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Vacancy diffusion driven surface rearrangement in the Cu3Au(0 0 1) and Ni3Al(0 0 1) surfaces
Applied Surface Science 205 (2003) 280±288
Vacancy diffusion driven surface rearrangement in the Cu3Au(0 0 1) and Ni3Al(0 0 1) surfaces Ch.E. Lekka, G.A. Evangelakis* Department of Physics, Solid State Division, University of Ioannina, P.O. Box 1186, Gr-45 110 Ioannina, Greece Received 12 June 2002; received in revised form 12 June 2002; accepted 28 September 2002
Abstract We present molecular dynamics (MD) simulation results on the role of vacancy diffusion in the surface order of the Cu3Au(0 0 l) and Ni3Al(0 0 l) faces. We found that in both systems vacancy diffusion proceeds preferably by hopping along the [1 1 0] direction. In the Cu3Au(0 0 l) case, successive vacancy diffusion events induce irreversible loss of the surface order, while in the Ni3Al(0 0 l) system each vacancy hop leads to unstable atomic arrangements having a clear preference of recovering the initial surface order. These ®ndings can be used for the understanding of the order±disorder transition occurring in the Cu3Au(0 0 l) surface and the order±order kinetics characterizing the Ni3Al(0 0 l) face. In addition, we found that in the former case the vacancy diffusion rate saturates quickly (in <1 ns) and that it is correlated with the surface order parameter which reveals important disorder already at 500 K. In the later system, Al and Ni vacancies have constant hopping frequencies that exhibit Arrhenius behavior, while surface disorder is also predicted at high temperatures. # 2002 Elsevier Science B.V. All rights reserved. PACS: 71.15.D; 66.30.F; 68.35D; 68.35.B; 68.35.R Keywords: Molecular dynamics calculations condensed matter; Self diffusion in metals and alloys; Defects in solid surfaces; Crystal structure of surfaces; Order±disorder transition
1. Introduction The L12 binary intermetallic Cu3Au and Ni3Al alloys have been the subjects of numerous experimental and theoretical investigations. It is suggested that the Ni3Al is a directly ordered alloy, with a virtual transition temperature from atomic order to atomic disorder, equal to or even higher than the melting temperature and that it belongs to the group of the so-called super*
Corresponding author. Tel.: 30-6510-98590; fax: 30-6510-45631. E-mail address: [email protected] (G.A. Evangelakis).
alloys whose mechanical features improve at high temperatures [1±3]. In addition, from residual resistivity isochrone measurements [4,5] and by theoretical calculation [2] it is found that the high activation energy for vacancy formation makes the total activation energy for ``order±order'' kinetics very high. Furthermore, it came out that there are two ordering processes with different relaxation times associated to the jumps of vacancies between the two sub-lattices of the L12 structure. Turning on the Cu3Au case, it is found that this material is a sequentially ordered alloy with an order±disorder transition at Tc 663 K, well below the melting point [6±13]. Residual resistivity measure-
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 2 ) 0 1 1 1 8 - 2
Ch.E. Lekka, G.A. Evangelakis / Applied Surface Science 205 (2003) 280±288
ments (isothermal order±order annealing) [1] revealed that changes in the long-range order (LRO) parameter are mainly caused by jumps between vacancies and anti-site atoms within ordered domains and that in the related ordering relaxation two processes with different relaxation times are involved, as in the Ni3Al case. Focusing on the (0 0 1) surface, it is found that both Ni3Al and Cu3Au faces are rumpled, while to our knowledge there are no detailed studies on the role of the surface vacancy diffusivity of these systems. The LRO parameter of the Cu3Au(0 0 1) surface has been determined by low-energy electron diffraction (LEED) [14±16] equal to unity at temperatures below 500 K, to drop to zero at Tc 663 K. In addition, recent He diffraction experiments [17] provided a very reliable estimation of the long-range order parameter in the topmost atomic layers, consistent with the LEED data, revealing a continuous temperature dependence of the order parameter. Moreover, within the top surface layer, the short-range order (SRO) is also known to decrease with temperature. Low energy ion scattering (LEIS) measurements showed that starting at about 500 K, Au and Cu atoms change positions in the topmost atomic layer, while their concentrations are relatively constant up to the critical temperature [18]. However, recent molecular dynamics (MD) simulations on the adatom diffusion processes on the Cu3Au(0 0 1) surface suggested that the presence of Cu or Au adatoms on this face stimulates phenomena that act as precursors of the order±disorder transition, especially above 500 K [19]. This result is based on the ®nding of adatom activated exchange mechanisms, in which second layer atoms are also involved and on adatom induced vacancy creation, initiated by the correlated motion of Cu and Au surface atoms along the [1 1 0] direction. Both phenomena result in the rearrangement of the atoms in anti-sites and the loss of local order. These predictions were forti®ed recently by medium-energy ion-scattering spectroscopy (MEIS) measurements. Indeed, it was found that the atomic concentration of Au in the ®rst layer decreases from 50% at room temperature to 32% at 720 K [20]. In addition, in the same study, it was suggested that the different results found by LEIS and MEIS, concerning the surface concentration of Au and Cu atoms, are due to the production of surface vacancies and adatoms in the top layer of Cu3Au(0 0 l) surface at elevated temperatures, defects that have no signi®cant effect on the yield of LEIS experiments but are visible
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by MEIS [20]. This means that the presence of surface vacancies and adatoms in the Cu3Au(0 0 1) surface is related to the order±disorder transition that occurs by 150 K earlier than the bulk critical temperature, in agreement with the simulation results. It is therefore, very interesting to investigate and to try to understand the role of vacancy diffusion in the surface order of these alloys. It is the aim of the present work to study the interrelation of the surface vacancy diffusion with the surface order of the Cu3Au(0 0 1), that is characterized by an order±disorder transition and of the Ni3Al(0 0 1), which is characterized by order±order kinetics. 2. Computational details We performed simulations in the isothermal canonical ensemble using the Nose demon to control the temperature [21]. We used a system made up of 3800 atoms (1000 Au/Al and 2800 Cu/Ni) arranged on a FCC lattice corresponding to the stoichiometic and ordered Cu3Au and Ni3Al alloys. Periodic boundary conditions were applied in the three directions to mimic an in®nite system. The free surfaces were produced by ®xing the dimensions of the simulation box at a value twice as large as the thickness of the crystal along the z-direction. An in®nite slab of 19 atomic layers parallel to the (0 0 1) planes is thus produced with 200 atoms each, delimited by two free mixed surfaces, termination that is known to be energetically favored for both alloy surfaces [6±8,22±26] against pure Cu or Ni faces. To describe the attractive part of the atomic interactions we used an effective potential model in analogy to the tight binding scheme in the second moment approximation, while a Born±Mayer type was used for the repulsive contribution [27,28]. Accordingly, the total potential energy can be written as: 8 Nb Na < X X XX ab f Aab e pab rij =dab 1 : b j 1;j 6i a ia 1 a b b v9 u Nb = uX X ab t (1) x2ab e 2pab rij =dab 1 ; b j 1;j 6i b
b
a
where a and b refer to the two different kind of atoms, rijab jr~ia r~jb j is the distance between the ia and jb
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particles, while the indexes ia and jb run over all the particles. The interactions are computed up to the ®fth neighbor's distance. In the case of Cu3Au the parameters pab, qab, Aab, dab, and xab we used, have been determined using results of total-energy calculations as a function of volume, by means of the AugmentedPlane-Wave (APW) method, for the pure Cu, Au and the L12 Cu3Au and Au3Cu alloys [27]. This potential model is found to reproduce well the elastic constants of pure bulk elements and alloys, as well as the bulk modulus, the lattice expansion, atomic mean square displacements (MSDs) and phonon dispersion curves (DOS) of the ordered alloys. In addition, it is found [26] that it reproduces the surface energies, the rippling effect, the atomic MSDs and phonon DOSs for the low-index Cu3Au surfaces and it has been used for the study of the vibrational and diffusive [19,29] properties of Au and Cu adatoms. In the case of Ni3Al, the parameters pab, qab, Aab, dab and xab have been ®tted to the experimental values of cohesive energy, lattice parameters, and elastic constants for the metals and the alloys, in the appropriate crystal structure at T 0 K temperature [28]. The equations of motion were integrated by means of the Verlet's algorithm with a time step of dt 5 10 15 s. One Au/Al (or Cu/Ni) surface atom was removed from each Cu3Au(0 0 1) and Ni3Al(0 0 1) surface and the systems were equilibrated at the desired temperatures for 10 000 time-steps. 2 ns simulations were performed for the ®rst case at 300, 400, 450, 475, 500, 525, 550, 575 and 600 K and from 300 to 1000 K with a step of 100 K for the second one. At each temperature, we used the lattice constants that resulted to zero pressure for the bulk systems. The diffusion coef®cient D, was calculated using the jump frequency, G n=t, that is easily estimated by counting the number of hopping events, n, the surface atoms perform in the time interval t. D
lGd2 2z
(2)
where z is the dimensionality of the diffusion space (two in our case), l the number of jump directions (four in the present case) and d stands for the jump distance. The diffusion migration energy (DE) can be deduced using the standard result of the transition state theory: D Do e
DE=kT
(3)
Do, is the pre-exponential factor and k is the Boltzmann's constant. For the energy relaxation calculations, we used a quasi-dynamic minimization procedure integrated in the Molecular Dynamic code [30]. 3. Results and discussion 3.1. Static calculations-migration paths We calculated the Au and Cu surface vacancy formation energies by minimising the corresponding energies of the ordered Cu3Au(0 0 1) system (at T 0 K) and we found 0.99 and 0.50 eV, respectively. In the case of the ordered Ni3Al(0 0 1) surface, the corresponding formation energies for Al and Ni surface vacancies are 0.92 and 0.77 eV, respectively. It comes out that Cu and Ni vacancies are the energetically favored defects, in agreement with theoretical and experimental results referring to the Ni3Al case [2]. We note that for the correct description of the binding of the systems, we considered the contributions of the species A and B in the cohesion of the A3B alloy instead of the cohesive energies of the pure elements. In addition, although it is known that small deviation from stoichiometry can seriously affect the vacancy formation energies [31], in our case the corresponding changes are minimal and therefore the effect has to be insigni®cant. Concerning the surface vacancy diffusion, there are in principle two in-plane directions along which a hopping event can occur. In Fig. 1a and b we give a schematic representation of the two simple hopping mechanisms, along the [1 1 0] (SH) and the [1 0 0] direction (SH100), respectively. We note that the distances covered when a vacancy performs these two simple hopping mechanisms are different and equal to the nearest neighbor distance and the lattice constant, respectively. In addition, in a SH event of A type vacancy, a B surface atom jumps into the vacancy, Fig. la, resulting in a B vacancy and a B surface atom in anti-site. In a SH100 process the motion of an A surface atom along the [1 0 0] direction takes place in the A-sublattice and has no effect in the local order, Fig. 1b. Using constrained energy minimization, we calculated for both alloys the energy barriers required for A
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Fig. 1. Schematic representation of the vacancy simple hopping mechanisms: (a) along the [1 1 0] direction (SH) and (b) along the [1 0 0] (SH100) direction. Big and small circles stand for the A and B surface atoms of the A3B(0 0 1) surface, while empty site (square) stands for the A vacancy. Arrows show the vacancy hop direction.
Fig. 2. Potential energy barriers of three consecutive vacancy simple hops along the [1 1 0] direction: (a) initially Cu (circles) or Au (squares) vacancies in the Cu3Au(0 0 1) surface and (b) initially Ni (circles) or Al (squares) vacancies in the Ni3Al(0 0 1) surface.
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and B type vacancies to realize these hopping processes. The starting con®guration in all cases was a perfectly ordered surface. Since after a SH hop the local environment changes, in order to follow the evolution of the system in a succession of this kind of events, we calculated the energy barriers for three consecutive jumps of the vacancies. In Fig. 2a and b, we show these quantities for the Cu (circles) and Au (squares) vacancies in the Cu3Au(0 0 1) and the Ni (circles) and Al (squares) vacancies in the Ni3Al(0 0 1) surface, respectively. The initial energy level, zero energy values in Fig. 2, corresponds to the formation energies of the vacancies. It is very interesting to observe the completely different evolution of the energetic requirements for successive hops in the two systems: in the Ni3Al case the energy of the system increases after a jump, while vacancy hops in the Cu3Au(0 0 1) face lead the system to lower energy levels. This means that since the energy needed for the Ni or Al vacancy to perform a second or third hop towards the same direction is higher than for backward movements, these vacancies would prefer to return in their initial perfect positions. On the contrary, in the Cu3Au case the local conformations after a jump are energetically more stable (lower energy); Cu and Au vacancies are, therefore, more likely to continue diffusing, inducing thereby local disorder in the surface. Turning on the SH100 mechanism, we give in Fig. 3, the energy barriers needed for the Al (open diamonds),
Ni (®lled diamonds), Cu (open circles) and Au (®lled circles) vacancies to perform a hop in the [1 0 0] direction. We observe that in all cases the energy barriers required for the SH[1 0 0] mechanism are much higher than the energy barriers needed for the SH process, situated at the values of 2.00 and 1.50 eV, respectively. These results indicate that in-plane diffusion is favored in the [1 1 0] direction. In particular, vacancy diffusion in the Cu3Au(0 0 1) surface is expected to produce re-arrangement of the surface atoms in anti-sites, and loss of the local order, leaving the system in lower energy level and suggests that these processes must have an irreversible tendency explaining the order±disorder transition observed in this surface. In the Ni3Al(0 0 1) case the atomic arrangements after a vacancy diffusion event are less stable and the system relaxes at higher energy, a result that could explain the order±order kinetics that characterize this material. Nevertheless, the energetic requirements for the SH process in this alloy are not very high (<1.0 eV for the third successive hop) and therefore vacancy induced disordering has to be expected at high temperatures. We note here that our results are based on a semiempirical potential model and therefore the calculated values have to be considered with caution. However, we believe that these models are capable in reproducing correctly at least the main features of the vacancy behavior.
Fig. 3. Potential energy barriers for vacancy simple hopping mechanism along the [1 0 0] direction: (a) Cu (open circles) or Au (®lled circles) vacancies in the Cu3Au(0 0 1) surface and (b) Al (open diamonds) or Ni (®lled diamonds) vacancies in the Ni3Al(0 0 1) surface.
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Fig. 4. Time evolution of the surface order (SLRO) parameter along with the number of diffusion events (Nevents) at T 550 K. Dashed line stands for an initial Cu vacancy and solid line for an initial Au vacancy in the Cu3Au(0 0 1) surface.
3.2. Early state vacancy diffusion-surface rearrangement We tried to verify the above predictions by MD simulations. Starting from ordered surfaces containing a vacancy, we monitored the evolution of the vacancy hopping events, n, along with the surface order parameter (SLRO) for several temperatures. The SLRO parameter is calculated as the percentage of surface atoms being in anti-sites. In Fig. 4, we present for the case of a Cu (dashed line) or a Au (solid line) vacancy at T 550 K, the time evolution of the number of SH vacancy diffusion events in the [1 1 0] direction, n, normalized by the total number of hops, along with the corresponding SLRO. We observe that the number of diffusion events and the SLRO parameter are correlated and saturate attaining a steady state after 1 ns. In addition, although the diffusion events and the SLRO parameter of the two vacancies are distinguishable below 0.25 ns, they become very quickly identical, indicating that after an amount of reordering of the surface atoms, the identity of the vacancies is irrelevant. These results are in agreement with residual resistivity measurements in the Cu3Au system [1] from which it was found that changes in the LRO parameter are mainly caused by jumps between vacancies and antisite atoms within ordered domains. Moreover, in the same study it was also suggested that ``if two ordering
processes, related to the two sublattices of the L12structure, are required to explain the experimental results, only one ordering process can be expected above the critical temperature, because in this case no distinction of the sublattices is possible''. This result is in agreement with our static calculation results, according to which a hop of a Au-vacancy along the [1 1 0] direction, results in the production of a Cuvacancy and a Cu atom in anti-site inducing disorder. Similarly, for a Cu vacancy hop along the same direction, a Au atom jumps into the vacant position, resulting in a Au vacancy and a Au atom in anti-site. In addition, we can observe in the same ®gure that the SLRO parameter reaches the value of 0.85 at the steady state, suggesting that there are 15% surface atoms in anti-sites already at 550 K, in agreement with LEIS and MEIS experimental results. In addition, this result indicates that the loss of surface order starts below the bulk critical temperature, in agreement with the experimental data giving an order parameter value of 0.95 for the bulk at this temperature [14]. The vacancy diffusive behavior is similar for all temperatures studied resulting in the loss of surface order, the effect being accentuating as going close to the critical temperature. Turning on the Ni3Al case, we present in Fig. 5a and b, the time evolution of the SLRO parameter along with the number of vacancy hopping events,
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Fig. 5. Time evolution of the surface order (SLRO) parameter along with the number of diffusion events (Nevents) for an initial Ni vacancy (dashed line) and an initial Al vacancy (solid line) in the Ni3Al(0 0 1) surface: (a) T 700 K and (b) T 900 K.
n, calculated at T 700 and 900 K, respectively. As we can see, the diffusive behavior of both Ni and Al vacancies is totally different from the Cu3Au(0 0 1) surface, the number of events exhibiting linear time dependence (constant rate of hopping events). In addition, the order parameter is practically unchanged at T 700 K, while loss of surface order appears in the corresponding curve of T 900 K. We note also that the number of hopping events is by far greater than in the Cu3Au case.
These ®ndings con®rm the different vacancy diffusive behavior, predicted by the energetic requirements for hopping and suggest that the Ni3Al(0 0 1) face disorders also below the melting point. 3.3. Steady state calculations As mentioned before, the analysis of our MD trajectories revealed that Cu or Au vacancies, once formed, diffuse very fast and induce surface disorder,
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Fig. 6. Arrhenius diagrams of the SH diffusion coef®cients (along the [1 1 0] direction) for the Ni (circles) and Al (diamonds) vacancies in the Ni3Al(0 0 1) surface.
which, in turn, is responsible for the decay of their hopping rate. At the steady state, very few hopping events occur. Hence, the study of the vacancy diffusive behavior in the disordered surface was not possible due to bad statistics. On the contrary, in the Ni3Al case, the hopping rates of the vacancy diffusion events are constant, so we were able to calculate them at various temperatures. We found that the most frequent vacancy diffusion mechanism is the SH along the [1 1 0] direction exhibiting Arrhenius behavior up to Ts 900 K. In Fig. 6, we present the corresponding diagrams of the diffusion coef®cients referring to Al and Ni SH vacancy mechanisms, diamonds and circles, respectively. As we can see, below Ts, Al and Ni vacancies have different energetic requirements 0.13 and 0.23 eV, respectively, while above this temperature the identity of Al and Ni vacancies becomes meaningless due to the initiation of disordering process in the Ni3Al(0 0 1) surface and the two diffusion coef®cients saturate at the same value. 4. Conclusions In this communication we present Molecular Dynamics simulation results on the role of vacancies in the surface order of the Cu3Au and Ni3Al(0 0 1) surface. We found that the energetically favored vacancies for these two systems are the Cu and the Ni vacancies, respectively, in agreement with theoretical
and experimental results referring to the Ni3Al case [2]. In addition, we found that from the two available inplane directions for diffusion (the [1 1 0] and the [1 0 0] directions), the ®rst one has the smaller energetic requirements for vacancy diffusion in both alloys. Interestingly, in the case of the Cu3Au(0 0 1) surface, successive vacancy diffusion events induce loss of the surface order with an irreversible probability, while in the Ni3Al(0 0 1) case each vacancy hop lead to unstable atomic arrangements having a clear preference of recovering the initial surface order. These results can be used to explain the order±disorder transition occurring in the Cu3Au(0 0 1) surface and the order±order kinetics characterizing the Ni3Al(0 0 1) face. It has to be noted however, that vacancy-induced disorder has to be expected also in this later case, although at high temperatures. Moreover, from the trajectory analysis we found that in the Cu3Au(0 0 1) case, surface order parameter and the vacancy diffusion events are correlated. It comes out that already at 500 K, 15% surface atoms are located at anti-sites, while no distinction between the initial Cu or Au vacancies is possible due to the presence of substitutional defects induced by their diffusion. These results are in agreement with LEIS [18] and MEIS [20] experimental data and MD [19] calculations suggesting that the loss of surface order starts at T 500 K. From a similar treatment in the Ni3Al(0 0 1) case, we found that contrary to the previous system, the
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vacancy-hopping rate is constant, while important surface disorder is found at 900 K. In addition, we found that the most frequent vacancy diffusion mechanism is the SH along the [1 1 0] direction for both alloys. In the Cu3Au(0 0 1) surface the number of diffusion events saturates very quickly (within a nanosecond), while above T 550 K the surface order parameter has reached almost its maximum value, with 30% of the surface atoms being in anti-sites. On the contrary, in the case of the Ni3Al(0 0 1) surface, Al and Ni vacancies have constant hopping rates, with Arrhenius type temperature dependence up to Ts 900 K, from where we deduced the corresponding migration energies, 0.13 and 0.23 eV, respectively. These ®ndings are in agreement with available experimental observations and can be used to explain the different behavior of these two alloys. Acknowledgements The work was partially supported by the HPRN-CT2000-00038. References [1] H. Lang, H. Uzawa, T. Mohri, W. Pfeiler, Intermetallics 9 (2001) 9. [2] R. Kozubski, Prog. Mater. Sci. 41 (1997) 1. [3] M. Sluiter, Y. Hashi, Y. Kawazoe, Comp. Mater. Sci. 14 (1999) 283. [4] B. Sitaud, X. Zhang, C. Dimitrov, O. Dimitrov, in: H.E. Exner, V. Schuhmacher (Eds.), Advanced Materials and Processes, DGM, Oberusel, 1990, p. 389. [5] C. Dimitrov, T. Tarfa, O. Dimitrov, in: A.R. Yavari (Ed.), Ordering and Disordering in Alloys, Elsevier Applied Science, Barking, 1992, p. 130. [6] T.M. Buck, G.H. Wheatley, L. Marchut, Phys. Rev. Lett. 51 (1983) 43.
[7] H. Niehus, C. Achete, Surf. Sci. 289 (1993) 19. [8] E.G. Mc Rae, T.M. Buck, R.A. Malic, W.E. Wallace, Surf. Sci. 238 (1990) L481. [9] K.D. Jamison, D.M. Lind, F.B. Dunning, G.K. Walters, Surf. Sci. 159 (1985) L451. [10] H. Dosch, L. Mailander, A. Lied, J. Peisl, F. Grey, R.L. Johnson, S. Krummacher, Phys. Rev. Lett. 60 (1988) 2382. [11] H. Dosch, L. Mailander, H. Reichert, J. Peisl, Phys. Rev. B43 (1991) 13172. [12] S. Katano, M. Iizumi, Y. Noda, J. Phys. F: Met. Phys. 18 (1988) 2195. [13] S.M. Foiles, Surf. Sci. 191 (1987) 329. [14] V. Sundaram, B. Farrell, R.S. Alben, W.D. Robertson, Phys. Rev. Lett. 31 (1973) 1136. [15] E.G. McRae, R.A. Malic, Surf. Sci. 148 (1984) 551. [16] J.M. Sanchey, Surf. Sci. 157 (1985) L297. [17] C. Mannoni, G. Boato, M. Ganepa, P. Cantini, L. Mattera, S. Terreni, Europhys. Lett. 45 (1999) 685. [18] F.M. Zhang, B.V. King, D.J. O'Connor, Phys. Rev. Lett. 75 (1995) 4646. [19] Ch.E. Lekka, G.A. Evangelakis, Surf. Sci. 473 (2001) 39. [20] D.H. Oh, H.J. Kang, K.H. Chae, C.N. Whang, B.V. King, D.J. O'Connor, D.W. Moon, Surf. Sci. 477 (2-3) (2001) L289. [21] S. Nose, J. Chem. Phys. 81 (1984) 511. [22] D. Sondericker, F. Jona, P.M. Marcus, Phys. Rev. B33 (1986) 900. [23] M. Hayoun, V. Pontikis, Ch. Winter, Surf. Sci. 398 (1998) 125. [24] W.E. Wallace, G.J. Ackland, Surf. Sci. 275 (1992) L685. [25] R.J. Kobistek, G. Bozzolo, J. Ferrante, H. Schlosser, Surf. Sci. 307±309 (1994) 390. [26] Ch.E. Lekka, N.I. Papanicolaou, G.A. Evangelakis, Surf. Sci. 479 (2001) 287. [27] N.I. Papanicolaou, G.C. Kallinteris, G.A. Evangelakis, D.A. Papaconstatopoulos, M.J. Mehl, J. Phys. Cond. Matter 10 (1998) 10979. [28] F. Cleri, V. Rosato, Phys. Rev. B48 (1993) 22. [29] Ch.E. Lekka, N.I. Papanicolaou, G.A. Evangelakis, Surf. Sci. 488 (2001) 269. [30] C.H. Bennett, in: A.S. Nowick, J.J. Burton (Eds.), Diffusion in Solids, Recent Developments, Academic Press, New York, 1975, p. 73. [31] M. Hagen, M.W. Finnis, Phil. Mag. A77 (1998) 447.
Vacancy diffusion driven surface rearrangement in the Cu3Au(0 0 1) and Ni3Al(0 0 1) surfaces Ch.E. Lekka, G.A. Evangelakis* Department of Physics, Solid State Division, University of Ioannina, P.O. Box 1186, Gr-45 110 Ioannina, Greece Received 12 June 2002; received in revised form 12 June 2002; accepted 28 September 2002
Abstract We present molecular dynamics (MD) simulation results on the role of vacancy diffusion in the surface order of the Cu3Au(0 0 l) and Ni3Al(0 0 l) faces. We found that in both systems vacancy diffusion proceeds preferably by hopping along the [1 1 0] direction. In the Cu3Au(0 0 l) case, successive vacancy diffusion events induce irreversible loss of the surface order, while in the Ni3Al(0 0 l) system each vacancy hop leads to unstable atomic arrangements having a clear preference of recovering the initial surface order. These ®ndings can be used for the understanding of the order±disorder transition occurring in the Cu3Au(0 0 l) surface and the order±order kinetics characterizing the Ni3Al(0 0 l) face. In addition, we found that in the former case the vacancy diffusion rate saturates quickly (in <1 ns) and that it is correlated with the surface order parameter which reveals important disorder already at 500 K. In the later system, Al and Ni vacancies have constant hopping frequencies that exhibit Arrhenius behavior, while surface disorder is also predicted at high temperatures. # 2002 Elsevier Science B.V. All rights reserved. PACS: 71.15.D; 66.30.F; 68.35D; 68.35.B; 68.35.R Keywords: Molecular dynamics calculations condensed matter; Self diffusion in metals and alloys; Defects in solid surfaces; Crystal structure of surfaces; Order±disorder transition
1. Introduction The L12 binary intermetallic Cu3Au and Ni3Al alloys have been the subjects of numerous experimental and theoretical investigations. It is suggested that the Ni3Al is a directly ordered alloy, with a virtual transition temperature from atomic order to atomic disorder, equal to or even higher than the melting temperature and that it belongs to the group of the so-called super*
Corresponding author. Tel.: 30-6510-98590; fax: 30-6510-45631. E-mail address: [email protected] (G.A. Evangelakis).
alloys whose mechanical features improve at high temperatures [1±3]. In addition, from residual resistivity isochrone measurements [4,5] and by theoretical calculation [2] it is found that the high activation energy for vacancy formation makes the total activation energy for ``order±order'' kinetics very high. Furthermore, it came out that there are two ordering processes with different relaxation times associated to the jumps of vacancies between the two sub-lattices of the L12 structure. Turning on the Cu3Au case, it is found that this material is a sequentially ordered alloy with an order±disorder transition at Tc 663 K, well below the melting point [6±13]. Residual resistivity measure-
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 2 ) 0 1 1 1 8 - 2
Ch.E. Lekka, G.A. Evangelakis / Applied Surface Science 205 (2003) 280±288
ments (isothermal order±order annealing) [1] revealed that changes in the long-range order (LRO) parameter are mainly caused by jumps between vacancies and anti-site atoms within ordered domains and that in the related ordering relaxation two processes with different relaxation times are involved, as in the Ni3Al case. Focusing on the (0 0 1) surface, it is found that both Ni3Al and Cu3Au faces are rumpled, while to our knowledge there are no detailed studies on the role of the surface vacancy diffusivity of these systems. The LRO parameter of the Cu3Au(0 0 1) surface has been determined by low-energy electron diffraction (LEED) [14±16] equal to unity at temperatures below 500 K, to drop to zero at Tc 663 K. In addition, recent He diffraction experiments [17] provided a very reliable estimation of the long-range order parameter in the topmost atomic layers, consistent with the LEED data, revealing a continuous temperature dependence of the order parameter. Moreover, within the top surface layer, the short-range order (SRO) is also known to decrease with temperature. Low energy ion scattering (LEIS) measurements showed that starting at about 500 K, Au and Cu atoms change positions in the topmost atomic layer, while their concentrations are relatively constant up to the critical temperature [18]. However, recent molecular dynamics (MD) simulations on the adatom diffusion processes on the Cu3Au(0 0 1) surface suggested that the presence of Cu or Au adatoms on this face stimulates phenomena that act as precursors of the order±disorder transition, especially above 500 K [19]. This result is based on the ®nding of adatom activated exchange mechanisms, in which second layer atoms are also involved and on adatom induced vacancy creation, initiated by the correlated motion of Cu and Au surface atoms along the [1 1 0] direction. Both phenomena result in the rearrangement of the atoms in anti-sites and the loss of local order. These predictions were forti®ed recently by medium-energy ion-scattering spectroscopy (MEIS) measurements. Indeed, it was found that the atomic concentration of Au in the ®rst layer decreases from 50% at room temperature to 32% at 720 K [20]. In addition, in the same study, it was suggested that the different results found by LEIS and MEIS, concerning the surface concentration of Au and Cu atoms, are due to the production of surface vacancies and adatoms in the top layer of Cu3Au(0 0 l) surface at elevated temperatures, defects that have no signi®cant effect on the yield of LEIS experiments but are visible
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by MEIS [20]. This means that the presence of surface vacancies and adatoms in the Cu3Au(0 0 1) surface is related to the order±disorder transition that occurs by 150 K earlier than the bulk critical temperature, in agreement with the simulation results. It is therefore, very interesting to investigate and to try to understand the role of vacancy diffusion in the surface order of these alloys. It is the aim of the present work to study the interrelation of the surface vacancy diffusion with the surface order of the Cu3Au(0 0 1), that is characterized by an order±disorder transition and of the Ni3Al(0 0 1), which is characterized by order±order kinetics. 2. Computational details We performed simulations in the isothermal canonical ensemble using the Nose demon to control the temperature [21]. We used a system made up of 3800 atoms (1000 Au/Al and 2800 Cu/Ni) arranged on a FCC lattice corresponding to the stoichiometic and ordered Cu3Au and Ni3Al alloys. Periodic boundary conditions were applied in the three directions to mimic an in®nite system. The free surfaces were produced by ®xing the dimensions of the simulation box at a value twice as large as the thickness of the crystal along the z-direction. An in®nite slab of 19 atomic layers parallel to the (0 0 1) planes is thus produced with 200 atoms each, delimited by two free mixed surfaces, termination that is known to be energetically favored for both alloy surfaces [6±8,22±26] against pure Cu or Ni faces. To describe the attractive part of the atomic interactions we used an effective potential model in analogy to the tight binding scheme in the second moment approximation, while a Born±Mayer type was used for the repulsive contribution [27,28]. Accordingly, the total potential energy can be written as: 8 Nb Na < X X XX ab f Aab e pab rij =dab 1 : b j 1;j 6i a ia 1 a b b v9 u Nb = uX X ab t (1) x2ab e 2pab rij =dab 1 ; b j 1;j 6i b
b
a
where a and b refer to the two different kind of atoms, rijab jr~ia r~jb j is the distance between the ia and jb
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particles, while the indexes ia and jb run over all the particles. The interactions are computed up to the ®fth neighbor's distance. In the case of Cu3Au the parameters pab, qab, Aab, dab, and xab we used, have been determined using results of total-energy calculations as a function of volume, by means of the AugmentedPlane-Wave (APW) method, for the pure Cu, Au and the L12 Cu3Au and Au3Cu alloys [27]. This potential model is found to reproduce well the elastic constants of pure bulk elements and alloys, as well as the bulk modulus, the lattice expansion, atomic mean square displacements (MSDs) and phonon dispersion curves (DOS) of the ordered alloys. In addition, it is found [26] that it reproduces the surface energies, the rippling effect, the atomic MSDs and phonon DOSs for the low-index Cu3Au surfaces and it has been used for the study of the vibrational and diffusive [19,29] properties of Au and Cu adatoms. In the case of Ni3Al, the parameters pab, qab, Aab, dab and xab have been ®tted to the experimental values of cohesive energy, lattice parameters, and elastic constants for the metals and the alloys, in the appropriate crystal structure at T 0 K temperature [28]. The equations of motion were integrated by means of the Verlet's algorithm with a time step of dt 5 10 15 s. One Au/Al (or Cu/Ni) surface atom was removed from each Cu3Au(0 0 1) and Ni3Al(0 0 1) surface and the systems were equilibrated at the desired temperatures for 10 000 time-steps. 2 ns simulations were performed for the ®rst case at 300, 400, 450, 475, 500, 525, 550, 575 and 600 K and from 300 to 1000 K with a step of 100 K for the second one. At each temperature, we used the lattice constants that resulted to zero pressure for the bulk systems. The diffusion coef®cient D, was calculated using the jump frequency, G n=t, that is easily estimated by counting the number of hopping events, n, the surface atoms perform in the time interval t. D
lGd2 2z
(2)
where z is the dimensionality of the diffusion space (two in our case), l the number of jump directions (four in the present case) and d stands for the jump distance. The diffusion migration energy (DE) can be deduced using the standard result of the transition state theory: D Do e
DE=kT
(3)
Do, is the pre-exponential factor and k is the Boltzmann's constant. For the energy relaxation calculations, we used a quasi-dynamic minimization procedure integrated in the Molecular Dynamic code [30]. 3. Results and discussion 3.1. Static calculations-migration paths We calculated the Au and Cu surface vacancy formation energies by minimising the corresponding energies of the ordered Cu3Au(0 0 1) system (at T 0 K) and we found 0.99 and 0.50 eV, respectively. In the case of the ordered Ni3Al(0 0 1) surface, the corresponding formation energies for Al and Ni surface vacancies are 0.92 and 0.77 eV, respectively. It comes out that Cu and Ni vacancies are the energetically favored defects, in agreement with theoretical and experimental results referring to the Ni3Al case [2]. We note that for the correct description of the binding of the systems, we considered the contributions of the species A and B in the cohesion of the A3B alloy instead of the cohesive energies of the pure elements. In addition, although it is known that small deviation from stoichiometry can seriously affect the vacancy formation energies [31], in our case the corresponding changes are minimal and therefore the effect has to be insigni®cant. Concerning the surface vacancy diffusion, there are in principle two in-plane directions along which a hopping event can occur. In Fig. 1a and b we give a schematic representation of the two simple hopping mechanisms, along the [1 1 0] (SH) and the [1 0 0] direction (SH100), respectively. We note that the distances covered when a vacancy performs these two simple hopping mechanisms are different and equal to the nearest neighbor distance and the lattice constant, respectively. In addition, in a SH event of A type vacancy, a B surface atom jumps into the vacancy, Fig. la, resulting in a B vacancy and a B surface atom in anti-site. In a SH100 process the motion of an A surface atom along the [1 0 0] direction takes place in the A-sublattice and has no effect in the local order, Fig. 1b. Using constrained energy minimization, we calculated for both alloys the energy barriers required for A
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Fig. 1. Schematic representation of the vacancy simple hopping mechanisms: (a) along the [1 1 0] direction (SH) and (b) along the [1 0 0] (SH100) direction. Big and small circles stand for the A and B surface atoms of the A3B(0 0 1) surface, while empty site (square) stands for the A vacancy. Arrows show the vacancy hop direction.
Fig. 2. Potential energy barriers of three consecutive vacancy simple hops along the [1 1 0] direction: (a) initially Cu (circles) or Au (squares) vacancies in the Cu3Au(0 0 1) surface and (b) initially Ni (circles) or Al (squares) vacancies in the Ni3Al(0 0 1) surface.
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and B type vacancies to realize these hopping processes. The starting con®guration in all cases was a perfectly ordered surface. Since after a SH hop the local environment changes, in order to follow the evolution of the system in a succession of this kind of events, we calculated the energy barriers for three consecutive jumps of the vacancies. In Fig. 2a and b, we show these quantities for the Cu (circles) and Au (squares) vacancies in the Cu3Au(0 0 1) and the Ni (circles) and Al (squares) vacancies in the Ni3Al(0 0 1) surface, respectively. The initial energy level, zero energy values in Fig. 2, corresponds to the formation energies of the vacancies. It is very interesting to observe the completely different evolution of the energetic requirements for successive hops in the two systems: in the Ni3Al case the energy of the system increases after a jump, while vacancy hops in the Cu3Au(0 0 1) face lead the system to lower energy levels. This means that since the energy needed for the Ni or Al vacancy to perform a second or third hop towards the same direction is higher than for backward movements, these vacancies would prefer to return in their initial perfect positions. On the contrary, in the Cu3Au case the local conformations after a jump are energetically more stable (lower energy); Cu and Au vacancies are, therefore, more likely to continue diffusing, inducing thereby local disorder in the surface. Turning on the SH100 mechanism, we give in Fig. 3, the energy barriers needed for the Al (open diamonds),
Ni (®lled diamonds), Cu (open circles) and Au (®lled circles) vacancies to perform a hop in the [1 0 0] direction. We observe that in all cases the energy barriers required for the SH[1 0 0] mechanism are much higher than the energy barriers needed for the SH process, situated at the values of 2.00 and 1.50 eV, respectively. These results indicate that in-plane diffusion is favored in the [1 1 0] direction. In particular, vacancy diffusion in the Cu3Au(0 0 1) surface is expected to produce re-arrangement of the surface atoms in anti-sites, and loss of the local order, leaving the system in lower energy level and suggests that these processes must have an irreversible tendency explaining the order±disorder transition observed in this surface. In the Ni3Al(0 0 1) case the atomic arrangements after a vacancy diffusion event are less stable and the system relaxes at higher energy, a result that could explain the order±order kinetics that characterize this material. Nevertheless, the energetic requirements for the SH process in this alloy are not very high (<1.0 eV for the third successive hop) and therefore vacancy induced disordering has to be expected at high temperatures. We note here that our results are based on a semiempirical potential model and therefore the calculated values have to be considered with caution. However, we believe that these models are capable in reproducing correctly at least the main features of the vacancy behavior.
Fig. 3. Potential energy barriers for vacancy simple hopping mechanism along the [1 0 0] direction: (a) Cu (open circles) or Au (®lled circles) vacancies in the Cu3Au(0 0 1) surface and (b) Al (open diamonds) or Ni (®lled diamonds) vacancies in the Ni3Al(0 0 1) surface.
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Fig. 4. Time evolution of the surface order (SLRO) parameter along with the number of diffusion events (Nevents) at T 550 K. Dashed line stands for an initial Cu vacancy and solid line for an initial Au vacancy in the Cu3Au(0 0 1) surface.
3.2. Early state vacancy diffusion-surface rearrangement We tried to verify the above predictions by MD simulations. Starting from ordered surfaces containing a vacancy, we monitored the evolution of the vacancy hopping events, n, along with the surface order parameter (SLRO) for several temperatures. The SLRO parameter is calculated as the percentage of surface atoms being in anti-sites. In Fig. 4, we present for the case of a Cu (dashed line) or a Au (solid line) vacancy at T 550 K, the time evolution of the number of SH vacancy diffusion events in the [1 1 0] direction, n, normalized by the total number of hops, along with the corresponding SLRO. We observe that the number of diffusion events and the SLRO parameter are correlated and saturate attaining a steady state after 1 ns. In addition, although the diffusion events and the SLRO parameter of the two vacancies are distinguishable below 0.25 ns, they become very quickly identical, indicating that after an amount of reordering of the surface atoms, the identity of the vacancies is irrelevant. These results are in agreement with residual resistivity measurements in the Cu3Au system [1] from which it was found that changes in the LRO parameter are mainly caused by jumps between vacancies and antisite atoms within ordered domains. Moreover, in the same study it was also suggested that ``if two ordering
processes, related to the two sublattices of the L12structure, are required to explain the experimental results, only one ordering process can be expected above the critical temperature, because in this case no distinction of the sublattices is possible''. This result is in agreement with our static calculation results, according to which a hop of a Au-vacancy along the [1 1 0] direction, results in the production of a Cuvacancy and a Cu atom in anti-site inducing disorder. Similarly, for a Cu vacancy hop along the same direction, a Au atom jumps into the vacant position, resulting in a Au vacancy and a Au atom in anti-site. In addition, we can observe in the same ®gure that the SLRO parameter reaches the value of 0.85 at the steady state, suggesting that there are 15% surface atoms in anti-sites already at 550 K, in agreement with LEIS and MEIS experimental results. In addition, this result indicates that the loss of surface order starts below the bulk critical temperature, in agreement with the experimental data giving an order parameter value of 0.95 for the bulk at this temperature [14]. The vacancy diffusive behavior is similar for all temperatures studied resulting in the loss of surface order, the effect being accentuating as going close to the critical temperature. Turning on the Ni3Al case, we present in Fig. 5a and b, the time evolution of the SLRO parameter along with the number of vacancy hopping events,
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Fig. 5. Time evolution of the surface order (SLRO) parameter along with the number of diffusion events (Nevents) for an initial Ni vacancy (dashed line) and an initial Al vacancy (solid line) in the Ni3Al(0 0 1) surface: (a) T 700 K and (b) T 900 K.
n, calculated at T 700 and 900 K, respectively. As we can see, the diffusive behavior of both Ni and Al vacancies is totally different from the Cu3Au(0 0 1) surface, the number of events exhibiting linear time dependence (constant rate of hopping events). In addition, the order parameter is practically unchanged at T 700 K, while loss of surface order appears in the corresponding curve of T 900 K. We note also that the number of hopping events is by far greater than in the Cu3Au case.
These ®ndings con®rm the different vacancy diffusive behavior, predicted by the energetic requirements for hopping and suggest that the Ni3Al(0 0 1) face disorders also below the melting point. 3.3. Steady state calculations As mentioned before, the analysis of our MD trajectories revealed that Cu or Au vacancies, once formed, diffuse very fast and induce surface disorder,
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Fig. 6. Arrhenius diagrams of the SH diffusion coef®cients (along the [1 1 0] direction) for the Ni (circles) and Al (diamonds) vacancies in the Ni3Al(0 0 1) surface.
which, in turn, is responsible for the decay of their hopping rate. At the steady state, very few hopping events occur. Hence, the study of the vacancy diffusive behavior in the disordered surface was not possible due to bad statistics. On the contrary, in the Ni3Al case, the hopping rates of the vacancy diffusion events are constant, so we were able to calculate them at various temperatures. We found that the most frequent vacancy diffusion mechanism is the SH along the [1 1 0] direction exhibiting Arrhenius behavior up to Ts 900 K. In Fig. 6, we present the corresponding diagrams of the diffusion coef®cients referring to Al and Ni SH vacancy mechanisms, diamonds and circles, respectively. As we can see, below Ts, Al and Ni vacancies have different energetic requirements 0.13 and 0.23 eV, respectively, while above this temperature the identity of Al and Ni vacancies becomes meaningless due to the initiation of disordering process in the Ni3Al(0 0 1) surface and the two diffusion coef®cients saturate at the same value. 4. Conclusions In this communication we present Molecular Dynamics simulation results on the role of vacancies in the surface order of the Cu3Au and Ni3Al(0 0 1) surface. We found that the energetically favored vacancies for these two systems are the Cu and the Ni vacancies, respectively, in agreement with theoretical
and experimental results referring to the Ni3Al case [2]. In addition, we found that from the two available inplane directions for diffusion (the [1 1 0] and the [1 0 0] directions), the ®rst one has the smaller energetic requirements for vacancy diffusion in both alloys. Interestingly, in the case of the Cu3Au(0 0 1) surface, successive vacancy diffusion events induce loss of the surface order with an irreversible probability, while in the Ni3Al(0 0 1) case each vacancy hop lead to unstable atomic arrangements having a clear preference of recovering the initial surface order. These results can be used to explain the order±disorder transition occurring in the Cu3Au(0 0 1) surface and the order±order kinetics characterizing the Ni3Al(0 0 1) face. It has to be noted however, that vacancy-induced disorder has to be expected also in this later case, although at high temperatures. Moreover, from the trajectory analysis we found that in the Cu3Au(0 0 1) case, surface order parameter and the vacancy diffusion events are correlated. It comes out that already at 500 K, 15% surface atoms are located at anti-sites, while no distinction between the initial Cu or Au vacancies is possible due to the presence of substitutional defects induced by their diffusion. These results are in agreement with LEIS [18] and MEIS [20] experimental data and MD [19] calculations suggesting that the loss of surface order starts at T 500 K. From a similar treatment in the Ni3Al(0 0 1) case, we found that contrary to the previous system, the
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vacancy-hopping rate is constant, while important surface disorder is found at 900 K. In addition, we found that the most frequent vacancy diffusion mechanism is the SH along the [1 1 0] direction for both alloys. In the Cu3Au(0 0 1) surface the number of diffusion events saturates very quickly (within a nanosecond), while above T 550 K the surface order parameter has reached almost its maximum value, with 30% of the surface atoms being in anti-sites. On the contrary, in the case of the Ni3Al(0 0 1) surface, Al and Ni vacancies have constant hopping rates, with Arrhenius type temperature dependence up to Ts 900 K, from where we deduced the corresponding migration energies, 0.13 and 0.23 eV, respectively. These ®ndings are in agreement with available experimental observations and can be used to explain the different behavior of these two alloys. Acknowledgements The work was partially supported by the HPRN-CT2000-00038. References [1] H. Lang, H. Uzawa, T. Mohri, W. Pfeiler, Intermetallics 9 (2001) 9. [2] R. Kozubski, Prog. Mater. Sci. 41 (1997) 1. [3] M. Sluiter, Y. Hashi, Y. Kawazoe, Comp. Mater. Sci. 14 (1999) 283. [4] B. Sitaud, X. Zhang, C. Dimitrov, O. Dimitrov, in: H.E. Exner, V. Schuhmacher (Eds.), Advanced Materials and Processes, DGM, Oberusel, 1990, p. 389. [5] C. Dimitrov, T. Tarfa, O. Dimitrov, in: A.R. Yavari (Ed.), Ordering and Disordering in Alloys, Elsevier Applied Science, Barking, 1992, p. 130. [6] T.M. Buck, G.H. Wheatley, L. Marchut, Phys. Rev. Lett. 51 (1983) 43.
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