The stability of vacancy clusters and their effect on helium behaviors in 3C-SiC

Journal of Nuclear Materials 503 (2018) 271e278

Contents lists available at ScienceDirect

Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat

The stability of vacancy clusters and their effect on helium behaviors in 3C-SiC Jingjing Sun a, b, B.S. Li c, Yu-Wei You a, Jie Hou a, b, Yichun Xu a, C.S. Liu a, *, Q.F. Fang a, Z.G. Wang c a b c

Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, P. O. Box 1129, Hefei 230031, PR China University of Science and Technology of China, Hefei 230026, PR China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 October 2017 Received in revised form 7 February 2018 Accepted 5 March 2018 Available online 9 March 2018

We have carried out systematical ab initio calculations to study the stability of vacancy clusters and their effect on helium behaviors in 3C-SiC. It is found that the formation energies of vacancy clusters containing only carbon vacancies are the lowest although the vacancies are not closest to each other, while the binding energies of vacancy clusters composed of both silicon and carbon vacancies in the closest neighbors to each other are the highest. Vacancy clusters can provide with free space for helium atoms to aggregate, while interstitial sites are not favorable for helium atoms to accumulate. The binding energies of vacancy clusters with helium atoms increase almost linearly with the ratio of helium to vacancy, n/m. The binding strength of vacancy cluster having the participation of the silicon vacancy with helium is relatively stronger than that without silicon vacancy. The vacancy clusters with more vacancies can trap helium atoms more tightly. With the presence of vacancy clusters in the material, the diffusivity of helium will be significantly reduced. Moreover, the three-dimension electron density is calculated to analyze the interplay of vacancy clusters with helium. © 2018 Elsevier B.V. All rights reserved.

1. Introduction 3C-SiC and its composites are attractive candidates as structural materials for advanced fission and future fusion reactors due to the favorable mechanical properties, excellent damage tolerance and low neutron absorption cross-sections [1e7]. The irradiation of high-energy neurons in the fission and fusion reactors can produce not only plenty of vacancies and self-interstitial atoms but also helium (He) and hydrogen gas atoms in the material because of nuclear transmutation reactions. The He production rate was reported to be about 2.5 appm/dpa in 3C-SiC exposed to typical fission neutron energy spectrum [8,9]. In contrast, the He production rate was reported to be much higher of 50e180 appm/dpa in 3C-SiC in the fusion environment [9,10]. The aggregation of He atoms in 3C-SiC will result in the formation of He bubbles and cavities, further lead to the volume swelling, and ultimately cause the degradation of mechanical properties of SiC materials [11e14].

* Corresponding author. E-mail address: [email protected] (C.S. Liu). https://doi.org/10.1016/j.jnucmat.2018.03.010 0022-3115/© 2018 Elsevier B.V. All rights reserved.

Many experiments have been carried out to study He bubbles, cavities and volume swelling in 3C-SiC implanted by He ions [15e21]. It was found that the presence of He bubbles can not be resolved in any of the He-implanted single crystal 3C-SiC at 277 +C, however He platelets and dislocation loops present in the single sample with the highest implantation fluence after 1 h annealing at 700 +C [20]. The He platelets appear in the form of He bubbles surrounded by several dislocation loops. The formations of He bubbles are often associated with the temperature and irradiation does [22e26]. Chen et al. found that the size distribution of He bubbles in nano-engineered 3C-SiC becomes narrower with increasing dose, while the averaged size of bubbles remains unchanged and the density of bubbles increases somewhat with dose [26]. Besides, it was found that no cavity appears at the low dose range, a low density of planar bubbles emerges at an intermediate does, and a high density of bubbles forms in 4H-SiC at high does [24]. As for volume swelling of 3C-SiC implanted by He, it was found that a fusion-relevant He production can significantly enhance the volume swelling between 673 and 1073 K, while it does not impose a strong effect if the temperature is higher than 1273 K [18]. The effect of He on the formation of bubbles, cavities

272

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278

and volume swelling is associated tightly with the basic occupy, migration, aggregation, and evolution of these processes in 3C-SiC. Linez et al. found that He atoms are mobile at 673 K and then mainly trapped by the di-vacancy (one silicon vacancy VSi and one carbon vacancy VC) [27]. In addition, nuclear reaction analysis showed that He atoms are strongly trapped by helium-vacancy clusters [28]. Thermal desorption spectrometry (TDS) showed that the low and high temperature desorption peaks of He atoms may be attributed to the He desorption from interstitial sites and helium-vacancy clusters [29], respectively. However, the experiments are difficult to access these micro-mechanisms associated with the complex interaction of He atoms with vacancy clusters. Theoretical simulation is a good way to access the micromechanisms regarding He migration, aggregation, and the interaction of He with vacancy clusters in 3C-SiC. It was found that theoretical diffusion activation energy of He in 3C-SiC is calculated to be 1.49~1.55 eV [30e32], which is very close to the experimental value of 1.14 eV [33]. The interstitial He atom was determined to prefer to occupy the tetrahedral site neighboring four silicon atoms (TSi) by density functional theoretical (DFT) method [30e32,34]. Li et al. and Sun et al. found that the vacancy has a good trapping capability for He atoms and provides sufficient space for He aggregation in 6H-SiC and 3C-SiC respectively [35,36]. He atoms trapped in VSi or VC can induce large internal pressure in the order of magnitude of GPa [36]. It was found theoretically that the formation energies of He in di-vacancy of VCVSi and VSi are lower than those of He in VC and TSi [32,34]. In spite of many theoretical studies concerning the He occupancy property in 3C-SiC, the influence of vacancy clusters on the energetic and kinetic properties of He is still not well understood. The reason why so many dislocation loops surround He bubbles in 3C-SiC is still mysterious. In this work, we thereby perform DFT calculations to firstly investigate the stability of interstitial He clusters and the vacancy clusters. Then, the interaction of He atoms with vacancy clusters is studied. In addition, the influence of vacancy clusters on He diffusion is calculated and compared with the experimental observations. Finally, physical reasons controlling the interaction between He atoms and vacancy clusters are analysed. 2. Computational method All density functional theory calculations are performed using Vienna ab initio simulation package (VASP) with the projector augmented wave potential method [37,38]. The electronic exchange and correlation effect is treated in the local density approximation [39]. We use a 3
3
3 supercell composed of 108 silicon (Si) and 108 carbon (C) atoms, and a kinetic energy cutoff of 520 eV is used for the plane-wave basics. A k-point grid density of 3
3
3 is produced by using the Monkhorst-Pack method [40]. The atomic positions and the supercell shape and volume are full relaxed and optimized. The structural optimization is truncated when the force converges to less than 0.01 eV/Å. The calculated equilibrium lattice constant of 3C-SiC is 4.33 Å, in good agreement with the experimental value of 4.36 Å [41]. 3. Results and discussion 3.1. Verification of the calculations To verify the accuracy of our calculations, we firstly investigate the solution and diffusion properties of He in perfect 3C-SiC crystal. Three types of interstitial sites, i.e., tetrahedral, hexagonal and bond center sites, are considered. There are two different tetrahedral sites, one is TSi and the other is the tetrahedral site neighboring four C atoms (TC). At a hexagonal site (Hex), He has three Si and

three C neighbors. In the bond center (BC) configuration, He occupies the center site of Si-C bond. The solution energy is calculated by:

Es ðHeÞ ¼ Etot ðHeÞ  Eper  EðHeÞ;

(1)

where Etot(He) is the total energy of the supercell with a He atom at an interstitial site, Eper is the energy of the perfect supercell, and E(He) is the energy of an isolated He atom in the vacuum. The obtained results are summarized in Table 1. It is found that the He atoms at TSi, TC and BC sites are stable, while the He atom at the Hex site moves to the TSi site during the structural optimization. The solution energies of He at the TSi and TC sites are 2.50 and 3.00 eV respectively, which are much lower than that of He at the BC site (6.86 eV). Therefore, He prefers to occupy the TSi and TC sites rather than BC site, which agrees well with the experiment that interstitial He atoms mainly occupy TSi and TC sites in SiC implanted with He [42]. Besides, the solution energies calculated here are in good agreement with previous DFT results by Zhou et al. [30] and Kim et al. [31]. However, our results do not agree well with the values by Charaf Eddin et al. [32]. The difference may result from the different treatment ways of electronic exchange and correlation effect. Charaf Eddin et al. performed calculations with Perdew-Burke-Ernzerhof GeneralizedGradient Approximation, while this work is performed using the Local Density Approximation. Nevertheless, the lowest-energy state of He in 3C-SiC is in good consistence. Then we investigate the diffusion behavior of He in perfect 3C-SiC crystal based on the climbing image nudged elastic band method [43,44]. The most favorable diffusion path is from one TSi site to the next TSi site passing through a TC site. The migration barriers are 1.52 eV from TSi to TC, and 1.03 eV from TC to another TSi. The diffusion activation energy of He in perfect 3C-SiC is in good agreement with the values obtained by Zhou et al. [30] and Charaf Eddin et al. [32]. 3.2. Stability of interstitial helium clusters in 3C-SiC The stability of interstitial He clusters in perfect 3C-SiC is calculated here. The He clusters containing two, three and four He atoms are considered. Since TSi is found to be the most stable interstitial site for He, all the possible TSi sites around the first He atom located at TSi are considered for the second He atom. By calculating the total energies of the systems, we determine the lowest-energy configuration for He-He cluster. In the same way, the ground-state configurations for the He clusters containing three and four He atoms are obtained. The ground-state configurations of He clusters containing n He atoms (Hen) are plotted in Fig. 1, and the binding energies of Hen representing the interaction among all He atoms in He clusters are labeled below the configurations. The binding energy is defined as:

Eb ðHen Þ ¼ nEg ðHeÞ  Eg ðHen Þ  ðn  1ÞEper ;

(2)

where Eg(He) is the energy of the supercell with an interstitial He Table 1 The solution energies (in eV) of He atoms at the interstitial sites of TSi, TC and BC are presented, and the available theoretical data for comparison are summarized. Ref

Sites

Es(He)

Es (He)

TSi TC BC

2.50 3.00 6.86

2.54a, 2.87b, 3.04c 3.04a, 3.31b, 3.51c 7.03c

a b c

Zhou et al. [30]. Kim et al. [31]. Charaf Eddin et al. [32].

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278

273

Fig. 1. The ground-state configurations of interstitial Hen clusters and their corresponding binding energies in 3C-SiC. The white solid balls represent He atoms at the TSi sites. The olive and red balls are the host Si and C atoms respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

atom at the ground state, and Eg(Hen) is the energy of the supercell containing n interstitial He atoms at the ground state. In the clusters of He2, He3, and He4, all the He atoms occupy the nearest neighbor sites of TSi with each other, and the average distances of them vary from 3.08 to 3.09 Å. The positions of the He atoms in each considered configurations are nearly not moved, which may be due to the weak interaction among the He atoms in the clusters and relatively large diffusion barrier of He in 3C-SiC. It is found that the binding energies of He2, He3 and He4 are only 0.01, 0.03 and 0.07 eV, respectively. The small binding energies of Hen indicate a weak attraction between the He atoms, and thus the interstitial He atoms are very difficult to form clusters in perfect 3CSiC.

Table 2 The formation and binding energies (in eV) of Vm are presented, and the available theoretical reference values are summarized.

3.3. Stability of vacancy clusters in 3C-SiC No matter in a fission or fusion environment, 3C-SiC will be exposed to high-energy neutrons and ions, which produces large amounts of point defects and point-defect clusters. Among the point-defect clusters, vacancy clusters (Vm) are general of concern to us since they can interact strongly with other point defects and transmutation He. There are two kinds of vacancies VC and VSi in 3C-SiC, resulting in complicated configurations of Vm. In order to avoid too much calculation cost, m is restricted to not more than four, and here we only consider the most compact configurations of Vm. By comparing the energies of all the possible configurations of Vm, the lowest energy configurations of Vm are obtained and shown in Fig. 2. The formation energy of Vm is defined as [45]:

Ef ðVm Þ ¼ EðVm Þ  Eper þ mC nC þ mSi nSi ;

m 1 1 2 2 2 3 3 3 4 4 4 a

Vacancies C

V VSi VC2 VCVSi VSi 2 VC3 VC2 VSi VCVSi 2 VC4 VC3 VSi VC2 VSi 2

Ef(Vm)

Ref

Ef

(Vm)

Eb(Vm)

a

4.56 8.71 7.34

4.02 8.88a 6.33a

1.78

8.70 17.19

8.65a 16.39a

4.57 0.24

8.95

4.73

9.01 14.81 10.69

8.70a 14.49a

8.82 7.17 7.55

10.67 14.71

9.42a

11.72 11.83

Gao et al. [46].

(3)

where E(Vm) is the energy of the supercell with m vacancies. nC and nSi are the numbers of VC and VSi in Vm respectively. mC and mSi are respectively the chemical potentials of C in the diamond and Si in the silicon perfect crystals. The binding energy of Vm can be calculated by:

    Eb ðVm Þ ¼ nC E V C þ nSi E V Si  EðVm Þ  ðn  1ÞEper ;

Fig. 2. The most stable configurations of Vm in 3C-SiC. The olive and red hollow balls are respectively VSi and VC, and the olive and red solid balls are the host Si and C atoms respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

(4)

where E(VC) and E(VSi) are the energies of the supercells with a VC and VSi respectively. The obtained results are summarized in Table 2. The formation energies of VC and VSi are respectively 4.56 and 8.71 eV, which agree well with the other theoretical results [46]. As for di-vacancy, there are three kinds of combinations, that is, VC2 , VCVSi and VSi 2 as shown in Fig. 2(a)-(c). The two vacancies are next closest to each other in VC2 and VSi 2 , and closest to each other in VCVSi. The formation energies of VC2 , VCVSi and VSi 2 are respectively 7.34, 8.70 and 17.19 eV, and the binding energies are 1.78, 4.57 and 0.24 eV, respectively. The formation energy of VC2 is the lowest, hence VC2 is the most favorable to form. The formation energy of

VCVSi is 1.36 eV higher than that of VC2 , while the binding energy of VCVSi is 2.79 eV larger than that of VC2 . In contrast, VSi 2 has the highest formation energy and the smallest binding energy among the three di-vacancies. Thus, VSi 2 is extremely difficult to form and most unstable. As for tri-vacancy, only VC3 , VC2 VSi and VCVSi 2 are investigated (Fig. 2(d)-(f)), and VSi 3 is not considered due to the large formation energy and small binding energy. In the configuration of VC3 , three vacancies are in the next neighbors to each other. In the C Si configurations of VC2 VSi and VCVSi 2 , V and V are in the nearest neighbors to each other. The formation energy of VC3 is 0.06 and 5.86 eV lower than those of VC2 VSi and VCVSi 2 respectively, and the binding energy of VC2 VSi is 4.09 and 1.65eV higher than those of VC3 C and VCVSi 2 , respectively. Hence, V3 is the most preferable to form, C Si and V2 V has the highest binding energy among the three trivacancies in 3C-SiC. As for tetra-vacancy, the configurations of VC4 , VC3 VSi and VC2 VSi 2 are studied and shown in Fig. 2(g)-(i). In the configuration of VC4 , four vacancies are in the next neighbors to each C other. Three VC are closest to VSi in VC3 VSi. In VC2 VSi 2 , one V is in the Si C third neighbor to one of V , and the other V stays closest to the two VSi. The formation energies of VC4 , VC3 VSi, and VC2 VSi 2 are 10.69, 10.67 and 14.71 eV, respectively. The binding energies of them are

274

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278

7.55, 11.72 and 11.83 eV, respectively. The formation energy of VC3 VSi is 0.02 and 4.04 eV lower than those of VC4 and VC2 VSi 2 , and the binding energy of VC2 VSi 2 is 0.11 and 4.28 eV higher than those of VC3 VSi and VC4 . Hence both VC4 and VC3 VSi are most favorite to form as the difference between their formation energies is very small, and VC2 VSi 2 has the strongest binding strength. Compared with the formation energies of Vm in previous works, our results here agree well with the values calculated by Iwata et al. [45] and Gao et al. [46]. However, they are somewhat deviated from that by Morishita et al. [47], where the results are calculated by empirical potential molecular dynamics and molecular static. From the above calculations and discussions of the formation and binding energies of Vm, it can be found that VmC is most favorable to form although the C vacancies in the configurations are not closest to each other. The binding strength of vacancy clusters containing both VC and VSi is always the strongest. The vacancy clusters containing both VC and VSi had been identified in SiC by experiments [48,49], and it was reported that they are the dominating vacancy defects in 3C-SiC [49]. This may come from that the vacancy clusters containing both VC and VSi have the strong binding strength. 3.4. Interaction of helium atoms with vacancy clusters It has been widely observed that He platelets appear in the form of He bubbles surrounded by several dislocation loops [12,20,22e24], and it is relatively difficult for He atoms to form interstitial clusters in the above discussions. We expect that He atoms may accommodate inside Vm, pushing the lattice atoms surrounding Vm away [50,51]. This may result in the formation of He bubbles surrounded by several dislocation loops. Therefore we consider the interaction of He atoms with Vm, and the influence of He atoms on the movement of the lattice atoms surrounding Vm. Multiple He atoms are brought one by one to Vm, however, there are too many possible configurations for helium-vacancy clusters (HenVm) as the increase of He atoms. Hence, we perform ab initio molecular dynamics simulations at 600 K with NVT ensemble in order to find the most stable configurations. The similar method was used by Becquart et al. to determine the most stable configurations of H and He in tungsten [52]. As many as ten initial configurations are used to find ground-state of HenVm. The binding energies for all the lowest-energy geometries of HenVm shown in Fig. 3(a) are calculated by:

Eb ðHen Vm Þ ¼ EðVm Þ þ nEg ðHeÞ  EðHen Vm Þ  nEper ;

(5)

where E(HenVm) is the energy of supercell containing a HenVm complex with n He atoms and m vacancies. The binding energies of HenVm represent the interaction between He atoms and vacancy clusters. In addition, the volume increases of HenVm induced by He

Fig. 3. The binding energies of the most stable configurations of HenVm (a). The volume increase induced by He atoms accumulating in Vm (b).

accumulation are shown in Fig. 3(b), which is given by: DVol ¼ Vol(HenVm)-Vol(Vm), where Vol(HenVm) and Vol(Vm) are the volumes of the supercells containing a HenVm complex and a Vm complex respectively. The binding energies of HenVC and HenVSi almost linearly increase with the ratio of He to vacancy (n/m), and the binding energy of HenVSi is always higher than that of HenVC. VC and VSi can only trap up to eleven He atoms, and thus the maximum values of n and m are eleven and four, respectively. The binding energy of HenVCVSi is always higher than that of HenVC2 as the function of n/m. The configuration of HenVSi 2 is not stable by structural optimization and transfers to HenVSiVCCSi. The binding energy of HenVSiVCCSi is the highest among the interaction of He with the di-vacancies. As for the interaction of He with the tri-vacancy and tetra-vacancy, the binding energies of HenVC2 VSi and HenVC3 VSi are calculated with the increase of n/m because the binding energies of HenVC2 VSi and HenVC3 VSi are higher than those of HenVC3 and HenVC4 respectively. The increase of C vacancy in the vacancy clusters has a weak influence on increasing the binding energies of HenVC2 VSi and HenVC3 VSi. On the whole, the binding strength of HenVm is larger than that of HenVm1 if the n/m is the same. The volume increase of HenVm in Fig. 3(b) has the similar tendency as the binding energy of HenVm. The increased volume indicates significant expansion and stress around HenVm. With the increasing of n/m, the lattice-point atoms around HenVm may be gradually pushed away from their original positions. The influence of He inside mono-vacancy on the emission of lattice atoms is also considered. The energy needed for a latticepoint atom to emit and diffuse away by a self-interstitial atom (SIA) is given by the expressions [51]:

Ee ðSIAÞ ¼ EðHen V2 Þ þ Eg ðSIAÞ  EðHen VÞ  Eper þ Em ðSIAÞ;

(6)

and the emission energy of He atom is expressed by Ref. [51]:

Ee ðHeÞ ¼ EðHen1 VÞ þ Eg ðHeÞ  EðHen VÞ  Eper þ Em ðHeÞ:

(7)

Here Eg(SIA) is the energy of the supercell with the ground state configuration of a SIA including CSIA or SiSIA. Em(SIA) and Em(He) are the diffusion barriers of the SIA and He respectively [53]. Fig. 4(a) and 4(b) present the emission energies of He, lattice-point C and Si atoms from HenVC and HenVSi. The emission energy of He is always much lower than that of lattice-point C or Si atom from the Hevacancy complex, suggesting that the emission of He from the VC and VSi is much more energetically favorable than that of latticepoint C or Si atom. As the emission energies of lattice-point C and Si atoms are much larger than that of He atoms, it is extremely difficult for C and Si atoms to emit from their lattice-point positions.

Fig. 4. The emission energies of CSIA, SiSIA and He from HenVC (a) and HenVSi (b).

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278

The formation of He bubbles and platelets may include the processes, ie., the emission of He from small vacancy clusters and the re-trapping of He by large vacancy-type defects. This situation may be contributed to the formation of helium platelets observed in the experiment by Chen et al. [20]. It was found that the presence of He bubbles can't be observed in the He-implanted single crystal 3C-SiC at 550 K with the implantation fluence of 1
1016cm2. However He platelets and dislocation loops appear in the sample after 1 h annealing at 973 K [20]. 973 K may be not high enough for the emission of lattice-point C and Si atoms from HenVC and HenVSi, while which can promote the emission of He atoms from vacancy clusters in 3C-SiC according to the TDS experiment [29]. 3.5. Influence of vacancy cluster on helium diffusion Vacancy clusters can trap He atoms, which will of course affect the migration of He in 3C-SiC. Thus we study the diffusion property of He in 3C-SiC. The diffusion coefficient can reflect the effect of vacancy clusters on He diffusion property. Generally, according to the classic MacNabb and Forester formula [54], the effective diffusivity in the vacancy cluster field can be described by:

Deff ¼

Dper

 VHe ; E 1 þ ðcV Þexp bkT

(8)

where Dper is the diffusivity of He at the perfect system without trap sites, and cV is the concentration of vacancy clusters. EbV He is the binding energy between the He atom and vacancy clusters. T is the temperature, and k is the Boltzmann constant. According to the Arrhenius diffusion equation, the Dper can be evaluated by: Dper ¼ D0exp(-Ea/kT), where D0 is the diffusion constant, and Ea is diffusion activation energy of He at the interstitial site. The value of Ea is calculated to be 1.52 eV. In 3C-SiC system, D0 is given by D0 ¼ a2n/6, where a is the jump length, and n is the vibration frequency approximately given by the Zener and Wert's theory [55]: n¼(2Ea/ma2)1/2 (m is the mass of diffusion atom). Taking the mass of He atom 6.64
1027 kg and the jump length a ¼ 2a0/2 (a0 is the lattice constant of 3C-SiC), we can obtain n ¼ 2.80
1013 s1, and we can also obtain D0 ¼ 4.37
107 m2s1. The diffusivity of He in perfect 3C-SiC is predicated to be: Dper ¼ 4.37
107exp(-1.52 eV/ kT) m2s1 as shown in Fig. 5, which is much higher than that of experimental observations [28,56,57]. The diffusion coefficients of

Fig. 5. The diffusion coefficient of He vs. the reciprocal of temperature in 3C-SiC. Data are from Miro et al. [28], Pramono et al. [57] and Cherniak et al. [56].

275

He are 7.46
1020, 1.34
1015 and 1.81
1013 m2s1 at the temperatures of 600, 900 and 1200 K, respectively. Taking Dper in Eq. (8), we can obtain the Deff. However, it is difficult to accurately determine cV in the experiments. Linez et al. reported the concentration of VCVSi to be 2
1019 cm3 (~2.03
104 VCVSi per SiC atom), which is about 3.5 times lower than the He concentration [27]. In the present work, it is assumed that cV and He concentration are the same. The cV is predicated to about 3.3
103 according to the He peak concentration of 3.3
103 in the work by Miro et al. [28]. As shown in Fig. 5, the Deff is calculated and fitted to the results obtained by Miro et al. [28] and Cherniak et al. [56], and EbV He is obtained to be 1.55 eV with the cV of 3.3
103. The value of EbVHe is in the interval of the binding energy of the He atom with vacancy clusters in Fig. 3(a), and the Deff is in good agreement with the diffusivity reported by Miro et al. [28] and Cherniak et al. [56] between 1000 and 1423 K. There is somewhat difference between the diffusivity calculated here and that by Pramono et al. [57]. The underlying reason may be resulted from the influence of other impurities (B4C, B) in the SiC. Compared with the Dper, Deff is much smaller due to the influence of vacancy clusters on He diffusion. This may be the reason why the diffusivity reported in previous experiments is smaller than that calculated by our theoretical simulation. The large concentration of vacancy clusters can significantly decrease the diffusivity of He, which results in the aggregation of He in 3C-SiC. 3.6. Physics reasons controlling the interaction of helium atoms with vacancy clusters in 3C-SiC The presence of He atoms inside vacancy cluster can usually cause the displacement of lattice-point atoms nearby the vacancy, inducing the lattice distortion. The local distortion of the lattice in turn influences the distribution of electron density around the helium-vacancy complex. The average displacement distances (DD) of the nearest and next nearest lattice-point atoms away from the vacancy center are therefore summarized and shown in Fig. 6. The DD of the four nearest Si atoms around the VC center (Si1NN-VC) increases quickly before n adds up to four, and then increases slowly with n. In contrast, the DD of the four nearest C atoms around the VSi center (C1NN-VSi) always increases slowly with n. The DD of the twelve C atoms next nearest to VC center or the twelve Si

Fig. 6. The average displacement distances of the nearest Si atoms and next nearest C atoms away from the center of VC, and the nearest C atoms and the next nearest Si atoms away from the center of VSi.

276

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278

atoms next nearest to VSi center is relatively small. He is a kind of closed-shell atom, thus it is favorable to occupy the positions with low-electron-density regions in the system. Fig. 7(a) and 7(b) present the low-electron-density regions of 0.01 electrons/bohr3 around HenVC and HenVSi respectively. Initially, the low-electron-density regions of VC are small, and they begin to increase quickly when VC traps He atoms until n adds up to four. This is due to that the nearest lattice atoms are displaced away as shown in Fig. 6. The enlarged low-electron-density regions exceed the demand of the aggregation He atoms in VC. Therefore, the lowelectron-density regions present an increasing tendency when VC traps one to four He atoms as shown by the second to fifth structures in Fig. 7(a). However, the DD of the closest lattice-point atoms increases slowly when the number of He atoms exceeds four (Fig. 6), and thus the enlargement of low-electron-density regions is slowed down. With the continual increase of He atom number inside VC, the newly increased low-electron-density regions can not meet the demand of He atoms. This is why the low-electrondensity regions are gradually decreased and ultimately disappeared when the He atom number exceeds four. As shown in Fig. 6, the DD of the closest lattice-point atoms surrounding HenVSi slowly increases with n, which results in the slow increase of the low-electron-density regions. The increased low-electron-density regions can not meet the demand of He atoms. Therefore, the low-electron-density regions present a gradual decrease tendency as the He atom number is increased. Moreover, the interplay among He atoms also contributes to aggregation of He atoms inside vacancy clusters. In vacuum, the interaction between He atoms is repulsive (Fig. 8(a)), and there is weak repulsive interaction between He atoms with the binding energy of 0.23 eV when two He atoms are at the distance of 1.6 Å. The repulsion is disappeared when the distance of He-He exceeds 2.0 Å. For the interaction of the

Fig. 8. The binding energy of two He atoms in vacuum as a function of their distance (a). The distances between He atoms in the ground-state configurations of HenVC (b) and HenVSi (c), and the minimum distance in each ground-state configuration of HenVC or HenVSi is connected by a black line.

Fig. 7. The low-electron-density isosurfaces for the ground-state configurations of HenVC (a) and HenVSi (b) with n from zero to eleven. The yellow regions denote the low-electrondensity isosurfaces of 0.01 electrons/bohr3. VC and VSi are in the centers of the unit cells in (a) and (b) respectively. The white balls mark the He atoms, and the olive and red solid balls represent the host Si and C atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278

interstitial He atoms in 3C-SiC, Lu et al. found the He-He repulsion by calculating crystal orbital Hamilton population using DFT [58]. In helium-vacancy clusters, it is found that the shortest distance between He atoms in VC and VSi is about 1.6 Å as shown in Fig. 8(b) and (c). From charge density analysis, there is nearly no hybridization between He atoms, therefore there may present weak repulsive interaction or no interaction between He atoms. This may be against the accumulation of He atoms in vacancy cluster, while the eventual attraction interaction may originate from the joint contributions of He-He and lattice-atom-He interactions. 4. Conclusions By performing first-principle calculations, we systematically investigate the stability of interstitial helium clusters and the vacancy clusters, the interaction of helium with vacancy clusters, and the influence of vacancy clusters on helium diffusion. Helium atoms are difficult to form interstitial clusters in 3C-SiC. The formation of vacancy clusters containing only carbon vacancies is most energetically favorable, while the binding strength of vacancy clusters containing both carbon and silicon vacancies is always the strongest. Vacancy clusters can provide with free space for helium aggregation, and the binding energies of helium with vacancy clusters almost linearly increase with the ratio of helium to vacancy, n/m. The binding strength of helium with vacancy clusters containing silicon vacancy is relatively stronger than that of helium with vacancy clusters without silicon vacancy, and the vacancy clusters with more vacancies can more tightly trap helium atoms. The presence of vacancy clusters in 3C-SiC can significantly reduce the helium diffusivity, and this is why the helium diffusivity observed in the experiments is much smaller than the theoretical value. Acknowledgements This work was supported by the National Key Research and Development Program of China (Grant No.: 2017YFA0402800), and the National Natural Science Foundation of China (Nos.: 11405202 and 11475214). The authors gratefully acknowledge the support of the National Natural Science Foundation of China under grant number 11475229, and the Institute of Modern Physics, Chinese Academy of Sciences. Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.jnucmat.2018.03.010. References [1] L.L. Snead, T. Nozawa, Y. Katoh, T.S. Byun, S. Kondo, D.A. Petti, Handbook of SiC properties for fuel performance modeling, J. Nucl. Mater. 371 (2007) 329e377. [2] J.B. Malherbe, Diffusion of fission products and radiation damage in SiC, J. Phys. D Appl. Phys. 46 (2013), 473001. [3] Y. Katoh, K. Ozawa, C. Shih, T. Nozawa, R.J. Shinavski, A. Hasegawa, L.L. Snead, Continuous SiC fiber, CVI SiC matrix composites for nuclear applications: properties and irradiation effects, J. Nucl. Mater. 448 (2014) 448e476. [4] S.J. Zinkle, L.L. Snead, Designing radiation resistance in materials for fusion energy, Annu. Rev. Mater. Res. 44 (2014) 241e267. [5] Y. Katoh, L.L. Snead, I. Szlufarska, W.J. Weber, Radiation effects in SiC for nuclear structural applications, Curr. Opin. Solid State Mater. Sci. 16 (2012) 143e152. [6] Y. Katoh, L.L. Snead, C.H. Henager Jr., T. Nozawa, T. Hinoki, A. Ivekovi c, S. Novak, S.M. Gonzalez de Vicente, Current status and recent research achievements in SiC/SiC composites, J. Nucl. Mater. 445 (2014) 387e397. [7] K. Yueh, K.A. Terrani, Silicon carbide composite for light water reactor fuel assembly applications, J. Nucl. Mater. 448 (2014) 380e388. [8] H.L. Heinisch, L.R. Greenwood, W.J. Weber, R.E. Williford, Displacement damage in silicon carbide irradiated in fission reactors, J. Nucl. Mater. 327 (2004) 175e181. [9] M.E. Sawan, Y. Katoh, L.L. Snead, Transmutation of silicon carbide in fusion

277

nuclear environment, J. Nucl. Mater. 442 (2013) S370eS375. [10] M.E. Sawan, N.M. Ghoniem, L. Snead, Y. Katoh, Damage production and accumulation in SiC structures in inertial and magnetic fusion systems, J. Nucl. Mater. 417 (2011) 445e450. [11] S.J. Zinkle, Effect of H and He irradiation on cavity formation and blistering in ceramics, Nucl. Instrum. Meth. Phys. Res. B 286 (2012) 4e19. [12] P. Jung, H. Klein, J. Chen, A comparison of defects in helium implanted in aand b-SiC, J. Nucl. Mater. 283e287 (2000) 806e810. my, M.F. Beaufort, C. Tromas, J.F. Barbot, Swelling of SiC [13] S. Leclerc, A. Decle under helium implantation, J. Appl. Phys. 98 (2005), 113506. [14] L.L. Snead, R. Scholz, A. Hasegawa, A.F. Rebelo, Experimental simulation of the effect of transmuted helium on the mechanical properties of silicon carbide, J. Nucl. Mater. 307e311 (2002) 1141e1145. [15] S. Igarashi, S. Muto, T. Tanabe, Surface blistering of ion irradiated SiC studied by grazing incidence electron microscopy, J. Nucl. Mater. 307e311 (2002) 1126e1129. [16] A. Hasegawa, M. Saito, S. Nogami, K. Abe, R.H. Jones, H. Takahashi, Heliumbubble formation behavior of SiCf/SiC composites after helium implation, J. Nucl. Mater. 246 (1999) 355e358. [17] J. Chen, P. Jung, Effect of helium on radiation damage in a SiC/C composites, Ceram. Int. 26 (2000) 513e516. [18] Y. Katoh, H. Kishinoto, A. Kohyama, The influences of irradiation temperatures and helium production on the dimensional stability of silicon carbide, J. Nucl. Mater. 307e311 (2002) 1221e1226. [19] S. Kondo, T. Hinoki, A. Kohyama, Synergistic effects of heavy ion and helium irradiation on microstructural and dimensional change in b-SiC, Mater. Trans. 46 (2005) 1388e1392. [20] C.H. Chen, Y. Zhang, E. Fu, Y. Wang, M.L. Crespillo, C. Liu, S. Shannon, W.J. Weber, Irradiation-induced microstructural change in helium-implanted single crystal and nano-engineered SiC, J. Nucl. Mater. 453 (2014) 280e286. [21] M.F. Beaufort, M. Vallet, J. Nicolaï, E. Oliviero, J.F. Barbot, In-situ evolution of helium bubbles in SiC under irradiation, J. Appl. Phys. 118 (2015), 205904. [22] J. Chen, P. Jung, H. Trinkaus, Evolution of helium platelets and associated dislocation loops in a-SiC, Phys. Rev. Lett. 82 (1999) 2709e2712. [23] J. Chen, P. Jung, H. Trinkaus, Microstructural evolution of helium-implante aSiC, Phys. Rev. B 61 (2000) 12923e12932. [24] C.H. Zhang, S.E. Donnelly, V.M. Vishnyakov, J.H. Evans, Dose dependence of formation of nanoscale cavities in helium-implanted 4H-SiC, J. Appl. Phys. 94 (2003) 6017. [25] C. Zhang, Y. Song, Y. Yang, C. Zhou, L. Wei, H. Ma, Evolution of defects in silicon carbide implanted with helium ions, Nucl. Instrum. Meth. Phys. Res. B 326 (2014) 345e350. [26] C.H. Chen, Y. Zhang, Y. Wang, M.L. Crespillo, C.L. Fontana, J.T. Graham, G. Duscher, S.C. Shannon, W.J. Weber, Dose dependence of helium bubble formation in nano-engineered SiC at 700  C, J. Nucl. Mater. 472 (2016) 153e160. [27] F. Linez, E. Gilabert, A. Debelle, P. Desgardin, M.F. Barthe, Helium interaction with vacancy-type defects created in silicon carbide single crystal, J. Nucl. Mater. 436 (2013) 150e517. [28] S. Miro, J.M. Costantini, J. Haussy, L. Beck, S. Vaubaillon, S. Pellegrino, C. Meis, J.J. Grobe, Y. Zhang, W.J. Weber, Nuclear reaction analysis of helium migration in silicon carbide, J. Nucl. Mater. 415 (2011) 5e12. [29] E. Oliviero, M.F. Beaufort, J.F. Barbot, A. van Veen, A.V. Fedorov, Helium implantation defects in SiC: a thermal helium desorption spectrometry investigation, J. Appl. Phys. 93 (2003) 231. [30] Y. Zhou, Q. Liu, H. Xiao, X. Xiang, X. Zu, S. Li, Mechanism for hydrogenpromoted information of helium polymer in silicon carbide material: a diffusion study, J. Alloy. Comp. 647 (2015) 167e171. [31] J.H. Kim, Y.D. Kwon, P. Yonathan, I. Hidayat, J.G. Lee, J.H. Choi, S.C. Lee, The energetics of helium and hydrogen atoms in b-SiC: an ab initio approach, J. Mater. Sci. 44 (2009) 1828e1833. [32] A. Charaf Eddin, L. Pizzagalli, First principles calculation of noble gas atoms properties in 3C-SiC, J. Nucl. Mater. 429 (2012) 329e334. [33] P. Jung, Diffusion and retention of helium in graphite and silicon carbide, J. Nucl. Mater. 191e194 (1992) 377e381. [34] R.M. Van Ginhoven, A. Chartier, C. Meis, W.J. Weber, L.R. Corrales, Theoretical study of helium insertion and diffusion in 3C-SiC, J. Nucl. Mater. 348 (2006) 51e59. [35] R. Li, W. Li, C. Zhang, P. Zhang, H. Fang, D. Liu, L. Vitos, J. Zhao, He-vacancy interaction and multiple He trapping in small void of silicon carbide, J. Nucl. Mater. 457 (2015) 36e41. [36] D. Sun, R. Li, J. Ding, P. Zhang, Y. Wang, J. Zhao, Interaction between helium and intrinsic point defects in 3C-SiC single crystal, J. Appl. Phys. 121 (2017), 225111. [37] G. Kresse, J. Hafner, Abinitio molecular dynamics for liquid metals, Phys. Rev. B 47 (1993) 558. [38] G. Kresse, J. Furthmüller, Efficient iterative schemes for abinitio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169. [39] W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965). A1133. [40] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188. [41] K.J. Chang, M.L. Cohen, Abinitio pseudopotential study of structural and highpressure properties of SiC, Phys. Rev. B 35 (1987) 8196. [42] F. Linez, F. Garrido, H. Erramli, T. Sauvage, B. Courtois, P. Desgardin,

278

[43]

[44]

[45]

[46]

[47]

[48]

[49]

J. Sun et al. / Journal of Nuclear Materials 503 (2018) 271e278 M.F. Barthe, Experimental location of helium atoms in 6H-SiC crystal lattice after implantation and after annealing at 400  C, J. Nucl. Mater. 459 (2015) 62e69. nsson, A climbing image nudged elastic G. Henkelman, B.P. Uberuaga, H. Jo band method for finding saddle points and minimum energy paths, J. Chem. Phys. 113 (2000) 9901.  nsson, Improved tangent estimate in the nudged elastic G. Henkelman, H. Jo band method for finding minimum energy paths and saddle points, J. Chem. Phys. 113 (2000) 9978. J.I. Iwata, C. Shinei, A. Oshiyama, Density-functional study of atomic and electronic structures of multivacancies in silicon carbide, Phys. Rev. B 93 (2016), 125202. F. Gao, W.J. Weber, H.Y. Xiao, X.T. Zu, Formation and properties of defects and small vacancy clusters in SiC: abinitio calculations, Nucl. Instrum. Meth. Phys. Res. B 267 (2009) 2995e2998. K. Morishita, Y. Watanabe, A. Kohyama, H.L. Heinisch, F. Gao, Nucleation and growth of vacancy clusters in b-SiC during irradiation, J. Nucl. Mater. 386e388 (2009) 30e32. J. Wiktor, X. Kerbiriou, G. Jomard, S. Esnouf, M.F. Barthe, M. Bertolus, Positron annihilation spectroscopy investigation of vacancy clusters in silicon carbide: combining experiments and electronic structure calculations, Phys. Rev. B 89 (2014), 155203. X. Hu, T. Koyanagi, Y. Katoh, B.D. Wirth, Positron annihilation spectroscopy investigation of vacancy defects in neutron-irradiated 3C-SiC, Phys. Rev. B 95 (2017), 104103.

[50] Y.W. You, X.S. Kong, X.B. Wu, C.S. Liu, J.L. Chen, G.N. Luo, Bubble growth from clustered hydrogen and helium atoms in tungsten under a fusion environment, Nucl. Fusion 57 (2017), 016006. [51] Y.W. You, D.D. Li, X.S. Kong, X.B. Wu, C.S. Liu, Q.F. Fang, B.C. Pan, J.L. Chen, G.N. Luo, Clustering of H and He, and their effects on vacancy evolution in tungsten in a fusion environment, Nucl. Fusion 54 (2014), 103007. [52] C.S. Becquart, C. Domain, A density functional theory assessment of the clustering behaviour of He and H in tungsten, J. Nucl. Mater. 386e388 (2009) 109e111. [53] D. Guo, I. Maryin-Bragado, C. He, H. Zang, P. Zhang, Modeling of long-term defect evolution in heavy-ion irradiated 3C-SiC: mechanism for thermal annealing and influences of spatial correlation, J. Appl. Phys. 116 (2014), 204901. [54] R.A. Oriani, The diffusion and trapping of hydrogen in steel, Acta Metall. 18 (1970) 147. [55] R. Frauenfelder, Solution and diffusion of hydrogen in tungsten, J. Vac. Sci. Technol. 6 (1969) 388. [56] D.J. Cherniak, E.B. Watson, R. Trappisch, J.B. Thomas, D. Chaussende, Diffusion of helium in SiC and implications for retention of cosmogenic He, Geochim. Cosmochim. Acta 192 (2016) 248e257. [57] Y. Pramono, K. Sasaki, T. Yano, Release and diffusion rate of helium in neutron-irradiated SiC, J. Nucl. Sci. Technol. 41 (2004) 751e755. [58] X.K. Lu, T.Y. Xin, Q. Zhang, Y.J. Feng, X.L. Ren, Y.X. Wang, Decentral distribution of helium in b-SiC: studied by density functional theory, Nucl. Mater. Energy 13 (2017) 35e41.