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Formation and properties of defects and small vacancy clusters in SiC: Ab initio calculations
Nuclear Instruments and Methods in Physics Research B 267 (2009) 2995–2998
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Formation and properties of defects and small vacancy clusters in SiC: Ab initio calculations F. Gao a,*, W.J. Weber a, H.Y. Xiao b, X.T. Zu b a b
Pacific Northwest National Laboratory, P.O. Box 999, Richland, WA 99352, USA Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu, 610054, China
a r t i c l e
i n f o
Article history: Available online 14 June 2009 PACS: 61.72.-y 31.15.A61.80.-x 61.72.uj Keywords: Defect properties Ab initio simulation Defect clusters Silicon carbide
a b s t r a c t Large-scale ab initio simulation methods have been employed to investigate the configurations and properties of defects in SiC. Atomic structures, formation energies and binding energies of small vacancy clusters have also been studied as a function of cluster size, and their relative stabilities are determined. The calculated formation energies of point defects are in good agreement with previously theoretical calculations. The results show that the di-vacancy cluster consists of two C vacancies located at the second nearest neighbor sites is stable up to 1300 K, while a di-vacancy with two Si vacancies is not stable and may dissociate at room temperature. In general, the formation energies of small vacancy clusters increase with size, but the formation energies for clusters with a Si vacancy and nC vacancies (VSi–nVC) are much smaller than those with a C vacancy and nSi vacancies (VC–nVSi). These results demonstrate that the VSi–nVC clusters are more stable than the VC–nVSi clusters in SiC, and provide possible nucleation sites for larger vacancy clusters or voids to grow. For these small vacancy clusters, the binding energy decreases with increasing cluster size, and ranges from 2.5 to 4.6 eV. These results indicate that the small vacancy clusters in SiC are stable at temperatures up to 1900 K, which is consistent with experimental observations. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Studies of irradiation effects in silicon carbide (SiC) have been carried out for the last decade due to its possible applications in high-temperature, high-frequency and high-power electronics and as structural components for use in gas-cooled fission reactors and fusion environments. In particular, there is a strong interest in understanding defect formation and properties because these defects and their aggregation will affect many of the macroscopic properties of electronic devices and nuclear components, as well as its overall performance under hash environments. Determination of defect formation and energetics is therefore crucial for understanding the response of SiC to irradiation damage and ion implantation. There have been many studies of defect properties in SiC using both ab initio methods [1,2] and empirical potential calculations [3,4]. These calculations provide significant insights into defect stability, defect configuration and formation energy, but only point defects have been investigated. As shown previously, a displacement cascade predominantly produces point defects, and only about 19% of the interstitial population is contained in clusters [5]. The size of the interstitial clus* Corresponding author. Tel.: +1 509 371 6490; fax: +1 509 371 6242. E-mail address: [email protected] (F. Gao). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.018
ters is small, with the largest clusters containing only four interstitial atoms for 50 keV Si displacement cascades. However, these interstitials, particular C interstitials, have relative low energy barriers for migration, as compared with vacancies [6,7], and the migration of carbon interstitials seems quite feasible at room temperature, leading to interstitial aggregations or defect recovery in SiC. As the temperature increases above the critical amorphization temperature, the number of defects surviving in the cascade are reduced, and the mobility of both silicon and carbon interstitials become significant. However, experimental studies have noted the presence of both Frank loops and tiny voids at temperatures approaching 1300 K, indicating limited mobility of vacancies [8]. The apparent increases in swelling in the temperature range of between 1300 and 1800 K may be associated with the production of voids, which are observed to continue to grow in SiC irradiated to 1900 K. This phenomenon may suggest that the diffusion energies of the Si and C vacancies must be very high, as are the binding energies for clustered vacancies. Using isochronal annealing and positron lifetime analysis, Lam [9] has shown a carbon–silicon vacancy complex to dissociate above 1800 K in 6H–SiC. Despite of the evidence for aggregation of interstitials at low temperatures and void swelling at higher temperatures, knowledge about the behavior of interstitial and vacancy clusters in SiC
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is scare. In addition, modeling of microstructural evolution under irradiation requires detailed understanding regarding the nature and structures of interstitial and vacancy clusters. This paper systematically investigates the formation and binding properties of defects and vacancy clusters in 3C–SiC. On the basis of the simulations, the results are discussed and compared with the experimental measurements. 2. Computational approach All the calculations were performed using the density functional theory (DFT) method as implemented in the SIESTA code [10], while geometry optimization and energy calculations were carried out within the framework of LDA using the parametrized functional of Perdew and Zunger for electron exchange and correlation [11]. The electron–ion interactions were described by normconserving Troullier–Martins [12] pseudopotentials factorized in the Kleinman–Bylander form, where the core radii in the construction of the pseudopotentials were 1.89 and 1.25 Å for Si and C, respectively. The valence wave functions were expanded in a basis set of localized atomic orbitals, and both single-f (SZ) and double-f polarized (DZP) basis sets were used for the purpose of comparison. The calculations were carried out in a supercell containing 128 atoms (DZP) and 512 atoms (SZ). The convergence of the calculations has been tested, and we found a 2 2 2 k-point grid and a 90 Ry cutoff for the real space mesh can give well converged results. With the double-f polarized basis set, the calculated lattice parameter, a0 = 4.36 Å, and the bulk modulus, B = 226 GPa, were found to be in good agreement with the experimental values of 4.36 Å and 224 GPa [13], respectively. Fig. 1 shows the total energy as a function of the lattice parameter of 3C–SiC using different basis sets. It can be seen that the lattice parameter determined by the triple-f (TZ) basis set and double-f basis set plus polarization (DZP) orbitals is in excellent agreement with the experimental values of 4.36 Å, while single-f (SZ) and DZ basis sets give slightly larger lattice parameter (4.45 Å). Although the DZP and TZ basis sets provide a highly accurate lattice parameter, the SZ basis set offers already quite well converged results, comparable to those used in practice in most plane-wave calculations. The structures of point defects and defect clusters were relaxed using conjugate gradient (CG) coordinate optimization until the forces on each atom were <0.01 eV/Å. 3. Results and discussion As described in [2], there are a number of possible configurations of intrinsic defects in 3C–SiC, and the fundamental properties
Total energy (eV)
-260
SZ TZ DZP DZ
-261
of these defects have been investigated using both ab initio calculations [2,3,14] and empirical potentials [2,15]. In this paper, only a subset of intrinsic defects are investigated and compared with the previous calculations since the present study mainly deals with defect clusters. The point defects considered include a C vacancy, a Si vacancy, an antisite pair (two nearest neighboring atoms exchange their positions), a C interstitial and a Si interstitial. It should be noted that there are a number of possible configurations for an interstitial [2], but only their most stable configurations are considered (i.e. the C–C dumbbell along the <1 0 0> direction and Si tetrahedral surrounded by four C atoms). The geometrical structures of these defects are relaxed completely and optimized, and the corresponding defect formation energies were calculated. The formation energy of a defect is defined by the following formula [4]:
1 1 Ef ¼ Edtot ðnSi þ nC ÞlSiC ðnSi nC Þ 2 2
1 2
lbSi lbC ðnSi nC ÞDl: ð1Þ
Edtot
Here is the calculated total energy of a supercell containing nSi silicon and nC carbon atoms with a defect, and lbSi and lbC represents the chemical potential of bulk Si and C crystal, respectively. Dl is related to the deviation from ideal stoichiometry, varying from Crich kDl = DY| to Si-rich kDl = DY|. The calculated defect formation energies obtained by SZ and DZP basis sets are presented in Table 1 for the ideal stoichiometry condition, along with other theoretical calculations [1,3,14,16,17] for comparison. In addition, a di-vacancy defect is included, where two vacancies located at the first or second nearest neighbor distance are considered. In general, the formation energies calculated with DZP and SZ basis sets are in good agreement with those obtained by GGA and LDA calculations. The formation energies of C and Si vacancies calculated with DZP basis sets are 3.74 and 8.30 eV, respectively, and the formation energies of CC <1 0 0> dumbbell and SiTC interstitials are 6.71 and 7.22 eV, respectively. It should be noticed that the formation energies obtained by SZ basis set is systematically higher than those calculated with DZP basis set. Using the formation energies, the binding energy of a cluster can be calculated and the results indicate that a di-vacancy (di-VC–Si) with a C vacancy and a Si vacancy at the nearest neighbors is the most stable di-vacancy configuration, followed by that with two C vacancies located at the second nearest neighbors (di-VC). The formation energy of a di-vacancy cluster with two Si vacancies at second nearest neighbors (di-VSi) is very high, but with a small binding energy of about 0.21, which indicates that its nucleation at temperatures below room temperature may be impossible. The binding energy for a di-VC–Si cluster is 4.29 eV, which indicates that this vacancy cluster can be stable at hightemperatures, and it may act as a nucleation site for the growth of larger vacancy clusters. Also, a di-VC cluster, with a binding energy of 1.15 eV, can be stable up to 1300 K and may be dissociated at higher temperatures.
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Table 1 Calculated formation energies of intrinsic defects in the neutral charge state, along with other theoretical calculations for comparison.
-263
Defect formation energy Ef (eV) Defect DZP SZ 3.74 4.02 VC 8.30 8.88 VSi CC <1 0 0> 6.71 7.02 7.22 9.20 SiTC Antisite pair 4.43 3.95 di-V (1 n n) 7.75 8.65 16.39 – di-VSi (2 n n) 6.33 – di-VC (2 n n)
-264 4.1
4.2
4.3
4.4
4.5
4.6
4.7
a (A) Fig. 1. Total energy of a unit cell versus lattice parameter for bulk 3C–SiC for different basis sets.
[14] 3.63 7.48 6.65 7.04 – – – –
[3] 3.74 8.38 – 7.02 – – – –
[16] 4.2 8.1 8.9 – – – – –
[17] 4.3 8.45
– – – –
[1] – – 6.7 6.0 – – – –
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Eb ðnÞ ¼ Ef ðn 1Þ þ Eaf ð1Þ Ef ðnÞ;
ð2Þ
where a is either a C or Si vacancy, which depends on the cluster type. The binding energy of the vacancy clusters is shown in Fig. 3 as a function of the cluster size. The results show that the binding energy decreases with increasing the cluster size, in contrast to that in pure metals, where the binding energy of vacancy clusters generally increases with increasing cluster size, but it is much larger than that in metals. Values reported here are only for the compact vacancy clusters, as described above. It should be pointed out that sampling of all possible configurations is complicated and it is possible that lower energy configurations than those reported here could be found. It is of interest to note that the small vacancy clusters (n < 10) formed from successive shells of neighbors
Table 2 Calculated formation energies of small vacancy clusters in the neutral charge state. Defect formation energy Ef (eV) VC–VSi Defect VC Ef (eV) 4.02 8.65 VSi–VC Defect VSi Ef (eV) 8.88 8.65
VC–2VSi 14.49 VSi–2VC 8.70
VC–3VSi 21.16 VSi–3VC 9.42
VC–4VSi 26.59 VSi–4VC 10.84
4.5
CV+nSiV SiV+nCV
4.0
E b (eV)
In the present study, several small vacancy clusters are investigated, but only compact configurations are considered, i.e. a vacancy in cluster has at least one vacancy at the nearest neighbor distance. Fig. 2 shows an example of a four-vacancy cluster (a Si and three C vacancies), where (a) represents the atomic configuration before relaxation and (b) exhibits the atomic arrangements after relaxation. It is of interest to find that the relaxations around the vacancy cluster are very small. The first nearest neighbor Si atoms to the C vacancies on the same (1 1 1) plane with the Si vacancy relax towards the vacancies, but the displacement from their original positions is only about 0.14 Å. The first nearest neighbor Si atoms on the plane below the C vacancies relax outwards from the vacancies with a displacement of about 0.016 Å. These small relaxations can not explain a large reduction of the total energy observed after relaxation, which may be associated with charge redistribution and will be discussed in details elsewhere. The formation energies of the small vacancy clusters are calculated using Eq. (1), and listed in Table 2. It can be seen that the formation energies of the small vacancy clusters generally increase with their size, but the formation energies for the VSi–nVC clusters are much smaller than those for the VC–nVSi clusters, and thus, the VSi–nVC clusters is more feasible to provide possible nucleation sites for larger vacancy clusters or voids to grow. It is of interest to note that the formation energy of the VSi–VC cluster is even smaller than that of a single Si vacancy (VSi), and this di-vacancy cluster can be easily formed during irradiation. Previously theoretical calculations argued that the VSi is not stable [18] and a C neighbor exchanges place with the vacancy producing a close-by pair VC–CSi (a C vacancy plus a Si antisite defect), and the high formation energy of a single Si vacancy is consistent with this previous observation. From the formation energies, the binding properties of vacancy clusters can be determined. The binding energy, Eb(n), of a defect cluster with n vacancies can be defined as:
3.5 3.0 2.5 2.0
1
2
3
4
5
6
Size N Fig. 3. Binding energy of the small vacancy clusters as a function of the cluster size in SiC.
of a given lattice site in Si [19] show a similar behavior to the binding energies observed in the present study. Although the binding energy of small vacancy clusters decreases with increasing cluster size, it is generally larger than 2.5 eV, which may imply that these small vacancy clusters are stable at high-temperatures. The dissociation energy of a defect from a cluster is determined by the sum of the binding energy and the migration of that defect. The migration energies of C and Si vacancies are 4.2 and 3.2 eV, respectively, obtained by ab initio method [18] and 4.1 and 2.4 eV determined by empirical potential methods [7], respectively. The migration energy of a Si vacancy given by empirical potential (2.4 eV) is reasonable, as compared with the void formation observed at 1300 K, where at least one of vacancies becomes mobile. The dissociation energy of a vacancy from the cluster is on the order of 5–6 eV, which suggests that vacancy clusters in SiC are stable to high-temperatures of between 1600 and 1900 K, in agreement with experimental observations [8]. It should be point out that the large migration energies of vacancies may play a significant role on the formation and dissociation of vacancy clusters. Previously, swelling experiments demonstrated that interstitial loops are formed even at room temperature, while vacancy clusters or voids are formed only above 1300 K [20]. This indicates that at least one type of interstitial is mobile at room temperature, while vacancies start to migrate at 1300 K. However, it is of interest to further study the structural and binding properties of interstitial clusters in SiC. 4. Conclusions
Fig. 2. Atomic arrangements of a four-vacancy cluster (one Si and three C vacancies), where red spheres represent Si atoms, green spheres C atoms, a small dark gray sphere Si vacancy and small light gray spheres C vacancies; (a) initial atomic configuration and (b) atomic configuration after relaxation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The atomic structures, formation energies and binding properties of point defects and small vacancy clusters in SiC are investigated by large-scale ab initio simulation methods. The calculated formation energies of point defects using both the DZP and SZ basis sets are in good agreement with the previously theoretical calculations, but the SZ basis set gives slightly higher formation energies. One of the interesting results is that a di-vacancy cluster, consisting of two C vacancies located at the second nearest neighbor sites
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F. Gao et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2995–2998
is stable up to 1300 K, while a di-vacancy with two Si vacancies is not stable and may dissociate even at room temperature. The formation energies of small vacancy clusters increase with their size, but the formation energies for the VSi–nVC clusters are generally smaller than those for the VC-nVSi clusters, which may suggest that VSi–nVC clusters are more stable than VC–nVSi clusters. The binding energies of the small vacancy clusters decrease with increasing the cluster size in SiC, which is unexpected, in contrast to those in metals. The calculated values of the binding energies ranged from 2.5 to 4.6 eV imply that these small vacancy clusters can be stable up to 1900 K, which is consistent with experimental observations. Acknowledgements This research is supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy under Contract DE-AC05-76RL01830. The authors also wish to thank the Molecular Science Computing Facility in the Environmental Molecular Sciences Laboratory for a grant of computer time at (PNNL), a multiprogram national laboratory operated by Battelle for the US Department of Energy under Contract DE-AC0576RL01830.
References [1] J.M. Lento, L. Torpo, T.E.M. Staab, R.M. Nieminen, J. Phys.: Condens. Matter 16 (2004) 1053. [2] F. Gao, E.J. Bylaska, W.J. Weber, L.R. Corrales, Phys. Rev. B 64 (2001) 245208. [3] M. Salvador, J.M. Perlado, A. Mattoni, F. Bernardini, L. Colombo, J. Nucl. Matter. 329–333 (2004) 1219. [4] M. Posselt, F. Gao, W.J. Weber, V. Belko, J. Phys.: Condens. Matter 16 (2004) 1307. [5] F. Gao, W.J. Weber, Phys. Rev. 63 (2000) 054101. [6] M. Bockstedte, M. Heid, A. Mattausch, O. Pankratov, Mater. Sci. Forum 389–393 (2002) 471. [7] F. Gao, W.J. Weber, M. Posselt, V. Belko, Phys. Rev. 69 (2004) 245205. [8] L.L. Snead, Y. Katoh, S. Connery, J. Nucl. Mater. 367–370 (2007) 677. [9] C.H. Lam et al., Mater. Res. Soc. Symp. Proc. 792 (2004) R3.19.1. [10] J.M. Soler, E. Artacho, J.D. Gale, A. Garcia, J. Junquera, P. Ordejon, D.S. Portal, J. Phys.: Condens. Matter 14 (2002) 2745. [11] J. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [12] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [13] D.H. Yean, J.R. Riter Jr,, J. Phys. Chem. Solids 32 (1971) 653. [14] G. Lucas, L. Pizzagalli, Nucl. Instr. and Meth. B 255 (2007) 124. [15] J.M. Perlado, L. Malerba, A. Sanchz-Rubio, T. Diaz de la Rubia, J. Nucl. Mater. 276 (2001) 2303. [16] A. Gali, P. Dek, E. Rauls, N.T. Son, I.G. Ivanov, F.H.C. Carlsson, E. Janzn, W.J. Choyke, Phys. Rev. B 67 (2003) 155203. [17] A. Zywietz, J. Furthmüller, F. Bechstedt, S.T. Picrauz, Phys. Rev. B 59 (1999) 15166. [18] E. Rauls, T. Frauenheim, A. Cali, P. Deak, Phys. Rev. B 68 (2003) 155208. [19] A. Bongiorno, L. Colombo, T. Diaz De la Rubia, Europhys. Lett. 43 (1998) 695. [20] H. Miyazaki, T. Suzuki, T. Yano, T. Iseki, J. Nucl. Sci. Technol. 29 (1992) 656.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Formation and properties of defects and small vacancy clusters in SiC: Ab initio calculations F. Gao a,*, W.J. Weber a, H.Y. Xiao b, X.T. Zu b a b
Pacific Northwest National Laboratory, P.O. Box 999, Richland, WA 99352, USA Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu, 610054, China
a r t i c l e
i n f o
Article history: Available online 14 June 2009 PACS: 61.72.-y 31.15.A61.80.-x 61.72.uj Keywords: Defect properties Ab initio simulation Defect clusters Silicon carbide
a b s t r a c t Large-scale ab initio simulation methods have been employed to investigate the configurations and properties of defects in SiC. Atomic structures, formation energies and binding energies of small vacancy clusters have also been studied as a function of cluster size, and their relative stabilities are determined. The calculated formation energies of point defects are in good agreement with previously theoretical calculations. The results show that the di-vacancy cluster consists of two C vacancies located at the second nearest neighbor sites is stable up to 1300 K, while a di-vacancy with two Si vacancies is not stable and may dissociate at room temperature. In general, the formation energies of small vacancy clusters increase with size, but the formation energies for clusters with a Si vacancy and nC vacancies (VSi–nVC) are much smaller than those with a C vacancy and nSi vacancies (VC–nVSi). These results demonstrate that the VSi–nVC clusters are more stable than the VC–nVSi clusters in SiC, and provide possible nucleation sites for larger vacancy clusters or voids to grow. For these small vacancy clusters, the binding energy decreases with increasing cluster size, and ranges from 2.5 to 4.6 eV. These results indicate that the small vacancy clusters in SiC are stable at temperatures up to 1900 K, which is consistent with experimental observations. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Studies of irradiation effects in silicon carbide (SiC) have been carried out for the last decade due to its possible applications in high-temperature, high-frequency and high-power electronics and as structural components for use in gas-cooled fission reactors and fusion environments. In particular, there is a strong interest in understanding defect formation and properties because these defects and their aggregation will affect many of the macroscopic properties of electronic devices and nuclear components, as well as its overall performance under hash environments. Determination of defect formation and energetics is therefore crucial for understanding the response of SiC to irradiation damage and ion implantation. There have been many studies of defect properties in SiC using both ab initio methods [1,2] and empirical potential calculations [3,4]. These calculations provide significant insights into defect stability, defect configuration and formation energy, but only point defects have been investigated. As shown previously, a displacement cascade predominantly produces point defects, and only about 19% of the interstitial population is contained in clusters [5]. The size of the interstitial clus* Corresponding author. Tel.: +1 509 371 6490; fax: +1 509 371 6242. E-mail address: [email protected] (F. Gao). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.018
ters is small, with the largest clusters containing only four interstitial atoms for 50 keV Si displacement cascades. However, these interstitials, particular C interstitials, have relative low energy barriers for migration, as compared with vacancies [6,7], and the migration of carbon interstitials seems quite feasible at room temperature, leading to interstitial aggregations or defect recovery in SiC. As the temperature increases above the critical amorphization temperature, the number of defects surviving in the cascade are reduced, and the mobility of both silicon and carbon interstitials become significant. However, experimental studies have noted the presence of both Frank loops and tiny voids at temperatures approaching 1300 K, indicating limited mobility of vacancies [8]. The apparent increases in swelling in the temperature range of between 1300 and 1800 K may be associated with the production of voids, which are observed to continue to grow in SiC irradiated to 1900 K. This phenomenon may suggest that the diffusion energies of the Si and C vacancies must be very high, as are the binding energies for clustered vacancies. Using isochronal annealing and positron lifetime analysis, Lam [9] has shown a carbon–silicon vacancy complex to dissociate above 1800 K in 6H–SiC. Despite of the evidence for aggregation of interstitials at low temperatures and void swelling at higher temperatures, knowledge about the behavior of interstitial and vacancy clusters in SiC
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F. Gao et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2995–2998
is scare. In addition, modeling of microstructural evolution under irradiation requires detailed understanding regarding the nature and structures of interstitial and vacancy clusters. This paper systematically investigates the formation and binding properties of defects and vacancy clusters in 3C–SiC. On the basis of the simulations, the results are discussed and compared with the experimental measurements. 2. Computational approach All the calculations were performed using the density functional theory (DFT) method as implemented in the SIESTA code [10], while geometry optimization and energy calculations were carried out within the framework of LDA using the parametrized functional of Perdew and Zunger for electron exchange and correlation [11]. The electron–ion interactions were described by normconserving Troullier–Martins [12] pseudopotentials factorized in the Kleinman–Bylander form, where the core radii in the construction of the pseudopotentials were 1.89 and 1.25 Å for Si and C, respectively. The valence wave functions were expanded in a basis set of localized atomic orbitals, and both single-f (SZ) and double-f polarized (DZP) basis sets were used for the purpose of comparison. The calculations were carried out in a supercell containing 128 atoms (DZP) and 512 atoms (SZ). The convergence of the calculations has been tested, and we found a 2 2 2 k-point grid and a 90 Ry cutoff for the real space mesh can give well converged results. With the double-f polarized basis set, the calculated lattice parameter, a0 = 4.36 Å, and the bulk modulus, B = 226 GPa, were found to be in good agreement with the experimental values of 4.36 Å and 224 GPa [13], respectively. Fig. 1 shows the total energy as a function of the lattice parameter of 3C–SiC using different basis sets. It can be seen that the lattice parameter determined by the triple-f (TZ) basis set and double-f basis set plus polarization (DZP) orbitals is in excellent agreement with the experimental values of 4.36 Å, while single-f (SZ) and DZ basis sets give slightly larger lattice parameter (4.45 Å). Although the DZP and TZ basis sets provide a highly accurate lattice parameter, the SZ basis set offers already quite well converged results, comparable to those used in practice in most plane-wave calculations. The structures of point defects and defect clusters were relaxed using conjugate gradient (CG) coordinate optimization until the forces on each atom were <0.01 eV/Å. 3. Results and discussion As described in [2], there are a number of possible configurations of intrinsic defects in 3C–SiC, and the fundamental properties
Total energy (eV)
-260
SZ TZ DZP DZ
-261
of these defects have been investigated using both ab initio calculations [2,3,14] and empirical potentials [2,15]. In this paper, only a subset of intrinsic defects are investigated and compared with the previous calculations since the present study mainly deals with defect clusters. The point defects considered include a C vacancy, a Si vacancy, an antisite pair (two nearest neighboring atoms exchange their positions), a C interstitial and a Si interstitial. It should be noted that there are a number of possible configurations for an interstitial [2], but only their most stable configurations are considered (i.e. the C–C dumbbell along the <1 0 0> direction and Si tetrahedral surrounded by four C atoms). The geometrical structures of these defects are relaxed completely and optimized, and the corresponding defect formation energies were calculated. The formation energy of a defect is defined by the following formula [4]:
1 1 Ef ¼ Edtot ðnSi þ nC ÞlSiC ðnSi nC Þ 2 2
1 2
lbSi lbC ðnSi nC ÞDl: ð1Þ
Edtot
Here is the calculated total energy of a supercell containing nSi silicon and nC carbon atoms with a defect, and lbSi and lbC represents the chemical potential of bulk Si and C crystal, respectively. Dl is related to the deviation from ideal stoichiometry, varying from Crich kDl = DY| to Si-rich kDl = DY|. The calculated defect formation energies obtained by SZ and DZP basis sets are presented in Table 1 for the ideal stoichiometry condition, along with other theoretical calculations [1,3,14,16,17] for comparison. In addition, a di-vacancy defect is included, where two vacancies located at the first or second nearest neighbor distance are considered. In general, the formation energies calculated with DZP and SZ basis sets are in good agreement with those obtained by GGA and LDA calculations. The formation energies of C and Si vacancies calculated with DZP basis sets are 3.74 and 8.30 eV, respectively, and the formation energies of CC <1 0 0> dumbbell and SiTC interstitials are 6.71 and 7.22 eV, respectively. It should be noticed that the formation energies obtained by SZ basis set is systematically higher than those calculated with DZP basis set. Using the formation energies, the binding energy of a cluster can be calculated and the results indicate that a di-vacancy (di-VC–Si) with a C vacancy and a Si vacancy at the nearest neighbors is the most stable di-vacancy configuration, followed by that with two C vacancies located at the second nearest neighbors (di-VC). The formation energy of a di-vacancy cluster with two Si vacancies at second nearest neighbors (di-VSi) is very high, but with a small binding energy of about 0.21, which indicates that its nucleation at temperatures below room temperature may be impossible. The binding energy for a di-VC–Si cluster is 4.29 eV, which indicates that this vacancy cluster can be stable at hightemperatures, and it may act as a nucleation site for the growth of larger vacancy clusters. Also, a di-VC cluster, with a binding energy of 1.15 eV, can be stable up to 1300 K and may be dissociated at higher temperatures.
-262
Table 1 Calculated formation energies of intrinsic defects in the neutral charge state, along with other theoretical calculations for comparison.
-263
Defect formation energy Ef (eV) Defect DZP SZ 3.74 4.02 VC 8.30 8.88 VSi CC <1 0 0> 6.71 7.02 7.22 9.20 SiTC Antisite pair 4.43 3.95 di-V (1 n n) 7.75 8.65 16.39 – di-VSi (2 n n) 6.33 – di-VC (2 n n)
-264 4.1
4.2
4.3
4.4
4.5
4.6
4.7
a (A) Fig. 1. Total energy of a unit cell versus lattice parameter for bulk 3C–SiC for different basis sets.
[14] 3.63 7.48 6.65 7.04 – – – –
[3] 3.74 8.38 – 7.02 – – – –
[16] 4.2 8.1 8.9 – – – – –
[17] 4.3 8.45
– – – –
[1] – – 6.7 6.0 – – – –
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F. Gao et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2995–2998
Eb ðnÞ ¼ Ef ðn 1Þ þ Eaf ð1Þ Ef ðnÞ;
ð2Þ
where a is either a C or Si vacancy, which depends on the cluster type. The binding energy of the vacancy clusters is shown in Fig. 3 as a function of the cluster size. The results show that the binding energy decreases with increasing the cluster size, in contrast to that in pure metals, where the binding energy of vacancy clusters generally increases with increasing cluster size, but it is much larger than that in metals. Values reported here are only for the compact vacancy clusters, as described above. It should be pointed out that sampling of all possible configurations is complicated and it is possible that lower energy configurations than those reported here could be found. It is of interest to note that the small vacancy clusters (n < 10) formed from successive shells of neighbors
Table 2 Calculated formation energies of small vacancy clusters in the neutral charge state. Defect formation energy Ef (eV) VC–VSi Defect VC Ef (eV) 4.02 8.65 VSi–VC Defect VSi Ef (eV) 8.88 8.65
VC–2VSi 14.49 VSi–2VC 8.70
VC–3VSi 21.16 VSi–3VC 9.42
VC–4VSi 26.59 VSi–4VC 10.84
4.5
CV+nSiV SiV+nCV
4.0
E b (eV)
In the present study, several small vacancy clusters are investigated, but only compact configurations are considered, i.e. a vacancy in cluster has at least one vacancy at the nearest neighbor distance. Fig. 2 shows an example of a four-vacancy cluster (a Si and three C vacancies), where (a) represents the atomic configuration before relaxation and (b) exhibits the atomic arrangements after relaxation. It is of interest to find that the relaxations around the vacancy cluster are very small. The first nearest neighbor Si atoms to the C vacancies on the same (1 1 1) plane with the Si vacancy relax towards the vacancies, but the displacement from their original positions is only about 0.14 Å. The first nearest neighbor Si atoms on the plane below the C vacancies relax outwards from the vacancies with a displacement of about 0.016 Å. These small relaxations can not explain a large reduction of the total energy observed after relaxation, which may be associated with charge redistribution and will be discussed in details elsewhere. The formation energies of the small vacancy clusters are calculated using Eq. (1), and listed in Table 2. It can be seen that the formation energies of the small vacancy clusters generally increase with their size, but the formation energies for the VSi–nVC clusters are much smaller than those for the VC–nVSi clusters, and thus, the VSi–nVC clusters is more feasible to provide possible nucleation sites for larger vacancy clusters or voids to grow. It is of interest to note that the formation energy of the VSi–VC cluster is even smaller than that of a single Si vacancy (VSi), and this di-vacancy cluster can be easily formed during irradiation. Previously theoretical calculations argued that the VSi is not stable [18] and a C neighbor exchanges place with the vacancy producing a close-by pair VC–CSi (a C vacancy plus a Si antisite defect), and the high formation energy of a single Si vacancy is consistent with this previous observation. From the formation energies, the binding properties of vacancy clusters can be determined. The binding energy, Eb(n), of a defect cluster with n vacancies can be defined as:
3.5 3.0 2.5 2.0
1
2
3
4
5
6
Size N Fig. 3. Binding energy of the small vacancy clusters as a function of the cluster size in SiC.
of a given lattice site in Si [19] show a similar behavior to the binding energies observed in the present study. Although the binding energy of small vacancy clusters decreases with increasing cluster size, it is generally larger than 2.5 eV, which may imply that these small vacancy clusters are stable at high-temperatures. The dissociation energy of a defect from a cluster is determined by the sum of the binding energy and the migration of that defect. The migration energies of C and Si vacancies are 4.2 and 3.2 eV, respectively, obtained by ab initio method [18] and 4.1 and 2.4 eV determined by empirical potential methods [7], respectively. The migration energy of a Si vacancy given by empirical potential (2.4 eV) is reasonable, as compared with the void formation observed at 1300 K, where at least one of vacancies becomes mobile. The dissociation energy of a vacancy from the cluster is on the order of 5–6 eV, which suggests that vacancy clusters in SiC are stable to high-temperatures of between 1600 and 1900 K, in agreement with experimental observations [8]. It should be point out that the large migration energies of vacancies may play a significant role on the formation and dissociation of vacancy clusters. Previously, swelling experiments demonstrated that interstitial loops are formed even at room temperature, while vacancy clusters or voids are formed only above 1300 K [20]. This indicates that at least one type of interstitial is mobile at room temperature, while vacancies start to migrate at 1300 K. However, it is of interest to further study the structural and binding properties of interstitial clusters in SiC. 4. Conclusions
Fig. 2. Atomic arrangements of a four-vacancy cluster (one Si and three C vacancies), where red spheres represent Si atoms, green spheres C atoms, a small dark gray sphere Si vacancy and small light gray spheres C vacancies; (a) initial atomic configuration and (b) atomic configuration after relaxation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The atomic structures, formation energies and binding properties of point defects and small vacancy clusters in SiC are investigated by large-scale ab initio simulation methods. The calculated formation energies of point defects using both the DZP and SZ basis sets are in good agreement with the previously theoretical calculations, but the SZ basis set gives slightly higher formation energies. One of the interesting results is that a di-vacancy cluster, consisting of two C vacancies located at the second nearest neighbor sites
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is stable up to 1300 K, while a di-vacancy with two Si vacancies is not stable and may dissociate even at room temperature. The formation energies of small vacancy clusters increase with their size, but the formation energies for the VSi–nVC clusters are generally smaller than those for the VC-nVSi clusters, which may suggest that VSi–nVC clusters are more stable than VC–nVSi clusters. The binding energies of the small vacancy clusters decrease with increasing the cluster size in SiC, which is unexpected, in contrast to those in metals. The calculated values of the binding energies ranged from 2.5 to 4.6 eV imply that these small vacancy clusters can be stable up to 1900 K, which is consistent with experimental observations. Acknowledgements This research is supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy under Contract DE-AC05-76RL01830. The authors also wish to thank the Molecular Science Computing Facility in the Environmental Molecular Sciences Laboratory for a grant of computer time at (PNNL), a multiprogram national laboratory operated by Battelle for the US Department of Energy under Contract DE-AC0576RL01830.
References [1] J.M. Lento, L. Torpo, T.E.M. Staab, R.M. Nieminen, J. Phys.: Condens. Matter 16 (2004) 1053. [2] F. Gao, E.J. Bylaska, W.J. Weber, L.R. Corrales, Phys. Rev. B 64 (2001) 245208. [3] M. Salvador, J.M. Perlado, A. Mattoni, F. Bernardini, L. Colombo, J. Nucl. Matter. 329–333 (2004) 1219. [4] M. Posselt, F. Gao, W.J. Weber, V. Belko, J. Phys.: Condens. Matter 16 (2004) 1307. [5] F. Gao, W.J. Weber, Phys. Rev. 63 (2000) 054101. [6] M. Bockstedte, M. Heid, A. Mattausch, O. Pankratov, Mater. Sci. Forum 389–393 (2002) 471. [7] F. Gao, W.J. Weber, M. Posselt, V. Belko, Phys. Rev. 69 (2004) 245205. [8] L.L. Snead, Y. Katoh, S. Connery, J. Nucl. Mater. 367–370 (2007) 677. [9] C.H. Lam et al., Mater. Res. Soc. Symp. Proc. 792 (2004) R3.19.1. [10] J.M. Soler, E. Artacho, J.D. Gale, A. Garcia, J. Junquera, P. Ordejon, D.S. Portal, J. Phys.: Condens. Matter 14 (2002) 2745. [11] J. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [12] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [13] D.H. Yean, J.R. Riter Jr,, J. Phys. Chem. Solids 32 (1971) 653. [14] G. Lucas, L. Pizzagalli, Nucl. Instr. and Meth. B 255 (2007) 124. [15] J.M. Perlado, L. Malerba, A. Sanchz-Rubio, T. Diaz de la Rubia, J. Nucl. Mater. 276 (2001) 2303. [16] A. Gali, P. Dek, E. Rauls, N.T. Son, I.G. Ivanov, F.H.C. Carlsson, E. Janzn, W.J. Choyke, Phys. Rev. B 67 (2003) 155203. [17] A. Zywietz, J. Furthmüller, F. Bechstedt, S.T. Picrauz, Phys. Rev. B 59 (1999) 15166. [18] E. Rauls, T. Frauenheim, A. Cali, P. Deak, Phys. Rev. B 68 (2003) 155208. [19] A. Bongiorno, L. Colombo, T. Diaz De la Rubia, Europhys. Lett. 43 (1998) 695. [20] H. Miyazaki, T. Suzuki, T. Yano, T. Iseki, J. Nucl. Sci. Technol. 29 (1992) 656.