nucl. Sci., 2021, 68, 312–317.
62. B. Dou, S. Falletta, J. Neugebauer, C. Freysoldt, X. Zhang and S.-H. Wei, Phys. Rev. Appl., 2023, 19, 054054.
63. G. Zhang, Y. Cheng, J.-P. Chou and A. Gali, Appl. Phys. Rev., 2020, 7, 031308.
64. K. Momma and F. Izumi, J. Appl. Crystallogr., 2011, 44, 1272–1276.
65. V. Wang, N. Xu, J.-C. Liu, G. Tang and W.-T. Geng, Comput. Phys. Commun., 2021, 267, 108033.
66. S. R. Kavanagh, A. G. Squires, A. Nicolson, I. Mosquera-Lois, A. M. Ganose, B. Zhu, K. Brlec, A. Walsh and D. O. Scanlon, Journal of Open Source Software, 2024, 9, 6433.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc04558k

---