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Molecular dynamics study of the transport and structural properties of the Cu3Au and Ni3Al(110) surface
Surface Science 518 (2002) 111–125 www.elsevier.com/locate/susc
Molecular dynamics study of the transport and structural properties of the Cu3Au and Ni3Alð1 1 0Þ surface Ch.E. Lekka a, D.G. Papageorgiou b, G.A. Evangelakis
c,*
a
c
Center of Computational Material Science, Naval Research Laboratory, Washington, DC 20375-5345, USA b Department of Materials Science, University of Ioannina, P.O. Box 1186, Gr-45110 Ioannina, Greece Department of Physics, Solid State Division, University of Ioannina, P.O. Box 1186, Gr-45110 Ioannina, Greece Received 14 June 2002; accepted for publication 25 July 2002
Abstract We present molecular dynamics results concerning the transport and the structural properties of the Cu3 Au and Ni3 Al(1 1 0) surfaces in presence of adatoms and vacancies. In the Cu3 Au case we found that below 500 K the Cu adatom occupies preferably the so-called ‘‘dumb-bell’’ position, while the Au adatom penetrates into the second pure Cu layer via an exchange type mechanism inducing thereby local disorder. At higher temperatures the Cu adatom resides almost exclusively at fourfold positions, while spontaneous creation of adatom–vacancy pairs is also present. Vacancy diffusion processes induce surface disorder that starts by as much as 150 K below the bulk order–disorder temperature. In the case of Ni3 Al(1 1 0) surface, we found a new Al adatom position situated between two Ni surface atoms. It came out that this position plays an important role in the adatom’s diffusive behavior. Above 800 K adatom– vacancy pairs are spontaneously created and in conjunction with exchange type diffusion mechanisms they affect seriously the surface order. In addition, we found that the Ni adatoms are more active than the Al adatoms, this difference in the adatom diffusivity being accentuated by exchange events, leading very quickly the surface in a disordered state. In conclusion, it comes out that the Cu3 Au and Ni3 Al(1 1 0) surfaces disorder well below the bulk transition temperature (663 K, Cu3 Au) and the melting point (1663 K, Ni3 Al) via diffusion processes of the adatoms and/or vacancies. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Surface diffusion; Surface structure, morphology, roughness and topography; Alloys; Adatoms
1. Introduction The last decade the Ni3 Al and the Cu3 Au intermetallic L12 alloys have been the subject of numerous experimental and theoretical investigations because they are related to materials pos-
*
Corresponding author. Tel.: +30-6510-98590; fax: +306510-45631. E-mail address: [email protected] (G.A. Evangelakis).
sessing improved mechanical properties and having important technological applications in catalysis, coating, high temperature engineering, etc. It is found that the Ni3 Al is a directly ordered alloy while the Cu3 Au is a sequentially ordered alloy with an order–disorder transition at Tr ¼ 663 K, well below the melting point [1–8]. Although these L12 alloys exhibit different behaviour in the bulk system, some similarities have been found for their low-index surfaces. Indeed, the energetically favoured surfaces are the Au (Al) rich faces, while
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 2 1 1 6 - 7
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their low-indexed surfaces are rumpled, with the Au (Al) atoms lying above the Cu (Ni) atoms [9– 20]. Focusing on the (1 1 0) surface, low energy electron diffraction (LEED) and low energy ion scattering (LEIS) observations revealed that in the Cu3 Au case the surface undergoes a broadened discontinuous transition below Tr and that above 400 K the compositional fraction of Au atoms changes in the first two atomic layers [21]. In addition, LEED experiments revealed nucleation of islands on the Cu3 Au(1 1 0) with local 2 1 periodicity on a 2 1 ordered substrate, while the apparent activation energy of ordering was found close to that of diffusion in Cu3 Au, suggesting that island formation is controlled by diffusion [22]. However, it is still not clear what is the precursor of the observed changes and which are the exact diffusion mechanisms that are responsible for the surface loss of order on the Cu3 Au(1 1 0) surface and what is happening in the Ni3 Al(1 1 0) case. It is therefore, very interesting to investigate and to try to understand the role of adatoms or vacancies in the early stages of phenomena like order–disorder transition of Cu3 Au and order kinetics of the Ni3 Al(1 1 0) face. In the aim of having a better insight of the transport properties of adatoms and vacancies on these systems as well as their role in the surface order, we performed typical 1 ns molecular dynamics simulations (MD).
2. Computational details
system having 19 atomic layers of 280 atoms each showed that the main quantities we are interested in (surface relaxation, surface energies, vacancy formation energies, etc.) are not affected seriously by the small size of our system. An important issue is the terminal layer for each face. For the (1 1 0) face there is the possibility of having either mixed or pure Cu terminal layer. From experimental and theoretical studies it is now established that in the ordered Cu3 Au a mixed terminal layer is preferred for the three lowindexed surfaces [10,15,21,24–26]. Similar findings are available also in the ordered Ni3 Al [16–18]. We verified by energy minimization that the potential we adopted reproduces these results for both alloys and we used these terminations for the surface layers in our simulations. 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 contributions [27,28]. Accordingly, the total potential energy can be written as: 8 > !! Nb Na > b jb¼1 dab a ia ¼1 > : jb6¼i a 9 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !!> u > Nb ab = uX X 2 rij u t nab exp 2qab 1 > dab > jb ¼1 b ; j b6¼ia
The simulations were performed in the isothermal canonical ensemble using the Nose demon to control the temperature [23]. The system was made up of 2730 atoms (700 Au and 2030 Cu) arranged on a FCC lattice corresponding to the stoichiometric and ordered Cu3 Au and Ni3 Al alloys. Periodic boundary conditions were applied in the three space directions to mimic an infinite system. By fixing the dimension of the simulation box at a value twice as large as the thickness of the crystal along the z-direction we simulated an infinite slab of 39 atomic layers parallel to (1 1 0) planes (each containing 70 particles) delimited by two free surfaces normal to the corresponding direction. Some preliminary calculations with a
ð1Þ where a and b refer to the two different kind of ! atoms, rijab ¼ jr! ia rjb j is the distance between the ia and jb particles, while the indexes ia and jb run over all the particles. The interactions are computed up to the fifth neighbor’s distance. In the Cu3 Au case the parameters pab , qab , Aab , dab and nab , have been determined using results of total energy calculations as a function of volume, by means of the augmented-plane-wave (APW) method, for the pure Cu, Au and the Ll2 Cu3 Au and Au3 Cu alloys [27]. This potential model is found to reproduce well the elastic constants of pure bulk elements and
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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 [28] that it reproduces well surface energies, rippling effect, atomic MSDs and phonon DOSs for the lowindexed Cu3 Au surfaces and it has been used successfully for the study of the vibrational and diffusive [29,30] properties of Au and Cu adatoms. Similar potential models have been also used successfully in simulations of the [0 0 1] grain boundary of this alloy [31,32]. In the case of Ni3 Al, the corresponding potential parameters have been fitted 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 ¼ 0 K [28]. The equations of motion were integrated by means of the Verlet’s algorithm with a time step of dt ¼ 5 1015 s. In order to simulate the surface in presence of an adatom or a vacancy, one Au/Al (or Cu/Ni) adatom was placed or removed from each Cu3 Au and Ni3 Al surface and the systems were equilibrated at the desired temperatures for 10 000 timesteps. 1 ns simulations were performed covering the temperature range 300–600 K for the Cu3 Au and 300–1000 K for the Ni3 Al with a step of 100 K, respectively. At each temperature, we used the lattice constants that resulted to zero pressure for the bulk systems. For the energy relaxation calculations we used a quasi-dynamic minimization procedure integrated in the molecular dynamic code [33].
3. Results 3.1. Energetic considerations referring to Au and Cu adatoms or vacancies on the Cu3 Au(1 1 0) (static calculations) The geometric anisotropy of the (1 1 0) surface plays an important role in the adatom deposition and the epitaxial growth. This anisotropy is reflected in the potential energy maps in presence of Cu and Au adatoms obtained by relaxing the system with the adatoms at different positions (Fig. 1a and b respectively). The color scale sig-
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nifies (going from blue to red) the gradual energy increase. We can easily recognize the characteristic channels of the (1 1 0) face (red areas) and the different energetic content of the two in-plane directions. We observed that the energetically favoured position for the Cu adatom on the Cu3 Au(1 1 0) surface is between two consecutive Cu surface atoms, dark blue areas in Fig. 1a. These two Cu surface atoms are displaced slightly from their normal lattice sites accommodating the adatom between them, distorting locally the surface layer. Due to this distortion the adatom and a Cu surface atom reside into two consecutive channels, a feature that is distinctive of this surface. It is worth also noting that at this site the Cu adatom belongs in fact in the surface layer since it relaxes at the same distance from the second atomic layer with the Cu surface atoms. This position is known as ‘‘dumb-bell’’ (DB) position [30] and it is found to play important role in the transport phenomena happening on the FCC(1 1 0) surface. Indeed, the DB site is found by MD simulations of different metals as an intermediate metastable position in the exchange type diffusion process [34,35] and also experimentally in the Ag(1 1 0) case when an adatom is displaced by the STM tip [36]. However, in the present case, the DB position is not just a meta-stable site but the energetically favoured adatom position, the usual fourfold position (FF) being a less stable alternative for the Cu adatom. The presence of these two Cu atoms in DB positions has very important consequence in the diffusion processes happening on the surface, stimulating complicated and correlated hopping mechanisms. One of the simplest hopping mechanisms involves a hop of these Cu atoms to the next DB position along the [0 0 1] direction. This is the cross channel direction, along which a usual jump is known to be energetically very expensive. According to this mechanism the Cu adatom situated in the DB1 position moves towards the DB2 site pushing the Cu atom that is located at this position to the next dumb-bell location DB3 (Fig. 1a). We call this mechanism dumb-bell–Cu–dumb-bell (DBCuDB) mechanism. As we shall see in the Section 3.3, the trajectory analysis revealed this process as one of the important mechanisms for surface diffusion in this surface.
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Fig. 1. Potential energy map for various adatom positions on the Cu3 Au (1 1 0) surface: (a) Cu adatom and (b) Au adatom. The color scale indicates (going from blue to red) the gradual increase of the energy (eV). FF refers to the fourfold adatom positions, DB1;2;3 refer to the dumb-bell adatom position and Au (Cu) represent the corresponding surface atoms.
Contrary to the Cu adatom, we found that in the case of Au adatom the DB position is not energetically favoured, the FF position being the only available adatom site (Fig. 1b). From the geometry of this energy map we can predict in principle three possible hopping mechanisms from one FF position to the next one. There may be two hopping mechanisms along the [1 1 0] and another one along the [0 0 1] direction. (a) Between two Au surface atoms requiring an energy barrier of 0.27 eV (path FFAFF), (b) passing between two Cu surface atoms with energy barrier 0.38 eV (path FFBFF) and (c) the less favoured jump crossing the channel with energy barrier equal to 1.06 eV. Turning our attention to the vacancy cases, we estimated by energy minimization at 0 K the formation energies of Au surface vacancy and Cu surface vacancy at 0.81 and 0.12 eV, respectively. The corresponding values for the bulk system are estimated at 1.48 and 0.84 eV. In addition, we found that the presence of a vacancy in the or-
dered Cu3 Au(1 1 0) surface affects the relaxed positions of the neighboring surface atoms. We quantified this effect by calculating for the surface atoms and in presence of a Cu vacancy the inplane (%) deviations of the atomic distances from the perfect lattice spacing (DAD). In Fig. 2, we show this quantity computed along the [0 0 1] and the [1 1 0] directions for the in-plane and normal to the surface direction (normalized with respect to the bulk interlayer spacing). In Fig. 2a, we observe that the Cu vacancy induces expansion to the closest Cu surface atoms along the [0 0 1] in-plane direction, resulting in strained row along that direction, while in the [1 1 0] direction the effect is insignificant. This expansion is about þ2% and involves mainly the first neighbors. On the contrary, normal to the surface direction, we observe that the closest Au surface atoms along the [1 1 0] direction are contracted by )2.89%. Concluding, it seams that the presence of the Cu vacancy affects mainly the first nearest neighbors of both in-plane
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Fig. 2. Difference (%) between the relaxed positions of the surface atoms with and without vacancy as a function of the vacancy–atom distance for the in-plane directions and the normal to the surface direction with respect to the bulk interlayer spacing: (a) Cu vacancy and (b) Au vacancy. Filled squares and circles stand for the Cu and Au surface atoms along the [1 1 0] direction, while open diamonds stand for the atoms along the [0 0 1] direction.
directions, a result that is different from the behavior of the Cu vacancy in the Cu3 Au(0 0 1) surface [37]. Focusing on the Au vacancy (Fig. 2b), we found expansion along the [1 1 0] and [0 0 1] in-plane directions by 0.75% (Cu atoms) and 0.20% (Au atoms) for the first neighboring atoms. In the normal to the surface direction these atoms are attracted towards the second layer by 1.62% (Cu atoms) and 0.75% (Au atoms). These relaxation effects are important since they can be related with the diffussive behaviour of the vacancies. Indeed, we would expect that diffusion of the Cu vacancy takes place in both in-plane directions, while inplane diffusion along the [1 1 0] direction is expected for the Au vacancy contrary to the (0 0 1) surface where the vacancy diffusion was found to take place only along the [1 1 0] direction [37]. Consequently, there is in principle one possible vacancy hopping mechanism along the [1 1 0] direction corresponding to simple (SH) vacancy jump and another one (simple hopping) along the
[0 0 1] direction (SH001). In Fig. 3 we give a schematic representation of the various vacancy hopping mechanisms studied. We note that in the SH and the SH001 mechanisms the distances that the vacancy covers are different and equal to the nearest neighbour distance and the lattice constant, respectively. In the SH event, a Au surface atom is jumping into the Cu vacancy (Fig. 3a), resulting in local disorder. On the contrary, in the SH001 process the movement of a Cu surface atom along the [0 0 1] direction has no effect in the local order, since in this direction only Cu atoms are participating (Fig. 3b). It is also worth noting that the SH mechanism leads to an energetically lower configuration, while the SH001 process leaves the system energetically unchanged. This can be seen in Fig. 4a and b where we give the energy required for a SH and SH001 hopping mechanism of the Cu vacancy (tagged squares) and the Au vacancy (filled squares). Starting from the same initial configuration, after an SH hop we end up with a configuration with smaller energy, )0.14 eV for the
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Fig. 3. Schematic representation of the various vacancy hopping mechanisms studied: (a) simple hopping mechanism along the [1 1 0] direction and (b) simple hopping mechanism along the [0 0 1] direction. Big and small circles stand for the Au and Cu surface atoms, while open squares stand for the Cu vacancy. Arrows show the hopping direction of the vacancy.
Cu vacancy and )0.09 eV for the Au vacancy (Fig. 4a), while the energy barrier is around 0.35 eV for both cases. The zero level in this figure corresponds to the vacancy formation energy. In addition, after the accomplishment of a SH001 event there is no energy change (Fig. 4b), while the energy barrier is 0.45 eV for the Cu vacancy and 0.88 eV for the Au vacancy. It has to be noted that the energy required for a Cu vacancy to perform a SH001 is equal to the corresponding barrier for a SH mechanism indicating that both hopping mechanisms should have equal frequencies in both in-plane directions. However, after an SH jump along the [1 1 0] direction (Fig. 4a), the system is left at lower energy estimated at )0.29 and )0.21 eV for the Cu and Au vacancy, respectively. In addition, in the SH mechanism the Cu and/or Au surface atoms are involved in a random sequence, resulting in rearrangement of the surface atoms that are located in anti-sites, yielding configurations with lower energies and local disorder. This result is inevitably related to the Cu3 Au orderdisorder transition.
Fig. 4. Potential energy barriers for the Cu vacancy (tagged squares) and Au vacancy (filled squares) hopping mechanisms studied: (a) simple hopping (SH) mechanism along the [1 1 0] direction and (b) simple hopping (SH001) mechanism along the [0 0 1] direction.
3.2. Static calculations for Al and Ni adatoms or vacancies on the Ni3 Al(1 1 0) The behaviour of Ni and Al adatoms on the Ni3 Al(1 1 0) surface is completely different from the one described above concerning the Cu3 Au case. The available adatom positions are shown in the energy maps (Fig. 5a and b). The color scale indicates (going from blue to red) the gradual energetic increase. We found that for both Ni and Al adatoms the energetically favoured position is the FF. A similar to the DB position over a row of Ni atoms is also found for both adatoms. However, in the present case the adatoms do not exhibit the
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Fig. 5. Potential energy map for various adatom positions on the Ni3 Al(1 1 0) surface: (a) Ni adatom and (b) Al adatom. The color scale indicates (going from blue to red) the gradual increase of the energy (eV). FF refers to the fourfold adatom positions, BNA refers to the ‘‘between Ni surface atoms’’ adatom position, A refers to the adatom position between two Al surface atoms and Ni (Al) refer to the corresponding surface atoms.
important contraction we found in the Cu3 Au case, in which there were found to reside practically at the same level with the surface atoms. This new adatom position is situated along the channel between two Ni surface atoms, (BNA) position hereafter. Starting from the FF position, the Ni adatom could hop to the next FF position along the channel, between two Ni surface atoms (path FF-BNA-FF with energy barrier 0.15 eV) or between two Al surface atoms (path FFAFF with energy barrier 0.43 eV) (Fig. 5a). A hopping mechanism of the Ni adatom along the [0 0 1] direction requires 1.10 eV. Concerning the Al adatom (Fig. 5b), the energetically favoured hopping mechanism is the one from a BNA position to the closest FF position (energy barrier 0.18 eV). In addition, there are two other possible hopping mechanisms, one starting from a FF position to the next FF between two Al surface atoms (path FFAFF, energy barrier 0.55 eV) and another one
crossing the channel from a BNA to the next BNA (path BNA-Ni-BNA, energy barrier 1.29 eV). Concerning the vacancy cases for this material, we found from energy minimization at T ¼ 0 K that the formation energies of an Al and a Ni surface vacancy are 0.62 and 0.48 eV, respectively. These values have to be compared with the corresponding values for the bulk system that are estimated at 2.04 and 1.94 eV. Moreover, similarly to the Cu3 Au(1 1 0) surface, we found that the presence of a vacancy in the ordered Ni3 Al(1 1 0) surface affects the relaxed positions of the neighboring surface atoms. To demonstrate this effect for the Ni vacancy, we show in Fig. 6a the DAD computed along [0 0 1] and [1 1 0] directions for the in-plane and normal to the surface direction (normalized with respect to the bulk interlayer spacing). Moreover, in Fig. 6b we show the corresponding quantities for the Al vacancy. In both cases, we observe that in the in-plane directions the
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Fig. 6. Difference (%) between the relaxed positions of the surface atoms with and without vacancy as a function of the vacancy–atom distance for the in-plane direction and for the normal to the surface direction with respect to the bulk interlayer spacing: (a) Ni vacancy and (b) Al vacancy. Filled squares and circles stand for the Ni and Al surface atoms along the [1 1 0] direction, while open diamonds stand for the atoms along the [0 0 1] direction.
vacancy attracts the two nearest neighbor atoms along the [1 1 0] direction (the first one by )3.33%), while the first neighboring atoms along the [0 0 1] direction are expanded (1.77% for the Ni vacancy and 1.55% for the Al vacancy). From these findings one would expect simple or double vacancy hops to occur along this direction. In the normal to the surface direction (Fig. 6a), we found that the presence of a Ni vacancy induces small contraction ()0.64%) in the first neighboring atoms along the [1 1 0] direction. In addition, in the normal to the surface direction (Fig. 6b), the Al vacancy attracts the first atoms in both directions ()0.16% for the [0 0 1] direction and )0.14% for the [1 1 0] direction). Concluding, it seems that the vacancy presence affects mainly the relaxed positions of the surface atoms in the in-plane [1 1 0] direction, while it expands the surface atoms in the in-plane [0 0 1] direction. In addition, there is not enough influence in the normal to the surface direction. This suggests that the energetically favored hopping mechanisms would be along the in-plane [1 1 0]
direction, a result that is similar with the findings concerning the vacancy diffusion in the Ni3 Al(0 0 1) surface. These differences in the relaxation of both adatoms and vacancies between the two systems we are studying may have serious consequences in the surface kinetics. The understanding of their physical origin requires detailed and accurate electronic structure calculations. In our case all electronic contributions are incorporated into the effective interatomic model we are using. Therefore, no information can be extracted about the electronic bands for the various cases we studied. However, the present results could be understood taking into account the different bonding character of these alloys. Indeed, on one hand we know that Cu3 Au is a ‘‘common d-band alloy’’ where the Au bands are concentrated at the bottom of the d band and the Cu bands are concentrated at the top of the d band. Upon formation of the Cu3 Au the Au 5d band is depleted [38]. This spatial compression on the 5d shells of Au is most likely due to the sp
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electrons compressing to a smaller atomic volume [39]. In order to decrease this spatial compression on 5d electrons, the surface Au atoms are displaced outward [40]. On the other hand the bonding charge density of the Ni3 Al alloy can be described by the combination of charge transfer from Al to Ni and a strong Al p-Ni d hybridisation effect [41,42]. Furthermore, using constrained energy minimization, we calculated the energy barriers required for the Ni and the Al vacancy to perform a SH and a SH001 hopping mechanisms. Similarly to the Cu3 Au(1 1 0) surface, after a SH hop along the [1 1 0] direction, the local environment changes (Fig. 3). In order to follow the evolution of the system in a succession of such events, we calculated the energy barriers for two consecutive jumps of the vacancies. In Fig. 7a, we show these quantities for the SH hop mechanism of the Ni vacancy (tagged squares) and the Al vacancy (filled circles), respectively. Again, the initial energy level, zero energy values in this figure, corresponds to the formation energies of the vacancies. It is very interesting to observe that the energy needed for the Ni or Al vacancy to perform a first (0.50 or 0.43 eV) or a second (0.62 or 0.52 eV) hop towards the [1 1 0] direction is higher than for backward movements (0.48 or 0.41 eV for the first hop). This indicates that these vacancies would prefer to return in their initial perfect positions than perform a second hop towards the [1 1 0] direction. On the contrary, in the Cu3 Au case, Fig. 4a the local configurations after a successful 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 along the [0 0 1] direction, we give in Fig. 7b, the energy barriers required for the Al (filled circles) and Ni (tagged squares) vacancies to perform a hop in the [0 0 1] direction. We observe that in all cases the energy barriers required for the SH001 mechanism are much higher than the energy barriers needed for the SH process, situated at the values of 0.87 and 0.73 eV, respectively. These results indicate that the in-plane diffusion is favored in the [1 1 0] direction. In particular, vacancy diffusion in the Cu3 Au(1 1 0) surface is expected to produce re-
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Fig. 7. Potential energy barriers for the Ni vacancy (tagged squares) and Al vacancy (filled squares) hopping mechanisms studied: (a) simple hopping (SH) mechanism along the [1 1 0] direction and (b) simple hopping (SH001) mechanism along the [0 0 1] direction.
arrangement of the surface atoms in anti-sites, and loss of the local order, leaving the system in lower energy level. In addition, it appears that these processes must have an irreversible tendency explaining the order–disorder transition observed in this surface. On the contrary, in the Ni3 Al(1 1 0) 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 (less than 1.0 eV for the second successive hop) and therefore vacancy induced disordering has to be
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expected at high temperatures. We note here that our results are based on a semi-empirical potential model and therefore the calculated values have to be considered with caution. 3.3. Transport properties of Cu and Au adatoms and vacancies on the Cu3 Au(1 1 0) surface As we have seen in the previous section, static calculations predict that the energetically favoured position of the Cu adatom on the Cu3 Au(1 1 0) surface is the ‘‘dumb-bell’’ (DB) position which plays an important role in the transport phenomena occurring on this surface. In addition, we found that the usual FF position, although less energetically favoured, is also an equilibrium site for the Cu adatom. Contrary to the Cu adatom, the DB position is not energetically favoured for the Au adatom, the FF position being therefore the only available adatom site in this case. Interestingly, we found that the diffusion events are rather scarce for both Cu and Au adatoms when residing at the FF position. Specifically, above the room temperature the Au adatom enters into the second atomic layer via an exchange type mechanism similar to the on found in the case of the (0 0 1) surface, namely the so-called second layer exchange (SLE) [29]. We conclude therefore that the high diffusivity that this face exhibits is attributed to Cu adatoms. In order to quantify the stability of the Cu adatom in the DB position and compare with the corresponding occupation of the FF position, we calculated the average life-time of the Cu adatoms in both positions. It comes out that at low temperatures the DB position is clearly favoured. As the temperature increases the adatom spends most of the time on the FF position (Fig. 8). This phenomenon is accompanied by the spontaneous creation of adatoms that relax almost exclusively at the FF position, especially at temperatures higher than 500 K. Spontaneous creation of adatoms has been observed on the Cu(1 1 0) surface as well [43,44]. In Fig. 9 we show the concentration of Cu adatoms on the two available positions as a function of temperature. It can be seen that the concentration of Cu adatoms on the DB position is practically unchanged for a large temperature
Fig. 8. Average life-time of the Cu adatoms at DB (open circles) and FF (filled circles) positions as a function of temperature.
Fig. 9. Arrhenius diagram of the concentration of the Cu adatoms at DB (open circles) and FF (filled circles) positions.
range. On the contrary, the concentration of Cu adatoms at the FF position is almost zero at low temperatures to saturate at a value of 10% above 500 K. In this region the spontaneous creation of adatoms appears to follow Arrhenius behaviour, from where we deduced the corresponding energetic requirement for the creation of an adatom– vacancy pair, 0.9 eV. These findings are compatible with the results concerning the relative relaxed positions and suggest modifications in the perfect order of surface atoms. Indeed, we found that the long-range order (LRO) diverges significantly from the value corresponding to perfectly ordered surface at the same temperature (500 K) (Fig. 10).
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Fig. 10. Time evolution of the Cu3 Au(1 1 0) surface order (LRO) parameter along with the number of diffusion events (Nevents ) of the adatoms and vacancies for three different temperatures.
It is worth noting also that this change of surface order takes place very quickly. Indeed, the process seems to be concluded after 0.5 ns and the system appears in a stationary regime after that time. In order to estimate the correlation between the diffusion events and the surface order, we counted the number of diffusion events (Nevents ) as a function of time, the system being in thermodynamic equilibrium. The results for different temperatures, normalized by the maximum number of events at T ¼ 500 K for comparison reasons, are presented in the same figure (Fig. 10). It is clearly seen that the rate of diffusion events is positive for the first two temperatures, while it is almost zero at T ¼ 600 K, temperature at which the LRO parameter reaches its highest value. This result suggests that the surface disordering occurs via diffusion events. In Fig. 11 we show the temperature dependence of the LRO parameter. As we can see, the LRO decreases dramatically above T ¼ 450 K, suggesting that the surface looses its order above 500 K, much earlier than the bulk system (Tr ¼ 663 K [45]), in agreement with Monte Carlo simulations [46]. We conclude therefore, that surface disorder is a direct result of the diffusion of adatoms and vacancies. At temperatures lower than 500 K the Cu adatom diffuses from a DB position to another DB position (DBCuDB mechanism), while at
Fig. 11. Temperature dependence of the Cu3 Au(1 1 0) surface order (LRO) parameter.
higher temperatures Cu adatom diffusion takes place almost exclusively between FF positions. We must emphasize that the DBCuDB diffusion mechanism happens with the simultaneous movement of two Cu atoms that lie in two surface channels. The equivalent movement of a dimer over a surface channel is a mechanism with high energetic requirements. It is worth noting that the DB mechanism does not alter the ordering of atoms on the surface since it occurs along a row of Cu atoms, the surface order being in fact a prerequisite for this mechanism. In addition, it
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Fig. 12. Dumb-bell–dumb-bell (DB–DB) hopping mechanism of two Cu atoms at 450 K: one of the Cu atoms (B labeled particle) hops towards the second Cu atom (A labeled atom) forcing it into the channel, t ¼ 1:1 ps. Subsequently, the A atom hops to the next DB position popping a Cu surface atom to an adatom position, t ¼ 2:4 ps.
appears that the surface stoichiometry is decisive for the existence of the mechanism, this kind of process not being observed in pure metals (e.g. Cu). Indeed, in this last case the dimers on the corresponding surface, although very active, were not found to move in any way perpendicularly to the [1 1 0] direction (the channel direction), due to the large energetic requirement for such a diffusion event. However, the DB position of the adatom has been found to play an important role in inchannel diffusion mechanisms. A typical example of a diffusion event between two DB positions (DBCuDB mechanism) at T ¼ 450 K is shown in a succession of snapshots in Fig. 12. The system being in a steady state, at least as far as surface structural changes are concerned, we counted the diffusion events for all temperatures (Fig. 13). It can be seen that the frequency of the
Fig. 13. Arrhenius diagram for the DBC uDB Cu adatom diffusion mechanism on the Cu3 Au(1 1 0) face.
diffusion events follows Arrhenius behaviour up to 500 K, from where we deduced activation energy of 0.18 eV. At higher temperatures the frequency of diffusion is clearly reduced (almost by an order of magnitude), a result that can be understood taking into account that at this temperature region the surface is disordered. Turning our attention to the behaviour of vacancies, we recall that an important characteristic of this surface is their spontaneous creation at relatively low temperatures. A detailed trajectory analysis revealed that these vacancies are very active and their diffusion changes locally the surface order. In addition, we observed that especially the Cu vacancies could diffuse in both in plane directions, in agreement with the static calculations. Another important result is that they tend to combine forming di-vacancies, which contrary to simple vacancies are very stable and practically immobile, at least within the time scale of our simulations. Therefore, although the presence of vacancies is one of the main reasons for the loss of the surface order, when this happens, the majority of vacancies form clusters that do not show any significant activity, rendering a further study of the vacancy diffusion mechanisms meaningless. 3.4. Transport properties of Ni and Al adatoms and vacancies on the Ni3 Al(1 1 0) surface The static calculations revealed a new adatom position on the Ni3 Al(1 1 0) surface, which we called BNA. This position, in conjunction with the FF positions at both sides of the channel allows for the possibility of an oscillatory motion of the adatoms. This movement is typical for both species, although Al adatoms appear more stable in this position. From the trajectory analysis it came out that up to 800 K both adatoms diffuse mainly with simple hops, almost exclusively along the surface channels, with the Ni adatom diffusing more frequently. Spontaneous creation of adatoms has not been observed, contrary to the Cu3 Au case where adatom–vacancy pairs were found to exist above room temperature. In addition, we found that diffusion events over a row of Al atoms are rare. We can conclude therefore that diffusion takes place only along the channel of two consecutive Al
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atom rows. This finding has the important consequence that both adatoms are essentially trapped in the positions defined by their corresponding sub-lattices, yielding a surface that maintains its perfect order up to this temperature. At temperatures higher than 900 K, spontaneous creation of adatom–vacancy pairs does occur. However, these additional vacancies do not have significant effect to the surface activity and structure, since their concentration is rather limited, they are short lived and combining between them they form di-vacancies that are particularly stable and hard to move. On the contrary, exchange type phenomena between adatoms and surface atoms are important and decisive, since they induce an immediate change to the surface order. Such phenomena were observed in all possible combinations, namely, exchange of a Ni or Al adatom with the same kind of surface atom, as well as exchanges between different kinds of atoms, like exchanges of Ni adatoms with a surface Al and vice versa. Therefore above 900 K the surface disorders through these exchange type mechanisms. Interestingly, the local loss of the surface order alters the diffusive behaviour of the adatoms. Indeed, the adatoms that were trapped between two successive Al rows can now escape passing over the points where the Al row is broken. It follows that they can diffuse in longer distances and create through a new exchange mechanism another locally disordered area. We found that this process is repeated leading to an intense diffusive activity and surface disordering. It is worth also noting that the initially greater activity of Ni compared to Al adatoms, is further increased through the exchange events which especially for this adatom can be multiple (double or triple exchanges) leading the surface faster in a disordered state. Hence, as far as that surface order and diffusivity are concerned, the effect of an Al adatom exchange with a surface Ni, is greater than that of a Ni adatom with a surface Al. These results are in agreement with recent experimental data in the Ni3 Al system which revealed that in the temperature range between 1423 and 1523 K, the intrinsic diffusion coefficients of the Ni and Al atoms are almost the same and that the diffusion mechanisms incorporate anti-site defects [47].
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Fig. 14. Simple exchange mechanism of Al adatom (atom A) with a surface Ni (atom B) at 900 K. The procedure starts with the Al atom relaxing at the normal equilibrium position, over a row of Ni atoms. After 1 ps, it moves towards the position of a neighbouring surface atom, with which it exchanges positions. Grey and black particles stand for Al and Ni atoms.
In Fig. 14, we present a typical example of a simple exchange mechanism between an Al adatom (atom A) with a surface Ni (atom B) at 900 K. The process starts with the Al atom relaxing in its equilibrium position over a row of Ni atoms that moves towards the position of a neighboring surface atom, with which it exchanges positions after 1.1 ps. Grey atoms are Al atoms, while black atoms correspond to Ni. The diffusion events of both kinds of adatoms were counted for various temperatures and the corresponding hopping frequencies were obtained. We found that they exhibit Arrhenius behaviour from where we deduced the corresponding activation energies, ENi ¼ 0:09 eV and EAl ¼ 0:19 eV, in good agreement with the static calculations values. At temperatures higher than 900 K, the surface starts to disorder and the diffusion events decrease. In Fig. 15 we present the time evo lution of the LRO parameter along with the number of diffusion events (Nevents ) for three different temperatures 700, 800 and 900 K. It can be seen that the surface exhibits significant loss of order at 900 K, while the number of diffusion events is gradually decreased, the higher temperature exhibiting less diffusion
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Fig. 15. Time evolution of the Ni3 Al(1 1 0) surface order (LRO) parameter along with the number of adatoms and vacancies diffusion events (Nevents ) for three different temperatures.
Fig. 16. Temperature dependence of the Ni3 Al(1 1 0) surface order (LRO) parameter.
events (up to 40% less than those counted in 700 K). We calculated the LRO parameter for various temperatures and the results are shown in Fig. 16. It can be seen that at 900 K this parameter exhibits a rapid change, yielding a value that is comparable to the one found for the Cu3 Au case.
4. Conclusions In this communication we present results concerning the structural and transport properties of the (1 1 0) surface of Cu3 Au and Ni3 Al in presence of Cu/Au and Ni/Al adatoms and vacancies respectively. We found that below 500 K the Cu
adatoms occupy preferably the so-called ‘‘dumbbell’’ position, while at higher temperatures they reside almost exclusively at fourfold positions. This change in the occupation of the available Cu adatom position is accompanied by the spontaneous creation of adatom–vacancy pairs. In addition, we found that the Au adatoms relax mostly at the FF sites. Interestingly, both adatoms when starting from this position, exhibit limited number of hopping type events, while the Au adatom is found to penetrate into the second pure Cu layer via an exchange type mechanism inducing thereby local disorder. Loss of order is induced also in the surface layer mainly via the diffusion of vacancies that aggregate very quickly forming stable and rather immobile vacancy clusters. These diffusion processes result in a disordered surface layer well below the bulk order–disorder temperature by as much as 150 K. In the case of Ni/Al adatoms on the Ni3 Al(1 1 0) surface we found a new adatom position over Ni rows (a site that is energetically favoured for the Al adatom) that is however different from the DB position we found in the Cu3 Au case. Indeed, in this case the Ni and Al adatoms do not exhibit the characteristic strong contraction that made Cu and Au indistinguishable from the surface atoms. The trajectory analysis revealed that up to 800 K both adatoms diffuse mainly by simple hopping mechanism taking place almost exclusively along the channels in between the surface Ni atoms. Consequently, they appear as being trapped in positions defined by their sub-lattices, maintaining therefore the surface in a perfect order. Above 800 K spontaneous creation of adatom–vacancy pairs takes place that in conjunction with exchange type diffusion mechanisms affect seriously the surface order. In addition, the enhanced activity of Ni adatoms, compared to that of Al adatoms (multiple exchanges are found to be quite often), is accentuated by these exchange events leading very quickly the surface in a disordered state.
Acknowledgements The present work was partially supported by the project HPRN-CT-2000-00038.
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Molecular dynamics study of the transport and structural properties of the Cu3Au and Ni3Alð1 1 0Þ surface Ch.E. Lekka a, D.G. Papageorgiou b, G.A. Evangelakis
c,*
a
c
Center of Computational Material Science, Naval Research Laboratory, Washington, DC 20375-5345, USA b Department of Materials Science, University of Ioannina, P.O. Box 1186, Gr-45110 Ioannina, Greece Department of Physics, Solid State Division, University of Ioannina, P.O. Box 1186, Gr-45110 Ioannina, Greece Received 14 June 2002; accepted for publication 25 July 2002
Abstract We present molecular dynamics results concerning the transport and the structural properties of the Cu3 Au and Ni3 Al(1 1 0) surfaces in presence of adatoms and vacancies. In the Cu3 Au case we found that below 500 K the Cu adatom occupies preferably the so-called ‘‘dumb-bell’’ position, while the Au adatom penetrates into the second pure Cu layer via an exchange type mechanism inducing thereby local disorder. At higher temperatures the Cu adatom resides almost exclusively at fourfold positions, while spontaneous creation of adatom–vacancy pairs is also present. Vacancy diffusion processes induce surface disorder that starts by as much as 150 K below the bulk order–disorder temperature. In the case of Ni3 Al(1 1 0) surface, we found a new Al adatom position situated between two Ni surface atoms. It came out that this position plays an important role in the adatom’s diffusive behavior. Above 800 K adatom– vacancy pairs are spontaneously created and in conjunction with exchange type diffusion mechanisms they affect seriously the surface order. In addition, we found that the Ni adatoms are more active than the Al adatoms, this difference in the adatom diffusivity being accentuated by exchange events, leading very quickly the surface in a disordered state. In conclusion, it comes out that the Cu3 Au and Ni3 Al(1 1 0) surfaces disorder well below the bulk transition temperature (663 K, Cu3 Au) and the melting point (1663 K, Ni3 Al) via diffusion processes of the adatoms and/or vacancies. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Surface diffusion; Surface structure, morphology, roughness and topography; Alloys; Adatoms
1. Introduction The last decade the Ni3 Al and the Cu3 Au intermetallic L12 alloys have been the subject of numerous experimental and theoretical investigations because they are related to materials pos-
*
Corresponding author. Tel.: +30-6510-98590; fax: +306510-45631. E-mail address: [email protected] (G.A. Evangelakis).
sessing improved mechanical properties and having important technological applications in catalysis, coating, high temperature engineering, etc. It is found that the Ni3 Al is a directly ordered alloy while the Cu3 Au is a sequentially ordered alloy with an order–disorder transition at Tr ¼ 663 K, well below the melting point [1–8]. Although these L12 alloys exhibit different behaviour in the bulk system, some similarities have been found for their low-index surfaces. Indeed, the energetically favoured surfaces are the Au (Al) rich faces, while
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 2 1 1 6 - 7
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their low-indexed surfaces are rumpled, with the Au (Al) atoms lying above the Cu (Ni) atoms [9– 20]. Focusing on the (1 1 0) surface, low energy electron diffraction (LEED) and low energy ion scattering (LEIS) observations revealed that in the Cu3 Au case the surface undergoes a broadened discontinuous transition below Tr and that above 400 K the compositional fraction of Au atoms changes in the first two atomic layers [21]. In addition, LEED experiments revealed nucleation of islands on the Cu3 Au(1 1 0) with local 2 1 periodicity on a 2 1 ordered substrate, while the apparent activation energy of ordering was found close to that of diffusion in Cu3 Au, suggesting that island formation is controlled by diffusion [22]. However, it is still not clear what is the precursor of the observed changes and which are the exact diffusion mechanisms that are responsible for the surface loss of order on the Cu3 Au(1 1 0) surface and what is happening in the Ni3 Al(1 1 0) case. It is therefore, very interesting to investigate and to try to understand the role of adatoms or vacancies in the early stages of phenomena like order–disorder transition of Cu3 Au and order kinetics of the Ni3 Al(1 1 0) face. In the aim of having a better insight of the transport properties of adatoms and vacancies on these systems as well as their role in the surface order, we performed typical 1 ns molecular dynamics simulations (MD).
2. Computational details
system having 19 atomic layers of 280 atoms each showed that the main quantities we are interested in (surface relaxation, surface energies, vacancy formation energies, etc.) are not affected seriously by the small size of our system. An important issue is the terminal layer for each face. For the (1 1 0) face there is the possibility of having either mixed or pure Cu terminal layer. From experimental and theoretical studies it is now established that in the ordered Cu3 Au a mixed terminal layer is preferred for the three lowindexed surfaces [10,15,21,24–26]. Similar findings are available also in the ordered Ni3 Al [16–18]. We verified by energy minimization that the potential we adopted reproduces these results for both alloys and we used these terminations for the surface layers in our simulations. 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 contributions [27,28]. Accordingly, the total potential energy can be written as: 8 > !! Nb Na >
The simulations were performed in the isothermal canonical ensemble using the Nose demon to control the temperature [23]. The system was made up of 2730 atoms (700 Au and 2030 Cu) arranged on a FCC lattice corresponding to the stoichiometric and ordered Cu3 Au and Ni3 Al alloys. Periodic boundary conditions were applied in the three space directions to mimic an infinite system. By fixing the dimension of the simulation box at a value twice as large as the thickness of the crystal along the z-direction we simulated an infinite slab of 39 atomic layers parallel to (1 1 0) planes (each containing 70 particles) delimited by two free surfaces normal to the corresponding direction. Some preliminary calculations with a
ð1Þ where a and b refer to the two different kind of ! atoms, rijab ¼ jr! ia rjb j is the distance between the ia and jb particles, while the indexes ia and jb run over all the particles. The interactions are computed up to the fifth neighbor’s distance. In the Cu3 Au case the parameters pab , qab , Aab , dab and nab , have been determined using results of total energy calculations as a function of volume, by means of the augmented-plane-wave (APW) method, for the pure Cu, Au and the Ll2 Cu3 Au and Au3 Cu alloys [27]. This potential model is found to reproduce well the elastic constants of pure bulk elements and
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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 [28] that it reproduces well surface energies, rippling effect, atomic MSDs and phonon DOSs for the lowindexed Cu3 Au surfaces and it has been used successfully for the study of the vibrational and diffusive [29,30] properties of Au and Cu adatoms. Similar potential models have been also used successfully in simulations of the [0 0 1] grain boundary of this alloy [31,32]. In the case of Ni3 Al, the corresponding potential parameters have been fitted 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 ¼ 0 K [28]. The equations of motion were integrated by means of the Verlet’s algorithm with a time step of dt ¼ 5 1015 s. In order to simulate the surface in presence of an adatom or a vacancy, one Au/Al (or Cu/Ni) adatom was placed or removed from each Cu3 Au and Ni3 Al surface and the systems were equilibrated at the desired temperatures for 10 000 timesteps. 1 ns simulations were performed covering the temperature range 300–600 K for the Cu3 Au and 300–1000 K for the Ni3 Al with a step of 100 K, respectively. At each temperature, we used the lattice constants that resulted to zero pressure for the bulk systems. For the energy relaxation calculations we used a quasi-dynamic minimization procedure integrated in the molecular dynamic code [33].
3. Results 3.1. Energetic considerations referring to Au and Cu adatoms or vacancies on the Cu3 Au(1 1 0) (static calculations) The geometric anisotropy of the (1 1 0) surface plays an important role in the adatom deposition and the epitaxial growth. This anisotropy is reflected in the potential energy maps in presence of Cu and Au adatoms obtained by relaxing the system with the adatoms at different positions (Fig. 1a and b respectively). The color scale sig-
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nifies (going from blue to red) the gradual energy increase. We can easily recognize the characteristic channels of the (1 1 0) face (red areas) and the different energetic content of the two in-plane directions. We observed that the energetically favoured position for the Cu adatom on the Cu3 Au(1 1 0) surface is between two consecutive Cu surface atoms, dark blue areas in Fig. 1a. These two Cu surface atoms are displaced slightly from their normal lattice sites accommodating the adatom between them, distorting locally the surface layer. Due to this distortion the adatom and a Cu surface atom reside into two consecutive channels, a feature that is distinctive of this surface. It is worth also noting that at this site the Cu adatom belongs in fact in the surface layer since it relaxes at the same distance from the second atomic layer with the Cu surface atoms. This position is known as ‘‘dumb-bell’’ (DB) position [30] and it is found to play important role in the transport phenomena happening on the FCC(1 1 0) surface. Indeed, the DB site is found by MD simulations of different metals as an intermediate metastable position in the exchange type diffusion process [34,35] and also experimentally in the Ag(1 1 0) case when an adatom is displaced by the STM tip [36]. However, in the present case, the DB position is not just a meta-stable site but the energetically favoured adatom position, the usual fourfold position (FF) being a less stable alternative for the Cu adatom. The presence of these two Cu atoms in DB positions has very important consequence in the diffusion processes happening on the surface, stimulating complicated and correlated hopping mechanisms. One of the simplest hopping mechanisms involves a hop of these Cu atoms to the next DB position along the [0 0 1] direction. This is the cross channel direction, along which a usual jump is known to be energetically very expensive. According to this mechanism the Cu adatom situated in the DB1 position moves towards the DB2 site pushing the Cu atom that is located at this position to the next dumb-bell location DB3 (Fig. 1a). We call this mechanism dumb-bell–Cu–dumb-bell (DBCuDB) mechanism. As we shall see in the Section 3.3, the trajectory analysis revealed this process as one of the important mechanisms for surface diffusion in this surface.
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Fig. 1. Potential energy map for various adatom positions on the Cu3 Au (1 1 0) surface: (a) Cu adatom and (b) Au adatom. The color scale indicates (going from blue to red) the gradual increase of the energy (eV). FF refers to the fourfold adatom positions, DB1;2;3 refer to the dumb-bell adatom position and Au (Cu) represent the corresponding surface atoms.
Contrary to the Cu adatom, we found that in the case of Au adatom the DB position is not energetically favoured, the FF position being the only available adatom site (Fig. 1b). From the geometry of this energy map we can predict in principle three possible hopping mechanisms from one FF position to the next one. There may be two hopping mechanisms along the [1 1 0] and another one along the [0 0 1] direction. (a) Between two Au surface atoms requiring an energy barrier of 0.27 eV (path FFAFF), (b) passing between two Cu surface atoms with energy barrier 0.38 eV (path FFBFF) and (c) the less favoured jump crossing the channel with energy barrier equal to 1.06 eV. Turning our attention to the vacancy cases, we estimated by energy minimization at 0 K the formation energies of Au surface vacancy and Cu surface vacancy at 0.81 and 0.12 eV, respectively. The corresponding values for the bulk system are estimated at 1.48 and 0.84 eV. In addition, we found that the presence of a vacancy in the or-
dered Cu3 Au(1 1 0) surface affects the relaxed positions of the neighboring surface atoms. We quantified this effect by calculating for the surface atoms and in presence of a Cu vacancy the inplane (%) deviations of the atomic distances from the perfect lattice spacing (DAD). In Fig. 2, we show this quantity computed along the [0 0 1] and the [1 1 0] directions for the in-plane and normal to the surface direction (normalized with respect to the bulk interlayer spacing). In Fig. 2a, we observe that the Cu vacancy induces expansion to the closest Cu surface atoms along the [0 0 1] in-plane direction, resulting in strained row along that direction, while in the [1 1 0] direction the effect is insignificant. This expansion is about þ2% and involves mainly the first neighbors. On the contrary, normal to the surface direction, we observe that the closest Au surface atoms along the [1 1 0] direction are contracted by )2.89%. Concluding, it seams that the presence of the Cu vacancy affects mainly the first nearest neighbors of both in-plane
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Fig. 2. Difference (%) between the relaxed positions of the surface atoms with and without vacancy as a function of the vacancy–atom distance for the in-plane directions and the normal to the surface direction with respect to the bulk interlayer spacing: (a) Cu vacancy and (b) Au vacancy. Filled squares and circles stand for the Cu and Au surface atoms along the [1 1 0] direction, while open diamonds stand for the atoms along the [0 0 1] direction.
directions, a result that is different from the behavior of the Cu vacancy in the Cu3 Au(0 0 1) surface [37]. Focusing on the Au vacancy (Fig. 2b), we found expansion along the [1 1 0] and [0 0 1] in-plane directions by 0.75% (Cu atoms) and 0.20% (Au atoms) for the first neighboring atoms. In the normal to the surface direction these atoms are attracted towards the second layer by 1.62% (Cu atoms) and 0.75% (Au atoms). These relaxation effects are important since they can be related with the diffussive behaviour of the vacancies. Indeed, we would expect that diffusion of the Cu vacancy takes place in both in-plane directions, while inplane diffusion along the [1 1 0] direction is expected for the Au vacancy contrary to the (0 0 1) surface where the vacancy diffusion was found to take place only along the [1 1 0] direction [37]. Consequently, there is in principle one possible vacancy hopping mechanism along the [1 1 0] direction corresponding to simple (SH) vacancy jump and another one (simple hopping) along the
[0 0 1] direction (SH001). In Fig. 3 we give a schematic representation of the various vacancy hopping mechanisms studied. We note that in the SH and the SH001 mechanisms the distances that the vacancy covers are different and equal to the nearest neighbour distance and the lattice constant, respectively. In the SH event, a Au surface atom is jumping into the Cu vacancy (Fig. 3a), resulting in local disorder. On the contrary, in the SH001 process the movement of a Cu surface atom along the [0 0 1] direction has no effect in the local order, since in this direction only Cu atoms are participating (Fig. 3b). It is also worth noting that the SH mechanism leads to an energetically lower configuration, while the SH001 process leaves the system energetically unchanged. This can be seen in Fig. 4a and b where we give the energy required for a SH and SH001 hopping mechanism of the Cu vacancy (tagged squares) and the Au vacancy (filled squares). Starting from the same initial configuration, after an SH hop we end up with a configuration with smaller energy, )0.14 eV for the
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Fig. 3. Schematic representation of the various vacancy hopping mechanisms studied: (a) simple hopping mechanism along the [1 1 0] direction and (b) simple hopping mechanism along the [0 0 1] direction. Big and small circles stand for the Au and Cu surface atoms, while open squares stand for the Cu vacancy. Arrows show the hopping direction of the vacancy.
Cu vacancy and )0.09 eV for the Au vacancy (Fig. 4a), while the energy barrier is around 0.35 eV for both cases. The zero level in this figure corresponds to the vacancy formation energy. In addition, after the accomplishment of a SH001 event there is no energy change (Fig. 4b), while the energy barrier is 0.45 eV for the Cu vacancy and 0.88 eV for the Au vacancy. It has to be noted that the energy required for a Cu vacancy to perform a SH001 is equal to the corresponding barrier for a SH mechanism indicating that both hopping mechanisms should have equal frequencies in both in-plane directions. However, after an SH jump along the [1 1 0] direction (Fig. 4a), the system is left at lower energy estimated at )0.29 and )0.21 eV for the Cu and Au vacancy, respectively. In addition, in the SH mechanism the Cu and/or Au surface atoms are involved in a random sequence, resulting in rearrangement of the surface atoms that are located in anti-sites, yielding configurations with lower energies and local disorder. This result is inevitably related to the Cu3 Au orderdisorder transition.
Fig. 4. Potential energy barriers for the Cu vacancy (tagged squares) and Au vacancy (filled squares) hopping mechanisms studied: (a) simple hopping (SH) mechanism along the [1 1 0] direction and (b) simple hopping (SH001) mechanism along the [0 0 1] direction.
3.2. Static calculations for Al and Ni adatoms or vacancies on the Ni3 Al(1 1 0) The behaviour of Ni and Al adatoms on the Ni3 Al(1 1 0) surface is completely different from the one described above concerning the Cu3 Au case. The available adatom positions are shown in the energy maps (Fig. 5a and b). The color scale indicates (going from blue to red) the gradual energetic increase. We found that for both Ni and Al adatoms the energetically favoured position is the FF. A similar to the DB position over a row of Ni atoms is also found for both adatoms. However, in the present case the adatoms do not exhibit the
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Fig. 5. Potential energy map for various adatom positions on the Ni3 Al(1 1 0) surface: (a) Ni adatom and (b) Al adatom. The color scale indicates (going from blue to red) the gradual increase of the energy (eV). FF refers to the fourfold adatom positions, BNA refers to the ‘‘between Ni surface atoms’’ adatom position, A refers to the adatom position between two Al surface atoms and Ni (Al) refer to the corresponding surface atoms.
important contraction we found in the Cu3 Au case, in which there were found to reside practically at the same level with the surface atoms. This new adatom position is situated along the channel between two Ni surface atoms, (BNA) position hereafter. Starting from the FF position, the Ni adatom could hop to the next FF position along the channel, between two Ni surface atoms (path FF-BNA-FF with energy barrier 0.15 eV) or between two Al surface atoms (path FFAFF with energy barrier 0.43 eV) (Fig. 5a). A hopping mechanism of the Ni adatom along the [0 0 1] direction requires 1.10 eV. Concerning the Al adatom (Fig. 5b), the energetically favoured hopping mechanism is the one from a BNA position to the closest FF position (energy barrier 0.18 eV). In addition, there are two other possible hopping mechanisms, one starting from a FF position to the next FF between two Al surface atoms (path FFAFF, energy barrier 0.55 eV) and another one
crossing the channel from a BNA to the next BNA (path BNA-Ni-BNA, energy barrier 1.29 eV). Concerning the vacancy cases for this material, we found from energy minimization at T ¼ 0 K that the formation energies of an Al and a Ni surface vacancy are 0.62 and 0.48 eV, respectively. These values have to be compared with the corresponding values for the bulk system that are estimated at 2.04 and 1.94 eV. Moreover, similarly to the Cu3 Au(1 1 0) surface, we found that the presence of a vacancy in the ordered Ni3 Al(1 1 0) surface affects the relaxed positions of the neighboring surface atoms. To demonstrate this effect for the Ni vacancy, we show in Fig. 6a the DAD computed along [0 0 1] and [1 1 0] directions for the in-plane and normal to the surface direction (normalized with respect to the bulk interlayer spacing). Moreover, in Fig. 6b we show the corresponding quantities for the Al vacancy. In both cases, we observe that in the in-plane directions the
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Fig. 6. Difference (%) between the relaxed positions of the surface atoms with and without vacancy as a function of the vacancy–atom distance for the in-plane direction and for the normal to the surface direction with respect to the bulk interlayer spacing: (a) Ni vacancy and (b) Al vacancy. Filled squares and circles stand for the Ni and Al surface atoms along the [1 1 0] direction, while open diamonds stand for the atoms along the [0 0 1] direction.
vacancy attracts the two nearest neighbor atoms along the [1 1 0] direction (the first one by )3.33%), while the first neighboring atoms along the [0 0 1] direction are expanded (1.77% for the Ni vacancy and 1.55% for the Al vacancy). From these findings one would expect simple or double vacancy hops to occur along this direction. In the normal to the surface direction (Fig. 6a), we found that the presence of a Ni vacancy induces small contraction ()0.64%) in the first neighboring atoms along the [1 1 0] direction. In addition, in the normal to the surface direction (Fig. 6b), the Al vacancy attracts the first atoms in both directions ()0.16% for the [0 0 1] direction and )0.14% for the [1 1 0] direction). Concluding, it seems that the vacancy presence affects mainly the relaxed positions of the surface atoms in the in-plane [1 1 0] direction, while it expands the surface atoms in the in-plane [0 0 1] direction. In addition, there is not enough influence in the normal to the surface direction. This suggests that the energetically favored hopping mechanisms would be along the in-plane [1 1 0]
direction, a result that is similar with the findings concerning the vacancy diffusion in the Ni3 Al(0 0 1) surface. These differences in the relaxation of both adatoms and vacancies between the two systems we are studying may have serious consequences in the surface kinetics. The understanding of their physical origin requires detailed and accurate electronic structure calculations. In our case all electronic contributions are incorporated into the effective interatomic model we are using. Therefore, no information can be extracted about the electronic bands for the various cases we studied. However, the present results could be understood taking into account the different bonding character of these alloys. Indeed, on one hand we know that Cu3 Au is a ‘‘common d-band alloy’’ where the Au bands are concentrated at the bottom of the d band and the Cu bands are concentrated at the top of the d band. Upon formation of the Cu3 Au the Au 5d band is depleted [38]. This spatial compression on the 5d shells of Au is most likely due to the sp
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electrons compressing to a smaller atomic volume [39]. In order to decrease this spatial compression on 5d electrons, the surface Au atoms are displaced outward [40]. On the other hand the bonding charge density of the Ni3 Al alloy can be described by the combination of charge transfer from Al to Ni and a strong Al p-Ni d hybridisation effect [41,42]. Furthermore, using constrained energy minimization, we calculated the energy barriers required for the Ni and the Al vacancy to perform a SH and a SH001 hopping mechanisms. Similarly to the Cu3 Au(1 1 0) surface, after a SH hop along the [1 1 0] direction, the local environment changes (Fig. 3). In order to follow the evolution of the system in a succession of such events, we calculated the energy barriers for two consecutive jumps of the vacancies. In Fig. 7a, we show these quantities for the SH hop mechanism of the Ni vacancy (tagged squares) and the Al vacancy (filled circles), respectively. Again, the initial energy level, zero energy values in this figure, corresponds to the formation energies of the vacancies. It is very interesting to observe that the energy needed for the Ni or Al vacancy to perform a first (0.50 or 0.43 eV) or a second (0.62 or 0.52 eV) hop towards the [1 1 0] direction is higher than for backward movements (0.48 or 0.41 eV for the first hop). This indicates that these vacancies would prefer to return in their initial perfect positions than perform a second hop towards the [1 1 0] direction. On the contrary, in the Cu3 Au case, Fig. 4a the local configurations after a successful 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 along the [0 0 1] direction, we give in Fig. 7b, the energy barriers required for the Al (filled circles) and Ni (tagged squares) vacancies to perform a hop in the [0 0 1] direction. We observe that in all cases the energy barriers required for the SH001 mechanism are much higher than the energy barriers needed for the SH process, situated at the values of 0.87 and 0.73 eV, respectively. These results indicate that the in-plane diffusion is favored in the [1 1 0] direction. In particular, vacancy diffusion in the Cu3 Au(1 1 0) surface is expected to produce re-
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Fig. 7. Potential energy barriers for the Ni vacancy (tagged squares) and Al vacancy (filled squares) hopping mechanisms studied: (a) simple hopping (SH) mechanism along the [1 1 0] direction and (b) simple hopping (SH001) mechanism along the [0 0 1] direction.
arrangement of the surface atoms in anti-sites, and loss of the local order, leaving the system in lower energy level. In addition, it appears that these processes must have an irreversible tendency explaining the order–disorder transition observed in this surface. On the contrary, in the Ni3 Al(1 1 0) 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 (less than 1.0 eV for the second successive hop) and therefore vacancy induced disordering has to be
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expected at high temperatures. We note here that our results are based on a semi-empirical potential model and therefore the calculated values have to be considered with caution. 3.3. Transport properties of Cu and Au adatoms and vacancies on the Cu3 Au(1 1 0) surface As we have seen in the previous section, static calculations predict that the energetically favoured position of the Cu adatom on the Cu3 Au(1 1 0) surface is the ‘‘dumb-bell’’ (DB) position which plays an important role in the transport phenomena occurring on this surface. In addition, we found that the usual FF position, although less energetically favoured, is also an equilibrium site for the Cu adatom. Contrary to the Cu adatom, the DB position is not energetically favoured for the Au adatom, the FF position being therefore the only available adatom site in this case. Interestingly, we found that the diffusion events are rather scarce for both Cu and Au adatoms when residing at the FF position. Specifically, above the room temperature the Au adatom enters into the second atomic layer via an exchange type mechanism similar to the on found in the case of the (0 0 1) surface, namely the so-called second layer exchange (SLE) [29]. We conclude therefore that the high diffusivity that this face exhibits is attributed to Cu adatoms. In order to quantify the stability of the Cu adatom in the DB position and compare with the corresponding occupation of the FF position, we calculated the average life-time of the Cu adatoms in both positions. It comes out that at low temperatures the DB position is clearly favoured. As the temperature increases the adatom spends most of the time on the FF position (Fig. 8). This phenomenon is accompanied by the spontaneous creation of adatoms that relax almost exclusively at the FF position, especially at temperatures higher than 500 K. Spontaneous creation of adatoms has been observed on the Cu(1 1 0) surface as well [43,44]. In Fig. 9 we show the concentration of Cu adatoms on the two available positions as a function of temperature. It can be seen that the concentration of Cu adatoms on the DB position is practically unchanged for a large temperature
Fig. 8. Average life-time of the Cu adatoms at DB (open circles) and FF (filled circles) positions as a function of temperature.
Fig. 9. Arrhenius diagram of the concentration of the Cu adatoms at DB (open circles) and FF (filled circles) positions.
range. On the contrary, the concentration of Cu adatoms at the FF position is almost zero at low temperatures to saturate at a value of 10% above 500 K. In this region the spontaneous creation of adatoms appears to follow Arrhenius behaviour, from where we deduced the corresponding energetic requirement for the creation of an adatom– vacancy pair, 0.9 eV. These findings are compatible with the results concerning the relative relaxed positions and suggest modifications in the perfect order of surface atoms. Indeed, we found that the long-range order (LRO) diverges significantly from the value corresponding to perfectly ordered surface at the same temperature (500 K) (Fig. 10).
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Fig. 10. Time evolution of the Cu3 Au(1 1 0) surface order (LRO) parameter along with the number of diffusion events (Nevents ) of the adatoms and vacancies for three different temperatures.
It is worth noting also that this change of surface order takes place very quickly. Indeed, the process seems to be concluded after 0.5 ns and the system appears in a stationary regime after that time. In order to estimate the correlation between the diffusion events and the surface order, we counted the number of diffusion events (Nevents ) as a function of time, the system being in thermodynamic equilibrium. The results for different temperatures, normalized by the maximum number of events at T ¼ 500 K for comparison reasons, are presented in the same figure (Fig. 10). It is clearly seen that the rate of diffusion events is positive for the first two temperatures, while it is almost zero at T ¼ 600 K, temperature at which the LRO parameter reaches its highest value. This result suggests that the surface disordering occurs via diffusion events. In Fig. 11 we show the temperature dependence of the LRO parameter. As we can see, the LRO decreases dramatically above T ¼ 450 K, suggesting that the surface looses its order above 500 K, much earlier than the bulk system (Tr ¼ 663 K [45]), in agreement with Monte Carlo simulations [46]. We conclude therefore, that surface disorder is a direct result of the diffusion of adatoms and vacancies. At temperatures lower than 500 K the Cu adatom diffuses from a DB position to another DB position (DBCuDB mechanism), while at
Fig. 11. Temperature dependence of the Cu3 Au(1 1 0) surface order (LRO) parameter.
higher temperatures Cu adatom diffusion takes place almost exclusively between FF positions. We must emphasize that the DBCuDB diffusion mechanism happens with the simultaneous movement of two Cu atoms that lie in two surface channels. The equivalent movement of a dimer over a surface channel is a mechanism with high energetic requirements. It is worth noting that the DB mechanism does not alter the ordering of atoms on the surface since it occurs along a row of Cu atoms, the surface order being in fact a prerequisite for this mechanism. In addition, it
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Fig. 12. Dumb-bell–dumb-bell (DB–DB) hopping mechanism of two Cu atoms at 450 K: one of the Cu atoms (B labeled particle) hops towards the second Cu atom (A labeled atom) forcing it into the channel, t ¼ 1:1 ps. Subsequently, the A atom hops to the next DB position popping a Cu surface atom to an adatom position, t ¼ 2:4 ps.
appears that the surface stoichiometry is decisive for the existence of the mechanism, this kind of process not being observed in pure metals (e.g. Cu). Indeed, in this last case the dimers on the corresponding surface, although very active, were not found to move in any way perpendicularly to the [1 1 0] direction (the channel direction), due to the large energetic requirement for such a diffusion event. However, the DB position of the adatom has been found to play an important role in inchannel diffusion mechanisms. A typical example of a diffusion event between two DB positions (DBCuDB mechanism) at T ¼ 450 K is shown in a succession of snapshots in Fig. 12. The system being in a steady state, at least as far as surface structural changes are concerned, we counted the diffusion events for all temperatures (Fig. 13). It can be seen that the frequency of the
Fig. 13. Arrhenius diagram for the DBC uDB Cu adatom diffusion mechanism on the Cu3 Au(1 1 0) face.
diffusion events follows Arrhenius behaviour up to 500 K, from where we deduced activation energy of 0.18 eV. At higher temperatures the frequency of diffusion is clearly reduced (almost by an order of magnitude), a result that can be understood taking into account that at this temperature region the surface is disordered. Turning our attention to the behaviour of vacancies, we recall that an important characteristic of this surface is their spontaneous creation at relatively low temperatures. A detailed trajectory analysis revealed that these vacancies are very active and their diffusion changes locally the surface order. In addition, we observed that especially the Cu vacancies could diffuse in both in plane directions, in agreement with the static calculations. Another important result is that they tend to combine forming di-vacancies, which contrary to simple vacancies are very stable and practically immobile, at least within the time scale of our simulations. Therefore, although the presence of vacancies is one of the main reasons for the loss of the surface order, when this happens, the majority of vacancies form clusters that do not show any significant activity, rendering a further study of the vacancy diffusion mechanisms meaningless. 3.4. Transport properties of Ni and Al adatoms and vacancies on the Ni3 Al(1 1 0) surface The static calculations revealed a new adatom position on the Ni3 Al(1 1 0) surface, which we called BNA. This position, in conjunction with the FF positions at both sides of the channel allows for the possibility of an oscillatory motion of the adatoms. This movement is typical for both species, although Al adatoms appear more stable in this position. From the trajectory analysis it came out that up to 800 K both adatoms diffuse mainly with simple hops, almost exclusively along the surface channels, with the Ni adatom diffusing more frequently. Spontaneous creation of adatoms has not been observed, contrary to the Cu3 Au case where adatom–vacancy pairs were found to exist above room temperature. In addition, we found that diffusion events over a row of Al atoms are rare. We can conclude therefore that diffusion takes place only along the channel of two consecutive Al
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atom rows. This finding has the important consequence that both adatoms are essentially trapped in the positions defined by their corresponding sub-lattices, yielding a surface that maintains its perfect order up to this temperature. At temperatures higher than 900 K, spontaneous creation of adatom–vacancy pairs does occur. However, these additional vacancies do not have significant effect to the surface activity and structure, since their concentration is rather limited, they are short lived and combining between them they form di-vacancies that are particularly stable and hard to move. On the contrary, exchange type phenomena between adatoms and surface atoms are important and decisive, since they induce an immediate change to the surface order. Such phenomena were observed in all possible combinations, namely, exchange of a Ni or Al adatom with the same kind of surface atom, as well as exchanges between different kinds of atoms, like exchanges of Ni adatoms with a surface Al and vice versa. Therefore above 900 K the surface disorders through these exchange type mechanisms. Interestingly, the local loss of the surface order alters the diffusive behaviour of the adatoms. Indeed, the adatoms that were trapped between two successive Al rows can now escape passing over the points where the Al row is broken. It follows that they can diffuse in longer distances and create through a new exchange mechanism another locally disordered area. We found that this process is repeated leading to an intense diffusive activity and surface disordering. It is worth also noting that the initially greater activity of Ni compared to Al adatoms, is further increased through the exchange events which especially for this adatom can be multiple (double or triple exchanges) leading the surface faster in a disordered state. Hence, as far as that surface order and diffusivity are concerned, the effect of an Al adatom exchange with a surface Ni, is greater than that of a Ni adatom with a surface Al. These results are in agreement with recent experimental data in the Ni3 Al system which revealed that in the temperature range between 1423 and 1523 K, the intrinsic diffusion coefficients of the Ni and Al atoms are almost the same and that the diffusion mechanisms incorporate anti-site defects [47].
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Fig. 14. Simple exchange mechanism of Al adatom (atom A) with a surface Ni (atom B) at 900 K. The procedure starts with the Al atom relaxing at the normal equilibrium position, over a row of Ni atoms. After 1 ps, it moves towards the position of a neighbouring surface atom, with which it exchanges positions. Grey and black particles stand for Al and Ni atoms.
In Fig. 14, we present a typical example of a simple exchange mechanism between an Al adatom (atom A) with a surface Ni (atom B) at 900 K. The process starts with the Al atom relaxing in its equilibrium position over a row of Ni atoms that moves towards the position of a neighboring surface atom, with which it exchanges positions after 1.1 ps. Grey atoms are Al atoms, while black atoms correspond to Ni. The diffusion events of both kinds of adatoms were counted for various temperatures and the corresponding hopping frequencies were obtained. We found that they exhibit Arrhenius behaviour from where we deduced the corresponding activation energies, ENi ¼ 0:09 eV and EAl ¼ 0:19 eV, in good agreement with the static calculations values. At temperatures higher than 900 K, the surface starts to disorder and the diffusion events decrease. In Fig. 15 we present the time evo lution of the LRO parameter along with the number of diffusion events (Nevents ) for three different temperatures 700, 800 and 900 K. It can be seen that the surface exhibits significant loss of order at 900 K, while the number of diffusion events is gradually decreased, the higher temperature exhibiting less diffusion
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Fig. 15. Time evolution of the Ni3 Al(1 1 0) surface order (LRO) parameter along with the number of adatoms and vacancies diffusion events (Nevents ) for three different temperatures.
Fig. 16. Temperature dependence of the Ni3 Al(1 1 0) surface order (LRO) parameter.
events (up to 40% less than those counted in 700 K). We calculated the LRO parameter for various temperatures and the results are shown in Fig. 16. It can be seen that at 900 K this parameter exhibits a rapid change, yielding a value that is comparable to the one found for the Cu3 Au case.
4. Conclusions In this communication we present results concerning the structural and transport properties of the (1 1 0) surface of Cu3 Au and Ni3 Al in presence of Cu/Au and Ni/Al adatoms and vacancies respectively. We found that below 500 K the Cu
adatoms occupy preferably the so-called ‘‘dumbbell’’ position, while at higher temperatures they reside almost exclusively at fourfold positions. This change in the occupation of the available Cu adatom position is accompanied by the spontaneous creation of adatom–vacancy pairs. In addition, we found that the Au adatoms relax mostly at the FF sites. Interestingly, both adatoms when starting from this position, exhibit limited number of hopping type events, while the Au adatom is found to penetrate into the second pure Cu layer via an exchange type mechanism inducing thereby local disorder. Loss of order is induced also in the surface layer mainly via the diffusion of vacancies that aggregate very quickly forming stable and rather immobile vacancy clusters. These diffusion processes result in a disordered surface layer well below the bulk order–disorder temperature by as much as 150 K. In the case of Ni/Al adatoms on the Ni3 Al(1 1 0) surface we found a new adatom position over Ni rows (a site that is energetically favoured for the Al adatom) that is however different from the DB position we found in the Cu3 Au case. Indeed, in this case the Ni and Al adatoms do not exhibit the characteristic strong contraction that made Cu and Au indistinguishable from the surface atoms. The trajectory analysis revealed that up to 800 K both adatoms diffuse mainly by simple hopping mechanism taking place almost exclusively along the channels in between the surface Ni atoms. Consequently, they appear as being trapped in positions defined by their sub-lattices, maintaining therefore the surface in a perfect order. Above 800 K spontaneous creation of adatom–vacancy pairs takes place that in conjunction with exchange type diffusion mechanisms affect seriously the surface order. In addition, the enhanced activity of Ni adatoms, compared to that of Al adatoms (multiple exchanges are found to be quite often), is accentuated by these exchange events leading very quickly the surface in a disordered state.
Acknowledgements The present work was partially supported by the project HPRN-CT-2000-00038.
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