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Molecular dynamics simulations of self-diffusion of adatoms on tungsten surfaces
Nuclear Inst. and Methods in Physics Research B 456 (2019) 1–6
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Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Molecular dynamics simulations of self-diffusion of adatoms on tungsten surfaces Jiechao Cui, Min Li, Baoqin Fu, Qing Hou
T
⁎
Key Lab for Radiation Physics and Technology, Ministry of Education, Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, China
ARTICLE INFO
ABSTRACT
Keywords: Molecular dynamics simulations Self-diffusion Adatom cluster Tungsten surface
Under an extreme fusion environment, the self-diffusion behaviors of W adatoms on W surfaces are thought to have a very important influence on the modification of surface morphology of W. Thus, in the present paper, the self-diffusion behaviors of W adatom clusters on the three typical W surfaces ((0 0 1), (0 1 1) and (1 1 1)) were investigated using the parallel replica dynamics method and the conventional molecular dynamics method. The diffusion barrier was obtained by the nudged elastic band method, and the corresponding pre-factor was also calculated. The results show that adatom clusters on the (0 1 1) surface are easier to diffuse than those on the (0 0 1) surface, and the diffusion barriers of single adatoms are 0.46 eV and 1.61 eV, separately. Furthermore, for the clusters of different sizes on the (0 0 1) surface, all diffusion barriers are strongly affected by relative positions between the jump adatom and its neighboring adatoms, but little by farther adatoms; all the scales of the pre-factors are fs−1. However, on the (0 1 1) surface, the energy barrier is mainly determined by the whole structure of an adatom cluster, the diffusion pathways are much more complex, and even trimer-shearing may occur, while all the scales of the pre-factors are ns−1. On the (1 1 1) surface, surface atoms can move and cause surface undulation, the adatoms and substrate atoms are indistinguishable, so it is meaningless to just obtain diffusions of the adatoms. This work can direct the investigation of the adatom diffusion on W surfaces by larger simulation methods, such as kinetic Monte Carlo.
1. Introduction The diffusion of adatoms is considered as a key factor in most dynamical processes such as chemical reactions, growth of islands and epitaxial layers taking place on surfaces [1–3]. Furthermore, the knowledge of the self-diffusion properties of single adatoms and of clusters on crystal surfaces can be helpful for understanding the evolutionary process of surface nanostructure [4]. In general, the self-diffusion behavior of adatoms on crystal surfaces could be divided into migration involving two kinds of mechanisms: jump (hopping) mechanism and exchange mechanism. The former includes single jump (jump to the nearest surface site) and long jump [5], and the later includes single exchange and crowdion-mediated exchange. The relationship between the occurrence probabilities fb of the migration mechanisms mentioned above and the diffusion barrier Ea at a temperature T could be described by the Arrhenius relation [6]:
fb (T ) = Aexp
Ea kB T
(1)
where A is the pre-factor, and kB is the Boltzmann constant. Thus, ⁎
the occurrence probabilities fb at different temperature can be obtained by knowing the diffusion barrier Ea and the pre-factor A. Tungsten (W) is chosen as a promising plasma facing component (PFC) material in the ITER divertor region, because its excellent properties allow it to function in such extreme environments. However, high flux and fluence energetic ions and neutrons can still induce dramatic radiation damage, and affect the safe operation of the reactor. In this process, as the results of the rupture of helium bubbles [7], loop punching [8,9], and other physical behaviors, plenty of W adatoms are produced. The behaviors of these adatoms, including migration, are deemed as important factors that affect the morphological evolution of W surface [10,11]. Therefore, from the standpoint of not just the reactor application but also applications in other fields, the self-diffusion behaviors of W adatoms on W surfaces have been a subject of many excellent researches. For example, by using the field ion microscope (FIM), Antczak et al. [5], and Oh et al. [12] investigated a W adatom self-diffusing on the W (0 1 1) surface, and found that in addition to nearest-neighbor and second-nearest-neighbor transitions along the close-packed 〈1 1 1〉 directions, atoms also carry out jumps along the 〈0 0 1〉 and the 〈1 1 0〉 directions, and that the contribution from later
Corresponding author. E-mail address: [email protected] (Q. Hou).
https://doi.org/10.1016/j.nimb.2019.06.036 Received 7 August 2018; Received in revised form 19 June 2019; Accepted 24 June 2019 Available online 02 July 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 456 (2019) 1–6
J. Cui, et al.
processes increases rapidly with temperature. By using the molecular dynamics (MD) method, Chen et al. [13] investigated the self-diffusion of W adatom clusters on the W (1 1 0) surface, and observed that the close-packed adatom clusters are more stable with the relatively lower binding energies, and the dimer-shearing mechanism occurs inside the hexamer, while it occurs at the periphery of heptamer. For the (0 0 1) surface, Olewicz et al. [14] demonstrated the experimental evidence for the coexistence of the exchange diffusion mechanism and adatom jump. On the basis of ab initio plane-wave calculations, the study of the self–diffusion of tungsten adatoms on the W (1 1 2) surface by Fijak at al. [15] has shown that the contour of the barrier is asymmetric and the saddle point does not appear in the middle between equilibrium sites. However, there are still many issues that deserves a series of comprehensive and systematic study and analysis. For example, to offer data to simulation methods such as kinetic Monte Carlo (KMC), the dependence of the migration behavior on the cluster size should be clarified, and the occurrence probabilities fb of the migration behaviors are in demand. In fact, although both Ea and A are needed to obtain fb , many previous works just concerned on Ea but ignored A [13–16]. Thus, in the present paper, by using the parallel replica dynamics (ParRep) method [17] and the conventional molecular dynamics (CMD) method, we investigated the self-diffusion behavior of W adatom clusters on three typical W surfaces ((0 0 1), (0 1 1), (1 1 1)). The influence of the cluster size on the diffusion behavior was given the focus, and different diffusion pathways were identified by the nudged elastic band (NEB) method [18]. This work can be helpful for further understanding of the self-diffusion behavior of W adatom clusters, and can also offer data for larger simulation methods, such as KMC. 2. Computational methods A MD package MDPSCU (Molecular Dynamics Package of Sichuan University) [19] was applied in all simulations. To describe the interactions between W atoms, a Finnis–Sinclair-type (FS) potential obtained by Ackland and Thetford [20] was applied mainly and an embedded atom method (EAM) type potential EAM2 developed by Marinica et al. [21] was used locally for the comparison. In this paper, if there is no special note, FS potential is the default one. Before the MD simulation run, BCC tungsten boxes of dimensions 15a 0 × 15 a 0 × 14.5 a 0 for the (0 0 1) surface, 12.728a 0 × 12a 0 × 12.728 a 0 for the (0 1 1) surface, and 12.728a 0 × 14.697a 0 × 13.856 a 0 for the (1 1 1) surface were prepared separately, where α0 is the lattice constant of tungsten. Here, α0 = 3.1652 Å for the FS potential, and α0 = 3.14 Å for the EAM2 potential. Periodic boundary conditions were imposed in the xand y- directions, while free boundary conditions in the z- direction. To avoid the displacement of atoms, the two bottom layers of atoms were fixed at their original positions throughout the simulations. To set the adatom clusters, from one to five W atoms were placed just a little above the surface. The x and y positions as well as the shape of the cluster were random to avoid the possible impact. In the MD simulations, considering that the temperature of the divertor ranges from 300 K to 3000 K [22], an approximately intermediate value 1500 K was set as the system temperature. To track the diffusion trajectory and calculate the diffusion barrier, a self-diffusion event was identified when any adatom moved larger than 0.5a 0 . The ParRep method and the CMD method were used in the simulations. For the ParRep method, 100 independent simulation boxes were prepared to accelerate the cluster diffusion. As such the simulation can achieve a 100× speedup here and self-diffusion events could be obtained directly in the simulation process. For the CMD method, the output simulation boxes were quenched to 0 K to search the self-diffusion events. Finally, the NEB method was applied to calculate the barriers of all diffusion events. It should be mentioned that, although overall migrations of clusters were observed in our simulations, the time needed for the calculation of the diffusion coefficient using Einstein relation is very long. Thus, we focused on the individual self-diffusion events in which
Fig. 1. (a) The relationship between the number of diffusion events and corresponding diffusion barriers of a single adatom on the W (0 0 1) surface. (1), (2), (3) and (4) represent 4 types of diffusion behaviors. (b) and (c) indicate the diffusion pathways represented by (1) and (2) in (a) separately.
one or a small number of adatoms in the clusters may take part. These events will induce the change of the center of mass and their occurrence rates can be used for the calculation of the diffusion coefficients though such as KMC. 3. Simulation results 3.1. The W (0 0 1) surface First, a single W adatom on the W (0 0 1) surface is tracked by the ParRep method, and 141 diffusion events are searched in 78.98 ns. These events are then calculated and different diffusion pathways are identified by the NEB method; the obtained results are presented in Fig. 1. As shown in Fig. 1(a), the 141 events can be divided into 4 diffusion pathways, and each pathway has the corresponding diffusion 2
Nuclear Inst. and Methods in Physics Research B 456 (2019) 1–6
J. Cui, et al.
Table 1 The occurrence probabilities of self-diffusion behaviors of a single adatom on the W (0 0 1) surface obtained by Eq. (1). Type
1
2
3
4
Barrier Ea (eV) Pre-factor A (fs−1)
1.61 19.4
2.30 1.50 × 10−7
2.49 9.30
0.21 1.02 × 10−7
barrier. Fig. 1(b) describes the most predicted pathway, indicated as (1) in Fig. 1(a). This pathway (1) illustrates that when a single adatom diffuses on the W (0 0 1) surface, it tends to be exchanged with one nearest surface atom and the barrier is about 1.61 eV. It is worth mentioning that there is a sub-stable state in the middle. Of course, there is another important diffusion pathway existing, that is shown as (2) in Fig. 1(a), and this mechanism is described by Fig. 1(c) in detail. It is the single jump mechanism and the barrier is about 2.30 eV. This coexistence of two diffusion mechanisms is also explored by Olewicz et al. [14] experimentally. In addition, we find another behavior that is one surface atom may be dragged by the adatom from its initial position and leaves a vacancy. This event is (3) as shown in Fig. 1(a) and it needs high energy of about 2.49 eV. But it is not a stable structure, and the dragged atom can return to the surface easily, with barrier only of about 0.21 eV as shown as (4) in Fig. 1(a). Using Eq. (1), the occurrence probabilities of diffusion behaviors can be obtained, and are listed in Table 1. This table provides data for large simulation methods, such as KMC. When the number of W adatoms is 2, only one diffusion pathway on the W (0 0 1) surface is identified. It is also the single exchange mechanism, and the barrier energy is about 1.36 eV, significantly smaller than a single atom’. This seems the existing of another adatom nearby can decrease the single exchange barrier and promote the jump mechanism. Thus, the number of W adatoms is extended to 3, 4 and 5 further to validate this phenomenon. Here we just exhibit the result of the 4 adatom cluster as an example. According to Fig. 2, this adatom cluster has 5 main diffusion pathways, with barriers ranging 1.20–2.22 eV. All of these involve the single jump mechanism. Despite its big span, the diffusion barrier exhibits a disciplinary change. For the diffusion pathway (1), (3) and (5), when the diffusing adatom is moving, they all have one neighboring adatom. And they have very similar barriers, 1.44 eV, 1.40 eV and 1.44 eV separately. These values are all larger than the value of 2 adatom cluster. Moreover, for pathway (4), the diffusing atom has two neighboring adatoms and the barrier is much smaller, about 1.20 eV. When a diffusion adatom has 3 neighboring adatoms and the cluster has a quadrilateral-like structure, as in pathway (1), it is difficult to diffuse, with the barrier 2.22 eV. Therefore, the quadrilateral-like structure is the most stable configuration for a 4 adatom cluster. In contrast, when a 4 adatom cluster is dividing into 2 clusters, as we obverse, the energy needed may be only about 1.97 eV. As such when a 5 adatom cluster diffuse, most of the time, we can only see a peripheral adatom exchanges with the surface atom and the quadrilateral-like structure keeps immobile if the cluster has this structure. We also notice that for a 3 adatom cluster, the single jump mechanism is still existing, and the energy is 1.59 eV. If we ignore these infrequent behaviors like jump and division, and only focus on the main mechanism - exchange, we notice that only one adatom in the cluster takes part in each exchange and all the scales of the pre-factors obtained are fs−1. The results of the diffusion barriers of 1–5 adatom clusters are also summed up, as displayed in Table 2. By observing this table, we can conclude that the diffusion of an adatom on the W (0 0 1) surface is mainly influenced by its neighboring adatoms but little by other farther adatoms. For the most part, compared with the situation of a single adatom, the existence of neighboring adatoms can increase diffusion behaviors. However, the quadrilateral-like structure can restrain adatom diffusion remarkably.
Fig. 2. The 5 main self-diffusion pathways of a 4 adatom cluster on the W (0 0 1) surface and corresponding diffusion barriers. Arrow indicates the diffusion direction, and red number only marks the moving atom. Table 2 The relationship between the self-diffusion barrier and the number of the nearest neighbor adatoms NN for different adatom cluster sizes Na when a cluster diffusing on the W (0 0 1) surface. Na NN
1
2
3
4
5
0 1 2 3
1.161 eV \ \ \
\ 1.36 eV \ \
\ 1.42 eV 1.15 eV \
\ 1.39–1.44 eV 1.20 ev 2.22 ev
\ 1.35–1.44 eV 1.16–1.20 eV 2.11 eV
3.2. The W (0 1 1) surface A series of researches by Antczak et al. [5,23,24] have shown that because of the individual configuration of the surface atoms, the diffusion behavior of a single W adatom on the W (0 1 1) surface is very complex. In addition to the nearest-neighbor jumps α, there is the double jumps β spanning two nearest-neighbor distances along the 〈1 1 1〉 direction; the δx transitions along the 〈0 0 1〉 direction and the δy jumps along the 〈0 1 1〉 direction also appear. Here on the W (0 1 1) surface, we were also concerned about the self-diffusion of a single W adatom initially. Because of the limitation of the simulation time, only α and δx jumps are captured, as shown in Fig. 3. The α jump is the main self-diffusion pathway and the barrier energy is about 0.46 eV. This value is obviously smaller than that on the W (0 0 1) surface, so a single adatom diffuses more quickly on this surface. As shown in Table 3, the corresponding pre-factors were also calculated, and these values are much smaller than those on the (0 0 1) 3
Nuclear Inst. and Methods in Physics Research B 456 (2019) 1–6
J. Cui, et al.
Fig. 3. (a) The relationship between the number of diffusion events and corresponding diffusion barriers of a single adatom on the W (0 1 1) surface. The FS potential is applied. (b) Types of diffusion pathways according to (a). The result of the EAM2 potential is listed as a comparison. Table 3 The occurrence probabilities of self-diffusion behaviors of a single adatom on the W (0 1 1) surface obtained by Eq. (1). Type
1
2
Barrier Ea (eV) Pre-factor A(ns−1)
0.46 55.28
1.13 463.30
Fig. 4. The relationship between the number of diffusion events and corresponding diffusion barriers of adatom clusters on the W (0 1 1) surface. (a) 2 adatoms; (b) 3 adatoms; (c) 4 adatoms.
surface. The EAM2 potential is also applied, but only α jumps are observed in the simulation time, with the barrier energy is 0.50 eV. The obtained barriers of the above two potentials are both smaller than the results shown in Ref. [5,23,24]. Considering that this paper aims to qualitatively analyze the relationship between the self-diffusion behaviors of W adatoms and their sizes, we did not pay attention to specific and precise values here. When the number of adatoms is added to 2, as shown in Fig. 4(a), the number of recorded diffusion pathways increases to 4, and the barrier energy ranges from 0.36 eV to 1.72 eV. Therefore, it is obvious that the behavior of 2 adatoms on the W (0 1 1) surface is much more complex than on the W (0 0 1) surface. Compared to a single adatom, the existence of one more adatom can both increase and decrease the self-diffusion. Most of the behaviors are single jump mechanism but dimer-shearing can also occur. When the cluster sizes trend to 3 and 4, as indicated in Fig. 4(b) and (c), the number of diffusion pathways increases very obviously, and the energy ranges are 0.20–1.60 eV. For the 3 adatoms, the most main diffusion pathway is that one adatom keeps it position, and other two present the dimer-shearing.
When we observe the self-diffusion behaviors of the clusters with size 1–5 in detail, we can notice that with the increasing of the cluster size, diffusion pathways become more and more varied. That is because of the individual configuration of the surface atoms, and the ease of adatom jumps. Jumps α, β, δx, δy can occur for the cluster with the definite size; furthermore, dimer-shearing and even trimer- shearing may also appear. According to the diffusion barrier, it seems there is no relationship between small clusters and larger one, except all the scales of the pre-factors are ns−1. 3.3. The W (1 1 1) surface While focusing on the self-diffusion behaviors of a single W adatom on the W (1 1 1) surface, we find that the situations are very different from the other two surface orientations. After 103.4 ns simulating by the ParRep method, only one diffusion event of the adatom is recorded. Calculated by the NEB method, this event is the exchange behavior and 4
Nuclear Inst. and Methods in Physics Research B 456 (2019) 1–6
J. Cui, et al.
Fig.5. The surface topography on the W (1 1 1) surface at different simulating time. Red spheres: atoms initially on the first surface layer; gray spheres: atoms initially on the second surface layer; white spheres: inner atoms initially. (1) 0 ns; (2) 2 ns; (3) 5 ns; (4) 100 ns.
the barrier energy is 1.30 eV. Besides this event, others are the movements of surface atoms. Thus, it seems that surface atoms on the W (1 1 1) surface can move themselves and prevent the diffusion of the adatoms. In this case, the adatoms and substrate atoms are indistinguishable. Thus, just tracking the diffusion of the adatoms and calculating the barrier energy seem to be meaningless. Therefore, we turned to pay attention to the behaviors of surface atoms on the W (1 1 1) surface ignoring the existing of adatoms. The result is described by Fig. 5 and Table 4. At t = 0 ns, the surface is the BCC structure and all atoms have the regular arrangement. At t = 2 ns, plenty of atoms near the surface exchange with atoms on other layers, and some atoms initially on the first and second surface layer even occupy sites two layers below the initial surface. Also, other atoms move up from the surface, and a few atoms initially on the first layer occupy sites two layers above the initial surface. Besides, some vacancies appear on the first and second layer, so the surface becomes rough. Then at t = 5 ns and 100 ns, more surface atoms change their positions. Especially at t = 100 ns, atoms initially on the second surface layer spreads widely from three layers below the initial surface to two
layers above. However, comparing with the situation at t = 2 ns, the roughness does not increase obviously even at t = 100 ns. These phenomena further show the mobility of surface atoms on the W(1 1 1) surface. When we applied the potential EAM2 to repeat the simulation, similar phenomena appear. 4. Discussion and conclusion To summarize, using MD simulations, the self-diffusion behaviors of adatom clusters on the W (0 0 1), (0 1 1) and (1 1 1) surface were explored. For the case of self-diffusion of a single adatom, it has the most difficult likelihood to diffuse on the W (0 0 1) surface, as the minimum diffusion barrier is 1.61 eV. The microscopic mechanism is the single jump; it is the easiest to diffuse on the (0 1 1) surface, with the minimum energy barrier being 0.46 eV. For the self-diffusion of multiadatom cluster, the main mechanism on the (0 0 1) surface is single exchange, but there are also a few movements via single jump, which needs higher energy. The diffusion barrier is deeply influenced by relative positions between the jumping adatom and its neighbouring
Table 4 The atomic distribution near the W (1 1 1) surface at different simulating time. Ln denotes the initial first surface layer, Ln-1 is the initial second surface layer, and Ln+1 is the one a layer above Ln, and so on. W1: atoms initially on the first surface layer; W2: atoms initially on the second surface layer; W3: inner atoms initially; W0: no atom. The unit for the atomic distribution is%. Layer
Ln-3 Ln-2 Ln-1 Ln Ln+1 Ln+2
t = 2 ns
t = 5 ns
t = 100 ns
W0
W1
W2
W3
W0
W1
W2
W3
W0
W1
W2
W3
0 0 8.3 27.1 68.8 95.8
0 4.2 39.6 39.6 12.4 4.2
0 2.1 45.8 33.3 18.8 0
100 93.7 6.3 0 0 0
0 0 4.2 25 75 95.8
0 4.2 41.6 37.5 14.6 2.1
0 6.3 45.8 35.4 10.4 2.1
100 89.5 8.4 2.1 0 0
0 0 4.2 29.2 70.8 95.8
0 10.4 37.5 37.5 14.6 0
2.1 14.6 35.4 29.1 14.6 4.2
97.9 75 22.9 4.2 0 0
5
Nuclear Inst. and Methods in Physics Research B 456 (2019) 1–6
J. Cui, et al.
adatoms, but little by other farther ones. All scales of the pre-factors of the main diffusion pathways are fs−1. These results of small adatom clusters diffusion can direct larger ones. However, there is an obvious difference on the (0 1 1) surface. On this surface, the main mechanism for adatom diffusion is single jump and multiple jump. The energy barrier is highly influenced by the whole structure of an adatom cluster, while the pre-factors seem to be the same scale, ns−1. Therefore, there are some more things required to be explored to understand the relationship between small clusters and large ones. On the (1 1 1) surface, surface atoms can move and cause surface undulation, the adatoms and substrate atoms are indistinguishable, so it is meaningless to just obtain the self-diffusion behaviours of the adatoms. Thus, it is possible to investigate the adatoms diffusion on the (0 0 1) surface by KMC, but difficult on the other two surfaces. Considering this work only concerns small and monolayer clusters, the diffusion of large and multilayer adatoms will be explored in the future.
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Acknowledgements This work was partially supported by the National Natural Science Foundation of China (Grant No. 11505120 and 11775151). References [1] T. Ala-Nissila, R. Ferrando, S.C. Ying, Collective and single particle diffusion on surfaces, Adv. Phys. 51 (2002) 949–1078, https://doi.org/10.1080/ 00018730110107902. [2] A. Pimpinelli, J. Villain, Physics of crystal growth, Cambridge Univ. Press. (1998), https://doi.org/10.1016/S0025-5408(00)00215-4. [3] F. Schreiber, Structure and growth of self-assembling monolayers, Prog. Surf. Sci. (2000), https://doi.org/10.1016/S0079-6816(00)00024-1. [4] P. Jensen, Growth of nanostructures by cluster deposition: a review, 71, 1999, 1695–1735. doi:10.1103/RevModPhys.71.1695. [5] G. Antczak, G. Ehrlich, Long jumps in surface diffusion, J. Colloid Interface Sci. 276 (2004) 1–5, https://doi.org/10.1016/j.jcis.2003.11.035. [6] P. Hänggi, P. Talkner, M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. (1990), https://doi.org/10.1103/RevModPhys. 62.251. [7] J. Cui, Z. Wu, Q. Hou, Estimation of the lifetime of small helium bubbles near tungsten surfaces – a methodological study, Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 383 (2016) 136–142, https://doi.org/10.1016/j. nimb.2016.07.001. [8] J. Wang, L.-L. Niu, X. Shu, Y. Zhang, Stick–slip behavior identified in helium cluster growth in the subsurface of tungsten: effects of cluster depth, J. Phys. Condens.
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Molecular dynamics simulations of self-diffusion of adatoms on tungsten surfaces Jiechao Cui, Min Li, Baoqin Fu, Qing Hou
T
⁎
Key Lab for Radiation Physics and Technology, Ministry of Education, Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, China
ARTICLE INFO
ABSTRACT
Keywords: Molecular dynamics simulations Self-diffusion Adatom cluster Tungsten surface
Under an extreme fusion environment, the self-diffusion behaviors of W adatoms on W surfaces are thought to have a very important influence on the modification of surface morphology of W. Thus, in the present paper, the self-diffusion behaviors of W adatom clusters on the three typical W surfaces ((0 0 1), (0 1 1) and (1 1 1)) were investigated using the parallel replica dynamics method and the conventional molecular dynamics method. The diffusion barrier was obtained by the nudged elastic band method, and the corresponding pre-factor was also calculated. The results show that adatom clusters on the (0 1 1) surface are easier to diffuse than those on the (0 0 1) surface, and the diffusion barriers of single adatoms are 0.46 eV and 1.61 eV, separately. Furthermore, for the clusters of different sizes on the (0 0 1) surface, all diffusion barriers are strongly affected by relative positions between the jump adatom and its neighboring adatoms, but little by farther adatoms; all the scales of the pre-factors are fs−1. However, on the (0 1 1) surface, the energy barrier is mainly determined by the whole structure of an adatom cluster, the diffusion pathways are much more complex, and even trimer-shearing may occur, while all the scales of the pre-factors are ns−1. On the (1 1 1) surface, surface atoms can move and cause surface undulation, the adatoms and substrate atoms are indistinguishable, so it is meaningless to just obtain diffusions of the adatoms. This work can direct the investigation of the adatom diffusion on W surfaces by larger simulation methods, such as kinetic Monte Carlo.
1. Introduction The diffusion of adatoms is considered as a key factor in most dynamical processes such as chemical reactions, growth of islands and epitaxial layers taking place on surfaces [1–3]. Furthermore, the knowledge of the self-diffusion properties of single adatoms and of clusters on crystal surfaces can be helpful for understanding the evolutionary process of surface nanostructure [4]. In general, the self-diffusion behavior of adatoms on crystal surfaces could be divided into migration involving two kinds of mechanisms: jump (hopping) mechanism and exchange mechanism. The former includes single jump (jump to the nearest surface site) and long jump [5], and the later includes single exchange and crowdion-mediated exchange. The relationship between the occurrence probabilities fb of the migration mechanisms mentioned above and the diffusion barrier Ea at a temperature T could be described by the Arrhenius relation [6]:
fb (T ) = Aexp
Ea kB T
(1)
where A is the pre-factor, and kB is the Boltzmann constant. Thus, ⁎
the occurrence probabilities fb at different temperature can be obtained by knowing the diffusion barrier Ea and the pre-factor A. Tungsten (W) is chosen as a promising plasma facing component (PFC) material in the ITER divertor region, because its excellent properties allow it to function in such extreme environments. However, high flux and fluence energetic ions and neutrons can still induce dramatic radiation damage, and affect the safe operation of the reactor. In this process, as the results of the rupture of helium bubbles [7], loop punching [8,9], and other physical behaviors, plenty of W adatoms are produced. The behaviors of these adatoms, including migration, are deemed as important factors that affect the morphological evolution of W surface [10,11]. Therefore, from the standpoint of not just the reactor application but also applications in other fields, the self-diffusion behaviors of W adatoms on W surfaces have been a subject of many excellent researches. For example, by using the field ion microscope (FIM), Antczak et al. [5], and Oh et al. [12] investigated a W adatom self-diffusing on the W (0 1 1) surface, and found that in addition to nearest-neighbor and second-nearest-neighbor transitions along the close-packed 〈1 1 1〉 directions, atoms also carry out jumps along the 〈0 0 1〉 and the 〈1 1 0〉 directions, and that the contribution from later
Corresponding author. E-mail address: [email protected] (Q. Hou).
https://doi.org/10.1016/j.nimb.2019.06.036 Received 7 August 2018; Received in revised form 19 June 2019; Accepted 24 June 2019 Available online 02 July 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
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processes increases rapidly with temperature. By using the molecular dynamics (MD) method, Chen et al. [13] investigated the self-diffusion of W adatom clusters on the W (1 1 0) surface, and observed that the close-packed adatom clusters are more stable with the relatively lower binding energies, and the dimer-shearing mechanism occurs inside the hexamer, while it occurs at the periphery of heptamer. For the (0 0 1) surface, Olewicz et al. [14] demonstrated the experimental evidence for the coexistence of the exchange diffusion mechanism and adatom jump. On the basis of ab initio plane-wave calculations, the study of the self–diffusion of tungsten adatoms on the W (1 1 2) surface by Fijak at al. [15] has shown that the contour of the barrier is asymmetric and the saddle point does not appear in the middle between equilibrium sites. However, there are still many issues that deserves a series of comprehensive and systematic study and analysis. For example, to offer data to simulation methods such as kinetic Monte Carlo (KMC), the dependence of the migration behavior on the cluster size should be clarified, and the occurrence probabilities fb of the migration behaviors are in demand. In fact, although both Ea and A are needed to obtain fb , many previous works just concerned on Ea but ignored A [13–16]. Thus, in the present paper, by using the parallel replica dynamics (ParRep) method [17] and the conventional molecular dynamics (CMD) method, we investigated the self-diffusion behavior of W adatom clusters on three typical W surfaces ((0 0 1), (0 1 1), (1 1 1)). The influence of the cluster size on the diffusion behavior was given the focus, and different diffusion pathways were identified by the nudged elastic band (NEB) method [18]. This work can be helpful for further understanding of the self-diffusion behavior of W adatom clusters, and can also offer data for larger simulation methods, such as KMC. 2. Computational methods A MD package MDPSCU (Molecular Dynamics Package of Sichuan University) [19] was applied in all simulations. To describe the interactions between W atoms, a Finnis–Sinclair-type (FS) potential obtained by Ackland and Thetford [20] was applied mainly and an embedded atom method (EAM) type potential EAM2 developed by Marinica et al. [21] was used locally for the comparison. In this paper, if there is no special note, FS potential is the default one. Before the MD simulation run, BCC tungsten boxes of dimensions 15a 0 × 15 a 0 × 14.5 a 0 for the (0 0 1) surface, 12.728a 0 × 12a 0 × 12.728 a 0 for the (0 1 1) surface, and 12.728a 0 × 14.697a 0 × 13.856 a 0 for the (1 1 1) surface were prepared separately, where α0 is the lattice constant of tungsten. Here, α0 = 3.1652 Å for the FS potential, and α0 = 3.14 Å for the EAM2 potential. Periodic boundary conditions were imposed in the xand y- directions, while free boundary conditions in the z- direction. To avoid the displacement of atoms, the two bottom layers of atoms were fixed at their original positions throughout the simulations. To set the adatom clusters, from one to five W atoms were placed just a little above the surface. The x and y positions as well as the shape of the cluster were random to avoid the possible impact. In the MD simulations, considering that the temperature of the divertor ranges from 300 K to 3000 K [22], an approximately intermediate value 1500 K was set as the system temperature. To track the diffusion trajectory and calculate the diffusion barrier, a self-diffusion event was identified when any adatom moved larger than 0.5a 0 . The ParRep method and the CMD method were used in the simulations. For the ParRep method, 100 independent simulation boxes were prepared to accelerate the cluster diffusion. As such the simulation can achieve a 100× speedup here and self-diffusion events could be obtained directly in the simulation process. For the CMD method, the output simulation boxes were quenched to 0 K to search the self-diffusion events. Finally, the NEB method was applied to calculate the barriers of all diffusion events. It should be mentioned that, although overall migrations of clusters were observed in our simulations, the time needed for the calculation of the diffusion coefficient using Einstein relation is very long. Thus, we focused on the individual self-diffusion events in which
Fig. 1. (a) The relationship between the number of diffusion events and corresponding diffusion barriers of a single adatom on the W (0 0 1) surface. (1), (2), (3) and (4) represent 4 types of diffusion behaviors. (b) and (c) indicate the diffusion pathways represented by (1) and (2) in (a) separately.
one or a small number of adatoms in the clusters may take part. These events will induce the change of the center of mass and their occurrence rates can be used for the calculation of the diffusion coefficients though such as KMC. 3. Simulation results 3.1. The W (0 0 1) surface First, a single W adatom on the W (0 0 1) surface is tracked by the ParRep method, and 141 diffusion events are searched in 78.98 ns. These events are then calculated and different diffusion pathways are identified by the NEB method; the obtained results are presented in Fig. 1. As shown in Fig. 1(a), the 141 events can be divided into 4 diffusion pathways, and each pathway has the corresponding diffusion 2
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Table 1 The occurrence probabilities of self-diffusion behaviors of a single adatom on the W (0 0 1) surface obtained by Eq. (1). Type
1
2
3
4
Barrier Ea (eV) Pre-factor A (fs−1)
1.61 19.4
2.30 1.50 × 10−7
2.49 9.30
0.21 1.02 × 10−7
barrier. Fig. 1(b) describes the most predicted pathway, indicated as (1) in Fig. 1(a). This pathway (1) illustrates that when a single adatom diffuses on the W (0 0 1) surface, it tends to be exchanged with one nearest surface atom and the barrier is about 1.61 eV. It is worth mentioning that there is a sub-stable state in the middle. Of course, there is another important diffusion pathway existing, that is shown as (2) in Fig. 1(a), and this mechanism is described by Fig. 1(c) in detail. It is the single jump mechanism and the barrier is about 2.30 eV. This coexistence of two diffusion mechanisms is also explored by Olewicz et al. [14] experimentally. In addition, we find another behavior that is one surface atom may be dragged by the adatom from its initial position and leaves a vacancy. This event is (3) as shown in Fig. 1(a) and it needs high energy of about 2.49 eV. But it is not a stable structure, and the dragged atom can return to the surface easily, with barrier only of about 0.21 eV as shown as (4) in Fig. 1(a). Using Eq. (1), the occurrence probabilities of diffusion behaviors can be obtained, and are listed in Table 1. This table provides data for large simulation methods, such as KMC. When the number of W adatoms is 2, only one diffusion pathway on the W (0 0 1) surface is identified. It is also the single exchange mechanism, and the barrier energy is about 1.36 eV, significantly smaller than a single atom’. This seems the existing of another adatom nearby can decrease the single exchange barrier and promote the jump mechanism. Thus, the number of W adatoms is extended to 3, 4 and 5 further to validate this phenomenon. Here we just exhibit the result of the 4 adatom cluster as an example. According to Fig. 2, this adatom cluster has 5 main diffusion pathways, with barriers ranging 1.20–2.22 eV. All of these involve the single jump mechanism. Despite its big span, the diffusion barrier exhibits a disciplinary change. For the diffusion pathway (1), (3) and (5), when the diffusing adatom is moving, they all have one neighboring adatom. And they have very similar barriers, 1.44 eV, 1.40 eV and 1.44 eV separately. These values are all larger than the value of 2 adatom cluster. Moreover, for pathway (4), the diffusing atom has two neighboring adatoms and the barrier is much smaller, about 1.20 eV. When a diffusion adatom has 3 neighboring adatoms and the cluster has a quadrilateral-like structure, as in pathway (1), it is difficult to diffuse, with the barrier 2.22 eV. Therefore, the quadrilateral-like structure is the most stable configuration for a 4 adatom cluster. In contrast, when a 4 adatom cluster is dividing into 2 clusters, as we obverse, the energy needed may be only about 1.97 eV. As such when a 5 adatom cluster diffuse, most of the time, we can only see a peripheral adatom exchanges with the surface atom and the quadrilateral-like structure keeps immobile if the cluster has this structure. We also notice that for a 3 adatom cluster, the single jump mechanism is still existing, and the energy is 1.59 eV. If we ignore these infrequent behaviors like jump and division, and only focus on the main mechanism - exchange, we notice that only one adatom in the cluster takes part in each exchange and all the scales of the pre-factors obtained are fs−1. The results of the diffusion barriers of 1–5 adatom clusters are also summed up, as displayed in Table 2. By observing this table, we can conclude that the diffusion of an adatom on the W (0 0 1) surface is mainly influenced by its neighboring adatoms but little by other farther adatoms. For the most part, compared with the situation of a single adatom, the existence of neighboring adatoms can increase diffusion behaviors. However, the quadrilateral-like structure can restrain adatom diffusion remarkably.
Fig. 2. The 5 main self-diffusion pathways of a 4 adatom cluster on the W (0 0 1) surface and corresponding diffusion barriers. Arrow indicates the diffusion direction, and red number only marks the moving atom. Table 2 The relationship between the self-diffusion barrier and the number of the nearest neighbor adatoms NN for different adatom cluster sizes Na when a cluster diffusing on the W (0 0 1) surface. Na NN
1
2
3
4
5
0 1 2 3
1.161 eV \ \ \
\ 1.36 eV \ \
\ 1.42 eV 1.15 eV \
\ 1.39–1.44 eV 1.20 ev 2.22 ev
\ 1.35–1.44 eV 1.16–1.20 eV 2.11 eV
3.2. The W (0 1 1) surface A series of researches by Antczak et al. [5,23,24] have shown that because of the individual configuration of the surface atoms, the diffusion behavior of a single W adatom on the W (0 1 1) surface is very complex. In addition to the nearest-neighbor jumps α, there is the double jumps β spanning two nearest-neighbor distances along the 〈1 1 1〉 direction; the δx transitions along the 〈0 0 1〉 direction and the δy jumps along the 〈0 1 1〉 direction also appear. Here on the W (0 1 1) surface, we were also concerned about the self-diffusion of a single W adatom initially. Because of the limitation of the simulation time, only α and δx jumps are captured, as shown in Fig. 3. The α jump is the main self-diffusion pathway and the barrier energy is about 0.46 eV. This value is obviously smaller than that on the W (0 0 1) surface, so a single adatom diffuses more quickly on this surface. As shown in Table 3, the corresponding pre-factors were also calculated, and these values are much smaller than those on the (0 0 1) 3
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Fig. 3. (a) The relationship between the number of diffusion events and corresponding diffusion barriers of a single adatom on the W (0 1 1) surface. The FS potential is applied. (b) Types of diffusion pathways according to (a). The result of the EAM2 potential is listed as a comparison. Table 3 The occurrence probabilities of self-diffusion behaviors of a single adatom on the W (0 1 1) surface obtained by Eq. (1). Type
1
2
Barrier Ea (eV) Pre-factor A(ns−1)
0.46 55.28
1.13 463.30
Fig. 4. The relationship between the number of diffusion events and corresponding diffusion barriers of adatom clusters on the W (0 1 1) surface. (a) 2 adatoms; (b) 3 adatoms; (c) 4 adatoms.
surface. The EAM2 potential is also applied, but only α jumps are observed in the simulation time, with the barrier energy is 0.50 eV. The obtained barriers of the above two potentials are both smaller than the results shown in Ref. [5,23,24]. Considering that this paper aims to qualitatively analyze the relationship between the self-diffusion behaviors of W adatoms and their sizes, we did not pay attention to specific and precise values here. When the number of adatoms is added to 2, as shown in Fig. 4(a), the number of recorded diffusion pathways increases to 4, and the barrier energy ranges from 0.36 eV to 1.72 eV. Therefore, it is obvious that the behavior of 2 adatoms on the W (0 1 1) surface is much more complex than on the W (0 0 1) surface. Compared to a single adatom, the existence of one more adatom can both increase and decrease the self-diffusion. Most of the behaviors are single jump mechanism but dimer-shearing can also occur. When the cluster sizes trend to 3 and 4, as indicated in Fig. 4(b) and (c), the number of diffusion pathways increases very obviously, and the energy ranges are 0.20–1.60 eV. For the 3 adatoms, the most main diffusion pathway is that one adatom keeps it position, and other two present the dimer-shearing.
When we observe the self-diffusion behaviors of the clusters with size 1–5 in detail, we can notice that with the increasing of the cluster size, diffusion pathways become more and more varied. That is because of the individual configuration of the surface atoms, and the ease of adatom jumps. Jumps α, β, δx, δy can occur for the cluster with the definite size; furthermore, dimer-shearing and even trimer- shearing may also appear. According to the diffusion barrier, it seems there is no relationship between small clusters and larger one, except all the scales of the pre-factors are ns−1. 3.3. The W (1 1 1) surface While focusing on the self-diffusion behaviors of a single W adatom on the W (1 1 1) surface, we find that the situations are very different from the other two surface orientations. After 103.4 ns simulating by the ParRep method, only one diffusion event of the adatom is recorded. Calculated by the NEB method, this event is the exchange behavior and 4
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Fig.5. The surface topography on the W (1 1 1) surface at different simulating time. Red spheres: atoms initially on the first surface layer; gray spheres: atoms initially on the second surface layer; white spheres: inner atoms initially. (1) 0 ns; (2) 2 ns; (3) 5 ns; (4) 100 ns.
the barrier energy is 1.30 eV. Besides this event, others are the movements of surface atoms. Thus, it seems that surface atoms on the W (1 1 1) surface can move themselves and prevent the diffusion of the adatoms. In this case, the adatoms and substrate atoms are indistinguishable. Thus, just tracking the diffusion of the adatoms and calculating the barrier energy seem to be meaningless. Therefore, we turned to pay attention to the behaviors of surface atoms on the W (1 1 1) surface ignoring the existing of adatoms. The result is described by Fig. 5 and Table 4. At t = 0 ns, the surface is the BCC structure and all atoms have the regular arrangement. At t = 2 ns, plenty of atoms near the surface exchange with atoms on other layers, and some atoms initially on the first and second surface layer even occupy sites two layers below the initial surface. Also, other atoms move up from the surface, and a few atoms initially on the first layer occupy sites two layers above the initial surface. Besides, some vacancies appear on the first and second layer, so the surface becomes rough. Then at t = 5 ns and 100 ns, more surface atoms change their positions. Especially at t = 100 ns, atoms initially on the second surface layer spreads widely from three layers below the initial surface to two
layers above. However, comparing with the situation at t = 2 ns, the roughness does not increase obviously even at t = 100 ns. These phenomena further show the mobility of surface atoms on the W(1 1 1) surface. When we applied the potential EAM2 to repeat the simulation, similar phenomena appear. 4. Discussion and conclusion To summarize, using MD simulations, the self-diffusion behaviors of adatom clusters on the W (0 0 1), (0 1 1) and (1 1 1) surface were explored. For the case of self-diffusion of a single adatom, it has the most difficult likelihood to diffuse on the W (0 0 1) surface, as the minimum diffusion barrier is 1.61 eV. The microscopic mechanism is the single jump; it is the easiest to diffuse on the (0 1 1) surface, with the minimum energy barrier being 0.46 eV. For the self-diffusion of multiadatom cluster, the main mechanism on the (0 0 1) surface is single exchange, but there are also a few movements via single jump, which needs higher energy. The diffusion barrier is deeply influenced by relative positions between the jumping adatom and its neighbouring
Table 4 The atomic distribution near the W (1 1 1) surface at different simulating time. Ln denotes the initial first surface layer, Ln-1 is the initial second surface layer, and Ln+1 is the one a layer above Ln, and so on. W1: atoms initially on the first surface layer; W2: atoms initially on the second surface layer; W3: inner atoms initially; W0: no atom. The unit for the atomic distribution is%. Layer
Ln-3 Ln-2 Ln-1 Ln Ln+1 Ln+2
t = 2 ns
t = 5 ns
t = 100 ns
W0
W1
W2
W3
W0
W1
W2
W3
W0
W1
W2
W3
0 0 8.3 27.1 68.8 95.8
0 4.2 39.6 39.6 12.4 4.2
0 2.1 45.8 33.3 18.8 0
100 93.7 6.3 0 0 0
0 0 4.2 25 75 95.8
0 4.2 41.6 37.5 14.6 2.1
0 6.3 45.8 35.4 10.4 2.1
100 89.5 8.4 2.1 0 0
0 0 4.2 29.2 70.8 95.8
0 10.4 37.5 37.5 14.6 0
2.1 14.6 35.4 29.1 14.6 4.2
97.9 75 22.9 4.2 0 0
5
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adatoms, but little by other farther ones. All scales of the pre-factors of the main diffusion pathways are fs−1. These results of small adatom clusters diffusion can direct larger ones. However, there is an obvious difference on the (0 1 1) surface. On this surface, the main mechanism for adatom diffusion is single jump and multiple jump. The energy barrier is highly influenced by the whole structure of an adatom cluster, while the pre-factors seem to be the same scale, ns−1. Therefore, there are some more things required to be explored to understand the relationship between small clusters and large ones. On the (1 1 1) surface, surface atoms can move and cause surface undulation, the adatoms and substrate atoms are indistinguishable, so it is meaningless to just obtain the self-diffusion behaviours of the adatoms. Thus, it is possible to investigate the adatoms diffusion on the (0 0 1) surface by KMC, but difficult on the other two surfaces. Considering this work only concerns small and monolayer clusters, the diffusion of large and multilayer adatoms will be explored in the future.
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