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A molecular dynamics simulation of self-diffusion on Fe surfaces
Applied Surface Science 258 (2012) 4294–4300
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A molecular dynamics simulation of self-diffusion on Fe surfaces Changqing Wang a,b,∗ , Zhen Qin c , Yongsheng Zhang b , Qiang Sun a , Yu Jia a a b c
School of Physics and Engineering, Center for Clean Energy and Quantum Structures, Zhengzhou University, Zhengzhou 450052, China Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
a r t i c l e
i n f o
Article history: Received 19 October 2011 Received in revised form 17 December 2011 Accepted 18 December 2011 Available online 26 December 2011
a b s t r a c t Using embedded-atom-method (EAM) potential, an adatom and a vacancy diffusion processes have been simulated in detail by molecular dynamics on three Fe surfaces, Fe (1 1 0), Fe (1 0 0), and Fe (1 1 1). Our results reveal that adatom adsorption energies and diffusion migration energies on these surfaces have similar monotonic trend to the relative layer spacing relaxation, R(110) < R(100) < R(111) , adsorption energy, a a a m m m Ea(110) < Ea(100) < Ea(111) , diffusion migration energy, Ea(110) < Ea(100) < Ea(111) . However, for a vacancy, f
Keywords: EAM potential Molecular dynamics Diffusion Nudged elastic band (NEB) method Iron Adsorption Vacancy
f
f
formation and migration energies have a different trend, formation energy, Ev(111) < Ev(100) < Ev(110) , migration energy, Evm(111) < Evm(110) < Evm(100) . On the Fe (1 1 0) surface, simple jumping of an adatom (or a vacancy) is the main diffusion mechanism with relatively low migration energy barrier; nevertheless, exchange with a surface atom plays a dominant role in surface diffusion on the Fe (1 0 0) and Fe (1 1 1) surfaces. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The adatoms and vacancies are the main carriers in mass transport processes on metal surfaces. A detailed knowledge of the energy and mobility of the point defects (adatoms and vacancies) formed in the surface is important for understanding many surface phenomena such as oxidation, corrosion, catalysis, crystal growth, surface roughening, etc. On the atomic scale, investigating the diffusion behaviors of single adatom or vacancy on the surface, aids our understanding of the mechanism of these surface processes. These issues are of great interest for scientific reasons and technological applications. These problems have been intensively studied [1,2]. Atom diffusion on face-centered cubic (FCC) metal surfaces has been studied extensively [1,2]. Atom diffusion on the BCC Fe [3–8], Mo [9,10] surfaces was studied experimentally using the STM, the reflection high-energy electron diffraction (RHEED), ion scattering, spin polarized low energy electron microscope (SPLEEM) and low energy electron diffraction (LEED) techniques, but the dominant diffusion mechanism could not be identified. Theoretically, using First Principles [11], Molecular Dynamics [12–16] and Monte Carlo (MC) [17,18], it has been investigated intensively.
∗ Corresponding author at: Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China. E-mail address: [email protected] (C. Wang). 0169-4332/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2011.12.084
Iron is a very common metal for application. Abundant work has been done on atom diffusion on the Fe surfaces for comparison. STM observation at the temperature range of 20–250 ◦ C has been reported that the activation energy of a Fe atom diffusion on Fe (0 0 1) surface yields 0.45 eV [3]. Using EAM potential, Chamati et al. [12] carried out molecular dynamics studies on self-diffusion on the Fe (1 0 0) surface and gave adatom diagonal exchange diffusion mechanism. They investigated a vacancy diffusing on the Fe (1 0 0) surface by Molecular Dynamics technique, too [15]. However, little work has been done about adatoms or vacancies diffusion on Fe (1 1 0) and Fe (1 1 1) surfaces. Finite temperature molecular dynamics simulations are very helpful to study adatom diffusion, as they allow tracking the adatom on the surface. Moreover, it is convenient to find low energy diffusion paths and new diffusion mechanisms. In this paper, using EAM potential [12] of iron, molecular dynamics studies have been carried out on a point defect (adatom and vacancy) diffusion on three low index surfaces, Fe (1 1 0), Fe (0 0 1), Fe (1 1 1). Due to different structural stability of these three surfaces, it is well known that the Fe (1 1 0) surface is more stable than the Fe (1 0 0) surface, and the Fe (1 1 1) surface is the most unstable surface [12]. We have found that diffusion mechanism is diverse extremely by our molecular dynamics simulations. On the compact Fe (1 1 0) surface, the adatom or vacancy diffuses mainly by simple hopping mechanism. However, on the Fe (1 0 0) and Fe (1 1 1) surface, it is difficult to diffuse by simple hopping. Adatoms or vacancies exchange with surface atoms take place frequently on these two surfaces. The adatom and vacancy migration energies of
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self-diffusion on the Fe (1 1 1) surface have been firstly calculated by using molecular dynamics combining with EAM potential. Moreover, possible diffusion mechanism has been given in the present work. In addition, comparing to the formation and migration energies on these three Fe surfaces with different relaxation has been presented in this paper.
2. Methods All of the simulations were performed by using embeddedatom-method (EAM) potential for BCC Fe [12,19]. Although the potential is semi-empirical, it can give reasonable results about the bulk Fe and Fe (1 0 0) surface [12,15], such as elastic constants, phonon dispersion curves, vacancy and adatom diffusion properties. All of calculations were carried out by using twenty 400-atom (total 8000 Fe atoms) layers to simulate the Fe surface, with periodic boundary conditions in the two directions parallel to the surface. Atoms in upper eighteen layers of the slab with a 20 A˚ vacuum to eliminate the interaction between its two surfaces are free to relax in the three directions. Nevertheless, atoms in the lowest two layers of the slab are fixed to the BCC lattice. To control temperature to be isothermal, the velocity scale constant temperature scheme was applied to the system. For the numerical integration, the Verlet algorithm was employed in our calculations at the constant temperature, with a time step of 2 fs. To simulate the surface in presence of an adatom or a vacancy, one Fe atom was placed on each relaxed surface or one surface atom was removed from each surface and the systems were equilibrated at the desired temperatures for 20 ps. The static diffusion barriers and the minimum energy paths were obtained using the nudged elastic band (NEB) method [20]. It uses both the initial and the final states of the system to calculate a chain of “images” that initially represents the intermediate configurations. The static diffusion barrier is then obtained from the image with the highest potential energy, which can be identified as the saddle point configuration. For all of static calculations, the system has been relaxed into the minimum energy at 0 K. For adatom diffusion processes, simulations for 4 ns were performed covering the temperature range of 300–600 K for the Fe (1 1 0) surface and 600–1000 K for the Fe (1 0 0) and Fe (1 1 1) surface with a step of 50 K, respectively. However, for surface vacancy diffusion, 4 ns simulations were carried out in the range of 300–650 K for the Fe (1 1 0) and Fe (1 1 1) surface and 850–1000 K for the Fe (1 0 0) surface with a step of 50 K, respectively. At each temperature, we used the lattice constant resulting to zero pressure for the system. The diffusion coefficient D, was calculated using the jump frequency, f = n/t. Here, n is the number of the adatom or vacancy hopping events in the time interval t = 4 ns, given by,
D=
fd2 2z
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3. Results and discussions 3.1. Relaxation of three Fe surfaces At first, we have calculated relative layer spacing relaxation of three low index surfaces, Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) at 0 K. The relative layer spacing relaxation is the percent calculated by the expression, (ds − db )/db . The difference ds and db represents the relaxed layer spacing of the surface and the corresponding bulk interlayer spacing, respectively. A positive value indicates an expansion; a negative value, a contraction. In Fig. 1, we give the relative layer spacing relaxation of these three surfaces. In this figure, some interesting results were showed. Firstly, the relative layer spacing relaxation decreased gradually with the increasing number of layers and converged finally into zero in the bulk. Between the first and second layer, the layer spacing relaxation is larger than other layers. Atoms near the surface have more relaxation to achieve minimum energy. Secondly, the amplitude of the relative layer spacing relaxation in the Fe (1 1 1) surface is more severe than other two surfaces. Similar to the surface energy, the amplitude of the relative layer spacing relaxation has a growing tend in a turn of (1 1 0), (1 0 0) and (1 1 1) surface, i.e., R(110) < R(100) < R(111) . Here, R represents the amplitude of the relative layer spacing relaxation. For the body-center cubic (BCC) lattice, the (1 1 0) surface atoms are the most compact and the (1 1 1) surface atoms are the loosest. The order among the low-index surfaces, R(110) < R(100) < R(111) is in agreement with experimental information [21]. Thirdly, atoms in the first layer perform an outward displacement towards the surface layer for the Fe (1 1 0) and Fe (1 0 0) surface because the relative layer spacing relaxation is positive, that is expansion. However, atoms in the Fe (1 1 1) surface layer have an inward displacement. It is well consistent with previous reported results [12,22–25]. 3.2. Point defect adsorption and formation energy Adatom adsorption energies on three Fe surfaces have been computed by using molecular dynamics relaxation technique at 0 K (static relaxation), given by the expression, Ea = NEsurf − (N + 1)EN+1 ,
(3)
where Ea , Esurf and EN+1 represent the adsorption energy, average atom energy of the N atoms slab, and average atom energy of the
(1)
where z is the dimensionality of the diffusion space (z = 2 for surface diffusion), is the number of jump directions ( = 4 for the Fe (1 1 0) and Fe (1 0 0), but = 3 for the Fe (1 1 1) surface) and d is the jump distance. The diffusion migration energy (E) can be deduced using the result of the transition state theory, D = D0 exp
−E kT
(2)
D0 is the pre-exponential factor and k is the Boltzmann’s constant.
Fig. 1. Relative layer spacing relaxation of three low index surfaces, Fe (1 1 0) (solid square), Fe (1 0 0) (solid circle) and Fe (1 1 1) (solid triangle), has been calculated using the EAM potential at 0 K. Ii,j stands for inter-layer number between the ith and jth layer.
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Table 1 Adatom Formation and adsorption energies and vacancy formation energies on three Fe surfaces were calculated by molecular dynamics with the EAM method. The number of broken 1NN bonds, when a vacancy has been formed in each surface, has been listed as well. DH and SH stand for deep and shallow hollow site of an f adatom adsorbed on the Fe (1 1 1) surface, respectively. Eaa and Ea represent adatom f
adsorption and formation energy, respectively. Ev is the vacancy formation energy. For comparison, the available values in references have been listed as well. Properties
Fe (1 1 0)
Fe (1 0 0)
Fe (1 1 1)
Eaa (eV) f Ea (eV) f Ev (eV) Number of broken NN bonds
3.26, 3.52 [14] 1.03 0.70 6
3.70 0.58 0.56 4
4.19DH , 3.27SH 0.09DH , 1.01SH 0.08, −0.0150 [13] 3
slab and an adatom, respectively. The formation energies on three surfaces can be calculated by the expression, Ef = (N ± 1)EN±1 − NEsurf ∓ Ec
(4)
where Ef , Esurf and Ec refer to the formation energy, the average atom energy of the N atoms slab, and the cohesive energy, respectively. Here Ec equals to −4.28 eV. EN±1 stands for the average atom energy of the N ± 1 atoms system (positive for an adatom and negative for a vacancy). In Table 1, adatom adsorption and formation energies on three Fe low miller surfaces, Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1), have been listed. We have also listed in Table 1 the vacancy formation energy and the number of broken nearest-neighbor (NN) bonds required to form a vacancy in each surface. Only stable adsorbed positions have been given here. For BCC lattice structure, an adatom is adsorbed on the 2-, 4-, and 3-fold symmetrical hollow site on the (1 1 0), (1 0 0) and (1 1 1) surface, respectively. There are two 3-fold symmetrical sites for Fe (1 1 1) surface. That are, a deep (DH) and a shallow (SH) hollow site. This distinction bases on whether there is an atom in the second layer substrate atoms under the adatom. From the calculated results, the adsorption energy of an adatom adsorbed on the SH site was lower, 0.92 eV, than that of one adsorbed on the DH site. It was revealed that the SH site was a metastable adsorbed position. It could be found that adatom adsorption energies on three surfaces increased, except the SH site a a a on the Fe (1 1 1) surface, in the sequence, Ea(110) < Ea(100) < Ea(111) . It turned out that the Fe adatom was more tightly bound with the Fe (1 1 1) surface than the other two surfaces. Accordingly, there was a reverse trend for adatom and vacancy formation energy, i.e., f f f f f f Ea(110) > Ea(100) > Ea(111) , Ev(110) > Ev(100) > Ev(111) . For FCC metals, similar tendency is also obtained by MEAM calculations [26]. This might be related to the surface atom density. The Fe (1 1 1) surface has the least surface atom density among three low-index surfaces and is the most open surface, so it is the easiest for an adatom or a
vacancy to form on it. On the contrary, it is the most difficult to form on the closest Fe (1 1 0) surface. The results also indicated that it was more difficult to form an adatom than form a vacancy in the same surface, especially for the Fe (1 1 0) surface. Generally, the vacancy formation energy increases as the number of bonds required to be broken increases. Six, four and three nearest-neighbor (NN) bonds must be broken to form a vacancy in the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, respectively. It can also result in above results. 3.3. Point defect diffusion migration energy The potential energy map of an adatom on each surface has been calculated at 0 K. In Fig. 2, the maps were presented. The color scale indicated (going from blue to red) the gradual increase of the energy. As we can see in this figure, two-fold position (H) is the most energetically favored position on the Fe (1 1 0) surface (Fig. 2(a)); on the Fe (1 0 0) and Fe (1 1 1) surface, four- and three-fold position, respectively (Fig. 2(b) and (c)). Dash line showed the adatom simple jumping diffusion path in this figure for each surface. On the Fe (1 1 0) surface, an adatom jumped from one two-fold position (H) to another nearest-neighbor position over a 0.24 eV energy barrier along the [−1 1 1] compact direction. It is consistent well with the√result, 0.30 eV, reported by Chen et al. [14]. Its jump distance is 3a0 /2. a0 is the lattice constant of bulk Fe. On the Fe (1 0 0) surface, an adatom need conquer a 0.84 eV energy barrier through simple jumping from a four-fold position to another across a a0 distance along the [1 0 0] direction. From Fig. 2(c), we can find a lowenergy path for an adatom diffusing on the Fe (1 1 1) surface, from a deep-hollow (DH) position to another across a shallow-hollow (SH) position. Through this zigzag path, an adatom need overcome a 0.97 eV energy barrier. All of diffusion migration energies of an adatom and a vacancy simple jumping mechanism on three surfaces have been listed in Table 2. A vacancy migration to its nearest-neighbor atom site is equivalent to the atom migration to the neighboring vacant site. Diffusion migration energy of an adatom exchanging with a surface atom on the Fe (1 0 0) and Fe (1 1 1) surface has been calculated and listed in Table 2 as well. Adatoms exchanging with the surface atoms can occur in the process of adatoms diffusion on FCC (1 0 0) surface [2]. The calculated adatom migration energy of diagonal exchange mechanism on the Fe (1 0 0) is 0.60 eV, which is consistent well with the experimental value 0.45 eV [3,4] measured by STM and RHEED. Because of high surface energy and adatom adsorption energy of the Fe (1 1 1) surface with three-fold symmetry surface atom structure, single adatom or vacancy transfers relatively difficultly on this surface. The migration energy of a single vacancy on the Fe (1 1 1) surface along the [1–10] direction through simple jumping, is very high, 0.99 eV, which is consistent well with the result reported in reference [13] 0.8780 eV. However, through
Fig. 2. Potential energy map for various adatom positions on three surfaces: (a) Fe (1 1 0), (b) Fe (1 0 0), (c) Fe (1 1 1). The color scale indicates (going from blue to red) the gradual increase of the energy (eV). H refers to the adatom adsorbed positions. DH refers to the deep hollow adatom position on the Fe (1 1 1) surface. SH is the shallow hollow site on the Fe (1 1 1) surface. Dash line shows the adatom simple-jump diffusion path. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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Table 2 An adatom and vacancy diffusion migration energies on three Fe surfaces have been computed by molecular dynamics with the EAM and NEB method. Eam and Evm represent the adatom and vacancy diffusion migration energy, respectively. SJ and EX refer to simple jumping and exchange mechanism, respectively. For comparison, the available values in references have been listed as well. Properties
Fe (1 1 0)
Fe (1 0 0)
Fe (1 1 1)
Eam (eV) Evm (eV)
0.24SJ , 0.30 [14] 0.30SJ
0.84SJ , 0.60EX , 0.45 [3,4] 0.90SJ , 0.90EX , 0.90 [15]
0.97SJ , 0.90EX 0.99SJ , 0.28EX , 0.8780 [13]
zigzag exchange mechanism with a surface atom, a vacancy diffuses with lower migration energy on the Fe (1 1 1) surface, 0.28 eV, calculated by the NEB method. From Table 2, only considering the simple jumping mechanism, we can find that adatom diffusion migration energy has a simim m m lar trend with adsorption energy, i.e., Ea(110) < Ea(100) < Ea(111) and vacancy migration energy, has a reverse tendency with formation energy, i.e., Evm(110) < Evm(100) < Evm(111) . However, considered the exchange mechanism, vacancy migration energy has a different order for three surfaces, i.e., Evm(111) < Evm(110) < Evm(100) . Therefore, it is important to consider exchange mechanism for point defect (adatom and vacancy) diffusion on the Fe (1 0 0) and Fe (1 1 1) surface. 3.4. Molecular dynamics simulations at finite temperatures To study self-diffusion of single adatom or vacancy on three surfaces, we have performed molecular dynamics simulations in the temperature range of 300–1000 K (different systems at different temperatures). From the analysis of trajectories of the adatom or vacancy, diffusion mechanisms observed are simple jumping, diagonal exchange and zigzag exchange on the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, respectively. Some longer hops are also observed. Such complicated events, however, are not very often even at higher temperatures. And therefore no further analysis is possible. On the Fe (1 1 0) surface, the most frequent diffusion mechanism is simple jumping, which is found also statistically as energetically favored. For an adatom, this diffusion mechanism simulated at 300 K is presented in Fig. 3, in which the adatom (black solid circle) hops from one two-fold minimum energy site (Fig. 3(a)) to another nearest-neighbor two-fold adsorbed position (Fig. 3(d)) over the two-fold bridge site (Fig. 3(c)). The whole diffusion process lasts for about 1.6 ps. For a vacancy, this diffusion mechanism simulated at 400 K is presented in Fig. 4, in which the surface vacancy (hollow circle labeled V) (Fig. 4(a)) migrates to the position of a nearestneighbor surface atom (Fig. 4(c)). The latter moves into the initial position of the vacancy over the two-fold bridge site (Fig. 4(d)). The duration of this mechanism is about 1.2 ps. On the Fe (1 0 0) surface, the adatom or a vacancy diffuses mainly by diagonal exchange mechanism performed at 700 and 900 K as showed in Fig. 5 and Fig. 6, respectively. In the adatom diffusion process, the adatom (black solid circle) moves from a four-fold hollow site (Fig. 5(a)) to the neighboring surface atom (circled light grey circle) (Fig. 5(b) and (c)) by replacing it. The latter moves to the four-fold adsorbed position along the diagonal (Fig. 5(d)) and finally becomes an adatom. This process wastes about 1.6 ps. For vacancy diffusion on this surface, an atom in the second layer (light grey circle labeled B), adjacent to the surface vacancy (hollow circle labeled V) (Fig. 6(a), leaves its position to the surface vacancy (Fig. 6(c)). The surface atom (black solid circle labeled A), in turn, hops into the second layer and takes the place of the second layer atom (labeled B) (Fig. 6(d)). Thus, the vacancy returns to the first layer. The duration of this process lasts about 2.0 ps. Due to the high adsorption energy (4.19 eV), the adatom diffuses difficultly on the Fe (1 1 1) surface. Molecular dynamics simulations have been carried out at 900 K to observe the process
Fig. 3. By simple jumping mechanism, Fe adatom diffuses on the Fe (1 1 0) surface at 300 K. Light gray and black solid circles represent substrate atoms and adatoms, respectively.
Fig. 4. By simple jumping mechanism, a vacancy diffuses on the Fe (1 1 0) surface at 400 K. Light gray and hollow circles represent substrate atoms and vacancies, respectively.
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Fig. 5. By diagonal exchange mechanism, Fe adatom diffuses on the Fe (1 0 0) surface at 700 K. Light gray and black solid circles represent substrate atoms and adatoms, respectively.
of Fe adatom diffusion on the Fe (1 1 1) surface as presented as Fig. 7. Zigzag exchange mechanism has been observed. In this exchange process, the adatom (black solid circle) migrates from a three-fold deep hollow site (Fig. 7(a)) to the neighboring surface atom (circled light grey circle) (Fig. 7(b) and (c)) by replacing it. The
Fig. 6. By diagonal exchange mechanism, a vacancy diffuses on the Fe (1 0 0) surface at 900 K. Black solid and light gray circles represent the first and second layer atoms in the Fe (1 0 0) surface, respectively.
Fig. 7. By zigzag exchange mechanism, Fe adatom diffuses on the Fe (1 1 1) surface at 900 K. Light grey and black solid circles represent substrate atoms and adatoms, respectively.
latter moves into the three-fold deep hollow position (Fig. 7(d)), and then becomes an adatom. This process lasts about 1.8 ps. About a vacancy on the Fe (1 1 1) surface, it has small formation energy (0.08 eV). Moreover, in previous static calculation at 0 K, the vacancy has high migration energy, 0.99 eV in our work and 0.8780 eV reported by Wen et al. [13]. It is supposed that the vacancy diffusion is quite difficult on this surface. However, in our molecular dynamics simulations, a frequent zigzag exchange mechanism was observed. We gave the simulations performed at 400 K as presented as Fig. 8. An atom in the second layer (light grey circle labeled B), adjacent to the surface vacancy (hollow circle labeled V) (Fig. 8(a), left its position to the surface vacancy (Fig. 8(c)). The surface atom (black solid circle labeled A), in turn, hopped into the second layer and took the place of the second layer atom (labeled B) (Fig. 8(d)). Therefore, the vacancy returned to the first layer. This diffusion process lasted about 2.0 ps. The above adatom and vacancy diffusion processes exhibit Arrhenius behavior from where we deduced the corresponding migration energies as showed as Figs. 9 and 10. For the adatom diffusion (Fig. 9), it was found that the migration energy required for the simple jumping mechanism on the Fe (1 1 0) surface is 0.20 eV (which is lower than that of the result, 0.40 eV, given in the Ref. [14]), the one for the diagonal exchange mechanism on the Fe (1 0 0) surface is 0.43 eV, and that of the zigzag exchange on the Fe (1 1 1) surface is 0.81 eV. Comparing with the static calculations, we can see an overall good agreement, with a small underestimate. For the vacancy diffusion, however, it is not as simple as the adatom. It is worthwhile to note that the value of the adatom diffusion migration energy on the Fe (1 0 0) is lower than that reported in the Ref. [12], although the same EAM potential has been used. This issue is probably attributed to the time step and total simulation time. In Ref. [12], they have simulated about 10–50 ns for the (1 0 0) surface with a 5 fs time step. Moreover they have focused on three diffusion mechanisms. We simulated for 4 ns with a time step of 2 fs. In our simulations, we have also observed the other mechanisms.
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Fig. 10. Arrhenius diagram for the vacancy diffusion on three Fe surfaces. For the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, simple jumping, diagonal exchange and zigzag exchange mechanism have been given, respectively. The corresponding migration energy has been deduced and given, as well.
Fig. 8. By zigzag exchange mechanism, a vacancy diffuses on the Fe (1 1 1) surface at 400 K. Black solid and light gray circles represent the first and second layer atoms in the Fe (1 1 1) surface, respectively.
But their diffusion rates are very small. It is not enough statistically. Maybe it is the explanation to a diffusion migration energy of Fe/Fe(1 0 0) system E = 0.43 eV, which is lower than that of Ref. [12] (E = 0.66 eV), although the use the same EAM potential. However, m m m the order of migration energy, Ea(110) < Ea(100) < Ea(111) , is true. As Fig. 10 shows, the vacancy migration energy required for the zigzag exchange on the Fe (1 1 1) is only 0.27 eV, the one for the simple jumping mechanism on the Fe (1 1 0) surface is 0.33 eV, and that of the diagonal exchange on the Fe (1 0 0) surface is 0.86 eV (which is consistent well with the result, 0.90 eV, that reported in Ref. [15]). Compared with the static calculations, they have an overall good agreement, with a small underestimate for zigzag and diagonal exchange mechanism. From the Arrhenius diagram (Fig. 9) of
the adatom diffusion, it revealed that the migration energy has a m m m sequence, Ea(110) < Ea(100) < Ea(111) , which is consistent with the static calculation result. However, for the vacancy diffusion, the Arrhenius diagram (Fig. 10) showed that the migration energy had an order of Evm(111) < Evm(110) < Evm(100) , which is not like the static calculation result of simple jumping, Evm(110) < Evm(100) < Evm(111) . This revealed that exchange mechanism play an important role in diffusion on the Fe (1 0 0) and Fe (1 1 1) surface. Let us revisit these three most favored mechanisms, simple jumping along the most [−1 1 1] direction on the (1 1 0) surface, diagonal exchange along the [1 1 −1] direction on the (1 0 0) surface, and zigzag exchange along the [1–11] direction, respectively. For BCC lattice, a series of 1 1 −1 directions are the most closely packed. All of these diffusion directions have the most compact atom density. For FCC lattice, atoms diffuse along the closest direction [1,2]. It is consistent with that the point defect (adatom and vacancy) migrates predominantly along the direction with the closest atom density. 4. Conclusions Single point defect (adatom and vacancy) adsorption and diffusion properties on the Fe (1 1 0), Fe (0 0 1), and Fe (1 1 1) surface have been studied by molecular dynamics simulations combining the EAM potential. Formation and diffusion migration energies of the adatom and vacancy on these three surfaces have been calculated. It is presented that adatom adsorption and diffusion migration energies increase in turn according to the three surfaces, a a a m m m Ea(110) < Ea(100) < Ea(111) , Ea(110) < Ea(100) < Ea(111) . However, for a vacancy, its formation and migration energies have different f f f order, i.e., formation energy Ev(111) < Ev(100) < Ev(110) , migration energy, Evm(111) < Evm(110) < Evm(100) . Our results present as well that the adatom and vacancy simple jumping is the main diffusion mechanism on the Fe (1 1 0) surface; however, the adatom and vacancy exchange mechanism with the surface atom plays a dominant role on the Fe (1 0 0) and Fe (1 1 1) surfaces. Acknowledgments
Fig. 9. Arrhenius diagram for the adatom diffusion on three Fe surfaces. For the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, simple jumping, diagonal exchange and zigzag exchange mechanism have been given, respectively. The corresponding migration energy has been deduced and given, as well.
The authors acknowledge the financial support from the NSF of China (grant no. 60876014), and the NSF of Henan province (grant nos. 072300410180, 092300410131). The project is also sponsored
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A molecular dynamics simulation of self-diffusion on Fe surfaces Changqing Wang a,b,∗ , Zhen Qin c , Yongsheng Zhang b , Qiang Sun a , Yu Jia a a b c
School of Physics and Engineering, Center for Clean Energy and Quantum Structures, Zhengzhou University, Zhengzhou 450052, China Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
a r t i c l e
i n f o
Article history: Received 19 October 2011 Received in revised form 17 December 2011 Accepted 18 December 2011 Available online 26 December 2011
a b s t r a c t Using embedded-atom-method (EAM) potential, an adatom and a vacancy diffusion processes have been simulated in detail by molecular dynamics on three Fe surfaces, Fe (1 1 0), Fe (1 0 0), and Fe (1 1 1). Our results reveal that adatom adsorption energies and diffusion migration energies on these surfaces have similar monotonic trend to the relative layer spacing relaxation, R(110) < R(100) < R(111) , adsorption energy, a a a m m m Ea(110) < Ea(100) < Ea(111) , diffusion migration energy, Ea(110) < Ea(100) < Ea(111) . However, for a vacancy, f
Keywords: EAM potential Molecular dynamics Diffusion Nudged elastic band (NEB) method Iron Adsorption Vacancy
f
f
formation and migration energies have a different trend, formation energy, Ev(111) < Ev(100) < Ev(110) , migration energy, Evm(111) < Evm(110) < Evm(100) . On the Fe (1 1 0) surface, simple jumping of an adatom (or a vacancy) is the main diffusion mechanism with relatively low migration energy barrier; nevertheless, exchange with a surface atom plays a dominant role in surface diffusion on the Fe (1 0 0) and Fe (1 1 1) surfaces. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The adatoms and vacancies are the main carriers in mass transport processes on metal surfaces. A detailed knowledge of the energy and mobility of the point defects (adatoms and vacancies) formed in the surface is important for understanding many surface phenomena such as oxidation, corrosion, catalysis, crystal growth, surface roughening, etc. On the atomic scale, investigating the diffusion behaviors of single adatom or vacancy on the surface, aids our understanding of the mechanism of these surface processes. These issues are of great interest for scientific reasons and technological applications. These problems have been intensively studied [1,2]. Atom diffusion on face-centered cubic (FCC) metal surfaces has been studied extensively [1,2]. Atom diffusion on the BCC Fe [3–8], Mo [9,10] surfaces was studied experimentally using the STM, the reflection high-energy electron diffraction (RHEED), ion scattering, spin polarized low energy electron microscope (SPLEEM) and low energy electron diffraction (LEED) techniques, but the dominant diffusion mechanism could not be identified. Theoretically, using First Principles [11], Molecular Dynamics [12–16] and Monte Carlo (MC) [17,18], it has been investigated intensively.
∗ Corresponding author at: Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China. E-mail address: [email protected] (C. Wang). 0169-4332/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2011.12.084
Iron is a very common metal for application. Abundant work has been done on atom diffusion on the Fe surfaces for comparison. STM observation at the temperature range of 20–250 ◦ C has been reported that the activation energy of a Fe atom diffusion on Fe (0 0 1) surface yields 0.45 eV [3]. Using EAM potential, Chamati et al. [12] carried out molecular dynamics studies on self-diffusion on the Fe (1 0 0) surface and gave adatom diagonal exchange diffusion mechanism. They investigated a vacancy diffusing on the Fe (1 0 0) surface by Molecular Dynamics technique, too [15]. However, little work has been done about adatoms or vacancies diffusion on Fe (1 1 0) and Fe (1 1 1) surfaces. Finite temperature molecular dynamics simulations are very helpful to study adatom diffusion, as they allow tracking the adatom on the surface. Moreover, it is convenient to find low energy diffusion paths and new diffusion mechanisms. In this paper, using EAM potential [12] of iron, molecular dynamics studies have been carried out on a point defect (adatom and vacancy) diffusion on three low index surfaces, Fe (1 1 0), Fe (0 0 1), Fe (1 1 1). Due to different structural stability of these three surfaces, it is well known that the Fe (1 1 0) surface is more stable than the Fe (1 0 0) surface, and the Fe (1 1 1) surface is the most unstable surface [12]. We have found that diffusion mechanism is diverse extremely by our molecular dynamics simulations. On the compact Fe (1 1 0) surface, the adatom or vacancy diffuses mainly by simple hopping mechanism. However, on the Fe (1 0 0) and Fe (1 1 1) surface, it is difficult to diffuse by simple hopping. Adatoms or vacancies exchange with surface atoms take place frequently on these two surfaces. The adatom and vacancy migration energies of
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self-diffusion on the Fe (1 1 1) surface have been firstly calculated by using molecular dynamics combining with EAM potential. Moreover, possible diffusion mechanism has been given in the present work. In addition, comparing to the formation and migration energies on these three Fe surfaces with different relaxation has been presented in this paper.
2. Methods All of the simulations were performed by using embeddedatom-method (EAM) potential for BCC Fe [12,19]. Although the potential is semi-empirical, it can give reasonable results about the bulk Fe and Fe (1 0 0) surface [12,15], such as elastic constants, phonon dispersion curves, vacancy and adatom diffusion properties. All of calculations were carried out by using twenty 400-atom (total 8000 Fe atoms) layers to simulate the Fe surface, with periodic boundary conditions in the two directions parallel to the surface. Atoms in upper eighteen layers of the slab with a 20 A˚ vacuum to eliminate the interaction between its two surfaces are free to relax in the three directions. Nevertheless, atoms in the lowest two layers of the slab are fixed to the BCC lattice. To control temperature to be isothermal, the velocity scale constant temperature scheme was applied to the system. For the numerical integration, the Verlet algorithm was employed in our calculations at the constant temperature, with a time step of 2 fs. To simulate the surface in presence of an adatom or a vacancy, one Fe atom was placed on each relaxed surface or one surface atom was removed from each surface and the systems were equilibrated at the desired temperatures for 20 ps. The static diffusion barriers and the minimum energy paths were obtained using the nudged elastic band (NEB) method [20]. It uses both the initial and the final states of the system to calculate a chain of “images” that initially represents the intermediate configurations. The static diffusion barrier is then obtained from the image with the highest potential energy, which can be identified as the saddle point configuration. For all of static calculations, the system has been relaxed into the minimum energy at 0 K. For adatom diffusion processes, simulations for 4 ns were performed covering the temperature range of 300–600 K for the Fe (1 1 0) surface and 600–1000 K for the Fe (1 0 0) and Fe (1 1 1) surface with a step of 50 K, respectively. However, for surface vacancy diffusion, 4 ns simulations were carried out in the range of 300–650 K for the Fe (1 1 0) and Fe (1 1 1) surface and 850–1000 K for the Fe (1 0 0) surface with a step of 50 K, respectively. At each temperature, we used the lattice constant resulting to zero pressure for the system. The diffusion coefficient D, was calculated using the jump frequency, f = n/t. Here, n is the number of the adatom or vacancy hopping events in the time interval t = 4 ns, given by,
D=
fd2 2z
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3. Results and discussions 3.1. Relaxation of three Fe surfaces At first, we have calculated relative layer spacing relaxation of three low index surfaces, Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) at 0 K. The relative layer spacing relaxation is the percent calculated by the expression, (ds − db )/db . The difference ds and db represents the relaxed layer spacing of the surface and the corresponding bulk interlayer spacing, respectively. A positive value indicates an expansion; a negative value, a contraction. In Fig. 1, we give the relative layer spacing relaxation of these three surfaces. In this figure, some interesting results were showed. Firstly, the relative layer spacing relaxation decreased gradually with the increasing number of layers and converged finally into zero in the bulk. Between the first and second layer, the layer spacing relaxation is larger than other layers. Atoms near the surface have more relaxation to achieve minimum energy. Secondly, the amplitude of the relative layer spacing relaxation in the Fe (1 1 1) surface is more severe than other two surfaces. Similar to the surface energy, the amplitude of the relative layer spacing relaxation has a growing tend in a turn of (1 1 0), (1 0 0) and (1 1 1) surface, i.e., R(110) < R(100) < R(111) . Here, R represents the amplitude of the relative layer spacing relaxation. For the body-center cubic (BCC) lattice, the (1 1 0) surface atoms are the most compact and the (1 1 1) surface atoms are the loosest. The order among the low-index surfaces, R(110) < R(100) < R(111) is in agreement with experimental information [21]. Thirdly, atoms in the first layer perform an outward displacement towards the surface layer for the Fe (1 1 0) and Fe (1 0 0) surface because the relative layer spacing relaxation is positive, that is expansion. However, atoms in the Fe (1 1 1) surface layer have an inward displacement. It is well consistent with previous reported results [12,22–25]. 3.2. Point defect adsorption and formation energy Adatom adsorption energies on three Fe surfaces have been computed by using molecular dynamics relaxation technique at 0 K (static relaxation), given by the expression, Ea = NEsurf − (N + 1)EN+1 ,
(3)
where Ea , Esurf and EN+1 represent the adsorption energy, average atom energy of the N atoms slab, and average atom energy of the
(1)
where z is the dimensionality of the diffusion space (z = 2 for surface diffusion), is the number of jump directions ( = 4 for the Fe (1 1 0) and Fe (1 0 0), but = 3 for the Fe (1 1 1) surface) and d is the jump distance. The diffusion migration energy (E) can be deduced using the result of the transition state theory, D = D0 exp
−E kT
(2)
D0 is the pre-exponential factor and k is the Boltzmann’s constant.
Fig. 1. Relative layer spacing relaxation of three low index surfaces, Fe (1 1 0) (solid square), Fe (1 0 0) (solid circle) and Fe (1 1 1) (solid triangle), has been calculated using the EAM potential at 0 K. Ii,j stands for inter-layer number between the ith and jth layer.
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Table 1 Adatom Formation and adsorption energies and vacancy formation energies on three Fe surfaces were calculated by molecular dynamics with the EAM method. The number of broken 1NN bonds, when a vacancy has been formed in each surface, has been listed as well. DH and SH stand for deep and shallow hollow site of an f adatom adsorbed on the Fe (1 1 1) surface, respectively. Eaa and Ea represent adatom f
adsorption and formation energy, respectively. Ev is the vacancy formation energy. For comparison, the available values in references have been listed as well. Properties
Fe (1 1 0)
Fe (1 0 0)
Fe (1 1 1)
Eaa (eV) f Ea (eV) f Ev (eV) Number of broken NN bonds
3.26, 3.52 [14] 1.03 0.70 6
3.70 0.58 0.56 4
4.19DH , 3.27SH 0.09DH , 1.01SH 0.08, −0.0150 [13] 3
slab and an adatom, respectively. The formation energies on three surfaces can be calculated by the expression, Ef = (N ± 1)EN±1 − NEsurf ∓ Ec
(4)
where Ef , Esurf and Ec refer to the formation energy, the average atom energy of the N atoms slab, and the cohesive energy, respectively. Here Ec equals to −4.28 eV. EN±1 stands for the average atom energy of the N ± 1 atoms system (positive for an adatom and negative for a vacancy). In Table 1, adatom adsorption and formation energies on three Fe low miller surfaces, Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1), have been listed. We have also listed in Table 1 the vacancy formation energy and the number of broken nearest-neighbor (NN) bonds required to form a vacancy in each surface. Only stable adsorbed positions have been given here. For BCC lattice structure, an adatom is adsorbed on the 2-, 4-, and 3-fold symmetrical hollow site on the (1 1 0), (1 0 0) and (1 1 1) surface, respectively. There are two 3-fold symmetrical sites for Fe (1 1 1) surface. That are, a deep (DH) and a shallow (SH) hollow site. This distinction bases on whether there is an atom in the second layer substrate atoms under the adatom. From the calculated results, the adsorption energy of an adatom adsorbed on the SH site was lower, 0.92 eV, than that of one adsorbed on the DH site. It was revealed that the SH site was a metastable adsorbed position. It could be found that adatom adsorption energies on three surfaces increased, except the SH site a a a on the Fe (1 1 1) surface, in the sequence, Ea(110) < Ea(100) < Ea(111) . It turned out that the Fe adatom was more tightly bound with the Fe (1 1 1) surface than the other two surfaces. Accordingly, there was a reverse trend for adatom and vacancy formation energy, i.e., f f f f f f Ea(110) > Ea(100) > Ea(111) , Ev(110) > Ev(100) > Ev(111) . For FCC metals, similar tendency is also obtained by MEAM calculations [26]. This might be related to the surface atom density. The Fe (1 1 1) surface has the least surface atom density among three low-index surfaces and is the most open surface, so it is the easiest for an adatom or a
vacancy to form on it. On the contrary, it is the most difficult to form on the closest Fe (1 1 0) surface. The results also indicated that it was more difficult to form an adatom than form a vacancy in the same surface, especially for the Fe (1 1 0) surface. Generally, the vacancy formation energy increases as the number of bonds required to be broken increases. Six, four and three nearest-neighbor (NN) bonds must be broken to form a vacancy in the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, respectively. It can also result in above results. 3.3. Point defect diffusion migration energy The potential energy map of an adatom on each surface has been calculated at 0 K. In Fig. 2, the maps were presented. The color scale indicated (going from blue to red) the gradual increase of the energy. As we can see in this figure, two-fold position (H) is the most energetically favored position on the Fe (1 1 0) surface (Fig. 2(a)); on the Fe (1 0 0) and Fe (1 1 1) surface, four- and three-fold position, respectively (Fig. 2(b) and (c)). Dash line showed the adatom simple jumping diffusion path in this figure for each surface. On the Fe (1 1 0) surface, an adatom jumped from one two-fold position (H) to another nearest-neighbor position over a 0.24 eV energy barrier along the [−1 1 1] compact direction. It is consistent well with the√result, 0.30 eV, reported by Chen et al. [14]. Its jump distance is 3a0 /2. a0 is the lattice constant of bulk Fe. On the Fe (1 0 0) surface, an adatom need conquer a 0.84 eV energy barrier through simple jumping from a four-fold position to another across a a0 distance along the [1 0 0] direction. From Fig. 2(c), we can find a lowenergy path for an adatom diffusing on the Fe (1 1 1) surface, from a deep-hollow (DH) position to another across a shallow-hollow (SH) position. Through this zigzag path, an adatom need overcome a 0.97 eV energy barrier. All of diffusion migration energies of an adatom and a vacancy simple jumping mechanism on three surfaces have been listed in Table 2. A vacancy migration to its nearest-neighbor atom site is equivalent to the atom migration to the neighboring vacant site. Diffusion migration energy of an adatom exchanging with a surface atom on the Fe (1 0 0) and Fe (1 1 1) surface has been calculated and listed in Table 2 as well. Adatoms exchanging with the surface atoms can occur in the process of adatoms diffusion on FCC (1 0 0) surface [2]. The calculated adatom migration energy of diagonal exchange mechanism on the Fe (1 0 0) is 0.60 eV, which is consistent well with the experimental value 0.45 eV [3,4] measured by STM and RHEED. Because of high surface energy and adatom adsorption energy of the Fe (1 1 1) surface with three-fold symmetry surface atom structure, single adatom or vacancy transfers relatively difficultly on this surface. The migration energy of a single vacancy on the Fe (1 1 1) surface along the [1–10] direction through simple jumping, is very high, 0.99 eV, which is consistent well with the result reported in reference [13] 0.8780 eV. However, through
Fig. 2. Potential energy map for various adatom positions on three surfaces: (a) Fe (1 1 0), (b) Fe (1 0 0), (c) Fe (1 1 1). The color scale indicates (going from blue to red) the gradual increase of the energy (eV). H refers to the adatom adsorbed positions. DH refers to the deep hollow adatom position on the Fe (1 1 1) surface. SH is the shallow hollow site on the Fe (1 1 1) surface. Dash line shows the adatom simple-jump diffusion path. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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Table 2 An adatom and vacancy diffusion migration energies on three Fe surfaces have been computed by molecular dynamics with the EAM and NEB method. Eam and Evm represent the adatom and vacancy diffusion migration energy, respectively. SJ and EX refer to simple jumping and exchange mechanism, respectively. For comparison, the available values in references have been listed as well. Properties
Fe (1 1 0)
Fe (1 0 0)
Fe (1 1 1)
Eam (eV) Evm (eV)
0.24SJ , 0.30 [14] 0.30SJ
0.84SJ , 0.60EX , 0.45 [3,4] 0.90SJ , 0.90EX , 0.90 [15]
0.97SJ , 0.90EX 0.99SJ , 0.28EX , 0.8780 [13]
zigzag exchange mechanism with a surface atom, a vacancy diffuses with lower migration energy on the Fe (1 1 1) surface, 0.28 eV, calculated by the NEB method. From Table 2, only considering the simple jumping mechanism, we can find that adatom diffusion migration energy has a simim m m lar trend with adsorption energy, i.e., Ea(110) < Ea(100) < Ea(111) and vacancy migration energy, has a reverse tendency with formation energy, i.e., Evm(110) < Evm(100) < Evm(111) . However, considered the exchange mechanism, vacancy migration energy has a different order for three surfaces, i.e., Evm(111) < Evm(110) < Evm(100) . Therefore, it is important to consider exchange mechanism for point defect (adatom and vacancy) diffusion on the Fe (1 0 0) and Fe (1 1 1) surface. 3.4. Molecular dynamics simulations at finite temperatures To study self-diffusion of single adatom or vacancy on three surfaces, we have performed molecular dynamics simulations in the temperature range of 300–1000 K (different systems at different temperatures). From the analysis of trajectories of the adatom or vacancy, diffusion mechanisms observed are simple jumping, diagonal exchange and zigzag exchange on the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, respectively. Some longer hops are also observed. Such complicated events, however, are not very often even at higher temperatures. And therefore no further analysis is possible. On the Fe (1 1 0) surface, the most frequent diffusion mechanism is simple jumping, which is found also statistically as energetically favored. For an adatom, this diffusion mechanism simulated at 300 K is presented in Fig. 3, in which the adatom (black solid circle) hops from one two-fold minimum energy site (Fig. 3(a)) to another nearest-neighbor two-fold adsorbed position (Fig. 3(d)) over the two-fold bridge site (Fig. 3(c)). The whole diffusion process lasts for about 1.6 ps. For a vacancy, this diffusion mechanism simulated at 400 K is presented in Fig. 4, in which the surface vacancy (hollow circle labeled V) (Fig. 4(a)) migrates to the position of a nearestneighbor surface atom (Fig. 4(c)). The latter moves into the initial position of the vacancy over the two-fold bridge site (Fig. 4(d)). The duration of this mechanism is about 1.2 ps. On the Fe (1 0 0) surface, the adatom or a vacancy diffuses mainly by diagonal exchange mechanism performed at 700 and 900 K as showed in Fig. 5 and Fig. 6, respectively. In the adatom diffusion process, the adatom (black solid circle) moves from a four-fold hollow site (Fig. 5(a)) to the neighboring surface atom (circled light grey circle) (Fig. 5(b) and (c)) by replacing it. The latter moves to the four-fold adsorbed position along the diagonal (Fig. 5(d)) and finally becomes an adatom. This process wastes about 1.6 ps. For vacancy diffusion on this surface, an atom in the second layer (light grey circle labeled B), adjacent to the surface vacancy (hollow circle labeled V) (Fig. 6(a), leaves its position to the surface vacancy (Fig. 6(c)). The surface atom (black solid circle labeled A), in turn, hops into the second layer and takes the place of the second layer atom (labeled B) (Fig. 6(d)). Thus, the vacancy returns to the first layer. The duration of this process lasts about 2.0 ps. Due to the high adsorption energy (4.19 eV), the adatom diffuses difficultly on the Fe (1 1 1) surface. Molecular dynamics simulations have been carried out at 900 K to observe the process
Fig. 3. By simple jumping mechanism, Fe adatom diffuses on the Fe (1 1 0) surface at 300 K. Light gray and black solid circles represent substrate atoms and adatoms, respectively.
Fig. 4. By simple jumping mechanism, a vacancy diffuses on the Fe (1 1 0) surface at 400 K. Light gray and hollow circles represent substrate atoms and vacancies, respectively.
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Fig. 5. By diagonal exchange mechanism, Fe adatom diffuses on the Fe (1 0 0) surface at 700 K. Light gray and black solid circles represent substrate atoms and adatoms, respectively.
of Fe adatom diffusion on the Fe (1 1 1) surface as presented as Fig. 7. Zigzag exchange mechanism has been observed. In this exchange process, the adatom (black solid circle) migrates from a three-fold deep hollow site (Fig. 7(a)) to the neighboring surface atom (circled light grey circle) (Fig. 7(b) and (c)) by replacing it. The
Fig. 6. By diagonal exchange mechanism, a vacancy diffuses on the Fe (1 0 0) surface at 900 K. Black solid and light gray circles represent the first and second layer atoms in the Fe (1 0 0) surface, respectively.
Fig. 7. By zigzag exchange mechanism, Fe adatom diffuses on the Fe (1 1 1) surface at 900 K. Light grey and black solid circles represent substrate atoms and adatoms, respectively.
latter moves into the three-fold deep hollow position (Fig. 7(d)), and then becomes an adatom. This process lasts about 1.8 ps. About a vacancy on the Fe (1 1 1) surface, it has small formation energy (0.08 eV). Moreover, in previous static calculation at 0 K, the vacancy has high migration energy, 0.99 eV in our work and 0.8780 eV reported by Wen et al. [13]. It is supposed that the vacancy diffusion is quite difficult on this surface. However, in our molecular dynamics simulations, a frequent zigzag exchange mechanism was observed. We gave the simulations performed at 400 K as presented as Fig. 8. An atom in the second layer (light grey circle labeled B), adjacent to the surface vacancy (hollow circle labeled V) (Fig. 8(a), left its position to the surface vacancy (Fig. 8(c)). The surface atom (black solid circle labeled A), in turn, hopped into the second layer and took the place of the second layer atom (labeled B) (Fig. 8(d)). Therefore, the vacancy returned to the first layer. This diffusion process lasted about 2.0 ps. The above adatom and vacancy diffusion processes exhibit Arrhenius behavior from where we deduced the corresponding migration energies as showed as Figs. 9 and 10. For the adatom diffusion (Fig. 9), it was found that the migration energy required for the simple jumping mechanism on the Fe (1 1 0) surface is 0.20 eV (which is lower than that of the result, 0.40 eV, given in the Ref. [14]), the one for the diagonal exchange mechanism on the Fe (1 0 0) surface is 0.43 eV, and that of the zigzag exchange on the Fe (1 1 1) surface is 0.81 eV. Comparing with the static calculations, we can see an overall good agreement, with a small underestimate. For the vacancy diffusion, however, it is not as simple as the adatom. It is worthwhile to note that the value of the adatom diffusion migration energy on the Fe (1 0 0) is lower than that reported in the Ref. [12], although the same EAM potential has been used. This issue is probably attributed to the time step and total simulation time. In Ref. [12], they have simulated about 10–50 ns for the (1 0 0) surface with a 5 fs time step. Moreover they have focused on three diffusion mechanisms. We simulated for 4 ns with a time step of 2 fs. In our simulations, we have also observed the other mechanisms.
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Fig. 10. Arrhenius diagram for the vacancy diffusion on three Fe surfaces. For the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, simple jumping, diagonal exchange and zigzag exchange mechanism have been given, respectively. The corresponding migration energy has been deduced and given, as well.
Fig. 8. By zigzag exchange mechanism, a vacancy diffuses on the Fe (1 1 1) surface at 400 K. Black solid and light gray circles represent the first and second layer atoms in the Fe (1 1 1) surface, respectively.
But their diffusion rates are very small. It is not enough statistically. Maybe it is the explanation to a diffusion migration energy of Fe/Fe(1 0 0) system E = 0.43 eV, which is lower than that of Ref. [12] (E = 0.66 eV), although the use the same EAM potential. However, m m m the order of migration energy, Ea(110) < Ea(100) < Ea(111) , is true. As Fig. 10 shows, the vacancy migration energy required for the zigzag exchange on the Fe (1 1 1) is only 0.27 eV, the one for the simple jumping mechanism on the Fe (1 1 0) surface is 0.33 eV, and that of the diagonal exchange on the Fe (1 0 0) surface is 0.86 eV (which is consistent well with the result, 0.90 eV, that reported in Ref. [15]). Compared with the static calculations, they have an overall good agreement, with a small underestimate for zigzag and diagonal exchange mechanism. From the Arrhenius diagram (Fig. 9) of
the adatom diffusion, it revealed that the migration energy has a m m m sequence, Ea(110) < Ea(100) < Ea(111) , which is consistent with the static calculation result. However, for the vacancy diffusion, the Arrhenius diagram (Fig. 10) showed that the migration energy had an order of Evm(111) < Evm(110) < Evm(100) , which is not like the static calculation result of simple jumping, Evm(110) < Evm(100) < Evm(111) . This revealed that exchange mechanism play an important role in diffusion on the Fe (1 0 0) and Fe (1 1 1) surface. Let us revisit these three most favored mechanisms, simple jumping along the most [−1 1 1] direction on the (1 1 0) surface, diagonal exchange along the [1 1 −1] direction on the (1 0 0) surface, and zigzag exchange along the [1–11] direction, respectively. For BCC lattice, a series of 1 1 −1 directions are the most closely packed. All of these diffusion directions have the most compact atom density. For FCC lattice, atoms diffuse along the closest direction [1,2]. It is consistent with that the point defect (adatom and vacancy) migrates predominantly along the direction with the closest atom density. 4. Conclusions Single point defect (adatom and vacancy) adsorption and diffusion properties on the Fe (1 1 0), Fe (0 0 1), and Fe (1 1 1) surface have been studied by molecular dynamics simulations combining the EAM potential. Formation and diffusion migration energies of the adatom and vacancy on these three surfaces have been calculated. It is presented that adatom adsorption and diffusion migration energies increase in turn according to the three surfaces, a a a m m m Ea(110) < Ea(100) < Ea(111) , Ea(110) < Ea(100) < Ea(111) . However, for a vacancy, its formation and migration energies have different f f f order, i.e., formation energy Ev(111) < Ev(100) < Ev(110) , migration energy, Evm(111) < Evm(110) < Evm(100) . Our results present as well that the adatom and vacancy simple jumping is the main diffusion mechanism on the Fe (1 1 0) surface; however, the adatom and vacancy exchange mechanism with the surface atom plays a dominant role on the Fe (1 0 0) and Fe (1 1 1) surfaces. Acknowledgments
Fig. 9. Arrhenius diagram for the adatom diffusion on three Fe surfaces. For the Fe (1 1 0), Fe (1 0 0) and Fe (1 1 1) surface, simple jumping, diagonal exchange and zigzag exchange mechanism have been given, respectively. The corresponding migration energy has been deduced and given, as well.
The authors acknowledge the financial support from the NSF of China (grant no. 60876014), and the NSF of Henan province (grant nos. 072300410180, 092300410131). The project is also sponsored
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C. Wang et al. / Applied Surface Science 258 (2012) 4294–4300
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