First principles study of foreign interstitial atom (carbon, nitrogen) interactions with intrinsic defects in tungsten

Journal of Nuclear Materials 430 (2012) 270–278

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First principles study of foreign interstitial atom (carbon, nitrogen) interactions with intrinsic defects in tungsten Xiang-Shan Kong a, Yu-Wei You a, Chi Song a, Q.F. Fang a, Jun-Ling Chen b, G.-N. Luo b, C.S. Liu a,⇑ a b

Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, P.O. Box 1129, Hefei 230031, PR China Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, PR China

h i g h l i g h t s " Interstitial C/N can reduce the vacancy formation energy nearby. " The interstitial C/N can attract other interstitial C/N. " The C/N is easily trapped by the vacancy. " The interstitial C/N enhances the self-interstitial cluster formation.

a r t i c l e

i n f o

Article history: Received 9 November 2011 Accepted 4 July 2012 Available online 20 July 2012

a b s t r a c t We performed a series of first-principles calculations to investigate the foreign interstitial atom (FIA) interactions with intrinsic defects in tungsten. We found the following: (i) The introduction of the FIA reduces the vacancy formation energy, resulting in the increase of the equilibrium concentration of vacancies. (ii) The positive binding energy between two FIAs suggests that the FIA can attract other FIAs. (iii) The FIA is easily trapped by the vacancy, and a single vacancy can accommodate up to 4 and 6 atoms in a stable manner for carbon and nitrogen, respectively. (iv) There is an attraction interaction between the FIA and the self-interstitial atom (SIA), and the FIA can reduce the SIA jump frequency and enhance the formation of SIA clusters in tungsten. Moreover, the difference between carbon and nitrogen are also discussed with respect to the formation of FIA–FIA covalent bond and the accumulation around the saturated FIAn —V complex. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Tungsten is considered to be the most promising candidate for the plasma-facing materials (PFM) in fusion reactors and has been selected as the PFM at the divertor baffles in the ITER design because of its high melting point, high thermal conductivity and low sputtering yield for light elements [1–4]. On the other hand, carbon and nitrogen are also introduced into the plasma device, where carbon is selected as the PFM in ITER and nitrogen is used as a seeding species to reduce the edge plasma temperature in the all-tungsten clad ASDEX Upgrade [2,5–10]. Therefore, carbon and nitrogen atoms are considered as two of the most frequent foreign interstitial atoms in tungsten in the plasma device. These impurities can significantly modify the properties of tungsten, which is a key issue for applications as the PFM. Experimental studies have been made to investigate the interaction of carbon or nitrogen with tungsten. When the concentration of carbon or nitrogen is much higher than their solubility ⇑ Corresponding author. Tel.: +86 551 5591062. E-mail address: [email protected] (C.S. Liu). 0022-3115/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2012.07.008

limit, tungsten carbides or tungsten nitrides are formed, which can change not only the thermo-mechanical properties, but also the hydrogen isotope ion trapping and retention properties of the plasma facing wall [5,8]. In our previous paper, we have studied the hydrogen behavior in tungsten carbides and found that the formation of tungsten carbides may enhance the retention of hydrogen in tungsten [11,12]. In the case of the concentration of carbon or nitrogen atoms below their solubility limit, even a very small amount of these impurity atoms in the interstitial site can have a drastic influence on the tungsten properties because of their strong interaction with the lattice defects. Hence, exploring the interaction between these impurity atoms and tungsten not only helps us to understand the effects of carbon or nitrogen on the structure and properties of tungsten, but also has a direct impact on the design and operation of a fusion reactor. So far, massive theoretical works have been done to investigate the interaction between carbon or nitrogen and some metals, especially for iron [13–18]. However, much less effort has been put into researching on the interaction between tungsten with these impurity atoms, especially for nitrogen. Recent calculated results [19– 21] show that the most stable interstitial site is the octahedral

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interstitial site (OIS) for carbon and nitrogen, and binding energies of carbon or nitrogen with a vacancy in tungsten are higher than other metals. Additionally, the interstitial carbon atom can bind strongly with the SIA [22]. However, the interaction between the carbon or nitrogen atom with intrinsic point defects in tungsten is still unclear, especially for the mechanism of interaction between these impurities and SIAs. In this work, we will systematically investigate the interaction of carbon or nitrogen in the tungsten as well as their interactions with the vacancy and SIA.

Table 1 Comparison of the results of DFT calculation with experiments: atomic volume X (Å3), vacancy (V) and self interstitial (I) formation energy EV;I (eV), migration energy f V;I EV;I in unit of atomic volume. m (eV), formation volume Xf Bcc-W

The present calculations are performed within density functional theory as implemented in the Vienna ab initio simulation package (VASP) [23,24]. The interaction between ions and electrons is described by the projector augmented wave potential (PAW) method [25]. Exchange and correlation functions are taken in a form proposed by Perdew and Wang within the generalized gradient approximation (GGA) [26]. The supercell composed of 128 lattice point ð4  4  4Þ is used. For tungsten, the four 5d electrons are considered as valence ones together with the two 6s (the 4 reference state is 5d 6s2 ). For carbon, four valence electrons are used: 2s and 2p (reference 2s2 2p2 ). For nitrogen, five valence electrons are used: 2s and 2p (reference 2s2 2p3 ). The relaxations of atomic position and optimizations of the shape and size of the supercell are performed. Ion relaxations were performed using the standard conjugated-gradient algorithms as implemented in the VASP code. The plane wave cutoff and k-point density, obtained using the Monkhorst–Pack method [27], are both checked for convergence for each system to be within 0.001 eV per atom. Following a series of test calculations a plane wave cutoff of 500 eV is used and a k-point grid density of 3  3  3 is employed. The structural optimization is truncated when the forces converge to less than 0.01 eV/Å. It should be pointed out that the valence charge density and density of states are calculated using 9  9  9 k-grides mesh obtained by the Monkhorst–Pack method. Additionally, in this paper, all transition states and barrier energies of diffusion paths are calculated by the nudged elastic band method associated with the Climbing Image (cNEB). Five images are used in all cNEB calculations. The formation energy of the point defect is calculated by

Edefect ¼ EðW N FIAn Þ  f

N EðW 128 Þ  nEðFIAisolated Þ; 128

ð1Þ

where EðW N FIAn Þ is the total energy of the defect supercell containing N tungsten atoms and n FIA, EðW 128 Þ is the total energy of tungsten perfect supercell, and EðFIAisolated Þ is the energy of the carbon or nitrogen isolated. The binding energy between two point defects ðA1 ; A2 Þ is calculated as follows:

EAb 1 —A2 ¼ EðA1 Þ þ EðA2 Þ  EðA1 þ A2 Þ  EðW 128 Þ;

ð2Þ

where EðA1 Þ and EðA2 Þ are the total energies of the supercell with A1 and A2 , respectively, and EðA1 þ A2 Þ is the total energy of the supercell containing both A1 and A2 . All supercells contain 128 lattice sites. Here, negative binding energy indicates repulsion between two point defects, while positive binding energy means attraction. 3. Results and discussion 3.1. Intrinsic defects in tungsten The properties of the tungsten intrinsic defects, i.e., the formation and migration energies, are first validated. The calculated results compared with experiments and other DFT calculations are

Present work

EVf

15.87a 3.68 ± 0.2a

15.96c 3.34c, 3.56d

16.02 3.20

EVm

1.78 ± 0.1a

1.71c, 1.78d

1.65

XVf EIf EIm XIf a b c d e f

DFT Other

X0

2. Computation method

Expt.

Ref. Ref. Ref. Ref. Ref. Ref.

0.62c,

0.71

9.06 ± 0.63b

9.86c, 9.6d

9.68

0.054e

0.005c, 0.05d, 0.002f

0.004

0.6c,

0.73

[28]. [29]. [30]. [31]. [32]. [33].

presented in Table 1. The calculated bcc lattice constant of 3.176 Å agrees well with the experimental value of 3.165 Å [28]. Our results for the intrinsic defects formation and migration energies agree with the experimental results and other DFT calculations [28–33]. The slight difference between our results and other DFT calculations is probably due to the difference in computation method, where a generalized gradient approximation in Perdew and Wang form is used here and the Perdew–Burke–Ernzerhof form was used in Ref. [30]. 3.2. Single FIA in bcc tungsten A single foreign atom defect consists of a substitutional or interstitial foreign atom. In bcc tungsten all substitutional sites are equivalent, while there are two possible interstitial sites, i.e., tetrahedral and octahedral positions. These interstitial sites are illustrated in Fig. 1. The FIA formation energies for substitutional, octahedral and tetrahedral positions are calculated using Eq. (1) and listed in Table 2. For both carbon and nitrogen atoms, the most stable position is the octahedral site, which is in agreement with the former calculation results [20,19]. Note that the energy difference between tetrahedral and octahedral sites is consistent with the results of Lin et al. [20], although the formation energies are different due to the different reference states. The relaxation of tungsten atoms around the octahedral interstitial carbon or nitrogen atom can be calculated as Mdi =di0 , where di0 is the ith nearest-neighbor (inn) solute-tungsten distance before relaxation and Mdi ¼ ðdi  di0 Þ is the change in distance due to relaxation. The calculated relaxations are presented in Table 3. The relaxations of 1nn of octahedral interstitial carbon and nitrogen atoms are 23.30% and 22.42%, respectively, which are greatly larger than the relaxations of other nearest-neighbor atoms (0.1– 2%). These results indicate that the influence range of FIA is very local. The lattice distortions introduced by the octahedral interstitial carbon or nitrogen atom can be characterized by determining the dipolar tensor from Kanzaki forces. Here, to obtain the dipolar tensor, we adopt a similar calculation procedure as used in Ref. [14], where the dipolar tensor P is calculated from the Kanzaki forces on all the tungsten atoms. The detailed procedure could be found in Ref. [14]. Due to the symmetry of the configuration, the dipolar tensor has two independent values: P11 and P33 , which are listed in Table 3. Similarly with Ref. [14], approximate values of dipolar tensor only can be obtained due to the small supercell. However, it appears clearly that the dipolar tensor is highly anisotropic. Whatever the FIA nature, the values of P33 are much larger than

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phenomenon has been found in other bcc metals, such as Fe [14], when a carbon or nitrogen atom is placed at the octahedral site. The vacancy formation energy is the energy required to take an atom from inside the crystal, and place it into a reservoir of the same atoms. In the perfect supercell, it can be calculated by Eq. (1). In the supercell with an FIA, the vacancy formation energy of the nearest-neighbors of octahedral interstitial carbon or nitrogen atom is defined as [34]:

EVf0 ¼ EðW 127 FIA1 Þ  EðW 128 FIA1 Þ þ

Fig. 1. The interstitial sites and their neighbor tungsten atoms: (a) tetrahedral site and (b) octahedral site. The FIA is represented by a small white ball. Its first, second, third and fourth nearest neighbors are represented by large light green, dark green, light cyan and dark cyan ball, respectively. 1nn stands for the first nearest neighbor, 2nn stands for the second nearest neighbor and so on. Two distinct fifth nearest neighbors for the octahedral site are labeled with 5nn-z and 5nn-o, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2 Formation energies for a single carbon or nitrogen atom positioned in a substitution site or in the two possible interstitial sites (in eV).

Carbon Nitrogen

ESub f

ETIS f

EOIS f

OIS ETIS f  Ef

2.32 0.13

4.45 3.06

5.29 3.79

1.47 0.73

the values of P 11 . The reason is that the coordination polyhedron of the octahedral site is not a regular octahedron. To summarize, the introduction of an FIA in an octahedral site in tungsten leads to a tetragonal distortion to the neighboring tungsten lattice. A similar

EðW 128 Þ : 128

ð3Þ

The calculated vacancy formation energies of the nearest-neighbors of the FIA are summarized in Table 3. The vacancy formation energies of the 1nn of the octahedral interstitial carbon and nitrogen atoms are 1.23 eV and 0.73 eV, respectively, which are remarkably lower than that in the perfect tungsten supercell (3.2 eV). For other nearest-neighbors, the vacancy formation energies are a bit lower than that in the perfect tungsten supercell. These results show that the octahedral interstitial carbon and nitrogen can decrease the energies needed to form a vacancy around them. The effect of vacancy formation energy on the equilibrium concentration of vacancies can be expressed as c0 ¼ expðEVf =kTÞ. Therefore, the introduction of carbon and nitrogen can increase the equilibrium concentration of vacancies. In order to explore the mechanisms of the FIA reducing the vacancy formation energy, the electronic charge density difference is plotted in the (1 1 0) plane with the FIA in the center, as presented in Fig. 2. The differential charge density is defined as the difference between the charge density of the interstitial-containing tungsten system and the superposition densities of tungsten host and the free interstitial. In Fig. 2, there are significant charge accumulations around the FIA and depletions on tungsten. This shows that the FIA not only attracts electrons from metals, but also induces electron redistribution nearby. The charge depletion on tungsten atoms around FIA leads to the weakening of the W–W bond, which in turn results in the decrease of vacancy formation energy. The charge transfer between the FIA and its nearest-neighbors is more significant for nitrogen than for carbon. This explains why nitrogen reduces the vacancy formation energy much more than carbon. To give a further confirmation, we list the charges of neighbor tungsten atoms from bader charge analysis [35,36] in Table 3. The bader charge analysis uses as sole input the charge density and considers the zero-flux surfaces as the borders between different atoms. Using these borders to define the volume related to each atom, the charge of each atom is obtained by integrating the density over this volume. Here, the bader charge is calculated using a grid based on the algorithm developed by Henkelman et al. [35,36] As can be seen, the charges of tungsten atoms nearest-neighbor to nitrogen are more positive than that to carbon. In addition, as shown in Fig. 2, for carbon, the charge transfer is only clearly visible between the FIA and its 1nn, while for nitrogen, both the 1nn and 2nn contribute remarkably. This implies that the influence range is larger for a nitrogen atom than for a carbon atom.

Table 3 Relaxation ðMdi =di0 Þ, dipolar tensor (P 11 ; P22 ; P33 , in eV), vacancy formation energy (EVf0 , eV) and bader charge (Q, in e) of tungsten atoms around FIA in the tungsten bcc structure. The second column indicates the atom number of its ith nearest-neighbor. Carbon

1nn 2nn 3nn 4nn 5nn-o 5nn-z

2 4 8 8 8 2

Nitrogen

Mdi =di0

P 11 ¼ P 22

P 33

EVf0

Q

Mdi =di0

P 11 ¼ P 22

P 33

EVf0

Q

23.30 1.91 0.51 2.01 0.10 1.93

0 0.40 4.46 6.27 9.30 9.30

16.20 16.20 19.99 18.33 18.94 19.05

1.23 2.74 2.91 2.76 3.19 2.82

0.51 0.24 0.00 -0.06 -0.07 0.16

22.42 -1.29 0.54 1.80 -0.02 2.02

0 0.46 5.35 5.83 7.59 7.59

16.02 16.02 19.59 17.01 16.39 16.17

0.73 2.26 2.83 2.80 3.20 2.78

0.77 0.35 0.00 -0.03 -0.06 0.17

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1.5

0.15

1.2

(a)

pure 1nn 2nn

0.9

1nn

0.6

2nn

2nn 1nn

001

0.3

0.08

0.0

(a)

0.00

-0.07

(b)

0.9 0.6 0.3 2.5 2.0

2nn 1nn

001

1.2

0.0

1nn 2nn

Density of state

1.5

110

(c)

Cp Np

1.5 1.0

110

-0.15

(b) Fig. 2. The differential charge density map for an FIA at the octahedral interstitial site in the ð1 1 0) plane. The white circles are tungsten atoms, the green (gray in black/white) stars are the FIAs; (a) the FIA is carbon and (b) the FIA is nitrogen. The units are eV per Å3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Besides the charge differences, the partial density of states for FIA (p-DOS) and its 1nn and 2nn tungsten atoms (d-DOS) are shown in Fig. 3. The depletion in the tungsten 5d states of the density of states is observed compared to that of pure bcc tungsten. There is a hybridization around 7 eV for carbon and 8 eV for nitrogen of the tungsten 5d states with FIA 2p states, suggesting the formation of an interaction between the FIA and its neighbors. The interaction is larger for its 1nn tungsten atom than its 2nn tungsten atom.

0.5 0.0 -10

-8

-6

-4

-2

0

2

4

Energy (eV) Fig. 3. The partial density of states in a bcc supercell containing 128 tungsten atoms and one FIA in an octahedral site. The black curves correspond to a tungsten atom 1nn to the FIA, the green (gray in black/white) curves to an atom 2nn to the FIA, and the red dashed line to a tungsten atom in perfect supercell; (a) the d states of tungsten atoms when the FIA is carbon, (b) the d states of tungsten atoms when the FIA is nitrogen, and (c) the p states of carbon (blue curve) and nitrogen (light magenta curve) in an octahedral site in a tungsten lattice. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.3. FIA–FIA interaction in bcc tungsten We have calculated the binding energy of double FIAs both positioned initially at octahedral sites according to the labeling of Fig. 4. The initial and finial distance between two FIAs and the corresponding binding energies are summarized in Table 4. For both C–C and N–N, the most stable configuration corresponds to the re3 laxed one of the case cfg FIA—FIA in Table 4 and has a positive binding energy (0.49 eV for C and 0.47 eV for N), indicating the presence of an attractive interaction between two FIAs. The situation is completely different for Fe, as in that case the interaction between two FIAs is repulsive. One possible explanation for the different behaviors can be that the lattice parameter of tungsten being larger than Fe (3.176 Å versus 2.87 Å), the octahedral site remains large enough for an FIA to stay in, even for the high strains induced by the FIA nearby. Such attractive interaction might lead to the increase of local carbon or nitrogen concentration and further form tungsten carbide or tungsten nitride. It should be pointed out that the detail structural evolution during the formation of tungsten carbides or tungsten nitride in tungsten is beyond our research scope of this paper and need further investigation. The carbon or nitrogen precipitation in tungsten and the formation of tungsten carbide or tungsten nitride has been observed in experiments [5,8]. Additionally, for most double FIAs configurations, the binding

Fig. 4. Possible configurations for two FIAs. The tungsten atom and FIA are represented by the large and the small ball, respectively. The first FIA is on the site i labeled FIA, the second one on the site labeled with i. Configuration cfg FIA—FIA of Table 4 corresponds to one FIA on the site labeled FIA and the other on the site labeled i.

energies of C–C pairs are slightly larger than that of N–N pairs. It means that the C–C interactions are more attractive than the N– N interactions. The similar phenomenon has been found in Fe by Domain et al. They thought that this phenomenon may come from that the interaction of metal with the FIA is less localized in energy for carbon than for nitrogen [14]. The interaction between two FIAs becomes repulsive when the distance between them is less than the most stable distance (2.852 Å for carbon and 2.796 Å for nitrogen). However, it should be noted that a C–C covalent-like bond is formed along the h1 0 0i direction with a bond length of 1.518 Å(see Fig. 5), which is slightly longer than the typical C–C covalent bond length (1.38–1.48 Å for sp2 ) [37]. The C–C covalent-bond is relatively unstable and has a

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Table 4 unrelax , in eV) between two FIAs. dFIA—FIA and The distance (in Å) and binding energies (EFIA—FIA b relax dFIA—FIA is the distance between two FIAs before and after relaxation, respectively. Configuration

dN—N

EN—N b

0.63

2.491

0.34

0.49

2.796

0.47

0.33

2.914

0.32

3.716

2.35

0.40

3.713

0.39

0.13

3.795

0.03

4.443

0.18

4.418

0.11

4.577

0.10

4.568

0.14

4.764

4.805

0.09

4.758

0.02

5.501

5.572

0.22

5.613

0.24

dFIA—FIA

dC—C

EC—C b

1

1.588

1.518

1.89

2

2.246

2.452

cfg FIA—FIA

3

2.750

2.852

4 cfg FIA—FIA 5 cfg FIA—FIA 6 cfg FIA—FIA 7 cfg FIA—FIA 8 cfg FIA—FIA 9 cfg FIA—FIA 10 cfg FIA—FIA 11 cfg FIA—FIA

3.176

3.218

3.176

3.806

3.551

3.704

3.890

3.864

4.492 4.492

unrelax

cfg FIA—FIA cfg FIA—FIA

relax

relax

2.5

1

Fig. 5. The charge density distribution map for cfg C—C in (1 0 0) plane.

Fig. 7. The FIA diffusion energy profile and the corresponding diffusion paths near a vacancy. The tungsten atom and the FIA are represented by the large cyan and the white small ball, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

negative binding energy (1.89 eV). The main reason might be that there is not enough space to provide for the ‘close-distance’ between carbon atoms in the intrinsic tungsten, leading to a larger local strain field surrounding the two atoms. However, such covalent-like bond does not appear when two FIAs are nitrogen atoms. This may because that the influence of nitrogen on charge redistribution nearby is much larger than carbon. 3.4. The interaction of one FIA with one vacancy We now turn our attention to the interaction of one FIA with one vacancy. Fig. 6 presents the binding energies of a carbon or nitrogen atom situated close to a vacancy. As seen from the figure,

Binding energy (eV)

2.5 2.0

Carbon Nitrogen

1.5 1.0 Fig. 8. The atomic configurations for multiple carbon atoms trapped in a tungsten vacancy. The tungsten atom and carbon atom are represented by the large cyan ball and the small white ball, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.5 0.0 -0.5 0

1

2

3

4

5

6

7

dFIA-V (Å) Fig. 6. Variation in the binding energy of one FIA and one vacancy as function of the FIA distance to the vacancy center. The dashed line is presented as guides to the eye.

the FIA is attracted by the vacancy and therefore the FIA-vacancy complex forms. It should be noted that the position of the FIA in the center of the vacancy is energetically unfavorable. Our calculations reveal the preferable position for FIA near the vacancy, at a distance of 1.3 Å from the vacancy center close to an OIS (site A in Fig. 7). The binding energy of the FIA with vacancy deceases to

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Fig. 9. The atomic configurations for multiple nitrogen atoms trapped in a tungsten vacancy. The tungsten atom and nitrogen atom are represented by the large cyan ball and the small white ball, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2

1

Etrap(n) (eV)

zero at the distance of 5 Å from the vacancy center and then remains zero with increasing the distance. It implies that the single vacancy trapping radius is 5 Å for the interstitial carbon or nitrogen atom. In addition, the binding energy of the nitrogen atom with a vacancy is slightly larger than that of the carbon atom, indicating that the nitrogen atom has a stronger interaction with a vacancy than the carbon atom. As mentioned above, when a vacancy and an FIA interacts, they can form a stable FIA-vacancy complex. To investigate the kinetic process of this complex formation, two migration assays are performed: first, the FIA jumps toward the vacancy along the h1 0 0iðE ! D ! C ! B ! A, Fig. 7); second, the vacancy jumps from the FIA’s 2nn to its 1nn. For the first migration assay, all energy barriers for the FIA jumping toward the vacancy are lower than that in the perfect tungsten supercell (1.5 eV for C and 0.73 eV for N), whereas all energy barriers for the FIA jumping away the vacancy are higher than that jumping in the perfect tungsten supercell. This means that the FIA is easily trapped by the vacancy and hardly detrapped from it. For the second migration assay, the energy barrier for the vacancy jumping from the FIA’s 2nn to its 1nn is 3.87 eV for carbon and 3.60 eV for nitrogen, and the reverse jump energy barriers are 2.35 eV and 2.05 eV for carbon and nitrogen, respectively. All of them are greatly larger than the energy barrier of vacancy diffusion in perfect tungsten supercell (1.65 eV, Table 1), suggesting that the vacancy around the FIA must overcome higher barriers to migrate from one site to another. Meanwhile the jumps towards the FIA ð2nn ! 1nnÞ involve an energy barrier lower than for a jump away ð1nn ! 2nnÞ. This implies that a vacancy returns back more frequently to the FIA from its neighbor sites than jumping away. Comparing these two migration assays energy barriers, we find that the energy barriers for vacancy jumping toward the FIA are greatly higher than that for the FIA jumping toward the vacancy. Hence, it is reasonable to expect that the FIA would jump toward the vacancy to form the FIA-vacancy complex in the area simultaneously with the FIA and vacancy. Besides, it should be expected that the vacancy diffusivity is significantly modified by the formation of FIAn —V complexes. Recent studies in Fe–C alloys showed that the diffusivity decreases with increasing carbon content and becomes negligible when the carbon concentration exceeds twice that of vacancies [18]. Here no significant difference is observed between carbon and nitrogen.

Carbon Nitrogen

0 -1

-2

-3

1

2

3

4

n

5

6

7

Fig. 10. The trapping energy for the nth FIA trapped by the FIAn1–V complex.

3.5. Multiple FIAs occupancy at a vacancy In this section, we study the process of multiple FIAs trapped into the vacancy to check the possibility for multiple FIAs occupancy at the vacancy in tungsten. First, we place one FIA at the most stable site in the vacancy, and then bring the FIA one by one into the vacancy and minimize the energy to find out the most stable configurations of multiple FIAs in the vacancy (FIAn —V complexes). In each step, we investigate up to fifteen possible configurations based on the most stable configurations of FIAn1 —V complex. The most stable configurations for Cn —V and Nn —V complex are presented in Figs. 8 and 9, respectively. The trapping energy for the nth FIA by the most stable FIAn1 —V complex is shown in Fig. 9, which is defined as [38]:

Etrap ðnÞ ¼ EðW 127 FIAn Þ  EðW 127 FIAn1 Þ  EðW 128 FIA1 Þ  EðW 128 Þ;

ð4Þ

when the number of impurity atoms is increased from n  1 to n. A negative value of Etrap ðnÞ indicates that it is energetically favorable to take an FIA and add it to a vacancy that already contains n-1 atoms, with jEtrap ðnÞj being the energy gained in that process. It is clearly seen from Figs. 8–10 that a single vacancy can accommodate

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1.5

Wd Cp

(a)

1.2 0.9

Density of state

0.6 0.3 0.0

(b)

1.2 0.9 0.6 0.3 0.0 -10

-8

-6

-4

-2

0

2

4

Energy Fig. 12. The partial density of states in a bcc supercell containing FIAn —V complex: (a) FIA1 —V; (b) FIA2 —V. The red (black in black/white) curve corresponds to a tungsten atom 1nn to the FIA in the vacancy, and the green (gray in black/white) curve corresponds to the FIA in the vacancy. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Binding energy (eV)

1.00

0.75

0.50

0.25 Carbon Nitrogen

0.00 Fig. 11. The charge density distribution map for FIAn —V complex: (a), (c), (e) and (g) are C1 —V; C2 —V; C3 —V; C4 —V in (1 0 0) plane, respectively; (b), (d), (f) and (h) are N1 —V; N2 —V; N3 —V and N4 —V in (1 1 0) plane, respectively.

up to 4 and 6 atoms in a stable manner for carbon and nitrogen, respectively. It should be noted that the trapping energy is still negative when a nitrogen atom is placed at the OIS nearby the saturated Nn —V complex while the trapping energy is positive in this case for carbon. This suggests that the nitrogen atom can still accumulate nearby the saturated Nn —V complex while the carbon atom cannot accumulate around the saturated Cn —V. We can also see a large different behavior between carbon and nitrogen. For carbon, all atoms prefer to octahedral sites 1nn to the vacancy, and the C–C covalent-like bonds are formed in Cn —V complex when nP2. Fig. 11 presents the charge density distribution map for FIAn —Vðn ¼ 1  4). The C–C covalent-like bond can be clearly seen from Figs. 11 c, e and g: the high electronic concentration between the two carbon atoms. The bond lengths between carbon atoms are 1.478 Å, 1.490 Å, and 1.579 Å for n = 2, 3 and 4 respectively. We see that the C–C bonds in C2 —V and C3 —V are the typical C–C covalent bond length of 1.38–1.48 Å for sp2 bonds [37]. Such strong covalent bond makes carbon atoms bind strongly at the vacancy. For nitrogen, the first two atoms occupy 1nn OIS of the vacancy and other atoms prefer to 2nn OIS of the vacancy. The

1

2

3

4

5

6

7

dFIA-SIA (Å) Fig. 13. Variation in the binding energy of an FIA and an SIA as function of the perpendicular distance of the FIA from the SIA. The dashed line is presented as guides to the eye.

nitrogen atoms in the Nn —V complex form strong covalent-like bond with their nearest tungsten atoms, and there is no direct interaction between two nearest-neighbor nitrogen atoms (Figs. 11 b, d, f and h). The covalent-like bond does not appear in all Nn —V complexes. The distance between two nearest-neighbor nitrogen atoms in all stable Nn —V complexes is 2.8 Å, which is slightly less than that in tungsten nitride (2.920 Å) [39]. This different behavior of carbon and nitrogen is due to the interaction of the FIA with its tungsten neighbors as well as with its FIA neighbors. The binding energy with the vacancy is higher for nitrogen than for carbon, as the attraction between two FIA atoms is higher for carbon than for nitrogen. It should be pointed out that the interaction between carbon and its 1nn tungsten atom is significantly weakened due to the formation of the C–C covalent bond. This can be clearly displayed by the partial density of states of carbon and its 1nn tungsten atom. As shown in Fig. 12, the hybridization

X.-S. Kong et al. / Journal of Nuclear Materials 430 (2012) 270–278

Fig. 14. The charge density distribution map for FIA in the vicinity of the h1 1 1i SIA crowdion in the ð1 1 0Þ plane; (a) the FIA is carbon and (b) the FIA is nitrogen. The units are eV per Å3.

Density of state

1.2

(a)

W 5d C 2p

277

ity of the SIA. On the other hand, we can also deduce that the SIA is trapped by the FIA. Fig. 14 shows the charge distribution map for FIA in the vicinity of the SIA. We can see that two strong covalent-like bonds are formed between FIA and tungsten atoms in the SIA and the bond lengths are 2.05 Å. Besides, the hybridization around 7 eV for carbon and 8 eV for nitrogen of the tungsten 5d states with FIA 2p suggests the formation of a covalentlike interaction between the FIA and the tungsten atom in the h1 1 1i SIA crowdion (Fig. 15). The formation of such strong covalent-like bonds would greatly increase the diffusion energy barrier of SIA because more energy is needed to break these covalent bonds in the SIA diffusion process. Becquart et al. [42] found that the release of the SIAs captured in the carbon needs the temperature above 200 K, which is much larger than the migration temperature of SIA in tungsten (about 1.5 K) [43]. Hence, we can expect that the FIA would reduce the jump frequency of the SIA and thus affect the long-range migration behavior of the SIA. As a result, the formation of SIA clusters is on a much finer scale than in the absence of FIA. The similar phenomenon has been observed in the iron experiments [44].

0.9

4. Conclusions

0.6

We use first-principles calculations to study the FIA interactions with intrinsic defects in tungsten. Our calculations confirm that the most stable position for a carbon and nitrogen interstitial atom is the octahedral site in tungsten. We found that the introduction of an FIA at an octahedral site leads to a tetragonal distortion to the neighboring tungsten lattice. Additionally, the FIA not only attracts electrons from metals, but also induces electron redistribution nearby. The charge depletion on the tungsten atoms around FIA leads to the weakening of the W–W bond, which in turn results in the decrease of vacancy formation energy. The binding energy between two FIAs is positive, suggesting that the FIA can attract other FIAs. This leads to the increase of local carbon or nitrogen concentration and further form tungsten carbide or tungsten nitride. The calculated results on the interaction between the FIA and vacancy show that the migrating FIA would jump toward the relatively stable vacancy to form the FIA-vacancy complex in the area simultaneously with the FIA and the vacancy and a single vacancy can accommodate up to 4 and 6 atoms in a stable manner for carbon and nitrogen, respectively. In addition, we find that there exists an attractive interaction between the FIA and the SIA. The FIA can trap the SIA, reducing the SIA jump frequency and enhancing the formation of SIA clusters. As concluded above, carbon and nitrogen have similar properties in terms of the migrations and binding with a vacancy or an SIA. There are some small difference between carbon and nitrogen. The migration energy of the FIA is higher for carbon than for nitrogen. The interaction between two interstitial carbon atoms is slightly more attractive than two interstitial nitrogen atoms. The binding energy with a vacancy is a bit smaller for carbon than for nitrogen, while the binding energy with an SIA is slightly higher for carbon than for nitrogen. Two significant differences between carbon and nitrogen in our studies are: (i) two carbon atoms can form the C–C covalent-like bond while nitrogen cannot; (ii) the nitrogen atom can accumulate around its saturated FIAn —V complex whereas the carbon atom cannot.

0.3 0.0 3.0 2.5

(b)

W 5d N 2p

2.0 1.5 1.0 0.5 0.0 -10

-8

-6

-4

-2

0

2

4

Energy (eV) Fig. 15. The partial density of states in a bcc supercell with an FIA in the vicinity of the h1 1 1i SIA crowdion. The black curves correspond to the tungsten atom in h1 1 1i SIA crowdion neighboring the FIA; (a) the d states of tungsten atoms when the FIA is carbon and (b) the d states of tungsten atoms when the FIA is nitrogen.

between the tungsten 5d states and carbon 2p states decreases remarkably in C2 —V complex. 3.6. The interaction of one FIA and one SIA Finally we focus on the interaction of carbon or nitrogen with SIAs. Previous studies have revealed that the h1 1 1i crowdion is the most stable SIA configuration in tungsten [40,41]. Hence, one h1 1 1i SIA crowdion is introduced in the supercell, and then one FIA is placed in the vicinity of the SIA. After full relaxation, the binding energy of one FIA and one SIA is calculated using the Eq. (2). Fig. 13 presents the evolution of binding energy of an FIA and an SIA with the perpendicular distance of the FIA from the SIA. It is clearly seen that both carbon and nitrogen have a positive binding energy with the SIA and the binding energy decreases with the increase of the perpendicular distance between the FIA and SIA. Positive binding energy indicates that there exists an attractive interaction between the FIA and SIA. The attraction is larger for the carbon atom than for the nitrogen atom. Such attractive interaction leads to the accumulation of carbon or nitrogen in the vicin-

Acknowledgements This work was supported by the National Magnetic Confinement Fusion Program (Grant Nos.: 2011GB108004 and 2010GB109004), the National Natural Science Foundation of China (Nos.: 91026002, 91126002) and the Strategic Priority Research

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