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Evolution of atoms with special coordination number in β-SiC with temperature
Journal of Nuclear Materials 435 (2013) 236–240
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Evolution of atoms with special coordination number in b-SiC with temperature Jianqi Xi, Chaohui He, Hang Zang, Daxi Guo, Tao Yang, Tao Bo, Peng Zhang ⇑ Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 27 August 2012 Accepted 3 January 2013 Available online 11 January 2013
a b s t r a c t The characteristics and mechanisms of evolution of Si and C atoms with different coordination number were studied using molecular dynamics and molecular static. Relaxation energy of these systems and the displacement of these special atoms from their original positions have been investigated. The results showed that some of these special Si atoms moved away from their lattice sites and may become Si interstitial atoms at 300 K. As temperature increased, more intrinsic defects were detected. In the case of the one-coordinated C atom, this atom did not move away from its lattice site until 600 K, turning into a C interstitial atom. Then it became a CSi antisite after 1200 K. In general, the relaxation energies and the displacement decrease with increasing coordination number. The differences between special Si and the equivalent special C systems strongly depend on the temperature. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Cubic silicon carbide (b-SiC) as a component of SiC/SiC composites has long been known for its potential for structural components in fusion reactors [1] and high-temperature reactors (HTRs) [2], cladding materials for gas-cooled fission reactors [3] due to its small cross section, low-activation and good thermal conductivity under neutron irradiation. In the harsh environment, many defects can be produced by displacement cascades occurring when particles irradiate b-SiC. They can lead to localized microstructural modification, and thus altering the mechanical property. During the displacement cascades, many atoms will displace away from their original lattice sites, creating several defects such Frenkel pairs, antisites. This phenomenon leaves several silicon and carbon atoms with different coordination numbers. Several studies of irradiation effects in b-SiC have been carried out for the last decades, yet there are still many problems unsolved. Especially, there is a strong desire to understand formation and evolution of defects, since these defects and their aggregation will affect many of macroscopic properties of nuclear components, as well as its overall performance. Investigations on the generation and evolution of defects were performed using ab initio calculations [4–7] as well as MD [8–13] and KMC simulation [14]. Specifically, it has been noted that Si atoms and C atoms with different coordination numbers play a unique role in the process of micro-structural modification. Jin and Niu [8] reported that the two-coordinated Si atom bond angle distribution was much wider and lower, which indicated that the two-coordinated Si atoms
contribute to the topological disorder. They also found that the whole lattice became amorphous when these defect concentrations reached 7.5%. Investigation on the formation of atoms with different coordination numbers in 3C–SiC was performed using Gao–Weber potential [15] by Morishita et al. [10]. They reported that the formation energies of atoms with different coordination numbers decreased with increasing coordination number. They also reported that after relaxation the one-coordinated C atom became a carbon antisite defect. However, both studies did not clearly show the evolution of these special atoms with change in different temperatures. At the same time, the atomistic mechanisms of those evolutions are not well understood. Thus further evidences for this as well as the mechanisms involved are needed. In this work, we investigate the evolution of the special Si atoms and C atoms with special coordination numbers in 3C–SiC at different temperatures using Tersoff/ZBL potential which can provide a good description of vacancy properties compared with Gao–Weber potential used in [10]. Meanwhile, the mechanisms of those evolutions at the atomic scale are analyzed. Additionally, the influence of these atoms on the vacancy migration and growth of vacancy cluster are studied. The bonding state of two atoms depends on their distance in the present work according to Jin and Niu [8]. Many experiments have shown that Si atoms at tetrahedral interstices will not bond with any neighboring atom, although the distance between them is less than the cutoff distance [8,16]. Therefore, in this work, we only study these coordination numbers, N, less than four, but larger than zero, as shown in Fig. 1.
2. Modeling methods ⇑ Corresponding author. Address: No. 28, Xianning West Road, Xi’an, Shaanxi 710049, PR China. E-mail address: [email protected] (P. Zhang). 0022-3115/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2013.01.001
All simulation reported herein were performed with the Sandia National Laboratories Large-scale Atomic/Molecular Massively
J. Xi et al. / Journal of Nuclear Materials 435 (2013) 236–240
237
Fig. 1. Snapshots of N-coordinated C and Si atoms. The first line from (a) to (c) represents the N-coordinated C systems, while the second line from (d) to (f) is the N-coordinated Si systems. The grey spheres represent C atoms, yellow spheres Si atoms, red spheres the special N-coordinated Si atoms, blue spheres the special N-coordinated C atoms, pink spheres C vacancies and green spheres Si vacancies. For example, in the (a) snapshot, the one-coordinated C atom means that three Si atoms around the selected C atom are deleted, making three Si vacancies. The coordination numbers increase from the left to right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Parallel Simulator code (LAMMPS) [17]. As an alternative mean of MD, molecular static (MS) can be employed to mimic the equilibrium state at 0 K by minimizing the potential energy of the system [18]. Therefore, the combined methods [10] can provide more relaxed configuration of defects than a simple static relaxation method [19]. The interatomic interactions with a hybrid Tersoff/ZBL potential, which was obtained by modifying the Tersoff potential [20] to improve the description of property of close-separation pairwise, were described in [21] with Tersoff potential parameters from [21]. This potential can provide a good description of vacancy properties compared with Gao–Weber potential. Firstly, a so-called N-coordinated Si or C atom was introduced into a perfect 3C–SiC crystal with constant volume and periodic boundary conditions by removing several nearest atoms bonded to the special atom. For example, the one-coordinated C atom was introduced by removing three random nearest Si atoms bonded to the selected C atom. The box contains 8000 lattice sites (10 10 10 unit cells, 4000 silicon lattice sites and 4000 carbon lattice sites), the lattice parameter is 0.436 nm based on the experimental value [22]. A four-step procedure was followed:
antisite have been calculated. The formation energies of these various defects are defined as follows [15]:
EVf ¼ DEV ðXÞ þ e
ð1Þ
For a vacancy;
Eaf ðX Y Þ
¼ DEa ðX Y Þ
ð2Þ
for an antisite, where X and Y represent the C or Si species, and e is the negative cohesive energy, 6.41 eV, in a perfect 3C–SiC at 0 K in this present work which is agreement with previous experimental values, 6.34 eV [22] and previous stimulation values, 6.39 eV [23]. DE denotes the total energy difference between the crystal containing a defect and the perfect crystal with the same number of lattice sites. The results are compared with those obtained previously by MD as well as ab initio calculations in Table 1. It can be seen that those results determined by our method are in agreement with the previous MD reference values [15,25] and ab initio calculation [25]. 3.2. Evolution of these special atoms
(1) Minimize the initial system at 0 K. (2) The system was then annealed at a given temperature for 80–100 ps to stabilize the defect configuration. Temperatures used here ranged from 300 K to 1500 K. It has been experimentally reported that the migration of vacancies and small vacancy clusters would create voids when irradiation temperature is greater than about 1300 K [23]. Therefore, in order to investigate the vacancy migration, the highest temperature of simulation was set at 1500 K. (3) After that, the system was quenched to 0 K for 20 ps, in order to obtain the defects configuration without interference from thermal activation [10,24]. (4) Finally, the system was relaxed to get the final value of total energy. 3. Results and discussion
Table 2 summarizes the results obtained at 300 K for the relaxation energies of different systems and the properties of these special N-coordinated Si and C atoms, including the distances when displaced away from the original sites. It is important to note that the special one-coordinated Si atom and two-coordinated Si atom had a clear movement from their original lattice sites, as seen in Fig. 2a and b, 53% and 34% of the Si–C bond length (0.189 nm), respectively. While for other special atoms, the displacement can be negligible. It indicates that the one-coordinated Si and the two-coordinated Si atom contributes more to structural disorder than other special atoms, which plays an important role in the process of amorphization as discussed in [8]. The binding energies, Eb(n) of those stable one-coordinated C system containing three Si vacancies and two-coordinated C system containing two Si vacancies have also been calculated using Eq. (3) as defined below:
3.1. SiC defect formation energies
Eb ðnÞ ¼ Ef ðn 1Þ þ ESi f ð1Þ Ef ðnÞ
In order to ensure that our method is valid and provide basic data for the study of the stability of defect clusters as discussed below, the formation energies of C and Si vacancy, CSi antisite and SiC
Both of the results were very small, about 0.10 eV, which are consistent with that in [4]. It indicates that their nucleation at room temperature may be impossible, namely, these Si vacancies may not create homogeneous vacancy clusters. From Table 2, it
ð3Þ
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J. Xi et al. / Journal of Nuclear Materials 435 (2013) 236–240 Table 1 The formation energy of vacancies and antisite defects in 3C–SiC. Parameter
Present work (eV)
Previous G–W potential value (eV)
Previous Tersoff/zbl potential value (eV)
Ab initio value (eV)
VC VSi CSi SiC
4.96 6.47 0.51 6.25
2.76a 3.30a 1.69a 7.79a
5.20b 6.31b 0.35b 6.10b
5.48b 6.64b 1.32b 7.20b
Ref. [15]. Ref. [25].
Table 2 The relaxation energy of different systems and the displacement of these special atoms from their original sites at 300 K. Parameter
Relaxation energy (eV)
Displacement (nm)
One-coordinated Si Two-coordinated Si Three-coordinated Si One-coordinated C Two-coordinated C Three-coordinated C
3.62 1.95 0.18
0.101 0.065 0.013
0.11 0.05 0.01
0.010 0.008 0.007
4.5 4.0
Change in potential energy (eV)
a b
3.5
one-coordinated Si two-coordinated Si three-coordinated Si one-coordinated C two-coordinated C three-coordinated C
3.0 2.5 2.0 1.5 1.0 0.5 0.0 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Temperature (K) Fig. 3. Change in potential energy of special atoms as a function of temperature.
7.0 6.5 6.0
Fig. 2. Snapshots of one-coordinated Si and two-coordinated Si atom at 300 K. (a) One-coordinated Si system, (b) two-coordinated Si system.
can be seen that the relaxation energies of N-coordinated Si systems are always larger than that of corresponding N-coordinated C systems. At the same time, the displacement of these special N-coordinated Si atoms are greater than that of corresponding N-coordinated C atoms. As the coordination numbers increase, both the relaxation energies of systems and the displacement decrease. It indicates that the larger the relaxation energy of system is, the greater the displacement of the atom. These asymmetric behaviors between N-coordinated Si systems and C systems are probably caused by the difference of atom size and the thermal fluctuation of atom, as discussed below. Cutoff distance which depends on the atom size can be scaled with the system volume to avoid uncertainties. For the potential we used, the largest cutoff distance of Si–Si is 0.3 nm, while for C–C it is 0.21 nm [21]. During the annealing, all atoms are vibrating randomly. In a perfect 3C–SiC system, each atom interacts with four nearest neighbor atoms. However, for systems with special N-coordinated Si atoms, the special Si atoms may interact with the nearest neighbor C atoms and simultaneously with the second-nearest-neighbor Si atoms due to the Si–Si cutoff that is slightly smaller than the second-nearest-neighbor distance (0.303 nm). Whereas for the N-coordinated C systems, the special C atoms only interact with the nearest neighbor Si atoms because
Relaxation energy (eV)
5.5 5.0
one-coordinated Si two-coordinated Si three-coordinated Si one-coordinated C two-coordinated C three-coordinated C
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Temperature (K) Fig. 4. Relaxation energy of systems as a function of temperature.
of the C–C cutoff that is much smaller than the second-nearestneighbor distance, as discussed in [22]. For one special N-coordinated Si atom, two modes of interaction are possible. Firstly, it can interact with 12 second-nearestneighbor Si atoms, making it move away from its lattice site. On the other hand, the interactions between the N-coordinated Si atom and those nearest neighbor C atoms can cause this Si atom to be stable in the lattice site, thus, the displacement of Si atom from its original position depends on the two aspects. However, for one special N-coordinated C atom, because this C atom only interacts with the nearest neighbor Si atoms, it always stays in the lattice site. Therefore, the relaxation energies of N-coordinated C systems are lower than that of corresponding N-coordinated Si systems, and the displacement of N-coordinated Si atoms are larger
J. Xi et al. / Journal of Nuclear Materials 435 (2013) 236–240
239
Fig. 5. The configuration of one-coordinated C system at (a) 300 K, (b) 600 K, and (c) 1200 K. (a) The one-coordinated C atom stays in its lattice site, (b) the displacement of special C atom toward the Si vacancy is about 84% and (c) the special C atom has occupied the Si site and become the CSi antisite.
than those of N-coordinated C atoms. With increasing coordination number, the interactions between nearest neighbor C or Si atoms are strengthened. This stabilizes this special N-coordinated Si or N-coordinated C atom in the lattice site leading to decrease in the relaxation energy, as shown in Table 2. How the secondnearest-neighbor interaction between Si–Si and C–C atoms causes the change in potential energy at 300 K is shown in Fig. 3. It can be seen that only the potential energies of one-coordinated Si atom and two-coordinated Si atom are changed due to the secondnearest-neighbor interactions between Si and Si at 300 K, indicating that the thermal fluctuation and the difference of atom size make the difference of evolution properties. Further studies were made at different temperatures ranging from 400 K to 1500 K. Surprising results were obtained when temperature increased, as shown in Figs. 3 and 4: both the change in potential energy of one-coordinated C atom and the relaxation energy are larger than those of one-coordinated Si atom at above 600 K due to the strengthening thermal fluctuation of C atoms. The slight fluctuation of change in potential energy of different atoms is due to the random vibration of atoms that may cause a change of interaction between atoms, but the trend is that the change in potential energy of N-coordinated Si atoms decreases with increasing coordination numbers and is less than that of one-coordinated C atom at high temperature. The special onecoordinated C atom relaxed toward the nearest Si vacancy with a displacement about 84% of the Si–C bond length, when temperature reached 600 K, as shown in Fig. 5b. It is also worth noting that this one-coordinated C atom jumped into the nearest neighbor Si vacancy and became one CSi antisite, as shown in Fig. 5c, with increasing potential energy and relaxation energy at 1200 K, as shown in Figs. 3 and 4, respectively. For the two-coordinated C or three-coordinated C atom, the relaxation energy and displacement from their original site is negligible, probably because of the intense interaction between the special C atom and its surrounding nearest neighbor Si atoms. Binding energy of that vacancy cluster (containing one C vacancy and two Si vacancies) in the changed one-coordinated C system was 2.79 eV which is dramatically different from the above result, 0.10 eV, at 300 K. This is because of the change in defect configuration from three Si vacancies to two Si vacancies, one C vacancy and a CSi antisite. The high binding energy implies that the small vacancy cluster is stable at high temperatures. It also indicates that at higher temperature, one-coordinated C atom provides possible nucleation sites for vacancy migration as well as small vacancy cluster and defect replacement, consistent with theoretical calculations [11] and experimental observations [2]. In addition, the special N-coordinated Si atoms and their surrounding atoms have undergone substantial annealing, when temperature was above 600 K. Note that there was no SiC antisite produced after N-coordinated Si atoms fully annealed; this is probably caused by the difference of formation properties between silicon antisites
and carbon antisites, as discussed in [10]. The above results show that the thermal fluctuation of atoms and the size asymmetry of silicon and carbon atoms make the difference in defect properties. 4. Conclusion MD and MS have been carried out to investigate the evolution of atoms with different coordination numbers in 3C–SiC. At low temperature (below 600 K), the fully annealed one-coordinated Si atom and two-coordinated Si atom moved from their original lattice sites, while other special atoms did not. These atoms may play an important role in the process of structural disorder and amorphization. The relaxation energies of N-coordinated Si systems at low temperature were larger than that of corresponding N-coordinated C systems. We found that the relaxation energy and the displacement of one-coordinated C atom from its original site became considerable at higher temperature because the enhanced thermal fluctuation of atoms resulted in larger movement and interatomic interaction. Furthermore, our study of one-coordinated C system annealed at high temperature (above 600 K) indicated that onecoordinated C atoms may provide possible nucleation sites for vacancy migration as well as vacancy clusters and defect replacement to create antisites at higher temperature. However, for N-coordinated Si, no SiC antisite was detected, even though those N-coordinated Si atoms had a large movement. It indicates that this was further evidence that the formation of CSi antisite is easier than that of SiC antisite at high temperature. Finally, based on our results, we expect that at higher temperature and longer timescales, these atoms specially coordinated may play a significant role in the nucleation and growth of small defect clusters. However, more investigations are needed to confirm that. Acknowledgements We acknowledge fruitful comments with Dr. Dane Morgan from University of Wisconsin-Madison. We also thank Mr. J.M. Kebwaro for helpful discussion. This work was financially supported by NSFC under Grant No. 11175138 and China Postdoctoral Science Foundation (2011M501450). P. Zhang was also supported by ‘‘the Fundamental Research Funds for the Central Universities’’. The computation work is partly supported by the cluster Hua-I in Xi’an Jiaotong University and the National Supercomputing Centre in Shenzhen. References [1] T. Nozawa, T. Hinoki, A. Hasegawa, A. Kohyama, Y. Katoh, L.L. Snead, C.H. Henager Jr., J.B.J. Hegeman, J. Nucl. Mater. 386–388 (2009) 622–627. [2] S. Sorieul, J.-M. Costantini, L. Gosmain, L. Thomé, J.-J. Grob, J. Phys.: Condens. Matter. 18 (2006) 5235–5251. [3] Lance L. Snead, Takashi Nozawa, Yutai Katoh, Thak-Sang Byun, Sosuke Kondo, David A. Petti, J. Nucl. Mater. 371 (2007) 329–377.
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[4] F. Gao, W.J. Weber, H.Y. Xiao, X.T. Zu, Nucl. Instrum. Methods Phys. Res. B 267 (2009) 2995–2998. [5] C. Wang, J. Bernholc, R.F. Davis, Phys. Rev. B 38 (1988) 12752. [6] Fabien Bruneval, Guido Roma, Phys. Rev. B 83 (2011) 144116. [7] Michel Bockstedte, Alexander Mattausch, Oleg Pankratov, Phys. Rev. B 69 (2004) 235202. [8] Enze Jin, Li-Sha Niu, Phys. B 406 (2011) 601–608. [9] F. Gao, W.J. Weber, Phys. Rev. B 69 (2004) 224108. [10] K. Morishita, Y. Watanabe, A. Kohyama, H.L. Heinisch, F. Gao, J. Nucl. Mater. 386–388 (2009) 30–32. [11] E. David, J. Nucl. Mater. 385 (2009) 572–581. [12] K. Morishita, Y. Watanabe, A. Kohyama, H.L. Heinisch, F. Gao, Fusion Sci. Technol. 56 (2009) 328–330. [13] S.T. Nakagawa, A. Okamoto, G. Betz, Nucl. Instrum. Methods Phys. Res. B 258 (2007) 194–198. [14] Yoshiyuki Watanabe, Kazunori Morishita, Yasunori Yamamoto, Nucl. Instrum. Methods Phys. Res. B 269 (2011) 1698–1701.
[15] F. Gao, W.J. Weber, Nucl. Instrum. Methods Phys. Res. B 191 (2002) 504–508. [16] N.T. Son, E. Janzen, J. Isoya, N. Morishia, H. Takizawa, T. Ohshima, A. Gali, Phys. Rev. B 80 (2009) 125201. [17] Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) website.
Contents lists available at SciVerse ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Evolution of atoms with special coordination number in b-SiC with temperature Jianqi Xi, Chaohui He, Hang Zang, Daxi Guo, Tao Yang, Tao Bo, Peng Zhang ⇑ Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 27 August 2012 Accepted 3 January 2013 Available online 11 January 2013
a b s t r a c t The characteristics and mechanisms of evolution of Si and C atoms with different coordination number were studied using molecular dynamics and molecular static. Relaxation energy of these systems and the displacement of these special atoms from their original positions have been investigated. The results showed that some of these special Si atoms moved away from their lattice sites and may become Si interstitial atoms at 300 K. As temperature increased, more intrinsic defects were detected. In the case of the one-coordinated C atom, this atom did not move away from its lattice site until 600 K, turning into a C interstitial atom. Then it became a CSi antisite after 1200 K. In general, the relaxation energies and the displacement decrease with increasing coordination number. The differences between special Si and the equivalent special C systems strongly depend on the temperature. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Cubic silicon carbide (b-SiC) as a component of SiC/SiC composites has long been known for its potential for structural components in fusion reactors [1] and high-temperature reactors (HTRs) [2], cladding materials for gas-cooled fission reactors [3] due to its small cross section, low-activation and good thermal conductivity under neutron irradiation. In the harsh environment, many defects can be produced by displacement cascades occurring when particles irradiate b-SiC. They can lead to localized microstructural modification, and thus altering the mechanical property. During the displacement cascades, many atoms will displace away from their original lattice sites, creating several defects such Frenkel pairs, antisites. This phenomenon leaves several silicon and carbon atoms with different coordination numbers. Several studies of irradiation effects in b-SiC have been carried out for the last decades, yet there are still many problems unsolved. Especially, there is a strong desire to understand formation and evolution of defects, since these defects and their aggregation will affect many of macroscopic properties of nuclear components, as well as its overall performance. Investigations on the generation and evolution of defects were performed using ab initio calculations [4–7] as well as MD [8–13] and KMC simulation [14]. Specifically, it has been noted that Si atoms and C atoms with different coordination numbers play a unique role in the process of micro-structural modification. Jin and Niu [8] reported that the two-coordinated Si atom bond angle distribution was much wider and lower, which indicated that the two-coordinated Si atoms
contribute to the topological disorder. They also found that the whole lattice became amorphous when these defect concentrations reached 7.5%. Investigation on the formation of atoms with different coordination numbers in 3C–SiC was performed using Gao–Weber potential [15] by Morishita et al. [10]. They reported that the formation energies of atoms with different coordination numbers decreased with increasing coordination number. They also reported that after relaxation the one-coordinated C atom became a carbon antisite defect. However, both studies did not clearly show the evolution of these special atoms with change in different temperatures. At the same time, the atomistic mechanisms of those evolutions are not well understood. Thus further evidences for this as well as the mechanisms involved are needed. In this work, we investigate the evolution of the special Si atoms and C atoms with special coordination numbers in 3C–SiC at different temperatures using Tersoff/ZBL potential which can provide a good description of vacancy properties compared with Gao–Weber potential used in [10]. Meanwhile, the mechanisms of those evolutions at the atomic scale are analyzed. Additionally, the influence of these atoms on the vacancy migration and growth of vacancy cluster are studied. The bonding state of two atoms depends on their distance in the present work according to Jin and Niu [8]. Many experiments have shown that Si atoms at tetrahedral interstices will not bond with any neighboring atom, although the distance between them is less than the cutoff distance [8,16]. Therefore, in this work, we only study these coordination numbers, N, less than four, but larger than zero, as shown in Fig. 1.
2. Modeling methods ⇑ Corresponding author. Address: No. 28, Xianning West Road, Xi’an, Shaanxi 710049, PR China. E-mail address: [email protected] (P. Zhang). 0022-3115/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2013.01.001
All simulation reported herein were performed with the Sandia National Laboratories Large-scale Atomic/Molecular Massively
J. Xi et al. / Journal of Nuclear Materials 435 (2013) 236–240
237
Fig. 1. Snapshots of N-coordinated C and Si atoms. The first line from (a) to (c) represents the N-coordinated C systems, while the second line from (d) to (f) is the N-coordinated Si systems. The grey spheres represent C atoms, yellow spheres Si atoms, red spheres the special N-coordinated Si atoms, blue spheres the special N-coordinated C atoms, pink spheres C vacancies and green spheres Si vacancies. For example, in the (a) snapshot, the one-coordinated C atom means that three Si atoms around the selected C atom are deleted, making three Si vacancies. The coordination numbers increase from the left to right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Parallel Simulator code (LAMMPS) [17]. As an alternative mean of MD, molecular static (MS) can be employed to mimic the equilibrium state at 0 K by minimizing the potential energy of the system [18]. Therefore, the combined methods [10] can provide more relaxed configuration of defects than a simple static relaxation method [19]. The interatomic interactions with a hybrid Tersoff/ZBL potential, which was obtained by modifying the Tersoff potential [20] to improve the description of property of close-separation pairwise, were described in [21] with Tersoff potential parameters from [21]. This potential can provide a good description of vacancy properties compared with Gao–Weber potential. Firstly, a so-called N-coordinated Si or C atom was introduced into a perfect 3C–SiC crystal with constant volume and periodic boundary conditions by removing several nearest atoms bonded to the special atom. For example, the one-coordinated C atom was introduced by removing three random nearest Si atoms bonded to the selected C atom. The box contains 8000 lattice sites (10 10 10 unit cells, 4000 silicon lattice sites and 4000 carbon lattice sites), the lattice parameter is 0.436 nm based on the experimental value [22]. A four-step procedure was followed:
antisite have been calculated. The formation energies of these various defects are defined as follows [15]:
EVf ¼ DEV ðXÞ þ e
ð1Þ
For a vacancy;
Eaf ðX Y Þ
¼ DEa ðX Y Þ
ð2Þ
for an antisite, where X and Y represent the C or Si species, and e is the negative cohesive energy, 6.41 eV, in a perfect 3C–SiC at 0 K in this present work which is agreement with previous experimental values, 6.34 eV [22] and previous stimulation values, 6.39 eV [23]. DE denotes the total energy difference between the crystal containing a defect and the perfect crystal with the same number of lattice sites. The results are compared with those obtained previously by MD as well as ab initio calculations in Table 1. It can be seen that those results determined by our method are in agreement with the previous MD reference values [15,25] and ab initio calculation [25]. 3.2. Evolution of these special atoms
(1) Minimize the initial system at 0 K. (2) The system was then annealed at a given temperature for 80–100 ps to stabilize the defect configuration. Temperatures used here ranged from 300 K to 1500 K. It has been experimentally reported that the migration of vacancies and small vacancy clusters would create voids when irradiation temperature is greater than about 1300 K [23]. Therefore, in order to investigate the vacancy migration, the highest temperature of simulation was set at 1500 K. (3) After that, the system was quenched to 0 K for 20 ps, in order to obtain the defects configuration without interference from thermal activation [10,24]. (4) Finally, the system was relaxed to get the final value of total energy. 3. Results and discussion
Table 2 summarizes the results obtained at 300 K for the relaxation energies of different systems and the properties of these special N-coordinated Si and C atoms, including the distances when displaced away from the original sites. It is important to note that the special one-coordinated Si atom and two-coordinated Si atom had a clear movement from their original lattice sites, as seen in Fig. 2a and b, 53% and 34% of the Si–C bond length (0.189 nm), respectively. While for other special atoms, the displacement can be negligible. It indicates that the one-coordinated Si and the two-coordinated Si atom contributes more to structural disorder than other special atoms, which plays an important role in the process of amorphization as discussed in [8]. The binding energies, Eb(n) of those stable one-coordinated C system containing three Si vacancies and two-coordinated C system containing two Si vacancies have also been calculated using Eq. (3) as defined below:
3.1. SiC defect formation energies
Eb ðnÞ ¼ Ef ðn 1Þ þ ESi f ð1Þ Ef ðnÞ
In order to ensure that our method is valid and provide basic data for the study of the stability of defect clusters as discussed below, the formation energies of C and Si vacancy, CSi antisite and SiC
Both of the results were very small, about 0.10 eV, which are consistent with that in [4]. It indicates that their nucleation at room temperature may be impossible, namely, these Si vacancies may not create homogeneous vacancy clusters. From Table 2, it
ð3Þ
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J. Xi et al. / Journal of Nuclear Materials 435 (2013) 236–240 Table 1 The formation energy of vacancies and antisite defects in 3C–SiC. Parameter
Present work (eV)
Previous G–W potential value (eV)
Previous Tersoff/zbl potential value (eV)
Ab initio value (eV)
VC VSi CSi SiC
4.96 6.47 0.51 6.25
2.76a 3.30a 1.69a 7.79a
5.20b 6.31b 0.35b 6.10b
5.48b 6.64b 1.32b 7.20b
Ref. [15]. Ref. [25].
Table 2 The relaxation energy of different systems and the displacement of these special atoms from their original sites at 300 K. Parameter
Relaxation energy (eV)
Displacement (nm)
One-coordinated Si Two-coordinated Si Three-coordinated Si One-coordinated C Two-coordinated C Three-coordinated C
3.62 1.95 0.18
0.101 0.065 0.013
0.11 0.05 0.01
0.010 0.008 0.007
4.5 4.0
Change in potential energy (eV)
a b
3.5
one-coordinated Si two-coordinated Si three-coordinated Si one-coordinated C two-coordinated C three-coordinated C
3.0 2.5 2.0 1.5 1.0 0.5 0.0 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Temperature (K) Fig. 3. Change in potential energy of special atoms as a function of temperature.
7.0 6.5 6.0
Fig. 2. Snapshots of one-coordinated Si and two-coordinated Si atom at 300 K. (a) One-coordinated Si system, (b) two-coordinated Si system.
can be seen that the relaxation energies of N-coordinated Si systems are always larger than that of corresponding N-coordinated C systems. At the same time, the displacement of these special N-coordinated Si atoms are greater than that of corresponding N-coordinated C atoms. As the coordination numbers increase, both the relaxation energies of systems and the displacement decrease. It indicates that the larger the relaxation energy of system is, the greater the displacement of the atom. These asymmetric behaviors between N-coordinated Si systems and C systems are probably caused by the difference of atom size and the thermal fluctuation of atom, as discussed below. Cutoff distance which depends on the atom size can be scaled with the system volume to avoid uncertainties. For the potential we used, the largest cutoff distance of Si–Si is 0.3 nm, while for C–C it is 0.21 nm [21]. During the annealing, all atoms are vibrating randomly. In a perfect 3C–SiC system, each atom interacts with four nearest neighbor atoms. However, for systems with special N-coordinated Si atoms, the special Si atoms may interact with the nearest neighbor C atoms and simultaneously with the second-nearest-neighbor Si atoms due to the Si–Si cutoff that is slightly smaller than the second-nearest-neighbor distance (0.303 nm). Whereas for the N-coordinated C systems, the special C atoms only interact with the nearest neighbor Si atoms because
Relaxation energy (eV)
5.5 5.0
one-coordinated Si two-coordinated Si three-coordinated Si one-coordinated C two-coordinated C three-coordinated C
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Temperature (K) Fig. 4. Relaxation energy of systems as a function of temperature.
of the C–C cutoff that is much smaller than the second-nearestneighbor distance, as discussed in [22]. For one special N-coordinated Si atom, two modes of interaction are possible. Firstly, it can interact with 12 second-nearestneighbor Si atoms, making it move away from its lattice site. On the other hand, the interactions between the N-coordinated Si atom and those nearest neighbor C atoms can cause this Si atom to be stable in the lattice site, thus, the displacement of Si atom from its original position depends on the two aspects. However, for one special N-coordinated C atom, because this C atom only interacts with the nearest neighbor Si atoms, it always stays in the lattice site. Therefore, the relaxation energies of N-coordinated C systems are lower than that of corresponding N-coordinated Si systems, and the displacement of N-coordinated Si atoms are larger
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Fig. 5. The configuration of one-coordinated C system at (a) 300 K, (b) 600 K, and (c) 1200 K. (a) The one-coordinated C atom stays in its lattice site, (b) the displacement of special C atom toward the Si vacancy is about 84% and (c) the special C atom has occupied the Si site and become the CSi antisite.
than those of N-coordinated C atoms. With increasing coordination number, the interactions between nearest neighbor C or Si atoms are strengthened. This stabilizes this special N-coordinated Si or N-coordinated C atom in the lattice site leading to decrease in the relaxation energy, as shown in Table 2. How the secondnearest-neighbor interaction between Si–Si and C–C atoms causes the change in potential energy at 300 K is shown in Fig. 3. It can be seen that only the potential energies of one-coordinated Si atom and two-coordinated Si atom are changed due to the secondnearest-neighbor interactions between Si and Si at 300 K, indicating that the thermal fluctuation and the difference of atom size make the difference of evolution properties. Further studies were made at different temperatures ranging from 400 K to 1500 K. Surprising results were obtained when temperature increased, as shown in Figs. 3 and 4: both the change in potential energy of one-coordinated C atom and the relaxation energy are larger than those of one-coordinated Si atom at above 600 K due to the strengthening thermal fluctuation of C atoms. The slight fluctuation of change in potential energy of different atoms is due to the random vibration of atoms that may cause a change of interaction between atoms, but the trend is that the change in potential energy of N-coordinated Si atoms decreases with increasing coordination numbers and is less than that of one-coordinated C atom at high temperature. The special onecoordinated C atom relaxed toward the nearest Si vacancy with a displacement about 84% of the Si–C bond length, when temperature reached 600 K, as shown in Fig. 5b. It is also worth noting that this one-coordinated C atom jumped into the nearest neighbor Si vacancy and became one CSi antisite, as shown in Fig. 5c, with increasing potential energy and relaxation energy at 1200 K, as shown in Figs. 3 and 4, respectively. For the two-coordinated C or three-coordinated C atom, the relaxation energy and displacement from their original site is negligible, probably because of the intense interaction between the special C atom and its surrounding nearest neighbor Si atoms. Binding energy of that vacancy cluster (containing one C vacancy and two Si vacancies) in the changed one-coordinated C system was 2.79 eV which is dramatically different from the above result, 0.10 eV, at 300 K. This is because of the change in defect configuration from three Si vacancies to two Si vacancies, one C vacancy and a CSi antisite. The high binding energy implies that the small vacancy cluster is stable at high temperatures. It also indicates that at higher temperature, one-coordinated C atom provides possible nucleation sites for vacancy migration as well as small vacancy cluster and defect replacement, consistent with theoretical calculations [11] and experimental observations [2]. In addition, the special N-coordinated Si atoms and their surrounding atoms have undergone substantial annealing, when temperature was above 600 K. Note that there was no SiC antisite produced after N-coordinated Si atoms fully annealed; this is probably caused by the difference of formation properties between silicon antisites
and carbon antisites, as discussed in [10]. The above results show that the thermal fluctuation of atoms and the size asymmetry of silicon and carbon atoms make the difference in defect properties. 4. Conclusion MD and MS have been carried out to investigate the evolution of atoms with different coordination numbers in 3C–SiC. At low temperature (below 600 K), the fully annealed one-coordinated Si atom and two-coordinated Si atom moved from their original lattice sites, while other special atoms did not. These atoms may play an important role in the process of structural disorder and amorphization. The relaxation energies of N-coordinated Si systems at low temperature were larger than that of corresponding N-coordinated C systems. We found that the relaxation energy and the displacement of one-coordinated C atom from its original site became considerable at higher temperature because the enhanced thermal fluctuation of atoms resulted in larger movement and interatomic interaction. Furthermore, our study of one-coordinated C system annealed at high temperature (above 600 K) indicated that onecoordinated C atoms may provide possible nucleation sites for vacancy migration as well as vacancy clusters and defect replacement to create antisites at higher temperature. However, for N-coordinated Si, no SiC antisite was detected, even though those N-coordinated Si atoms had a large movement. It indicates that this was further evidence that the formation of CSi antisite is easier than that of SiC antisite at high temperature. Finally, based on our results, we expect that at higher temperature and longer timescales, these atoms specially coordinated may play a significant role in the nucleation and growth of small defect clusters. However, more investigations are needed to confirm that. Acknowledgements We acknowledge fruitful comments with Dr. Dane Morgan from University of Wisconsin-Madison. We also thank Mr. J.M. Kebwaro for helpful discussion. This work was financially supported by NSFC under Grant No. 11175138 and China Postdoctoral Science Foundation (2011M501450). P. Zhang was also supported by ‘‘the Fundamental Research Funds for the Central Universities’’. The computation work is partly supported by the cluster Hua-I in Xi’an Jiaotong University and the National Supercomputing Centre in Shenzhen. References [1] T. Nozawa, T. Hinoki, A. Hasegawa, A. Kohyama, Y. Katoh, L.L. Snead, C.H. Henager Jr., J.B.J. Hegeman, J. Nucl. Mater. 386–388 (2009) 622–627. [2] S. Sorieul, J.-M. Costantini, L. Gosmain, L. Thomé, J.-J. Grob, J. Phys.: Condens. Matter. 18 (2006) 5235–5251. [3] Lance L. Snead, Takashi Nozawa, Yutai Katoh, Thak-Sang Byun, Sosuke Kondo, David A. Petti, J. Nucl. Mater. 371 (2007) 329–377.
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