- Email: [email protected]
First-principles study of hydrogen in perfect tungsten crystal
Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
First-principles study of hydrogen in perfect tungsten crystal Jingcheng Xu, Jijun Zhao * Laboratory of Materials Modification by Laser, Electron, and Ion Beams, School of Physics and Optoelectronic Technology, and College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Available online 16 June 2009 PACS: 61.72.S61.82.Bg Keywords: Hydrogen Tungsten Diffusion Interaction Binding energy
a b s t r a c t Tungsten-based materials are used as the first wall materials in ITER. Hydrogen impurities were introduced via bombarding with the reaction plasma, which are important for the behavior and stability of the tungsten wall. Using the first-principles density functional theory and planewave pseudopotential technique, we have simulated the behaviors of hydrogen atoms inside the perfect tungsten bcc lattice. The binding energies for different interstitial sites were compared to determine the optimal trapping site for the hydrogen atom inside the tungsten lattice. The diffusion barriers for hydrogen atom between nearby trapping sites and the interaction between two interstitial hydrogen atoms were also calculated. The implication of our theoretical results on the hydrogen diffusion and accumulation behavior was discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Because of high melting point, good thermal conductivity as well as low sputtering erosion [1], tungsten is one of the candidate materials for the plasma facing components in the divertor region of the International Thermonuclear Experimental Reactor (ITER) and the future fusion reactors. The divertor in ITER will be subjected to high particle fluxes from not only plasma background ions (H, D and T) but also intrinsic impurities (He, Be and C) [2,3]. Since hydrogen is one of the major plasma background ions, hydrogen irradiation may results in changes of mechanical properties of tungsten materials. It was known that helium irradiation leads to blister formation and subsequent degradation of the mechanical properties of metals [4–10]. Ye reported that the effects of hydrogen plasma bombarding on tungsten are different from helium [11]. Thus, it is important to explore the trapping and blister formation of hydrogen in tungsten via studying the behaviors of hydrogen impurities in tungsten. Previously, a number of experimental and theoretical studies were carried out to understand the interaction between the hydrogen impurity and the tungsten host lattice [7,8,12–23]. For example, Ye’s group presented the experimental results in a divertor plasma simulator NAGDIS-1, and observed formation of hydrogen bubbles on tungsten surface [21]. Arkhipov et al. also found that hydrogen could form blister at tungsten surface [20]. Henriksson and co-workers compared the differences in formation of hydrogen * Corresponding author. Tel.: +86 411 84709748; fax: +86 41 84706100. E-mail address: [email protected] (J. Zhao). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.072
and helium blister in tungsten using a combined simulation with density functional theory (DFT), empirical molecular dynamics (MD) and kinetic Monte Carlo [16]. Within the bcc tungsten lattice, they found that hydrogen atoms repulse with each other at H–H short distance; while the interaction of hydrogen atoms is attractive at the distance of 2.2 Å with tiny binding energy is (less than 0.1 eV by DFT calculation). Later, they investigated self-trapping of hydrogen and helium implanted into tungsten, and indicated that self-trapping plays no (or slight) roles in hydrogen bubbles formation [22]. In a recent first-principles study, Pan et al. investigated hydrogen absorption on tungsten–carbon nanotubes using DFT. They found that tungsten atop site is appropriate for absorption of hydrogen atom and hydrogen molecule, with binding distance of 1.7 and 1.85 Å, respectively [23]. Despite of the above efforts, a comprehensive and reliable understanding on the behaviors of hydrogen atom inside tungsten crystal has not been established yet. In this paper, using first-principles DFT method we have investigated the behavior of interstitial hydrogen atoms inside bcc tungsten host lattice via examining the possible trapping sites, diffusion barrier between neighboring sites and the interaction between two hydrogen impurities. These computational results might shed some light on the trapping and diffusion behavior of hydrogen atoms in the tungsten materials used in ITER and other fusion reactors. 2. Method First-principles calculations were performed using the density functional theory and the planewave pseudopotential technique
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
[24,25], as implemented in the Vienna Ab initio Simulation Package (VASP) [26,27]. The generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) [28] functional was used for describing exchange–correlation interaction and ultrasoft pseudopotentials [29] was employed for ion–electron interaction. The simulation supercell was built from a 5 5 5 bcc lattice, which contains 250 tungsten atoms. The Brillouin zone was sampled by the C-point. The cutoff energy for the planewave basis was chosen as 225 eV. We found that increasing the cutoff energy and k point grid only leads to little changes on the computational results. All atoms were adequately relaxed until the force on each atom is less than 0.01 eV/Å. Upon optimization, the dimension of cubic supercell of tungsten is 15.863 Å. Rigid lattice approximation was adopted to compute the site-dependent binding energy, diffusion barrier and the H–H interaction energy when we place one or two hydrogen atoms inside the supercell. Using the present computational scheme, the lattice parameter, cohesive energy and bulk modulus of the bcc crystal of tungsten is 3.168 Å, 5.95 eV/atom and 318.2 GPa, respectively, in good agreement with experimental values of 3.17 Å, 8.9 eV/atom and 323.2 GPa [30], respectively. The theoretical bond length, binding energy and vibrational frequency for an individual H2 molecule are 0.753 Å, 2.26 eV/atom and 4418 cm1, according well with the experimental value of 0.741Å, 2.24 eV/atom and 4401 cm1 [31], respectively. The above benchmark results clearly demonstrate the present computational scheme is appropriate for the systems we studied.
3. Results and discussion 3.1. Individual hydrogen atom inside tungsten Initially, we have considered eight kinds of possible sites for a single hydrogen atom trapping interstitially in the bcc lattice of tungsten. As shown in Fig. 1, site a locates at the center of octahedron, surrounded by six W atoms (No. 1–6); site b locates at the center of tetrahedron, surrounded by four W atoms (No. 1–4); site c locates at the center of another tetrahedron formed by three W atoms (No. 1, 3 and 6) and site a; site d locates at the one-fourth of edge between two W atoms (No. 1 and 5); site e locates at the middle of another edge between two W atoms (No. 1 and 4); site f locates at the center of the edge by the No. 1 W atom and site a; site g locates at the center of triangle formed by three W atoms
3171
(No. 1, 5 and 6); site h locates at the center of another triangle formed by two W atoms (No. 1 and 6) and site a. Among the eight sites considered, hydrogen atom can stay stably within only three of them after relaxation. As listed in Table 1, they are defined as tetrahedral site, diagonal site and octahedral site, respectively, corresponding to the sites b, d and a in Fig. 1, respectively. We compared the relative stability of these three sites by computing the binding energy for the trapped hydrogen atom. The binding energy EH of an individual hydrogen atom in host lattice is defined as:
EH ¼ EWH EW EHatom ;
ð1Þ
where EWH and EW are the total energy for the supercell of tungsten host lattice with and without the hydrogen atom, respectively. EHatom is the total energy of an isolated hydrogen atom. Here, negative binding energy indicates exothermic reaction, while positive denotes endothermic case. The theoretical results of binding energies are listed in Table 1. From our calculations, we found that the tetrahedral site with the lowest binding energy of 0.955 eV is the most favorable site for interstitial hydrogen atom inside tungsten lattice, in accordance with previous computational [16] and experimental [32– 35] results. The nearest W–H distances between hydrogen atom and surrounding tungsten lattice are 1.857 Å, similar to the DFT results reported by Pan et al. [23]. The second energetically preferred site is the diagonal site, with energy of 0.332 eV higher than the tetrahedral site. Compared with the tetrahedral and diagonal sites, the octahedral site with binding energy of 0.169 eV is still exothermic for incorporation of hydrogen atom, but it is much less favorable. Mulliken population analysis show that certain amount of electrons will transfer from tungsten to hydrogen. At the tetrahedral site, hydrogen atom will accept 0.28 electrons from the surrounding tungsten atoms; while the amount of charge transfer is insensitive to the interstitial site, i.e. 0.27 e at the octahedral site, and 0.25 e at the diagonal site. The negative charge on hydrogen atom inside tungsten lattice implies Coulomb repulsion between two interstitial hydrogen atoms, which will be discussed below. 3.2. Diffusion of hydrogen in tungsten To investigate the diffusion behaviors of hydrogen, we also computed the diffusion barrier for hydrogen atom hopping from one interstitial site to another inside the tungsten lattice.
Fig. 1. Eight kinds of trapping sites for hydrogen atom in the bcc tungsten lattice, with three orientations.
3172
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
Table 1 The final configurations and binding energies of four kinds of stable trapping sites. Number in parentheses means the number of the same value of distance. Site
Tetrahedral
Diagonal
Octahedral
0.955 0.28 1.857(4), 2.863(2)
0.623 0.25 1.674(2), 1.708, 2.307(2), 2.780
0.169 0.27 1.834(2), 2.199(4)
Final configuration
Binding energy (eV) Charge on H (e) W–H distances (Å)
According to the above theoretical results on the binding energy, hydrogen atom tends to occupy the tetrahedral sites. Hence, we concentrate on the diffusion between two tetrahedral sites. As shown in Fig. 2, we considered two diffusion paths: (a) path between two nearest tetrahedral sites (with distance of 1.122 Å) passing through one diagonal site in the middle; (b) path between two second nearest tetrahedral sites (with distance of 1.586 Å) passing through one octahedral site in the middle. Within the rigid lattice approximation, the binding energy curve for the hydrogen atom moving along these two paths are computed and shown in Fig. 2. It can be seen that the saddle points locate at the middle of both diffusion paths, corresponding to the diagonal and octahedral sites, respectively. The activation barriers defined as the energy difference between initial site and the saddle point are obtained as 0.332 eV for path (a) and 0.786 eV for path (b), respectively, indicating that hydrogen atom prefer to diffusion along path (a) with lower energy barrier and shorter diffusion distance. The present diffusion activation barrier of 0.332 eV for hydrogen diffusion in tungsten agrees well with the previously reported value of 0.39 eV in Frauenfelder’s experiments [36]. 3.3. H–H interaction To further explore the behavior of hydrogen atoms inside tungsten lattice, we have investigated the interaction between two individual hydrogen atoms. The large supercell dimension of 15.863 Å is helpful to eliminate the interaction between the hydrogen atoms and their periodic images. Within the same supercell, two hydrogen atoms were placed into two tetrahedral sites separated by different distances (from 1.12 to 3.55 Å). Single-point energy calculations were performed for each configuration to obtain the interaction energy EH–H between these two interstitial H atoms, which is defined as:
EHH
1 ¼ ½EWH2 EW EHatom ; 2
ð2Þ
where EWH2 and EW are the total energy of the host lattice with and without two hydrogen atoms, respectively. EH-atom is the total energy of an isolated H atom, EH is the binding energy for an individual hydrogen atom in the tetrahedral site of tungsten lattice, which has been obtain above. By such definition, positive interaction energy means repulsion between the two hydrogen atoms, while negative
energy indicates attraction. The computed H–H interaction energies are plotted as function of H–H distance in Fig. 3. Different from previous theoretical calculations [16], which suggested an attractive minimum for H–H interaction (at about 2.2 Å) inside the tungsten lattice, here we found that the H–H interaction is repulsive within all distance range. Generally speaking, there is strong repulsive interaction between two hydrogen atoms when they stay close to each other, since that each H atom in the tungsten lattice carries certain amount of negative charge. This repulsion interaction reduces rapidly when H–H distance increases. As the H–H distance exceeds 2.0 Å and increases further, the H–H interaction energy drops to nearly zero (0.005 eV). In other words, there is a so-called ‘‘blocking radius” for the H–H repulsive interaction [37]. The current ‘‘blocking radius” of 2.0 Å for tungsten has a good coherence with the previously reported results for system of 27 648 W atoms using empirical MD simulations [16,38]. Due to the strong repulsion between hydrogen atoms, inside the host lattice two hydrogen atoms will be separated with distance of 2.0 Å at least, which is much longer than the covalent H–H bond length in hydrogen molecule. Therefore, within a perfect bulk tungsten crystal without structural defects and surface/interface, it is difficult for the incorporated hydrogen atoms to accumulate and to form hydrogen bubble. We suspect that the hydrogen accumulation and formation of hydrogen bubble are associated with the structural defects of the tungsten crystal, for example, vacancies or voids. The theoretical study along this direction is still under way. Interestingly, we found that the H–H repulsive interaction in Fig. 3 fit well to a screened Coulomb potential [30],
VðqÞ ¼
q2 4pe0 r
eks r ;
ð3Þ
where q corresponds to an effective charge on each hydrogen atom, and ks is the so-called ‘‘screening constant”. Using the computed data for H–H interaction in Fig. 3, the fitted coefficients are q = 0.45 e and ks = 1.97 Å1. In the framework of free-electron gas model [30], we may also estimate the screening constant ks in tungsten:
K 2s ¼
e2
m
e0 p2 h2
ð3p2 nÞ1=3 :
ð4Þ
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
3173
Fig. 3. Interaction energy between two interstitial hydrogen atoms as function of H–H distance. Open stars are the theoretical data from DFT calculations; solid line is the fitted curve using the screening Coulomb potential by Eq. (3).
Considering that all 5d and 6s electrons in tungsten contribute to the electron density n, we obtained a screening constant of ks = 2.32 Å1, slightly higher than the fitted value of ks. This implies that in additional to the 6s electrons, only part of 5d valence electrons is responsible for the screening effect. 4. Conclusion We have carried out first-principles calculations to investigate the possible trapping sites, diffusion barrier and interatomic interaction of interstitial hydrogen atom inside the bcc tungsten crystal. We found that single hydrogen atom prefers to the tetrahedral interstitial site with binding energy of 0.955 eV and W–H distance of 1.857 Å, while the hydrogen atom can also stay in the other two absorption sites, i.e. diagonal and octahedral sites with less favorable binding energy. The energy barrier for hydrogen diffusion along the ‘‘tetrahedral ? diagonal ? tetrahedral” path is 0.332 eV, which is much lower than the barrier of 0.786 eV for the ‘‘tetrahedral ? octahedral ? tetrahedral” path. The intermediate height of energy barrier implies a reasonable diffusion rate of hydrogen inside tungsten at experimental conditions. The interaction between two hydrogen atoms within tungsten lattice can be considered as the Coulomb repulsion screened by the metal lattice. Within tungsten host lattice, the H–H blocking distance is about 2.0 Å, above which there is only little repulsion between the two hydrogen atoms (the interaction energy is 0.005 eV). The strong H–H repulsion within the blocking distance suggests that the hydrogen atoms trapped in a perfect tungsten crystal are difficult to accumulate with each other to form hydrogen blister. Acknowledgements This work was supported by National Basic Research Program of China (2008CB717801), the National Natural Science Foundation of China (10774019) and the Program for New Century Excellent Talents in University of China (NCET06-0281). References Fig. 2. Diffusion energy barriers of hydrogen atom in tungsten via different diffusion paths: (a) tetrahedral ? diagonal ? tetrahedral and (b) tetrahedral ? octahedral ? tetrahedral. The arrows show the corresponding diffusion paths in sketch maps.
[1] R. Causey, K. Wilson, T. Venhaus, W.R. Wampler, J. Nucl. Mater. 269 (1999) 467. [2] G. Federici, J.N. Brooks, D.P. Coster, G. Janeschitz, A. Kukuskhin, A. Loarte, H.D. Pacher, J. Stober, C.H. Wu, J. Nucl. Mater. 290 (2001) 260.
3174
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
[3] G. Federici, R.A. Anderl, et al., J. Nucl. Mater. 269 (1999) 14. [4] Q. Xu, N. Yoshida, T. Yoshiie, J. Nucl. Mater. 367 (2007) 806. [5] S.K. Das, M. Kaminsky, Radiation Effects on Solid Surfaces, American Chemical Society, 1976. [6] K. Tokunaga, T. Fujiwara, K. Ezato, S. Suzuki, M. Akiba, N. Yoshida, J. Nucl. Mater. 367 (2007) 812. [7] H.T. Lee, A.A. Haasz, J.W. Davis, R.G. Macaulay-Newcombe, D.G. Whyte, G.M. Wright, J. Nucl. Mater. 363 (2007) 898. [8] H.T. Lee, A.A. Haasz, J.W. Davis, R.G. Macaulay-Newcombe, J. Nucl. Mater. 360 (2007) 196. [9] K. Katayama, K. Imaoka, T. Okamura, M. Nishikawa, Fus. Eng. Des. 82 (2007) 1645. [10] A. Debelle, M.F. Barthe, T. Sauvage, R. Belamhawal, A. Chelgoum, P. Desgardin, H. Labrim, J. Nucl. Mater. 362 (2007) 181. [11] M.Y. Ye, Plasma Sci. Technol. 7 (2005) 2828. [12] H. Miura, Y. Kuroki, K. Yasui, M. Takata, T. Akahane, Thin Solid Films 516 (2008) 503. [13] P. Traskelin, N. Juslin, P. Erhart, K. Nordlund, Phys. Rev. B 75 (2007). [14] W.G. Zheng, A. Gallagher, Surf. Sci. 600 (2006) 2207. [15] K.O.E. Henriksson, K. Vortler, S. Dreissigacker, K. Nordlund, J. Keinonen, Surf. Sci. 600 (2006) 3167. [16] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Appl. Phys. Lett. 87 (2005) 163113. [17] T. Venhaus, R. Causey, R. Doerner, T. Abeln, J. Nucl. Mater. 290 (2001) 505. [18] T. Horikawa, B. Tsuchiya, K. Morita, J. Nucl. Mater. 269 (1999) 1091.
[19] P. Franzen, C. GarciaRosales, H. Plank, V.K. Alimov, J. Nucl. Mater. 241 (1997) 1082. [20] I.I. Arkhipov, S.L. Kanashenko, V.M. Sharapov, R.K. Zalavutdinov, A.E. Gorodetsky, J. Nucl. Mater. 363 (2007) 1168. [21] M.Y. Ye, H. Kanehara, S. Fukuta, N. Ohno, S. Takamura, J. Nucl. Mater. 313 (2003) 72. [22] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Fus. Sci. Technol. 50 (2006) 43. [23] H. Pan, Y.P. Feng, J.Y. Lin, Appl. Phys. Lett. 90 (2007) 223104. [24] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [25] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [26] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [27] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [28] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [29] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [30] C. Kittle, Introduction to Solid State Physics, 6th Ed., Wiley, New York, 1986. [31] B.G. Johnson, P.M.W. Gill, J.A. Pople, J. Chem. Phys. 98 (1993) 5612. [32] S. Nagata, K. Takahiro, J. Nucl. Mater. 283 (2000) 1038. [33] E. Ligeon, R. Danielou, J. Fontenille, R. Eymery, J. Appl. Phys. 59 (1986) 108. [34] S.T. Picraux, F.L. Vook, J. Nucl. Mater. 53 (1974) 246. [35] S.T. Picraux, F.L. Vook, Phys. Rev. Lett. 33 (1974) 1216. [36] R. Frauenfelder, J. Vac. Sci. Technol. 6 (1969) 388. [37] V.A. Somenkov, A.V. Irodova, J. Less-Common Met. 101 (1984) 481. [38] A.C. Switendick, SAND 78-0250 (1978) 1.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
First-principles study of hydrogen in perfect tungsten crystal Jingcheng Xu, Jijun Zhao * Laboratory of Materials Modification by Laser, Electron, and Ion Beams, School of Physics and Optoelectronic Technology, and College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Available online 16 June 2009 PACS: 61.72.S61.82.Bg Keywords: Hydrogen Tungsten Diffusion Interaction Binding energy
a b s t r a c t Tungsten-based materials are used as the first wall materials in ITER. Hydrogen impurities were introduced via bombarding with the reaction plasma, which are important for the behavior and stability of the tungsten wall. Using the first-principles density functional theory and planewave pseudopotential technique, we have simulated the behaviors of hydrogen atoms inside the perfect tungsten bcc lattice. The binding energies for different interstitial sites were compared to determine the optimal trapping site for the hydrogen atom inside the tungsten lattice. The diffusion barriers for hydrogen atom between nearby trapping sites and the interaction between two interstitial hydrogen atoms were also calculated. The implication of our theoretical results on the hydrogen diffusion and accumulation behavior was discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Because of high melting point, good thermal conductivity as well as low sputtering erosion [1], tungsten is one of the candidate materials for the plasma facing components in the divertor region of the International Thermonuclear Experimental Reactor (ITER) and the future fusion reactors. The divertor in ITER will be subjected to high particle fluxes from not only plasma background ions (H, D and T) but also intrinsic impurities (He, Be and C) [2,3]. Since hydrogen is one of the major plasma background ions, hydrogen irradiation may results in changes of mechanical properties of tungsten materials. It was known that helium irradiation leads to blister formation and subsequent degradation of the mechanical properties of metals [4–10]. Ye reported that the effects of hydrogen plasma bombarding on tungsten are different from helium [11]. Thus, it is important to explore the trapping and blister formation of hydrogen in tungsten via studying the behaviors of hydrogen impurities in tungsten. Previously, a number of experimental and theoretical studies were carried out to understand the interaction between the hydrogen impurity and the tungsten host lattice [7,8,12–23]. For example, Ye’s group presented the experimental results in a divertor plasma simulator NAGDIS-1, and observed formation of hydrogen bubbles on tungsten surface [21]. Arkhipov et al. also found that hydrogen could form blister at tungsten surface [20]. Henriksson and co-workers compared the differences in formation of hydrogen * Corresponding author. Tel.: +86 411 84709748; fax: +86 41 84706100. E-mail address: [email protected] (J. Zhao). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.072
and helium blister in tungsten using a combined simulation with density functional theory (DFT), empirical molecular dynamics (MD) and kinetic Monte Carlo [16]. Within the bcc tungsten lattice, they found that hydrogen atoms repulse with each other at H–H short distance; while the interaction of hydrogen atoms is attractive at the distance of 2.2 Å with tiny binding energy is (less than 0.1 eV by DFT calculation). Later, they investigated self-trapping of hydrogen and helium implanted into tungsten, and indicated that self-trapping plays no (or slight) roles in hydrogen bubbles formation [22]. In a recent first-principles study, Pan et al. investigated hydrogen absorption on tungsten–carbon nanotubes using DFT. They found that tungsten atop site is appropriate for absorption of hydrogen atom and hydrogen molecule, with binding distance of 1.7 and 1.85 Å, respectively [23]. Despite of the above efforts, a comprehensive and reliable understanding on the behaviors of hydrogen atom inside tungsten crystal has not been established yet. In this paper, using first-principles DFT method we have investigated the behavior of interstitial hydrogen atoms inside bcc tungsten host lattice via examining the possible trapping sites, diffusion barrier between neighboring sites and the interaction between two hydrogen impurities. These computational results might shed some light on the trapping and diffusion behavior of hydrogen atoms in the tungsten materials used in ITER and other fusion reactors. 2. Method First-principles calculations were performed using the density functional theory and the planewave pseudopotential technique
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
[24,25], as implemented in the Vienna Ab initio Simulation Package (VASP) [26,27]. The generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) [28] functional was used for describing exchange–correlation interaction and ultrasoft pseudopotentials [29] was employed for ion–electron interaction. The simulation supercell was built from a 5 5 5 bcc lattice, which contains 250 tungsten atoms. The Brillouin zone was sampled by the C-point. The cutoff energy for the planewave basis was chosen as 225 eV. We found that increasing the cutoff energy and k point grid only leads to little changes on the computational results. All atoms were adequately relaxed until the force on each atom is less than 0.01 eV/Å. Upon optimization, the dimension of cubic supercell of tungsten is 15.863 Å. Rigid lattice approximation was adopted to compute the site-dependent binding energy, diffusion barrier and the H–H interaction energy when we place one or two hydrogen atoms inside the supercell. Using the present computational scheme, the lattice parameter, cohesive energy and bulk modulus of the bcc crystal of tungsten is 3.168 Å, 5.95 eV/atom and 318.2 GPa, respectively, in good agreement with experimental values of 3.17 Å, 8.9 eV/atom and 323.2 GPa [30], respectively. The theoretical bond length, binding energy and vibrational frequency for an individual H2 molecule are 0.753 Å, 2.26 eV/atom and 4418 cm1, according well with the experimental value of 0.741Å, 2.24 eV/atom and 4401 cm1 [31], respectively. The above benchmark results clearly demonstrate the present computational scheme is appropriate for the systems we studied.
3. Results and discussion 3.1. Individual hydrogen atom inside tungsten Initially, we have considered eight kinds of possible sites for a single hydrogen atom trapping interstitially in the bcc lattice of tungsten. As shown in Fig. 1, site a locates at the center of octahedron, surrounded by six W atoms (No. 1–6); site b locates at the center of tetrahedron, surrounded by four W atoms (No. 1–4); site c locates at the center of another tetrahedron formed by three W atoms (No. 1, 3 and 6) and site a; site d locates at the one-fourth of edge between two W atoms (No. 1 and 5); site e locates at the middle of another edge between two W atoms (No. 1 and 4); site f locates at the center of the edge by the No. 1 W atom and site a; site g locates at the center of triangle formed by three W atoms
3171
(No. 1, 5 and 6); site h locates at the center of another triangle formed by two W atoms (No. 1 and 6) and site a. Among the eight sites considered, hydrogen atom can stay stably within only three of them after relaxation. As listed in Table 1, they are defined as tetrahedral site, diagonal site and octahedral site, respectively, corresponding to the sites b, d and a in Fig. 1, respectively. We compared the relative stability of these three sites by computing the binding energy for the trapped hydrogen atom. The binding energy EH of an individual hydrogen atom in host lattice is defined as:
EH ¼ EWH EW EHatom ;
ð1Þ
where EWH and EW are the total energy for the supercell of tungsten host lattice with and without the hydrogen atom, respectively. EHatom is the total energy of an isolated hydrogen atom. Here, negative binding energy indicates exothermic reaction, while positive denotes endothermic case. The theoretical results of binding energies are listed in Table 1. From our calculations, we found that the tetrahedral site with the lowest binding energy of 0.955 eV is the most favorable site for interstitial hydrogen atom inside tungsten lattice, in accordance with previous computational [16] and experimental [32– 35] results. The nearest W–H distances between hydrogen atom and surrounding tungsten lattice are 1.857 Å, similar to the DFT results reported by Pan et al. [23]. The second energetically preferred site is the diagonal site, with energy of 0.332 eV higher than the tetrahedral site. Compared with the tetrahedral and diagonal sites, the octahedral site with binding energy of 0.169 eV is still exothermic for incorporation of hydrogen atom, but it is much less favorable. Mulliken population analysis show that certain amount of electrons will transfer from tungsten to hydrogen. At the tetrahedral site, hydrogen atom will accept 0.28 electrons from the surrounding tungsten atoms; while the amount of charge transfer is insensitive to the interstitial site, i.e. 0.27 e at the octahedral site, and 0.25 e at the diagonal site. The negative charge on hydrogen atom inside tungsten lattice implies Coulomb repulsion between two interstitial hydrogen atoms, which will be discussed below. 3.2. Diffusion of hydrogen in tungsten To investigate the diffusion behaviors of hydrogen, we also computed the diffusion barrier for hydrogen atom hopping from one interstitial site to another inside the tungsten lattice.
Fig. 1. Eight kinds of trapping sites for hydrogen atom in the bcc tungsten lattice, with three orientations.
3172
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
Table 1 The final configurations and binding energies of four kinds of stable trapping sites. Number in parentheses means the number of the same value of distance. Site
Tetrahedral
Diagonal
Octahedral
0.955 0.28 1.857(4), 2.863(2)
0.623 0.25 1.674(2), 1.708, 2.307(2), 2.780
0.169 0.27 1.834(2), 2.199(4)
Final configuration
Binding energy (eV) Charge on H (e) W–H distances (Å)
According to the above theoretical results on the binding energy, hydrogen atom tends to occupy the tetrahedral sites. Hence, we concentrate on the diffusion between two tetrahedral sites. As shown in Fig. 2, we considered two diffusion paths: (a) path between two nearest tetrahedral sites (with distance of 1.122 Å) passing through one diagonal site in the middle; (b) path between two second nearest tetrahedral sites (with distance of 1.586 Å) passing through one octahedral site in the middle. Within the rigid lattice approximation, the binding energy curve for the hydrogen atom moving along these two paths are computed and shown in Fig. 2. It can be seen that the saddle points locate at the middle of both diffusion paths, corresponding to the diagonal and octahedral sites, respectively. The activation barriers defined as the energy difference between initial site and the saddle point are obtained as 0.332 eV for path (a) and 0.786 eV for path (b), respectively, indicating that hydrogen atom prefer to diffusion along path (a) with lower energy barrier and shorter diffusion distance. The present diffusion activation barrier of 0.332 eV for hydrogen diffusion in tungsten agrees well with the previously reported value of 0.39 eV in Frauenfelder’s experiments [36]. 3.3. H–H interaction To further explore the behavior of hydrogen atoms inside tungsten lattice, we have investigated the interaction between two individual hydrogen atoms. The large supercell dimension of 15.863 Å is helpful to eliminate the interaction between the hydrogen atoms and their periodic images. Within the same supercell, two hydrogen atoms were placed into two tetrahedral sites separated by different distances (from 1.12 to 3.55 Å). Single-point energy calculations were performed for each configuration to obtain the interaction energy EH–H between these two interstitial H atoms, which is defined as:
EHH
1 ¼ ½EWH2 EW EHatom ; 2
ð2Þ
where EWH2 and EW are the total energy of the host lattice with and without two hydrogen atoms, respectively. EH-atom is the total energy of an isolated H atom, EH is the binding energy for an individual hydrogen atom in the tetrahedral site of tungsten lattice, which has been obtain above. By such definition, positive interaction energy means repulsion between the two hydrogen atoms, while negative
energy indicates attraction. The computed H–H interaction energies are plotted as function of H–H distance in Fig. 3. Different from previous theoretical calculations [16], which suggested an attractive minimum for H–H interaction (at about 2.2 Å) inside the tungsten lattice, here we found that the H–H interaction is repulsive within all distance range. Generally speaking, there is strong repulsive interaction between two hydrogen atoms when they stay close to each other, since that each H atom in the tungsten lattice carries certain amount of negative charge. This repulsion interaction reduces rapidly when H–H distance increases. As the H–H distance exceeds 2.0 Å and increases further, the H–H interaction energy drops to nearly zero (0.005 eV). In other words, there is a so-called ‘‘blocking radius” for the H–H repulsive interaction [37]. The current ‘‘blocking radius” of 2.0 Å for tungsten has a good coherence with the previously reported results for system of 27 648 W atoms using empirical MD simulations [16,38]. Due to the strong repulsion between hydrogen atoms, inside the host lattice two hydrogen atoms will be separated with distance of 2.0 Å at least, which is much longer than the covalent H–H bond length in hydrogen molecule. Therefore, within a perfect bulk tungsten crystal without structural defects and surface/interface, it is difficult for the incorporated hydrogen atoms to accumulate and to form hydrogen bubble. We suspect that the hydrogen accumulation and formation of hydrogen bubble are associated with the structural defects of the tungsten crystal, for example, vacancies or voids. The theoretical study along this direction is still under way. Interestingly, we found that the H–H repulsive interaction in Fig. 3 fit well to a screened Coulomb potential [30],
VðqÞ ¼
q2 4pe0 r
eks r ;
ð3Þ
where q corresponds to an effective charge on each hydrogen atom, and ks is the so-called ‘‘screening constant”. Using the computed data for H–H interaction in Fig. 3, the fitted coefficients are q = 0.45 e and ks = 1.97 Å1. In the framework of free-electron gas model [30], we may also estimate the screening constant ks in tungsten:
K 2s ¼
e2
m
e0 p2 h2
ð3p2 nÞ1=3 :
ð4Þ
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
3173
Fig. 3. Interaction energy between two interstitial hydrogen atoms as function of H–H distance. Open stars are the theoretical data from DFT calculations; solid line is the fitted curve using the screening Coulomb potential by Eq. (3).
Considering that all 5d and 6s electrons in tungsten contribute to the electron density n, we obtained a screening constant of ks = 2.32 Å1, slightly higher than the fitted value of ks. This implies that in additional to the 6s electrons, only part of 5d valence electrons is responsible for the screening effect. 4. Conclusion We have carried out first-principles calculations to investigate the possible trapping sites, diffusion barrier and interatomic interaction of interstitial hydrogen atom inside the bcc tungsten crystal. We found that single hydrogen atom prefers to the tetrahedral interstitial site with binding energy of 0.955 eV and W–H distance of 1.857 Å, while the hydrogen atom can also stay in the other two absorption sites, i.e. diagonal and octahedral sites with less favorable binding energy. The energy barrier for hydrogen diffusion along the ‘‘tetrahedral ? diagonal ? tetrahedral” path is 0.332 eV, which is much lower than the barrier of 0.786 eV for the ‘‘tetrahedral ? octahedral ? tetrahedral” path. The intermediate height of energy barrier implies a reasonable diffusion rate of hydrogen inside tungsten at experimental conditions. The interaction between two hydrogen atoms within tungsten lattice can be considered as the Coulomb repulsion screened by the metal lattice. Within tungsten host lattice, the H–H blocking distance is about 2.0 Å, above which there is only little repulsion between the two hydrogen atoms (the interaction energy is 0.005 eV). The strong H–H repulsion within the blocking distance suggests that the hydrogen atoms trapped in a perfect tungsten crystal are difficult to accumulate with each other to form hydrogen blister. Acknowledgements This work was supported by National Basic Research Program of China (2008CB717801), the National Natural Science Foundation of China (10774019) and the Program for New Century Excellent Talents in University of China (NCET06-0281). References Fig. 2. Diffusion energy barriers of hydrogen atom in tungsten via different diffusion paths: (a) tetrahedral ? diagonal ? tetrahedral and (b) tetrahedral ? octahedral ? tetrahedral. The arrows show the corresponding diffusion paths in sketch maps.
[1] R. Causey, K. Wilson, T. Venhaus, W.R. Wampler, J. Nucl. Mater. 269 (1999) 467. [2] G. Federici, J.N. Brooks, D.P. Coster, G. Janeschitz, A. Kukuskhin, A. Loarte, H.D. Pacher, J. Stober, C.H. Wu, J. Nucl. Mater. 290 (2001) 260.
3174
J. Xu, J. Zhao / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3170–3174
[3] G. Federici, R.A. Anderl, et al., J. Nucl. Mater. 269 (1999) 14. [4] Q. Xu, N. Yoshida, T. Yoshiie, J. Nucl. Mater. 367 (2007) 806. [5] S.K. Das, M. Kaminsky, Radiation Effects on Solid Surfaces, American Chemical Society, 1976. [6] K. Tokunaga, T. Fujiwara, K. Ezato, S. Suzuki, M. Akiba, N. Yoshida, J. Nucl. Mater. 367 (2007) 812. [7] H.T. Lee, A.A. Haasz, J.W. Davis, R.G. Macaulay-Newcombe, D.G. Whyte, G.M. Wright, J. Nucl. Mater. 363 (2007) 898. [8] H.T. Lee, A.A. Haasz, J.W. Davis, R.G. Macaulay-Newcombe, J. Nucl. Mater. 360 (2007) 196. [9] K. Katayama, K. Imaoka, T. Okamura, M. Nishikawa, Fus. Eng. Des. 82 (2007) 1645. [10] A. Debelle, M.F. Barthe, T. Sauvage, R. Belamhawal, A. Chelgoum, P. Desgardin, H. Labrim, J. Nucl. Mater. 362 (2007) 181. [11] M.Y. Ye, Plasma Sci. Technol. 7 (2005) 2828. [12] H. Miura, Y. Kuroki, K. Yasui, M. Takata, T. Akahane, Thin Solid Films 516 (2008) 503. [13] P. Traskelin, N. Juslin, P. Erhart, K. Nordlund, Phys. Rev. B 75 (2007). [14] W.G. Zheng, A. Gallagher, Surf. Sci. 600 (2006) 2207. [15] K.O.E. Henriksson, K. Vortler, S. Dreissigacker, K. Nordlund, J. Keinonen, Surf. Sci. 600 (2006) 3167. [16] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Appl. Phys. Lett. 87 (2005) 163113. [17] T. Venhaus, R. Causey, R. Doerner, T. Abeln, J. Nucl. Mater. 290 (2001) 505. [18] T. Horikawa, B. Tsuchiya, K. Morita, J. Nucl. Mater. 269 (1999) 1091.
[19] P. Franzen, C. GarciaRosales, H. Plank, V.K. Alimov, J. Nucl. Mater. 241 (1997) 1082. [20] I.I. Arkhipov, S.L. Kanashenko, V.M. Sharapov, R.K. Zalavutdinov, A.E. Gorodetsky, J. Nucl. Mater. 363 (2007) 1168. [21] M.Y. Ye, H. Kanehara, S. Fukuta, N. Ohno, S. Takamura, J. Nucl. Mater. 313 (2003) 72. [22] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Fus. Sci. Technol. 50 (2006) 43. [23] H. Pan, Y.P. Feng, J.Y. Lin, Appl. Phys. Lett. 90 (2007) 223104. [24] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [25] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [26] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [27] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [28] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [29] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [30] C. Kittle, Introduction to Solid State Physics, 6th Ed., Wiley, New York, 1986. [31] B.G. Johnson, P.M.W. Gill, J.A. Pople, J. Chem. Phys. 98 (1993) 5612. [32] S. Nagata, K. Takahiro, J. Nucl. Mater. 283 (2000) 1038. [33] E. Ligeon, R. Danielou, J. Fontenille, R. Eymery, J. Appl. Phys. 59 (1986) 108. [34] S.T. Picraux, F.L. Vook, J. Nucl. Mater. 53 (1974) 246. [35] S.T. Picraux, F.L. Vook, Phys. Rev. Lett. 33 (1974) 1216. [36] R. Frauenfelder, J. Vac. Sci. Technol. 6 (1969) 388. [37] V.A. Somenkov, A.V. Irodova, J. Less-Common Met. 101 (1984) 481. [38] A.C. Switendick, SAND 78-0250 (1978) 1.