SCIENCE CHINA Physics, Mechanics & Astronomy
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1 SCIENCE CHINA Physics, Mechanics & Astronomy Article April 2012 Vol.55 No.4: doi: /s Stability and diffusion properties of self-interstitial atoms in tungsten: a first-principles investigation CHEN Long 1, LIU YueLin 2, ZHOU HongBo 1, JIN Shuo 1, ZHANG Ying 1 & LU GuangHong 1* 1 Department of Physics, Beihang University, Beijing , China; 2 Department of Physics, Yantai University, Yantai , China Received September 28, 2011; accepted December 13, 2011; published online March 8, 2012 Employing a first-principles method based on the density function theory, we systematically investigate the structures, stability and diffusion of self-interstitial atoms (SIAs) in tungsten (W). The <111> dumbbell is shown to be the most stable SIA defect configuration with the formation energy of ~9.43 ev. The on-site rotation modes can be described by a quite soft floating mechanism and a down-hill drift diffusion process from <110> dumbbell to <111> dumbbell and from <001> dumbbell to <110> dumbbell, respectively. Among different SIA configurations jumping to near neighboring site, the <111> dumbbell is more preferable to migrate directly to first-nearest-neighboring site with a much lower energy barrier of ev. These results provide a useful reference for W as a candidate plasma facing material in fusion Tokamak. tungsten, self-interstitial, stability, diffusion, first-principles PACS number(s): Bg, Ji, x, Dk Citation: Chen L, Liu Y L, Zhou H B, et al. Stability and diffusion properties of self-interstitial atoms in tungsten: a first-principles investigation. Sci China-Phys Mech Astron, 2012, 55: , doi: /s In metals, intrinsic defect vacancies generally play a crucial role in the nucleation, evolution, and kinetics of larger defects, which eventually govern the macroscopic deformation and failure mechanisms observed in a variety of metals [1 3]. As another typical defect in metals, self-interstitial atoms (SIAs) are relatively rarely studied in comparison with vacancies which have been considered to be responsible for the hydrogen blistering in tungsten (W) [4 6]. However, SIAs are easily produced as the same order of magnitude with vacancies under high-energy ions irradiation, especially under 14 MeV high-energy neutrons irradiation in a fusion reactor. On the one hand, the displacement cascade occurs during irradiation leading to the SIAs formation. On the other hand, the long time evolution of point defects such as vacancies and SIAs into voids and clusters can break the structure of materials so as to degrade their *Corresponding author ( lgh@buaa.edu.cn) lifetime. Thus, a better understanding of SIAs becomes a critical issue in the research and development of materials in a fusion reactor. Many efforts have been devoted to understanding the properties of SIAs in metals including energetics and kinetics. In the early experimental studies, the resistivity recovery curves showed that the SIAs are mobile at some temperature such as below 6 K for Nb and Ta [7], at 35 K for Mo [8], at 27 K for W [9], 120 K for -Fe [10], and 40 K for Cr [11]. Further, Atomic simulations in terms of empirical potentials have performed widely which predicts the behaviors of SIAs in V, -Fe and Fe-Cu alloys [12 15]. Recent first-principles calculations [16 20] have shown differences of, e.g., the formation energy of SIAs in α-fe, as compared with empirical potentials [17]. The fundamental behaviors of SIAs in W system is a research focus at present because W and W-alloys are considered as the most promising plasma facing materials (PFMs) Science China Press and Springer-Verlag Berlin Heidelberg 2012 phys.scichina.com
2 Chen L, et al. Sci China-Phys Mech Astron April (2012) Vol. 55 No due to their low sputtering erosion and good thermal properties such as high thermal conductivity and high melting temperature in nuclear fusion reaction. Some properties of SIAs in W have been investigated in the past few years. Ackland et al. [21] predicted the formation energies of point defects of SIAs in W based on semi-empirical potentials using atomistic simulation method, and found that the <111> crowd ion is the most stable configuration, while first-principles calculations [18,19] show that the <111> dumbbell is the lowest-energy configuration. In spite of many years of studies, these aspects mentioned above for SIAs in W are only for the energetic stabilities for different point defects. We see that there exist the discrepancies between semi-empirical potentials and first-principles calculations. Moreover, the mobility of the SIAs in W has not been systematically investigated using first-principles method. Up to now, the only work that the energy pathways including between <111> and <110> dumbbells and between <111> and <001> dumbbells have been studied by Nguyen-Manh et al. [18]. Here, we investigate the stability and diffusion of different SIAs in W using the first-principles method. The results will be quite helpful in not only understanding the interaction of SIAs with W, but also in providing a useful reference to the application of W as a PFM in a fusion reactor. 1 Computation methods Our first-principles calculations are performed using the VASP code [22,23] based on the density functional theory. We use the generalized gradient approximation of Perdew and Wang [24] and projector augmented wave potentials [25]. A plane wave energy cutoff of 350 ev is found to be sufficient to converge the total energy and geometry of W super-cell. The Brillouin zone is sampled in terms of the Monkhorst-Pack scheme [26] during the geometry optimization for bcc W super-cell. The calculated equilibrium lattice constant is 3.17 Å, which is in good agreement with the corresponding experimental value of 3.16 Å [27]. This demonstrates the accuracy of the current PAW method. As to the choice of k-points, we find that the 128 atoms super-cells with k-points are very well converged at the range of error. Thus, we choose the super-cells of 128 atoms with k-points for the present studies. During the calculations, the super-cell size, shape, and atomic positions are relaxed to equilibrium, and energy minimization is converged until the forces on all the atoms are less than 3 10 ev Å 1. The formation energy of SIA is calculated from E E ( n 1) E, (1) f SIA ( n 1)W W where E (n+1)w and E W are the energies of the super-cell with (n+1) W atoms with SIAs and single bulk W atom, respectively. 2 Results and discussion 2.1 The stability of SIAs in W We examined the stabilities of SIAs including the tetrahedral interstitial, the octahedral interstitial, <001>, <110>, and <111> dumbbells in W. Figure 1 plots the formation energies of these defects. In order to make a comparison with the present calculations, we also show the results from the previous semi-empirical potential (Finnis-Sinclair potential) [21] and first-principles calculations using PLATO (Package of linear combination of atomic type orbitals) [18] and VASP code [20] with the same super-cell, respectively. It can be seen that the formation energies of these SIAs exhibit the same change trend between the present calculations and the previous results, although the absolute values are a bit different. The semi-empirical potential method underestimates the formation energies in comparison with firstprinciples investigations. For the present calculations, the <111> dumbbell configuration in W is found to be the most stable configuration with the formation energy of 9.43 ev, followed by the <110> dumbbell, the tetrahedral interstitial, the <001> dumbbell, and the octahedral interstitial configurations, which are 9.82, 11.03, 11.58, and ev higher in energy, respectively. Nguyen-Manh et al. [18] also calculated the formation energy of the <111> crowd ion configuration, and found that the formation energy of the <111> crowd ion is slightly higher than the <111> dumbbell with the difference value of ~0.01 ev. We further found that the present results exhibit the same trend for the stabilities of SIAs in W compared with other nonmagnetic bcc metals such as V, Nb, Ta, Cr, and Mo [18]. However, the lowest-energy configuration is shown to contrast with the <110> dumbbell in ferro-magnetic metal such as -Fe [17]. The possible reason has been explained that the local anti- Figure 1 (Color online) The formation energies of SIAs in W in comparison with the previous semi-empirical potential (Finnis-Sinclair potential) [21] and the first-principles calculations using PLATO [18] and VASP code [20] with the same super-cell, respectively.
3 616 Chen L, et al. Sci China-Phys Mech Astron April (2012) Vol. 55 No. 4 ferro-magnetism might contribute to the energy stability. Figure 2 shows the charge density distribution maps for the three relaxed SIA structures, <001>, <110>, and <111> dumbbells in W, respectively. For the <001> and <110> dumbbell configurations, the two atoms share a lattice site with each other along the <001> and <110> directions, respectively. We see that there exists higher electronic density between the two atoms for the two respective dumbbell configurations, suggesting the significant directional-bonding character. The underlying physical mechanism can be understood as follows. When an additional W atom is embedded into the normal lattice site, the local strain field will be created surrounding the dumbbell resembling a uniaxial strain imposed along the symmetry direction of the dumbbell. For the <111> dumbbell configuration, the two atoms also share the same lattice site. This kind of structure is similar to the crowd ion configuration, where four atoms share three lattice sites. The deformation behaviors can be found to be confined to a single string of atoms running in the <111> direction, as shown in Figure 2. There also exists the directional-bonding character between atoms in the core of the <111> dumbbell due to the local pressing. Figure 2 (Color online) The charge density distribution maps of SIAs for the three relaxed dumbbell structures in W, respectively. 2.2 The mobility of SIAs in W We have studied systematically the mobility of SIAs in W by studying various migration jumps including two situations of rotation and translation. The activation energy barriers are calculated in W using a drag method: the SIA sites relative to the center of mass are constrained to relax in a hyper-plane perpendicular to the vector from the initial to the finial sites [17]. This method has been successfully applicable for the studies of diffusion path of intrinsic defects or impurity atom in metals. Here, the so-called rotation is on-site rotations from <111> dumbbell to <110> dumbbell and further from <110> dumbbell to <001> dumbbell, respectively. While translation shows that different SIA configurations jump to near neighboring sites, as shown in Figure 3. First, the on-site rotation cases are investigated between <111> dumbbell and <110> dumbbell, and between <110> dumbbell and <001> dumbbell, respectively. Figure 4(a) plots that <110> dumbbell jumps to <111> dumbbell via a mediate transition state which can be described by a quite soft floating process. As shown in Figure 4(b), the rotation path between <110> dumbbell and <001> dumbbell indicates that <001> dumbbell is rather unstable with an energy barrier of 1.80 ev translating directly into <110> dumbbell, which suggests a down-hill drift rotation process. This suggests that once <001> dumbbells are formed in W, and they will transform rapidly into <110> dumbbells. Next, we turn our attention to investigating the migration mechanism on <001>, <110>, and <111> dumbbells translating to their respective first-nearest-neighbor (1NN) and Figure 3 (Color online) Schematic illustration of translation jumps of SIAs investigated with their respective activation energy barriers. (a) <001> dumbbell translation to first-nearest-neighbor (1NN). (b) <110> dumbbell translation to 1NN. (c) <111> dumbbell translation to 1NN. (d) <001> dumbbell translation to 2NN. (e) <110> dumbbell translation to 2NN. (f) <111> dumbbell translation to 2NN.
4 Chen L, et al. Sci China-Phys Mech Astron April (2012) Vol. 55 No Figure 4 (b) in W. (Color online) Energy profile on-site rotations from <111> dumbbell to <110> dumbbell (a), and further from <110> dumbbell to <001> dumbbell 2NN, respectively. Figures 3(a) and (d) show that the <001> dumbbell translates to 1NN and 2NN, respectively. The respective activation energy barriers are and ev, which implies that the <001> dumbbell is not preferable to translate directly to 1NN but 2NN. As shown in Figures 3(b) and (e), the <110> dumbbell translates to 1NN and 2NN with their respective activation energy barriers of and ev. In comparison with the <001> dumbbell case, we see that the <110> dumbbell prefers to migrate to 1NN. This translation mechanism for the <110> dumbbell has also been proposed in other bcc metals [17,28]. Concerning the <111> dumbbell translation, it is important to note that it is more preferable to migrate directly to 1NN with a much lower energy barrier of ev than the value of ev of translating to 2NN. 3 Conclusions Using a first-principles pseudo-potential method based on the density functional theory with the generalized gradient approximation, we have investigated the structures, stability and diffusion properties of SIAs in W systematically. The <111> dumbbell is shown to be the most stable SIA configuration with the formation energy of ~9.43 ev. The on-site rotation modes can be described by a quite soft floating mechanism and a down-hill drift diffusion process from <110> dumbbell to <111> dumbbell and from <001> dumbbell to <110> dumbbell, respectively. Among different SIA configurations jumping to near neighboring site, the <111> dumbbell is more preferable to migrate directly to 1NN with a much lower energy barrier of ev, while the <001> dumbbell prefers to translate to 2NN instead of 1NN, and the <110> dumbbell prefers to migrate to 1NN. We believe that these results might provide a useful reference for W as a candidate PFM in fusion Tokamak. This work was supported by the National Natural Science Foundation of China (Grant No ) and the Fundamental Research Funds for the Central Universities. 1 Lu G, Kaxiras E. Can vacancies lubricate dislocation motion in aluminum? Phys Rev Lett, 2002, 89: Lu G, Kaxiras E. Hydrogen embrittlement of aluminum: The crucial role of vacancies. Phys Rev Lett, 2005, 94: Gavini V, Bhattacharya K, Ortiz M. Vacancy clustering and prismatic dislocation loop formation in aluminum. Phys Rev B, 2007, 76: Zhou H B, Liu Y L, Jin S, et al. Investigating behaviours of hydrogen in a tungsten grain boundary by first principles: From dissolution and diffusion to a trapping mechanism. Nucl Fusion, 2010, 50: Zhou H B, Liu Y L, Jin S, et al. Towards suppressing H blistering by investigating the physical origin of the H-He interaction in W. Nucl Fusion, 2010, 50: Liu Y L, Zhang Y, Zhou H B, et al. Vacancy trapping mechanism for hydrogen bubble formation in metal. Phys Rev B, 2009, 79: Ehrhart P, Jung P, Schults H, et al. Atomic Defects in Metals. Landolt-Bornstein New Series. Berlin: Springer-Verlag, Evans J H. Simulations of the effects of 1-d interstitial diffusion on void lattice formation during irradiation. Philos Mag A, 2005, 85: Dausinger F, Schults H. Long-range migration of self-interstitial atoms in tungsten. Phys Rev Lett, 1975, 35: Tanaka S, Fuss J, Kugler H, et al. The resistivity recovery of high purity and carbon doped iron following low temperature electron irradiation. Radiat Eff, 1983, 79: Israeloff N E, Weissman M B, Garfunkel G A. 1/f Noise in Cr Films from spin-density-wave polarization rotation. Phys Rev Lett, 1988, 60: Harder J M, Bacon D J. Point-defect and stacking-fault properties in body-centered-cubic metals with N-body interatomic potentials. Philos Mag A, 1986, 54: Adams J B, Foiles S M. Development of an embedded-atom potential for a bcc metal: Vanadium. Phys Rev B, 1990, 41: Wirth B D, Odette G R, Maroudas D, et al. Energetics of formation and migration of self-interstitials and self-interstitial clusters in α-iron. J Nucl Mater, 1997, 244: Marian J, Wirth B D, Caro A, et al. Dynamics of self-interstitial cluster migration in pure α-fe and Fe-Cu alloys. Phys Rev B, 2002, 65: Han S, Zepeda-Ruiz L A, Ackland G J, et al. Self-interstitials in V and Mo. Phys Rev B, 2002, 66: Fu C C, Willaime F, Ordejon P. Stability and mobility of mono- and Di-interstitials in α-fe. Phys Rev Lett, 2004, 92: Nguyen-Manh D, Horsfield A P, Dudarev S L. Self-interstitial atom defects in bcc transition metals: Group-specific trends. Phys Rev B, 2006, 73: Derlet P M, Nguyen-Manh D, Dudarev S L. Multiscale modeling of crowdion and vacancy defects in body-centered-cubic transition metals. Phys Rev B, 2007, 76:
5 618 Chen L, et al. Sci China-Phys Mech Astron April (2012) Vol. 55 No Becquart C S, Domain C. Ab initio calculations about intrinsic point defects and He in W. Nucl Instrum Methods Phys Res Sect B, 2007, 255: Ackland G J, Thetford R. An improved N-body semi-empirical model for body-centred cubic transition metals. Philos Mag A, 1987, 56: Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals. Phys Rev B, 1993, 47: Kresse G, Furthmuller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B, 1996, 54: Perdew J P, Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B, 1992, 45: Blochl P E. Projector augmented-wave method. Phys Rev B, 1994, 50: Monkhorst H J, Pack J D. Special points for Brillouin-zone integrations. Phys Rev B, 1976, 13: Kittel C. Introduction to Solid State Physics. 7th ed. New York: Wiley, Schilling W. Self-interstitial atoms in metals. J Nucl Mater, 1978, 69-70:
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