Journal of Nuclear Materials 417 (2011) 1119–1122

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Journal of Nuclear Materials

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Composition dependence of formation energy of self-interstitial atom clustersin b-SiC: Molecular dynamics and molecular statics calculations

Yoshiyuki Watanabe a,⇑, Kazunori Morishita b, Akira Kohyama c

a Graduate School of Energy Science, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japanb Institute of Advanced Energy, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japanc Department of Materials Science and Engineering, Muroran Institute of Technology, 27-1 Minamoto, Muroran, Hokkaido 050-8585, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Available online 24 December 2010

0022-3115/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.jnucmat.2010.12.087

⇑ Corresponding author. Tel.: +81 774 38 4541.E-mail address: y-watanabe@iae.kyoto-u.ac.jp (Y.

Molecular dynamics and molecular statics calculations have been performed using an empirical many-body interatomic potential to obtain the formation energy of relaxed configuration of self-interstitialatom (SIA) clusters with three chemical composition ratios of silicon- and carbon interstitials in cubic sil-icon carbide (b-SiC), which is necessary when nucleation and growth process of SIA-clusters are investi-gated. The formation energy of SIA-clusters in b-SiC depends on the size, composition and configurationof clusters. For each composition ratio, the formation energy has been described using a polynomial func-tion of cluster size. The resultant equations show the applicability to a wide range of cluster size, in whichthe rationale may be explained by the number of each type of dimer bond around an SIA in an SIA-cluster.

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1. Introduction

Cubic silicon carbide (b-SiC) is a component of SiC/SiC compos-ites that are candidates for blanket structural materials of nuclearfusion reactors because of high mechanical strength and low acti-vation properties, in which b-SiC has the zinc-blende crystal struc-ture with the stacking sequence of the silicon–carbon, described asABCABC. . . along the h1 1 1i direction. A variety of experimentshave been performed to evaluate irradiation effects in b-SiC. Someexperiments using Transmission Electron Microscopy (TEM) showthat self-interstitial atom (SIA) clusters with a diameter of 2.0–3.6 nm are observed in irradiated b-SiC at temperatures above600 �C [1–3]. Such microstructural changes cause the degradationof the material’s properties; therefore, they should be accuratelypredicted and controlled to suppress the degradation. However,the formation kinetics of SIA-clusters in b-SiC is not understoodwell yet. In our previous work [4], the energetics of SIA-clustersin b-SiC was investigated using atomistic calculations, which isimportant to clarify the formation kinetics of the clusters duringirradiation. In the previous work, the size dependence of the for-mation energy was investigated for SIA-clusters with stoichiome-tric composition. However, since silicon carbide is a binarychemical compound, SIA-clusters far from stoichiometric composi-tion could be also formed during irradiation.

In the present study, the composition dependence of formationenergy of SIA-clusters in b-SiC was investigated using atomisticcalculations.

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Watanabe).

2. Procedure

Molecular dynamics (MD) and molecular statics (MS) calcula-tions were performed with the empirical interatomic potential byGao and Weber [5]. This potential was developed to provide a gooddescription of self-interstitial’s properties in b-SiC. The MD and MScalculations were conducted to obtain the formation energy of SIA-clusters with the following three composition ratios of siliconinterstitials (ISi) and carbon interstitials (IC): (i) ISi/IC =1 (clustersconsisted of only silicon interstitials), (ii) ISi/IC = 0 (clusters con-sisted of only carbon interstitials) and (iii) ISi/IC = 1 (clusters con-sisted of the same number of silicon and carbon interstitials:stoichiometric composition). Notice that the SIA-clusters treatedhere are relatively large size (diameter d = 1.7–3.6 nm) which arevisible in TEM observation. The initially employed configurationof each SIA-cluster with the three composition ratios is in the formof hexagonal plate parallel to the close-packed (1 1 1) plane. Thisinitial configuration is determined by reference to a rough shapeof SIA-clusters observed in the TEM experiments [1–3].

First, an SIA-cluster with the initial configuration was intro-duced into a computational box with constant volume under 3-Dperiodic boundary conditions. The box size was chosen to be28a0 � 28a0 � 28a0 (for d = 1.7 nm) and 59a0 � 59a0 � 59a0 (ford = 3.6 nm) depending on cluster size, with a lattice constant ofa0 = 0.436 nm. The computational system was then fully relaxedfor 50 ps by keeping it at a finite temperature, in which the tem-perature was set at 100, 200, 300, 600, 800, 1000, 1300 and1500 K. And then, the system was quenched for 10 ps to 0 K to ob-tain the converged value of the total potential energy. It is notedthat this combined relaxation method of MD and MS techniques

1120 Y. Watanabe et al. / Journal of Nuclear Materials 417 (2011) 1119–1122

can provide a more relaxed configuration of defects than that pro-vided by simple static relaxation method, as described elsewhere[6].

Formation energy of an SIA-cluster is defined as the energy re-quired for introducing the cluster into an otherwise perfect crystal.When nSi

I and nCI denote the numbers of silicon- and carbon inter-

stitials in an SIA-cluster, respectively, the formation energy of thecluster is given by:

EF nSiI ;n

CI

� �¼ EtotðNSi;NCÞ � NSieSi � NCeC; ð1Þ

Fig. 1. Size dependence of the lowest calculated formation energy of SIA-clusters with the(d) ISi/IC = 1 (stoichiometric). The size is defined as n ¼ nSi

I þ nCI that is the total number o

(a) ISi/IC= (only silicon), n=127

(b) ISi/IC=0 (only carbon), n=108

(c) ISi/IC=1 (stoichiometric), n=122

[211][011]

[111]

Fig. 2. The most relaxed configuration of relatively large SIA-clusters with the three com(stoichiometric). Each diagram represents a cross section of the computational box inclu

where NSi and NC denote the total numbers of silicon atoms and car-bon atoms in the system, respectively. Etot(NSi, NC) is the total po-tential energy of the fully relaxed system mentioned above,eSi = �6.21 eV and eC = �6.61 eV are the calculated cohesive ener-gies of a silicon atom and a carbon atom in b-SiC at 0 K, respectively.

In the present study, the MD and MS calculations were con-ducted for relatively large SIA-clusters with the size n ranging from108 to 338, in which n ¼ nSi

I þ nCI that is the total number of SIAs in

an SIA-cluster.

three composition ratios: (j) ISi/IC =1 (only silicon), (N) ISi/IC = 0 (only carbon) andf SIAs in an SIA-cluster. The broken lines denote the fitting curves of EF = An1 + Bn1/2.

silicon carbon

silicon interstitial

carbon interstitial

silicon interstitial

carbon interstitial

position ratios: (a) ISi/IC =1 (only silicon), (b) ISi/IC = 0 (only carbon) and (c) ISi/IC = 1ding an SIA-cluster and atoms around the cluster.

Y. Watanabe et al. / Journal of Nuclear Materials 417 (2011) 1119–1122 1121

3. Results and discussion

The lowest calculated formation energy of relatively large SIA-clusters with the three composition ratios is plotted in Fig. 1, asa function of cluster size n. Notice that the formation energy of rel-atively small SIA-clusters with n = 1–6, obtained in our previouswork [4] using the same potential in the present study, is also plot-ted. The formation energy is an increasing function of n. As to therelatively large region, the formation energy of SIA-clusters withISi/IC = 0 (only carbon) is the highest, while the energy of clusterswith ISi/IC = 1 (stoichiometric) is the lowest. Fig. 2 shows the mostrelaxed configuration of relatively large SIA-clusters with the threecomposition ratios. Notice that each diagram represents a crosssection of the computational box including an SIA-cluster andatoms around the cluster. Although the configuration of eachSIA-cluster roughly maintains the hexagonal plate parallel to theclose-packed (1 1 1) plane, some atoms change their positionsfrom the initial those due to the relaxation, and the relaxed config-uration has a dilation field. As shown in Fig. 2a, the most relaxedconfiguration of a silicon interstitial in a cluster with ISi/IC =1(only silicon) is an SiTC configuration that is formed by a siliconatom located at the tetrahedral position, TC site, surrounded byfour regular carbon lattice sites. In this case, the silicon interstitialis also bonding with the three regular silicon lattice atoms. On theother hand, Fig. 2b shows that the most relaxed configuration of a

Fig. 3. Size dependence of the number of each type of dimer bond of Si–Si, C�C and Si�Csilicon), (b) ISi/IC = 0 (only carbon) and (c) ISi/IC = 1 (stoichiometric). The dimer bonds are

carbon interstitial in a cluster with ISi/IC = 0 (only carbon) is aCCh110i dumbbell configuration that is formed by two carbon atomsalong the h1 1 0i direction, where the two carbon atoms are sur-rounded by three regular silicon- and two regular carbon-latticeatoms in addition to a carbon which forms another CC<110> dumb-bell. And then, as represented in Fig. 2c, the most relaxed configu-ration of a silicon- and a carbon interstitials in a cluster with ISi/IC = 1 (stoichiometric) is similar to those in a perfect b-SiC crystal.

For each composition ratio, an attempt was made to describethe formation energy of SIA-clusters using the following equation:

EF ¼ pR2cSF þ 2pRcDL; ð2Þ

where an SIA-cluster is assumed to be a disc-shaped defect with ra-dius R, cSF is the stacking fault energy, and cDL is the energy per unitdislocation line. R has a relationship with n, as pR2b = nX, in whichX is the mean atomic volume, and b is the thickness of the disc inthe h1 1 1i direction. By the relationship between R and n, Eq. (2)is rewritten as follows:

EF ¼ An1 þ Bn1=2; ð3Þ

where A = cSFX/b and B = 2cDL(pX/b)1/2. As shown in Fig. 1, the fit-ting of Eq. (3) to the lowest calculated formation energy of the smalland large SIA-clusters with each composition ratio leads to valuesfor A and B, and each resultant fitting curve is also plotted as a bro-ken line.

around an SIA in an SIA-cluster with the three composition ratios: (a) ISi/IC =1 (onlycounted for interactions within the distance of 0.25 nm from each target atom.

1122 Y. Watanabe et al. / Journal of Nuclear Materials 417 (2011) 1119–1122

In order to validate the fitting curves, the most relaxed config-uration of SIA-clusters between small and large regions was inves-tigated focusing on atomic bonds. Fig. 3 shows the number of eachtype of dimer bond of Si–Si, C–C and Si–C around an SIA in an SIA-cluster with the three composition ratios: (a) ISi/IC =1 (only sili-con), (b) ISi/IC = 0 (only carbon) and (c) ISi/IC = 1 (stoichiometric),as a function of cluster size n. Notice that dimer bonds within0.25 nm from each target atom are counted; the value (0.25 nm)is the cutoff distance employed in the interatomic potential usedhere. For a perfect b-SiC crystal, the number of Si–C bonds aroundan atom is four, and the numbers of Si–Si and C–C bonds are zero;in other words, a regular silicon lattice atom is surrounded by onlyfour regular carbon-lattice atoms within the 0.25 nm from the sil-icon atom, while a regular carbon-lattice atoms is surrounded byonly four silicon regular lattice atoms within the 0.25 nm fromthe carbon atom. As shown in Fig. 3a for SIA-clusters consisted ofonly silicon interstitials, the numbers of Si–Si and Si–C bonds forthe cluster with n = 1 (isolated silicon interstitial) are six and four,respectively, in which the silicon interstitial is located at the TC sitesurrounded by four regular carbon-lattice atoms, in addition, thesilicon interstitial is surrounded by not three but six regular siliconlattice atoms. With increasing cluster size, the number of Si–Sibonds decreases, and the numbers of Si–Si and Si–C bonds forthe cluster with n = 5 are very close to those for much larger clus-ters. It indicates high similarity of the most relaxed configurationof clusters between small and large regions. Also, as to SIA-clustersconsisted of only carbon interstitials, good similarity of the mostrelaxed configuration of clusters between small and large regionsis indicated in Fig. 3b, where the number of C–C bonds for rela-tively small clusters increases with increasing cluster size, andthe numbers of C–C and Si–C bonds for the cluster with n = 5 arerelatively close to those for much larger clusters. And then, as rep-resented in Fig. 3c for SIA-clusters under stoichiometric condition,

relatively good similarity of the most relaxed configuration of clus-ters between small and large regions is indicated as well. It is notedthat with increasing cluster size, the number of Si–C bonds is get-ting close to four, and the numbers of the other bonds (Si–Si and C–C) are getting close to zero, resulting in the configuration similar tothat in a perfect b-SiC crystal.

4. Summary

The MD and MS calculations were performed using the Gao–Weber potential to obtain the formation energy of relaxed config-uration of SIA-clusters with the three chemical composition ratiosin b-SiC. The formation energy depends on the size, compositionand configuration of clusters. For each composition ratio, the for-mation energy was described using a polynomial function of clus-ter size. The resultant equations show the applicability to a widerange of cluster size, in which the rationale may be explained bythe number of each type of dimer bond around an SIA in an SIA-cluster. The equations derived here can be used to estimate thebiding energy of an SIA to an SIA-cluster, which is necessary whenformation kinetics of SIA-clusters such as nucleation and growthprocess is investigated.

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