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Island adsorption and adatom diffusion on 3D non-crystalline silver nanoclusters
Surface Science 490 (2001) 361±375
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Island adsorption and adatom diusion on 3D non-crystalline silver nanoclusters F. Baletto *, R. Ferrando Dipartimento di Fisica dell'Universita di Genova, INFM and CFSBT/CNR, via Dodecaneso 33, I-16146 Genova, Italy Received 2 March 2001; accepted for publication 8 June 2001
Abstract We report a systematic study of island adsorption and single-adatom diusion on free silver nanoclusters, and discuss the consequences on the growth. In our calculations, silver is modelled by semiempirical many-body potentials. We consider magic non-crystallographic structures at dierent sizes: icosahedra (Ih) at 55, 147 and 309 atoms; Markstruncated decahedra (m-Dh) at 75, 146 and 192 atoms. We calculate the map of adsorption sites and the energy barriers for the dierent diusion processes. We ®nd that, due to purely geometrical reasons, medium-size (from 6±8 to 30±40 atoms depending on the cluster) islands on the cluster (1 1 1) facets prefer the hcp stacking on both Ih and Dh structures, while both smaller and larger islands are better placed on fcc stacking. Interfacet diusion is easy on both Dh and Ih, indicating that large islands are easily grown; in particular, there are multi-atom diusion processes which allow fast diusion among the two caps of Dh clusters. For Dh clusters, islands on hcp stacking may lead to the appearance of new ®vefold symmetry points, and to the transformation of the cluster into an icosahedron. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Clusters; Silver; Molecular dynamics; Growth
1. Introduction Clusters are an important state of matter, being aggregates of atoms and molecules that are small enough not to have the same properties as the bulk liquid or solid. Quantum states in clusters are size dependent, leading to new electronic, optical, and magnetic properties. Clusters oer attractive possibilities for innovative technological applications in ever smaller devices, for example in the ®elds of
*
Corresponding author. E-mail addresses: baletto@®sica.unige.it (F. Baletto), ferrando@®sica.unige.it (R. Ferrando).
the controlled growth of nanostructures [1,2] and of catalysis [3]. In recent years, much experimental and theoretical eort has been devoted to the determination of the most favourable structures depending on the cluster size N. Small noble and transition metal clusters can present ``non-crystallographic'' structures, like icosahedra (Ih) and decahedra (Dh), especially Marks-truncated Dh (m-Dh) [4±14]. Atomic unsupported 3D clusters are formed under several dierent conditions, but the actual process is not known in detail. In typical experiments, metal clusters are grown in a gas phase of atoms; the metal vapour is carried in an inert gas beam into the condensation chamber, where the actual growth takes place [15±19]. The
0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 1 ) 0 1 3 5 5 - 3
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experimental conditions, such as the temperature and pressure of the carrier gas, the time the beam stays in the condensation room, in¯uence strongly the building up of the cluster morphology [16±18]. We have explicitly shown by molecular dynamics (MD) simulations that the ®nal cluster morphology depends on the interplay of kinetics and thermodynamics [20,21]. This is in agreement with experiments, which do not re¯ect the simple picture arising from equilibrium total-energy considerations, and gives strong support to kinetic modelling to explain experimental results about cluster growth [22]. Indeed, from experimental data and theoretical results, it is known that during growth [20,21] and solidi®cation from liquid droplets [23±25] small clusters can change shape with size among Ih, Dh and fcc structures. The coexistence of dierent structures, some of them far from the most stable ones, suggests that cluster growth is controlled by individual atomic-scale processes rather than by the minimization of the cluster total energy. Surface diusion has a great importance also in the coalescence of clusters [26, 27]. In this paper, we study island adsorption and single-adatom diusion on non-crystalline silver 3D nanoclusters by quenched MD. We consider, in particular, Ih and Dh structures, which are the most stable at these small sizes. The case of fcc truncated octahedra (TO) has been studied in Ref. [28]. We discuss how the diusion processes which allow an adatom to cross between dierent facets (interfacet processes) are fundamental during the building up of the cluster. Moreover, we calculate the energetics of adsorption on Ih and Dh clusters. For adatoms and islands on (1 1 1) facets, we show that dierent stackings are possible, with deep consequences on the cluster growth. In particular, we analyse the microscopic mechanism of the transformation of Dh into Ih, which has purely kinetic origin, being in the opposite direction to what equilibrium thermodynamics suggests [21]. The article is structured as follows. In the next section we report our theoretical model and methods. Section 3 deals with the adsorption energies of islands on Marks Dh, while in Section 4 we report the adsorption maps for an on Mackay Ih. Section 5 contains the results concerning the static energy
barriers on these non-crystallographic structures; the conclusions are outlined in Section 6. 2. Theoretical model In the following, we model silver by the manybody potential proposed by Rosato, Guillope and Legrand (RGL, see Ref. [29]; for a derivation of the second-moment approximation (SMA) for noble metals see Ref. [30]). The RGL potential is derived in the framework of the SMA to the density of states in the tight-binding model. In this framework, the band energy Ebi for a given atom i is proportional to the square root of the second moment of the local density of states. Ebi is an attractive many-body term; the stability of the cluster is insured by adding a phenomenological corerepulsion term Ebr of the Born±Mayer type. Thus, the potential energy of atom i is given by Eci Ebi Eri where the band energy Ebi can be written as: ( )1=2 X 2 rij i n exp 2q ; Eb 1 r0 j;rij
1
2
where n is an eective hopping integral, rij is the distance between the atoms i and j, rc is the cut-o radius for the interaction, r0 is the nearest-neighbour (NN) distance and q gives the distance dependence of the hopping integral. The second term Eri is given by: X rij i A exp p Er 1 : 3 r0 j;rij
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
data, ab initio and semi-empirical calculations are reported in Refs. [32,33]. It turns out that RGL results predict correctly the dominant diusion mechanisms on noble-metal surfaces; from the quantitative point of view, the agreement with experimental data and ®rst-principle calculations is very good, especially for silver. Comparison with density functional calculations for noble-metal clusters are found in Ref. [9], again with good results. In order to calculate the energy barriers for the diusion mechanisms, we have used the nudged elastic band (NEB) method [34]. The NEB method allows to identify the minimum energy path for a rearrangement of a group of atoms among two stable con®gurations and to estimate the activation energy. In this method, a chain of intermediate images in between the two stable con®gurations is built up; the images are connected by springs, and are optimized simultaneously (by quenched MD) while the initial and ®nal positions are kept ®xed. Details about the method are given in Ref. [28]. Here we simply add some information about the convergence of our NEB calculations, which has been checked as follows. First, we ®x the number of images in the NEB and quench the system (usually down to very low temperatures, of the order of 10 3 ±10 4 K) until we reach the convergence on four signi®cant digits in the energy of each image. Then we repeat the procedure with a larger number of images, until the barrier estimate converges to two signi®cant digits at least. Typically, the number of images is in between 10 and 20. 3. Adsorption sites and energies on decahedra In this section, we study the adsorption of an island of size n on the surface of a Dh. A Dh is a non-crystalline structure with a single ®vefold symmetry axis. The Dh can be constructed out of ®ve distorted tetrahedra which are packed together so to have a common edge. In this way the surface is close-packed. However, a Dh has a very large surface area in addition to internal strain, the latter due to the distortion of tetrahedra. For these reasons the Dh is not expected to be energetically favourable at any cluster size. But this is not the
363
Fig. 1. m-Dh at 75, 146 and 192 atoms: side view (on the left) and top view (on the right). For these magic clusters the sets m; n; p are 2; 2; 2, 3; 2; 2 and 3; 3; 2, respectively. m and n are the lengths of the edges of the (1 0 0)-like facets, p is the length of the Marks reentrance.
case for the truncated forms of the Dh, which are much more spherical structures. Marks [6] proposed a truncation which exposes reentrant (1 1 1) facets in addition to (1 0 0) facets, see Fig. 1. To describe the Marks Dh, one introduces three integers m, n, p which are the length of (1 0 0) edges and the length of the (1 1 1) reentrance, respectively. On a Dh, adatoms can be adsorbed on (1 1 1) and (1 0 0) facets, and in the Marks reentrance. The adsorption sites have dierent adsorption energies Eads . By counting the number of NN, it is easily seen that the adsorption sites on (1 0 0) facets are the most favourable, while the sites on (1 1 1) facets present the most unfavourable Eads . This is in agreement with the ®ndings in Refs. [35,36], where single-adatom adsorption and diffusion on fcc cuboctahedra of 13, 55 and 147
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
atoms and on the Dh of 13 atoms was studied for silver and other metals by Sutton-Chen and Murrell-Mottram semiempirical potentials. The sites in a Marks reentrance (R(110)-sites) are at intermediate energies for the Dh structures considered in the following. There, Eads is a few hundredths of eV higher than on (1 0 0) sites. In agreement with the data on ¯at surfaces, the energy dierence between the best sites on (1 0 0) and on (1 1 1) facets is DEads 0:24±0:30 eV. Comparing this with the analogous dierence for the fcc 201 TO, where DEads 0:22±0:28 eV [28], one could conclude that the internal strain of a Dh does not in¯uence very much the surface diusion barriers, which depend only on the size and the symmetry of the facet. On a Marks Dh, the most part of the surface is close-packed; thus big islands can nucleate only on one or more adjacent (1 1 1) facets. The biggest possible island covers the whole top cap or bottom cap of the Dh (both top and bottom caps are
formed by ®ve (1 1 1) facets). For these reasons, in the following we focus on adsorption on (1 1 1) facets for single adatoms and for islands of size n. Due to the ®nite sizes of these facets, there are several dierent kinds of adsorption sites, for example if the adatom is adsorbed close to an edge, to a vertex or on a central position. However, as it happens in the case of ¯at (1 1 1) surfaces, we can subdivide ®rst the adsorption sites in two groups (see the right column of Fig. 2), which are easily distinguished looking at the cluster along its ®vefold symmetry axis. From this viewpoint, the atoms of the Dh are perfectly arranged in columns. When an adatom is adsorbed on a (1 1 1) facet, it can either continue or break the decahedral arrangement in columns. In the former case, we have adsorption on a site in decahedral stacking; in the latter we have adsorption on a site in icosahedral stacking (see Fig. 2). By looking at a (1 1 1) facet along its normal, one can easily see that the decahedral stacking corresponds to the fcc stack-
Fig. 2. Adsorption sites on a (1 1 1) facet of a m-Dh: the views perpendicular to the facet are represented on the left; the views along the ®vefold axis are reported on the right. Islands on Dh and Ih stackings are shown in the upper and lower rows, respectively. It is clear that the Dh stacking corresponds to the fcc stacking on a ¯at (1 1 1) surface, while the Ih stacking is the hcp stacking.
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
ing on a ¯at (1 1 1) surface while the icosahedral stacking to the hcp stacking. According to our calculations, single adatoms prefer an fcc central position (in Fig. 2 the decahedral site d4) or just close to the (1 0 0) edge (in Fig. 2 the decahedral sites d1, d3 or the icosahedral site h1), but in any case the energy dierences are very small, of the order of few hundredths eV, see Table 1. Since we cannot expect that our calculations are sensitive down to such precision (this because of the approximate semiempirical model), we conclude that these sites are essentially equivalent. However, the important point is the following. For a single adatom, the fcc sites on a (1 1 1) edge (in Fig. 2 the Dh positions d5, d6, d7) and also the hcp sites like i6 are not stable minima. These sites become stable minima only if the adatom has already one or more neighbours on a adjacent (1 1 1) facet (for example on d4, d3 sites or i1, i3 sites). In the following, let NDh be the number of decahedral e central adsorption sites, NDh the fcc edge positions, which are not stable for a single adatom, and NIh
365
that of stable icosahedral adsorption sites. On any Marks Dh described by m, n, p, it is turn out that for a single facet:
e NDh
m2
7m 2mp p 2 m p 2;
NDh
NIh NDh m
2
2 ;
1:
In Fig. 2, a map of the adsorption sites on the e 3; 2; 2m-Dh (NDh 6, NDh 3, and NIh 8) is given, and in Table 1 we report the energy dierences DEads between the best site on a (1 1 1) facet and the other dierent adsorption sites. Let us now consider islands up to n 6 15 on a 146-Marks Dh, see Table 2. For any island size n 6 6 there are one or more decahedral con®gurations better than the best icosahedral con®guration. In any case, for n 6 6, since the energy dierences between the two stackings are always less than 0:1 eV, islands are likely to be placed on both stackings. As expected, at any size, the most stable con®guration is the most close-packed: so
Table 1 Adsorption energy for an adatom on a Marks Dh at dierent sizes N 75 146 192
(1 0 0) 0.304 0.270 0.246
R(1 1 1) 0.249 0.239 0.238
d2
d4
i1
d1
i2
d3
i5
i3
i4
0.000 0.000
0.0035 0.0035
0.000 0.010 0.007
0.009 0.010 0.010
0.024 0.018 0.018
0.020 0.022 0.022
0.031 0.031 0.030
0.031 0.030
0.037 0.036
DEads (eV) is calculated between the best site on a (1 1 1) facet and the others. Table 2 Adsorption energy for an island of size n on an (1 1 1) 3; 2; 2 m-Dh n
Dh con®guration
Ih con®guration
2 3 4 5 6 7 8 9 10 11 12 13 14 15
d1d2 (Id2 ) Id2 d3 (Id3 ) d2d3d3d4 (Id4 ) Id4 d1 (Id5 ) Id5 d1 (Id6 ) Id6 d6 (Id7 ) Id7 d7 (Id8 ) Id8 d3 Id6 d5d6 d1d3 (Id10 ) Id10 d2 (Id11 ) Id8 d5 d3d4 (Id12 ) Id12 d2 Id6 d5d6 Id6 Id6 d5d6d7 Id6
i1i1 (Ii2 ) Ii2 i3 (Ii3 ) Ii3 i2 (Ii4 ) Ii4 i2 (Ii5 ) Ii5 i4 (Ii6 ) Ii6 i4 (Ii7 ) Ii7 i5 (Ii8 ) Ii8 i6 Ii8 i4i5 (Ii10 ) Ii10 i2 (Ii11 ) Ii11 i3 (Ii12 ) Ii12 i4 (Ii13 ) Ii13 i1 (Ii14 ) Ii14 i1
The quantity DEads is calculated between the best con®guration on Dh stacking and on Ih stacking.
DEads (eV) 0.008 0.023 0.004 0.024 0.055 0.373 0.347 0.102 0.073 0.042 0.037 0.041 0.032 0.015
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
for trimers is a triangle, for tetramers is a rhombus etc. The situation is completely dierent when 7 6 n 6 8. At these sizes, the Ih stacking becomes by far more favourable than the Dh one, due to purely geometrical reasons. In fact, it is possible to place all the seven or eight atoms of the island on Ih sites of the same facet. On the contrary, only six atoms can be placed on good Dh sites while the remaining adatoms must be placed or on the Dh positions on a (1 1 1) edge (in Fig. 2, the sites d5, d6 or d7 which becomes stable when neighbour d4 or d3 are occupied), or on another (1 1 1) facet. These decahedral con®gurations present much worse adsorption energies than the island on Ih stacking (DEads 0:35±0:37 eV). For 9 6 n 6 15, on both stackings, it is necessary to place the adatoms on two (1 1 1) facets. We have found that there is always at least one Ih-stacking island better than any Dh island, even if the energy differences are only of few hundredths of eV in some cases, see Table 2. On Dh stacking, the best con®guration has two atoms on the common edge which are surrounded by neighbours on both facets. On Ih stacking, we can place one or two atoms in i6 sites (see Fig. 2) and the other atoms on an adjacent (1 1 1) facet close to the common edge. For large islands, however, it can be easily understood by counting stable sites that there should be also a size above which an aggregate is preferentially placed on Dh stacking. However, this happens when the island is almost covering the whole cap. In general, on a m; n; p m-Dh, the total number of Dh sites on each of the two caps is given by: TOT e NDh NV 5 NDh NDh 5NDh 5m 5p
9;
where N V 1 is the ®vefold vertex. On the other hand, the total number of Ih sites is (i6 sites in Fig. 2 are excluded, being always rather unfavourable) TOT NIh 5NIh 5 NDh m
1 5NDh 5m
4:
TOT TOT NIh For any choice of m; n; p, NDh TOT TOT 5p 4 > NIh or in other words, for NIh < n6 TOT NDh the island stays preferentially on decahedral
stacking. For example, let us consider a 146 m-Dh. TOT 40, so that we can place an In this case NIh island of 40 atoms on Ih stacking (see Fig. 3). This island is better than any island on Dh stacking of the same size, even if the dierence with the best Dh island is small (0:09 eV). At n 41 atoms, the best island on Ih stacking is obtained by adding one atom in the vertex position to the island of Fig. 3. However, even if this position is of coordination six, the six bonds are either stretched or compressed. Indeed, at 41 atoms, it is possible to ®nd better islands on Dh stacking, with an energy dierence of about 0:2 eV. The above ®ndings, and in particular the fact that islands prefer the Ih stacking, have important consequences on the growth of decahedra, as discussed in the following.
4. Adsorption sites and energies on icosahedra The Ih (see Fig. 4) has six ®vefold symmetry axes or, in other words, 12 vertexes with a ®vefold symmetry. An Ih consists of 20 tetrahedra sharing a common vertex. To build up this structure, the tetrahedra must distort as they are packed together, in order to ®ll out the entire volume of the Ih (for details see Refs. [4,5]). The icosahedral packing is characterized by a perfect shell structure. In fact, one can arrange 12 neighbouring atoms around a central atom at the corners of an Ih, and then it is possible continue to cover the particle by a second layer, then by a third layer forming always a perfect Ih (this is the construction of the Mackay Ih, see Fig. 4 for the ®rst four Mackay Ih). Moreover, each of its 20 triangular facets is (1 1 1)-like and close-packed. An Ih presents a very high internal strain due to the distortion of the tetrahedra. For this reason icosahedral packing is the best structure for suciently small clusters, having a high surface to volume ratio. In this section, we study the adsorption for an island as a function of its size n P 1 on a 147Mackay Ih, see Table 3. Let us consider a single adatom. As happens for decahedral clusters, also on (1 1 1) Ih facets there are two dierent kinds of adsorption sites, corresponding to fcc and hcp
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
367
Fig. 3. 146 3; 2; 2 m-Dh with an island of 40 atoms on Ih (hcp) stacking on a cap. The islands is composed by ®ve subunits of eight atoms, each covering a single facet. Neighbouring atoms on dierent adjacent subunits are separated by a distance which is only slightly against 2:89 A). Therefore the 15 bonds among the pairs of such atoms are not negligible; if more than the ®rst-neighbour distance (3:0 A neglected, ®rst-neighbour bond counting would favour islands on fcc stacking. On the contrary, total energy minimization show that this is the most favourable island of size 40, even if the dierence with the best island on fcc stacking is small (0:09 eV). On the other hand, island of size larger than 40 are better placed on Dh (fcc) stacking, because there are no more good Ih sites available on the cap.
stackings respectively (see Fig. 5). It was shown for example in Ref. [10] that an overlayer on a Mackay Ih can be placed in two ways: in the ®rst, the anti-Mackay overlayer, the atoms are added in hcp sites; in the second, the Mackay overlayer, the atoms are added in fcc sites, and this leads to the next Mackay Ih. In the following, we speak about fcc stacking for the Mackay arrangement and about hcp stacking for the anti-Mackay. As in the case of Dh, the fcc sites on the edge between two adjacent facets (sites f2 and f3 in Fig. 5) are not stable adsorption positions for a single adatom. Let Nf be the number of central fcc sites
(as f1 in Fig. 5) on a single facet and Nfe the number of unstable fcc sites on a single edge; let Nh the number of hcp sites on the same facet. The number of fcc sites is in relation to the number of atoms which forms a facet of the Ih. In fact, if a an edge consists of s atoms, we have: Nfe s Nf
s 2 X
1; i
s
2 s 2
i1
Nh
s 1 X i1
s s
1 2
1
;
Nf s
1:
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
Fig. 4. Mackay Ih at 13, 55, 147 and 309 atoms. At right, the clusters are represented along one of their ®vefold axes.
In Fig. 5, the map of adsorption sites is given for a 147-Mackay Ih (s 4, Nf 3, Nfe 3 and Nh 6) and the adsorption energy for an adatom and for islands of size n is reported in Table 3. For a single
adatom we have found that the central fcc sites (f1 in Fig. 5) and the hcp sites (h1 and h2) are have very close adsorption energies. On the contrary, fcc sites on the edge (f2 and f3) are unstable and
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375 Table 3 Adsorption energy for an island on a 147-Mackay Ih as a function of its size n n
Site
DEads (eV)
1 1 1
f1 h2 h1
0.000 0.027 0.034
n
fcc stacking
hcp stacking
2 3 4 5 6 7 8 9 10 11 12
f1 f1 If 2 f1 f1 f1 If 3 If 3 f3 If 4 If 2 f2f3 f1 If 5 f1 If 5 f1 If 3 f3 If 3 If 7 If 7 f2 If 8 If 8 f2 If 9 If 9 f2 If 7 f2f3 f1 If 9 f2f3 f1
h1 h2 Ih2 Ih2 h2 Ih2 Ih2 Ih4 Ih4 h1 Ih5 Ih5 h2 Ih6 Ih6 h2 Ih7 Ih7 h1 Ih8 Ih8 h2 Ih9 Ih9 h1 Ih10 Ih10 h1 Ih6 Ih6
DEads (eV) 0.026 0.062 0.162 0.325 0.300 0.059 0.064 0.016 0.267 0.421 0.415
The energy dierence DEads is referred to the best adsorption site on a facet.
369
become stable only if at least one neighbour f1 site is occupied. Also dimers and trimers can be placed on both stackings with small energy dierences. On the contrary, for 4 6 n 6 6 the hcp stacking is by far more favourable (with large energy dierences: DEads 0:30 eV for the hexamer, see Table 3), because up to six atoms can be placed on the same facet on good hcp sites, while it is possible to place only three atoms on good fcc sites, and remaining atoms have to be placed on a edge or on another facet. Adding atoms up to n 12, we have found that the hcp stacking is better only of a few hundredths for 6 6 n 6 9 but the energy dierence becomes again substantial (DEads 0:27±0:40 eV) for 10 6 n 6 12. The above results imply that, at the ®rst stage of the growth of a new shell on an Ih, the overlayer is preferentially placed on hcp stacking even if it covers some facets. Our results are in good
Fig. 5. Adsorption sites on a (1 1 1) facet of an Ih. In the upper row, there is an island on fcc stacking while in the lower row on hcp stacking.
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
agreement with the calculations in Ref. [10], where it was found that an incomplete external overlayers on a Mackay Ih preferentially stays on antiMackay stacking. But the situation changes for vary large islands. If NfTOT is the total number of fcc sites and NhTOT is the number of hcp sites, it turns out that: NfTOT 20Nf 30Nfe N V 10 s
1 s 1 12;
where N V 12 is the number of ®vefold vertexes, and NhTOT 20Nh NfTOT
10s
2:
Therefore, the atoms which can be placed on hcp sites are not sucient to cover the whole Ih, and an island covering several adjacent facets is better placed on fcc stacking. For example, for n 31 atoms on the 147-Ih, an island covering ®ve adjacent facets around a common vertex is better than any island on hcp stacking, as we have directly veri®ed. 5. Diusion on non-crystalline structures 5.1. Results for diusion on decahedral structures In this section, we report about the diusion of a single adatom on magic Dh at dierent sizes: 75 atoms 2; 2; 2 m-Dh, 146 atoms 3; 2; 2 m-Dh and 192 atoms 3; 3; 2 m-Dh (see Fig. 1). An adatom can diuse on a single facet (intrafacet diusion) and among dierent facets (interfacet diusion). Intrafacet diusion has approximately the same energy barriers as diusion on ¯at fcc surfaces [35]: on (1 1 1) facets, an adatom can move very fast even at low T [37±39], while on (1 0 0) facets diffusion is much slower [38]. The diusion along facet edges does not present appreciable dierences. Let us now focus on the interfacet diusion, because it plays an important role during the growth processes. In general, interfacet moves have higher barriers than intrafacet moves on the facet of departure. In fact, the adatom has to surmount an additional barrier DES , known as Ehrlich±
Schwoebel barrier [40,41], as happens for the diffusion across steps on surfaces. The possible diffusion processes between dierent facets depend strongly on the morphology of the structure, especially on the geometry and size of the single facet, and they can take place either by jump (hopping) or by exchange. An example of exchange interfacet diusion is given in Fig. 6. Diffusion between two neighbouring (1 1 1) takes place preferentially by jump (see Table 4), since the exchange mechanisms present much higher energy barriers, more than 0.5 eV, while diusion from a (1 1 1) to a (1 0 0) facet takes place preferentially by exchange (see Fig. 6). The barrier for this process increases with the size of the (1 0 0) facets, at least for small (1 0 0) facets (see Table 4); on the other hand the barrier for jumping is larger and does not depend on the facet size. These energy barriers are close to those for the analogous processes on fcc silver clusters [28], provided that (1 0 0) facets of the same sizes are considered. The reverse processes (from (1 0 0) to (1 1 1)) present in any case much higher energy barriers, essentially by the dierence in adsorption energy DEads between sites on the two facets (the same happens on fcc clusters, see Refs. [28,35]). Also the diusion from a (1 0 0) facet to the Marks reentrance has a very high energy barrier. Therefore adatoms adsorbed on (1 0 0) facets remain trapped there for long times. The diusion from a (1 1 1) to a Marks reentrance (R(110)) can take place either jump or by exchange with the common vertex with the neighbour (1 0 0) facet. The barriers do not depend on the cluster size, and are almost of the same magnitude for both jump and exchange. The reverse processes are quite dicult, because adsorption on the reentrance is more favourable than on (1 1 1) facets. Therefore, once the adatom falls in the Marks reentrance, it very likely remains there, so that diusion along the reentrance is not active for connecting the two caps of the Dh. The only mechanism connecting the caps is the chain process, a multiple exchange process [42] which connects two opposite (1 1 1) facets through the intermediate (1 0 0) facet [28,43±45]. An adatom on a (1 1 1) facet can push a central row (the chain process through a boundary row costs a lot, see Table 4) of the neighbour (1 0 0) facet until the
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
371
Fig. 6. Exchange connecting (1 1 1) and (1 0 0) facets of a 192-Marks Dh. On the left the light adatom moves from a icosahedral site close to the (1 0 0) edge and pushes the dark atom until the last goes onto the (1 0 0) facet.
Table 4 Energy barriers for an adatom on m-Dh (eV) Diusion mechanism
75 Dh
146 Dh
192 Dh
Exchange 1 1 1 ! 1 1 1 Exchange vertex Exchange T ! 1 0 0 Exchange 1 1 1 ! T via (1 0 0) vertex Exchange 1 1 1 ! T via (1 1 1) vertex Exchange 1 1 1 ! 1 0 0 Jump 1 1 1 ! 1 1 1 Jump 1 1 1 ! T Jump 1 1 1 ! 1 0 0
0.56 0.56 0.51 0.41
0.63 0.55 0.52 0.39
0.62 0.56 0.54 0.39
0.25 0.31 0.35 0.38
0.28 0.31 0.35 0.40
0.31 0.31 0.39 0.39
Chain 1 1 1 ! 1 0 0 ! 1 1 1 Via central row Via boundary row
n1 0.07
n2 0.13 0.28
n3 0.19
0.47
atom on the opposite (1 0 0)-edge falls down on the opposite (1 1 1) facet (see Fig. 7). The activation energy depends strongly on the length of the (1 0 0) central row: each additional atom along the central row increases the barrier of 0.06 eV (see Table 4). However, if the Marks integer n is less than 5, the chain process is more frequent than the jump between two adjacent (1 1 1) facets. From our results it turns out that barriers depend on the local arrangement of the surface, but not on internal strain. Processes with the same local geometry have practically the same barriers on both Dh and fcc structures. For example, the
jump between two (1 1 1) facets presents the same barrier on Dh and TO clusters, and this barrier is found also for step descent on a in®nite (1 1 1) crystal surface (see Ref. [46]); the chain mechanism has the same energy barrier on Dh and on TO clusters if the length of the central (1 0 0)-row is the same. Let us now analyse the consequences of our results on the growth of Dh clusters. A given Dh can grow to a larger Dh cluster both by the ®lling of the truncations and reentrances and by the superposition of an umbrella [4] covering either the top or the bottom cap of the cluster on decahedral stacking. This second growth mode gives a Dh with a ®vefold axis longer by one atom, and it is necessary to continue the growth of Dh structures because the number of atoms which can be placed in the truncations and reentrances is obviously limited. Our results show that adatoms can move easily towards truncations and reentrances, where they very likely remain trapped. Moreover, an adatom deposited on a (1 1 1) facet can move very fast between the two caps of the Dh by the chain process and quite easily between neighbouring (1 1 1) facets. This favours the building up of a big island either on the top or on the bottom cap: when the majority of the sites on (1 0 0) facets are ®lled, it is likely to build up a large island covering one of the caps. But this island is more likely grown on hcp sites, and this is like putting an umbrella over the Dh cluster [4], but on the wrong
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Fig. 7. Chain process connecting two opposite (1 1 1) facets of a 192-Marks Dh: on the left the light adatom on a (1 1 1) pushes the central row (dark atoms) of the neighbour (1 0 0), in the central ®gure the saddle point is represented and ®nally the atom on the opposite edge falls down onto the opposite (1 1 1) facet, on the right.
(icosahedral) stacking. While the umbrella on the decahedral stacking continues the growth of a Dh cluster, the umbrella on icosahedral stacking can initiate the structural transformation of the Dh into an Ih, as seen in MD simulations of growth [21]. Indeed, in this con®guration, further adatoms getting trapped into a Marks reentrance can form a new ®vefold symmetry point: the transformation from a Dh into an Ih is starting. 5.2. Diusion on icosahedral structures The results about single-adatom diusion on Ih clusters are reported in Table 5. We consider Ih of 55, 147 and 309 atoms (see Fig. 4). The surface of the Ih consists of distorted (1 1 1) facets, and in the following we consider both intrafacet and interfacet diusion (see Table 5). Intrafacet diusion takes place by jump, and the barrier is very close to the one on a perfect in®nite (1 1 1) surface, or on the Dh (1 1 1) facets. Again, the internal strain has a little eect on the diusion barriers. Concerning interfacet processes, the exchange with a vertex is much more dicult than the exchange with edge atoms. Because of that, interfacet diusion takes place mainly between adjacent facets sharing a common edge, and not a common vertex only. The diusion between adjacent (1 1 1) facets takes place by jump (see Fig. 8) if the number of atoms of the common (1 1 1) edge is
Table 5 Energy barriers for an adatom on Ih (eV) Diusion mechanism
55 Ih
147 Ih
309 Ih
Exchange 1 1 1 ! 1 1 1 Exchange vertex Jump 1 1 1 ! 1 1 1
0.15 0.44 0.32
0.34 0.47 0.31
0.44 0.50 0.30
larger than 3. Otherwise, the exchange process is more favourable. For example, on the 55 Ih, the common edge has three atoms, and the barrier for exchange is very low (see Table 5), because the facet is so small that the common edge can distort itself to accommodate the adatom (see Fig. 9, where we show the initial, saddle-point and ®nal positions). On larger Ih, this barrier increases a lot, and the jump process (whose barrier does not depend on the length of the common edge) becomes more favourable. Indeed, jumping between (1 1 1) facets has the same barrier for all structures we have considered here and also for fcc clusters [28]. What are the consequences of our ®ndings on the growth of Ih clusters? An Ih can grow shell by shell (SBS, see Ref. [21]) if the adatoms can pass easily among dierent facets in order to have only a single island covering several (1 1 1) facets: the fast diusion between facets guarantees the formation of overlayer structures. From our results it is clear that, at any size, the adatoms can move rather easily from a facet to another allowing to arrange themselves into a big island (this leads to
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
373
Fig. 8. Initial, saddle-point and ®nal con®gurations for the jump process connecting two neighbouring (1 1 1) facets on a 147-Mackay Ih: the dark atom hops on the common edge and then falls down on the opposite facet.
Fig. 9. Initial, saddle-point and ®nal con®gurations for the exchange mechanism between two neighbouring (1 1 1) facets of a 55Mackay Ih: the light adatom on a (1 1 1) pushes the central dark atoms onto a hcp site.
the umbrella growth model see Refs. [4,47]), which more likely grows on hcp stacking at intermediate sizes. This implies that the cluster grows via the SBS mode, provided that temperature is high enough to displace the island on fcc stacking. If not, defected structures are expected to grow. On the other hand, we have found that an Ih can transform itself into a Dh through quasi-melted structures (see Ref. [21]). This transformation can take place if, in conditions (cluster size and temperature) where the Dh structures are favoured, the cluster has sucient kinetic energy to rearrange all atoms around a single ®vefold axis. Because of
that, the Ih ! Dh transformation is more dicult than the Dh ! Ih transformation during growth processes. 6. Discussion and conclusions In this paper, we have studied the energetics of island adsorption and single-adatom diusion on non-crystalline silver clusters by MD simulations. In particular, we have focused our attention on how adatoms can be placed and move, discussing the implications on the cluster growth. Concerning
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island adsorption, we have found that islands can be placed on dierent stackings (fcc and hcp) on both Ih and Dh. The most favourable stacking changes depending on the size of the island. In particular, we have shown that there is always a island size range in which the hcp stacking is by far the best. A preference for hcp stacking was found experimentally on ¯at Ir surfaces (see Ref. [48]) for very small islands, like dimers and trimers, and with energy dierences of the order of a few hundredths of eV. What we ®nd here is completely dierent, and depends crucially on the ®nite extension of the clusters facets. Dimers and trimers stays slightly better on fcc sites, while the hcp stacking is better for islands of the order of 10 or more atoms, with considerable energy dierences (close to 0:5 eV). Here, the cause is not some local relaxation of the structure or some second-third neighbours eect, but is purely geometrical: there are more stable hcp sites than fcc sites on a single facet. Indeed, the fcc stacking becomes again favourable when islands are very large. In the case of Dh, this means almost the complete covering of one cap. This has deep consequences on the growth of Dh. If the island grows on fcc stacking, a larger Dh is obtained [4], with the length of the ®vefold axis increased by one atom. But islands very likely grow on hcp stacking at intermediate sizes, and if temperature is not high enough to let the island displace to fcc stacking, a structural transformation towards an Ih can begin, as seen in the growth simulations in Ref. [21]. Now the question is: how is it possible to build up large islands covering several (1 1 1) facets? The answer is found by inspecting the diusion of single adatoms: if adatoms are able to diuse easily among dierent facets, they can meet and form a single aggregate on the cluster surface; if interfacet diusion is slow, small islands nucleate on dierent facets. This is analogous to what happens in epitaxial crystal growth: fast adatom diusion causes the building up of few large islands. Our results have shown that diusion among dierent facets is rather easy on both Ih and Dh clusters, with the exception of the diusion from the truncations on Marks Dh. On the other hand, in the case of Dh clusters, there is a process (the chain, which is a very interesting example of a multiple exchange
process) which allows a very fast mobility between the two caps. Because of this process, it is likely to have a single island nucleating on only one of the caps. In the case of Ih clusters, interfacet diusion favours the formation of a single large island which grows to the completion of a further shell. Let us compare now adatom diusion on clusters of dierent symmetries: Ih, Dh and TO (for the latter see Ref. [28]). One of the main results of this work is that the diusion on non-crystalline Ag clusters is not very much in¯uenced by the internal strain which characterizes both Ih and Dh. This fact is con®rmed by dierent results about single-facet and interfacet diusion. Intrafacet processes have the same barriers on clusters of dierent structures, and these barriers are very close to those found on in®nite ¯at fcc surfaces. Interfacet jump processes have again the same barriers on dierent clusters, and the same happens for the chain process, whose barrier depends only on the size of the (1 0 0) facet where the chain is passing through, and not on the clusters being a Dh or a truncated octahedron. On the other hand, we have shown that the exchange processes depend strongly on size, local morphology and on the relative inclination between the facets. For any structure, the exchange between two adjacent (1 1 1) is strongly in¯uenced by the length of the (1 1 1) edge and it presents higher and higher barriers when the angle between the facets becomes ¯atter and ¯atter (from the step on a (1 1 1) surface to the exchange on a Dh). In conclusion, our results show that island adsorption and diusion on non-crystalline clusters present very interesting features, due to the peculiar geometry of these structures. The interplay between the energetics of adsorption, which is ruled by geometrical constraints, and the adatom mobility governs the formation of the new shells of atoms and therefore is expected to play a crucial role in the growth of these clusters in their solid state. Acknowledgements We acknowledge ®nancial support from the Italian MURST under the project N. 9902112831.
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References [1] P. Jensen, Rev. Mod. Phys. 71 (1999) 1695. [2] W. Harbich, Metal Clusters at Surfaces, Springer Series in Clusters Physics, Springer, Berlin 2000 (Chapter 4). [3] C.R. Henry, Surf. Sci. Rep. 31 (1998) 231. [4] T.P. Martin, Phys. Rep. 273 (1996) 199. [5] S. Ino, J. Phys. Soc. Jpn. 27 (1969) 941. [6] L.D. Marks, J. Cryst. Growth 61 (1984) 556. [7] L.D. Marks, Phys. Rep. 273 (1994) 199. [8] C.L. Cleveland, U. Landman, J. Chem. Phys. 94 (1991) 7376. [9] K. Michaelian, N. Rend on, I.L. Garzon, Phys. Rev. B 60 (1999) 2000. [10] J.P.K. Doye, D.J. Wales, New J. Chem. 22 (1998) 733. [11] J.P.K. Doye, D.J. Wales, Chem. Phys. Lett. 247 (1995) 339. [12] C. Mottet, G. Treglia, B. Legrand, Surf. Sci. 383 (1997) L719. [13] C. Barreteau, M.C. Desjonqueres, D. Spanjaard, Eur. Phys. D 11 (2000) 395. [14] S. Erkoc, T. Yilmaz, Physica E 5 (1999) 1. [15] W. de Heer, Rev. Mod. Phys. 65 (1993) 611. [16] B.D. Hall, et al., Phys. Rev. B 43 (1991) 3906. [17] D. Reinhard, et al., Phys. Rev. Lett. 79 (1997) 1459. [18] D. Reinhard, et al., Phys. Rev. B 55 (1997) 7868. [19] M. Hyslop, A. Wurl, S.A. Brown, B.D. Hall, R. Monot, preprint. [20] F. Baletto, C. Mottet, R. Ferrando, Phys. Rev. Lett. 84 (2000) 5544. [21] F. Baletto, C. Mottet, R. Ferrando, Phys. Rev. B 63 (2001) 155407. [22] B.W. van de Waal, J. Cryst. Growth 158 (1996) 153. [23] T. Ikeshoji, G. Torchet, M.-F. de Feraudy, K. Koga, Phys. Rev. E 63 (2001) 031101. [24] Y. Chushak, L.S. Bartell, J. Phys. Chem. A 104 (2000) 9328. [25] S.C. Hendy, B.D. Hall, preprint. [26] L.J. Lewis, P. Jensen, J.-L. Barrat, Phys. Rev B 56 (1997) 2248.
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[27] A.M. Mazzone, Philos. Mag. B 80 (2000) 95. [28] F. Baletto, C. Mottet, R. Ferrando, Surf. Sci. 446 (2000) 31. [29] V. Rosato, M. Guillope, B. Legrand, Philos. Mag. A 59 (1989) 321. [30] B. Legrand, M. Guillope, in: V. Vitek, D.J. Srolovitz (Eds.), Atomistic Simulation of Materials, Plenum Press, New York, 1989, p. 361. [31] C. Mottet, G. Treglia, B. Legrand, Phys. Rev. B 46 (1992) 16018. [32] F. Montalenti, R. Ferrando, Phys. Rev. B 59 (1999) 5881. [33] F. Montalenti, R. Ferrando, Phys. Rev. Lett. 82 (1999) 1498. [34] H. Jonsson, G. Mills, K.W. Jacobsen, Nudged elastic band for ®nding minimum energy paths of transitions, in: B.J. Berne, G. Ciccotti, D.F. Coker (Eds.), Classical and Quantum Dynamics in Condensed Phase Simulations, World Scienti®c, 1998. [35] D.J. Wales, L.J. Munro, J. Phys. Chem. 100 (1996) 2053. [36] D.J. Wales, L.J. Munro, J.P.K. Doye, J. Chem. Soc., Dalton Trans. (1996) 611. [37] H. Brune, K. Bromann, H. R oder, K. Kern, J. Jacobsen, P. Stolze, K. Jacobsen, J. Nùrskov, Phys. Rev. B 52 (1995) R14380. [38] N.I. Papanicolaou, G.A. Evangelakis, G.C. Kallinteris, Comput. Mat. Sci. 10 (1998) 105. [39] R. Ferrando, G. Treglia, Surf. Sci. 331±333 (1995) 920. [40] R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614. [41] G. Ehrlich, F.G. Hudda, J. Chem. Phys. 44 (1966) 1039. [42] J. Merikoski, T. Ala-Nissila, Phys. Rev. B 52 (1995) R8715. [43] S. Valkealahti, M. Manninen, Z. Phys. D 26 (1993) 255. [44] S. Valkealahti, M. Manninen, Z. Phys. D 40 (1997) 496. [45] S. Valkealahti, M. Manninen, Phys. Rev. B 57 (1998) 15533. [46] R. Ferrando, G. Treglia, Surf. Sci. 377±379 (1997) 843. [47] T.P. Martin, T. Bergmann, H. G ohlich, T. Lange, J. Phys. Chem. 95 (1991) 6421. [48] S.C. Wang, G. Ehrlich, Surf. Sci. 239 (1990) 301.
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Island adsorption and adatom diusion on 3D non-crystalline silver nanoclusters F. Baletto *, R. Ferrando Dipartimento di Fisica dell'Universita di Genova, INFM and CFSBT/CNR, via Dodecaneso 33, I-16146 Genova, Italy Received 2 March 2001; accepted for publication 8 June 2001
Abstract We report a systematic study of island adsorption and single-adatom diusion on free silver nanoclusters, and discuss the consequences on the growth. In our calculations, silver is modelled by semiempirical many-body potentials. We consider magic non-crystallographic structures at dierent sizes: icosahedra (Ih) at 55, 147 and 309 atoms; Markstruncated decahedra (m-Dh) at 75, 146 and 192 atoms. We calculate the map of adsorption sites and the energy barriers for the dierent diusion processes. We ®nd that, due to purely geometrical reasons, medium-size (from 6±8 to 30±40 atoms depending on the cluster) islands on the cluster (1 1 1) facets prefer the hcp stacking on both Ih and Dh structures, while both smaller and larger islands are better placed on fcc stacking. Interfacet diusion is easy on both Dh and Ih, indicating that large islands are easily grown; in particular, there are multi-atom diusion processes which allow fast diusion among the two caps of Dh clusters. For Dh clusters, islands on hcp stacking may lead to the appearance of new ®vefold symmetry points, and to the transformation of the cluster into an icosahedron. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Clusters; Silver; Molecular dynamics; Growth
1. Introduction Clusters are an important state of matter, being aggregates of atoms and molecules that are small enough not to have the same properties as the bulk liquid or solid. Quantum states in clusters are size dependent, leading to new electronic, optical, and magnetic properties. Clusters oer attractive possibilities for innovative technological applications in ever smaller devices, for example in the ®elds of
*
Corresponding author. E-mail addresses: baletto@®sica.unige.it (F. Baletto), ferrando@®sica.unige.it (R. Ferrando).
the controlled growth of nanostructures [1,2] and of catalysis [3]. In recent years, much experimental and theoretical eort has been devoted to the determination of the most favourable structures depending on the cluster size N. Small noble and transition metal clusters can present ``non-crystallographic'' structures, like icosahedra (Ih) and decahedra (Dh), especially Marks-truncated Dh (m-Dh) [4±14]. Atomic unsupported 3D clusters are formed under several dierent conditions, but the actual process is not known in detail. In typical experiments, metal clusters are grown in a gas phase of atoms; the metal vapour is carried in an inert gas beam into the condensation chamber, where the actual growth takes place [15±19]. The
0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 1 ) 0 1 3 5 5 - 3
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
experimental conditions, such as the temperature and pressure of the carrier gas, the time the beam stays in the condensation room, in¯uence strongly the building up of the cluster morphology [16±18]. We have explicitly shown by molecular dynamics (MD) simulations that the ®nal cluster morphology depends on the interplay of kinetics and thermodynamics [20,21]. This is in agreement with experiments, which do not re¯ect the simple picture arising from equilibrium total-energy considerations, and gives strong support to kinetic modelling to explain experimental results about cluster growth [22]. Indeed, from experimental data and theoretical results, it is known that during growth [20,21] and solidi®cation from liquid droplets [23±25] small clusters can change shape with size among Ih, Dh and fcc structures. The coexistence of dierent structures, some of them far from the most stable ones, suggests that cluster growth is controlled by individual atomic-scale processes rather than by the minimization of the cluster total energy. Surface diusion has a great importance also in the coalescence of clusters [26, 27]. In this paper, we study island adsorption and single-adatom diusion on non-crystalline silver 3D nanoclusters by quenched MD. We consider, in particular, Ih and Dh structures, which are the most stable at these small sizes. The case of fcc truncated octahedra (TO) has been studied in Ref. [28]. We discuss how the diusion processes which allow an adatom to cross between dierent facets (interfacet processes) are fundamental during the building up of the cluster. Moreover, we calculate the energetics of adsorption on Ih and Dh clusters. For adatoms and islands on (1 1 1) facets, we show that dierent stackings are possible, with deep consequences on the cluster growth. In particular, we analyse the microscopic mechanism of the transformation of Dh into Ih, which has purely kinetic origin, being in the opposite direction to what equilibrium thermodynamics suggests [21]. The article is structured as follows. In the next section we report our theoretical model and methods. Section 3 deals with the adsorption energies of islands on Marks Dh, while in Section 4 we report the adsorption maps for an on Mackay Ih. Section 5 contains the results concerning the static energy
barriers on these non-crystallographic structures; the conclusions are outlined in Section 6. 2. Theoretical model In the following, we model silver by the manybody potential proposed by Rosato, Guillope and Legrand (RGL, see Ref. [29]; for a derivation of the second-moment approximation (SMA) for noble metals see Ref. [30]). The RGL potential is derived in the framework of the SMA to the density of states in the tight-binding model. In this framework, the band energy Ebi for a given atom i is proportional to the square root of the second moment of the local density of states. Ebi is an attractive many-body term; the stability of the cluster is insured by adding a phenomenological corerepulsion term Ebr of the Born±Mayer type. Thus, the potential energy of atom i is given by Eci Ebi Eri where the band energy Ebi can be written as: ( )1=2 X 2 rij i n exp 2q ; Eb 1 r0 j;rij
1
2
where n is an eective hopping integral, rij is the distance between the atoms i and j, rc is the cut-o radius for the interaction, r0 is the nearest-neighbour (NN) distance and q gives the distance dependence of the hopping integral. The second term Eri is given by: X rij i A exp p Er 1 : 3 r0 j;rij
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
data, ab initio and semi-empirical calculations are reported in Refs. [32,33]. It turns out that RGL results predict correctly the dominant diusion mechanisms on noble-metal surfaces; from the quantitative point of view, the agreement with experimental data and ®rst-principle calculations is very good, especially for silver. Comparison with density functional calculations for noble-metal clusters are found in Ref. [9], again with good results. In order to calculate the energy barriers for the diusion mechanisms, we have used the nudged elastic band (NEB) method [34]. The NEB method allows to identify the minimum energy path for a rearrangement of a group of atoms among two stable con®gurations and to estimate the activation energy. In this method, a chain of intermediate images in between the two stable con®gurations is built up; the images are connected by springs, and are optimized simultaneously (by quenched MD) while the initial and ®nal positions are kept ®xed. Details about the method are given in Ref. [28]. Here we simply add some information about the convergence of our NEB calculations, which has been checked as follows. First, we ®x the number of images in the NEB and quench the system (usually down to very low temperatures, of the order of 10 3 ±10 4 K) until we reach the convergence on four signi®cant digits in the energy of each image. Then we repeat the procedure with a larger number of images, until the barrier estimate converges to two signi®cant digits at least. Typically, the number of images is in between 10 and 20. 3. Adsorption sites and energies on decahedra In this section, we study the adsorption of an island of size n on the surface of a Dh. A Dh is a non-crystalline structure with a single ®vefold symmetry axis. The Dh can be constructed out of ®ve distorted tetrahedra which are packed together so to have a common edge. In this way the surface is close-packed. However, a Dh has a very large surface area in addition to internal strain, the latter due to the distortion of tetrahedra. For these reasons the Dh is not expected to be energetically favourable at any cluster size. But this is not the
363
Fig. 1. m-Dh at 75, 146 and 192 atoms: side view (on the left) and top view (on the right). For these magic clusters the sets m; n; p are 2; 2; 2, 3; 2; 2 and 3; 3; 2, respectively. m and n are the lengths of the edges of the (1 0 0)-like facets, p is the length of the Marks reentrance.
case for the truncated forms of the Dh, which are much more spherical structures. Marks [6] proposed a truncation which exposes reentrant (1 1 1) facets in addition to (1 0 0) facets, see Fig. 1. To describe the Marks Dh, one introduces three integers m, n, p which are the length of (1 0 0) edges and the length of the (1 1 1) reentrance, respectively. On a Dh, adatoms can be adsorbed on (1 1 1) and (1 0 0) facets, and in the Marks reentrance. The adsorption sites have dierent adsorption energies Eads . By counting the number of NN, it is easily seen that the adsorption sites on (1 0 0) facets are the most favourable, while the sites on (1 1 1) facets present the most unfavourable Eads . This is in agreement with the ®ndings in Refs. [35,36], where single-adatom adsorption and diffusion on fcc cuboctahedra of 13, 55 and 147
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
atoms and on the Dh of 13 atoms was studied for silver and other metals by Sutton-Chen and Murrell-Mottram semiempirical potentials. The sites in a Marks reentrance (R(110)-sites) are at intermediate energies for the Dh structures considered in the following. There, Eads is a few hundredths of eV higher than on (1 0 0) sites. In agreement with the data on ¯at surfaces, the energy dierence between the best sites on (1 0 0) and on (1 1 1) facets is DEads 0:24±0:30 eV. Comparing this with the analogous dierence for the fcc 201 TO, where DEads 0:22±0:28 eV [28], one could conclude that the internal strain of a Dh does not in¯uence very much the surface diusion barriers, which depend only on the size and the symmetry of the facet. On a Marks Dh, the most part of the surface is close-packed; thus big islands can nucleate only on one or more adjacent (1 1 1) facets. The biggest possible island covers the whole top cap or bottom cap of the Dh (both top and bottom caps are
formed by ®ve (1 1 1) facets). For these reasons, in the following we focus on adsorption on (1 1 1) facets for single adatoms and for islands of size n. Due to the ®nite sizes of these facets, there are several dierent kinds of adsorption sites, for example if the adatom is adsorbed close to an edge, to a vertex or on a central position. However, as it happens in the case of ¯at (1 1 1) surfaces, we can subdivide ®rst the adsorption sites in two groups (see the right column of Fig. 2), which are easily distinguished looking at the cluster along its ®vefold symmetry axis. From this viewpoint, the atoms of the Dh are perfectly arranged in columns. When an adatom is adsorbed on a (1 1 1) facet, it can either continue or break the decahedral arrangement in columns. In the former case, we have adsorption on a site in decahedral stacking; in the latter we have adsorption on a site in icosahedral stacking (see Fig. 2). By looking at a (1 1 1) facet along its normal, one can easily see that the decahedral stacking corresponds to the fcc stack-
Fig. 2. Adsorption sites on a (1 1 1) facet of a m-Dh: the views perpendicular to the facet are represented on the left; the views along the ®vefold axis are reported on the right. Islands on Dh and Ih stackings are shown in the upper and lower rows, respectively. It is clear that the Dh stacking corresponds to the fcc stacking on a ¯at (1 1 1) surface, while the Ih stacking is the hcp stacking.
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
ing on a ¯at (1 1 1) surface while the icosahedral stacking to the hcp stacking. According to our calculations, single adatoms prefer an fcc central position (in Fig. 2 the decahedral site d4) or just close to the (1 0 0) edge (in Fig. 2 the decahedral sites d1, d3 or the icosahedral site h1), but in any case the energy dierences are very small, of the order of few hundredths eV, see Table 1. Since we cannot expect that our calculations are sensitive down to such precision (this because of the approximate semiempirical model), we conclude that these sites are essentially equivalent. However, the important point is the following. For a single adatom, the fcc sites on a (1 1 1) edge (in Fig. 2 the Dh positions d5, d6, d7) and also the hcp sites like i6 are not stable minima. These sites become stable minima only if the adatom has already one or more neighbours on a adjacent (1 1 1) facet (for example on d4, d3 sites or i1, i3 sites). In the following, let NDh be the number of decahedral e central adsorption sites, NDh the fcc edge positions, which are not stable for a single adatom, and NIh
365
that of stable icosahedral adsorption sites. On any Marks Dh described by m, n, p, it is turn out that for a single facet:
e NDh
m2
7m 2mp p 2 m p 2;
NDh
NIh NDh m
2
2 ;
1:
In Fig. 2, a map of the adsorption sites on the e 3; 2; 2m-Dh (NDh 6, NDh 3, and NIh 8) is given, and in Table 1 we report the energy dierences DEads between the best site on a (1 1 1) facet and the other dierent adsorption sites. Let us now consider islands up to n 6 15 on a 146-Marks Dh, see Table 2. For any island size n 6 6 there are one or more decahedral con®gurations better than the best icosahedral con®guration. In any case, for n 6 6, since the energy dierences between the two stackings are always less than 0:1 eV, islands are likely to be placed on both stackings. As expected, at any size, the most stable con®guration is the most close-packed: so
Table 1 Adsorption energy for an adatom on a Marks Dh at dierent sizes N 75 146 192
(1 0 0) 0.304 0.270 0.246
R(1 1 1) 0.249 0.239 0.238
d2
d4
i1
d1
i2
d3
i5
i3
i4
0.000 0.000
0.0035 0.0035
0.000 0.010 0.007
0.009 0.010 0.010
0.024 0.018 0.018
0.020 0.022 0.022
0.031 0.031 0.030
0.031 0.030
0.037 0.036
DEads (eV) is calculated between the best site on a (1 1 1) facet and the others. Table 2 Adsorption energy for an island of size n on an (1 1 1) 3; 2; 2 m-Dh n
Dh con®guration
Ih con®guration
2 3 4 5 6 7 8 9 10 11 12 13 14 15
d1d2 (Id2 ) Id2 d3 (Id3 ) d2d3d3d4 (Id4 ) Id4 d1 (Id5 ) Id5 d1 (Id6 ) Id6 d6 (Id7 ) Id7 d7 (Id8 ) Id8 d3 Id6 d5d6 d1d3 (Id10 ) Id10 d2 (Id11 ) Id8 d5 d3d4 (Id12 ) Id12 d2 Id6 d5d6 Id6 Id6 d5d6d7 Id6
i1i1 (Ii2 ) Ii2 i3 (Ii3 ) Ii3 i2 (Ii4 ) Ii4 i2 (Ii5 ) Ii5 i4 (Ii6 ) Ii6 i4 (Ii7 ) Ii7 i5 (Ii8 ) Ii8 i6 Ii8 i4i5 (Ii10 ) Ii10 i2 (Ii11 ) Ii11 i3 (Ii12 ) Ii12 i4 (Ii13 ) Ii13 i1 (Ii14 ) Ii14 i1
The quantity DEads is calculated between the best con®guration on Dh stacking and on Ih stacking.
DEads (eV) 0.008 0.023 0.004 0.024 0.055 0.373 0.347 0.102 0.073 0.042 0.037 0.041 0.032 0.015
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
for trimers is a triangle, for tetramers is a rhombus etc. The situation is completely dierent when 7 6 n 6 8. At these sizes, the Ih stacking becomes by far more favourable than the Dh one, due to purely geometrical reasons. In fact, it is possible to place all the seven or eight atoms of the island on Ih sites of the same facet. On the contrary, only six atoms can be placed on good Dh sites while the remaining adatoms must be placed or on the Dh positions on a (1 1 1) edge (in Fig. 2, the sites d5, d6 or d7 which becomes stable when neighbour d4 or d3 are occupied), or on another (1 1 1) facet. These decahedral con®gurations present much worse adsorption energies than the island on Ih stacking (DEads 0:35±0:37 eV). For 9 6 n 6 15, on both stackings, it is necessary to place the adatoms on two (1 1 1) facets. We have found that there is always at least one Ih-stacking island better than any Dh island, even if the energy differences are only of few hundredths of eV in some cases, see Table 2. On Dh stacking, the best con®guration has two atoms on the common edge which are surrounded by neighbours on both facets. On Ih stacking, we can place one or two atoms in i6 sites (see Fig. 2) and the other atoms on an adjacent (1 1 1) facet close to the common edge. For large islands, however, it can be easily understood by counting stable sites that there should be also a size above which an aggregate is preferentially placed on Dh stacking. However, this happens when the island is almost covering the whole cap. In general, on a m; n; p m-Dh, the total number of Dh sites on each of the two caps is given by: TOT e NDh NV 5 NDh NDh 5NDh 5m 5p
9;
where N V 1 is the ®vefold vertex. On the other hand, the total number of Ih sites is (i6 sites in Fig. 2 are excluded, being always rather unfavourable) TOT NIh 5NIh 5 NDh m
1 5NDh 5m
4:
TOT TOT NIh For any choice of m; n; p, NDh TOT TOT 5p 4 > NIh or in other words, for NIh < n6 TOT NDh the island stays preferentially on decahedral
stacking. For example, let us consider a 146 m-Dh. TOT 40, so that we can place an In this case NIh island of 40 atoms on Ih stacking (see Fig. 3). This island is better than any island on Dh stacking of the same size, even if the dierence with the best Dh island is small (0:09 eV). At n 41 atoms, the best island on Ih stacking is obtained by adding one atom in the vertex position to the island of Fig. 3. However, even if this position is of coordination six, the six bonds are either stretched or compressed. Indeed, at 41 atoms, it is possible to ®nd better islands on Dh stacking, with an energy dierence of about 0:2 eV. The above ®ndings, and in particular the fact that islands prefer the Ih stacking, have important consequences on the growth of decahedra, as discussed in the following.
4. Adsorption sites and energies on icosahedra The Ih (see Fig. 4) has six ®vefold symmetry axes or, in other words, 12 vertexes with a ®vefold symmetry. An Ih consists of 20 tetrahedra sharing a common vertex. To build up this structure, the tetrahedra must distort as they are packed together, in order to ®ll out the entire volume of the Ih (for details see Refs. [4,5]). The icosahedral packing is characterized by a perfect shell structure. In fact, one can arrange 12 neighbouring atoms around a central atom at the corners of an Ih, and then it is possible continue to cover the particle by a second layer, then by a third layer forming always a perfect Ih (this is the construction of the Mackay Ih, see Fig. 4 for the ®rst four Mackay Ih). Moreover, each of its 20 triangular facets is (1 1 1)-like and close-packed. An Ih presents a very high internal strain due to the distortion of the tetrahedra. For this reason icosahedral packing is the best structure for suciently small clusters, having a high surface to volume ratio. In this section, we study the adsorption for an island as a function of its size n P 1 on a 147Mackay Ih, see Table 3. Let us consider a single adatom. As happens for decahedral clusters, also on (1 1 1) Ih facets there are two dierent kinds of adsorption sites, corresponding to fcc and hcp
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
367
Fig. 3. 146 3; 2; 2 m-Dh with an island of 40 atoms on Ih (hcp) stacking on a cap. The islands is composed by ®ve subunits of eight atoms, each covering a single facet. Neighbouring atoms on dierent adjacent subunits are separated by a distance which is only slightly against 2:89 A). Therefore the 15 bonds among the pairs of such atoms are not negligible; if more than the ®rst-neighbour distance (3:0 A neglected, ®rst-neighbour bond counting would favour islands on fcc stacking. On the contrary, total energy minimization show that this is the most favourable island of size 40, even if the dierence with the best island on fcc stacking is small (0:09 eV). On the other hand, island of size larger than 40 are better placed on Dh (fcc) stacking, because there are no more good Ih sites available on the cap.
stackings respectively (see Fig. 5). It was shown for example in Ref. [10] that an overlayer on a Mackay Ih can be placed in two ways: in the ®rst, the anti-Mackay overlayer, the atoms are added in hcp sites; in the second, the Mackay overlayer, the atoms are added in fcc sites, and this leads to the next Mackay Ih. In the following, we speak about fcc stacking for the Mackay arrangement and about hcp stacking for the anti-Mackay. As in the case of Dh, the fcc sites on the edge between two adjacent facets (sites f2 and f3 in Fig. 5) are not stable adsorption positions for a single adatom. Let Nf be the number of central fcc sites
(as f1 in Fig. 5) on a single facet and Nfe the number of unstable fcc sites on a single edge; let Nh the number of hcp sites on the same facet. The number of fcc sites is in relation to the number of atoms which forms a facet of the Ih. In fact, if a an edge consists of s atoms, we have: Nfe s Nf
s 2 X
1; i
s
2 s 2
i1
Nh
s 1 X i1
s s
1 2
1
;
Nf s
1:
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
Fig. 4. Mackay Ih at 13, 55, 147 and 309 atoms. At right, the clusters are represented along one of their ®vefold axes.
In Fig. 5, the map of adsorption sites is given for a 147-Mackay Ih (s 4, Nf 3, Nfe 3 and Nh 6) and the adsorption energy for an adatom and for islands of size n is reported in Table 3. For a single
adatom we have found that the central fcc sites (f1 in Fig. 5) and the hcp sites (h1 and h2) are have very close adsorption energies. On the contrary, fcc sites on the edge (f2 and f3) are unstable and
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375 Table 3 Adsorption energy for an island on a 147-Mackay Ih as a function of its size n n
Site
DEads (eV)
1 1 1
f1 h2 h1
0.000 0.027 0.034
n
fcc stacking
hcp stacking
2 3 4 5 6 7 8 9 10 11 12
f1 f1 If 2 f1 f1 f1 If 3 If 3 f3 If 4 If 2 f2f3 f1 If 5 f1 If 5 f1 If 3 f3 If 3 If 7 If 7 f2 If 8 If 8 f2 If 9 If 9 f2 If 7 f2f3 f1 If 9 f2f3 f1
h1 h2 Ih2 Ih2 h2 Ih2 Ih2 Ih4 Ih4 h1 Ih5 Ih5 h2 Ih6 Ih6 h2 Ih7 Ih7 h1 Ih8 Ih8 h2 Ih9 Ih9 h1 Ih10 Ih10 h1 Ih6 Ih6
DEads (eV) 0.026 0.062 0.162 0.325 0.300 0.059 0.064 0.016 0.267 0.421 0.415
The energy dierence DEads is referred to the best adsorption site on a facet.
369
become stable only if at least one neighbour f1 site is occupied. Also dimers and trimers can be placed on both stackings with small energy dierences. On the contrary, for 4 6 n 6 6 the hcp stacking is by far more favourable (with large energy dierences: DEads 0:30 eV for the hexamer, see Table 3), because up to six atoms can be placed on the same facet on good hcp sites, while it is possible to place only three atoms on good fcc sites, and remaining atoms have to be placed on a edge or on another facet. Adding atoms up to n 12, we have found that the hcp stacking is better only of a few hundredths for 6 6 n 6 9 but the energy dierence becomes again substantial (DEads 0:27±0:40 eV) for 10 6 n 6 12. The above results imply that, at the ®rst stage of the growth of a new shell on an Ih, the overlayer is preferentially placed on hcp stacking even if it covers some facets. Our results are in good
Fig. 5. Adsorption sites on a (1 1 1) facet of an Ih. In the upper row, there is an island on fcc stacking while in the lower row on hcp stacking.
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
agreement with the calculations in Ref. [10], where it was found that an incomplete external overlayers on a Mackay Ih preferentially stays on antiMackay stacking. But the situation changes for vary large islands. If NfTOT is the total number of fcc sites and NhTOT is the number of hcp sites, it turns out that: NfTOT 20Nf 30Nfe N V 10 s
1 s 1 12;
where N V 12 is the number of ®vefold vertexes, and NhTOT 20Nh NfTOT
10s
2:
Therefore, the atoms which can be placed on hcp sites are not sucient to cover the whole Ih, and an island covering several adjacent facets is better placed on fcc stacking. For example, for n 31 atoms on the 147-Ih, an island covering ®ve adjacent facets around a common vertex is better than any island on hcp stacking, as we have directly veri®ed. 5. Diusion on non-crystalline structures 5.1. Results for diusion on decahedral structures In this section, we report about the diusion of a single adatom on magic Dh at dierent sizes: 75 atoms 2; 2; 2 m-Dh, 146 atoms 3; 2; 2 m-Dh and 192 atoms 3; 3; 2 m-Dh (see Fig. 1). An adatom can diuse on a single facet (intrafacet diusion) and among dierent facets (interfacet diusion). Intrafacet diusion has approximately the same energy barriers as diusion on ¯at fcc surfaces [35]: on (1 1 1) facets, an adatom can move very fast even at low T [37±39], while on (1 0 0) facets diffusion is much slower [38]. The diusion along facet edges does not present appreciable dierences. Let us now focus on the interfacet diusion, because it plays an important role during the growth processes. In general, interfacet moves have higher barriers than intrafacet moves on the facet of departure. In fact, the adatom has to surmount an additional barrier DES , known as Ehrlich±
Schwoebel barrier [40,41], as happens for the diffusion across steps on surfaces. The possible diffusion processes between dierent facets depend strongly on the morphology of the structure, especially on the geometry and size of the single facet, and they can take place either by jump (hopping) or by exchange. An example of exchange interfacet diusion is given in Fig. 6. Diffusion between two neighbouring (1 1 1) takes place preferentially by jump (see Table 4), since the exchange mechanisms present much higher energy barriers, more than 0.5 eV, while diusion from a (1 1 1) to a (1 0 0) facet takes place preferentially by exchange (see Fig. 6). The barrier for this process increases with the size of the (1 0 0) facets, at least for small (1 0 0) facets (see Table 4); on the other hand the barrier for jumping is larger and does not depend on the facet size. These energy barriers are close to those for the analogous processes on fcc silver clusters [28], provided that (1 0 0) facets of the same sizes are considered. The reverse processes (from (1 0 0) to (1 1 1)) present in any case much higher energy barriers, essentially by the dierence in adsorption energy DEads between sites on the two facets (the same happens on fcc clusters, see Refs. [28,35]). Also the diusion from a (1 0 0) facet to the Marks reentrance has a very high energy barrier. Therefore adatoms adsorbed on (1 0 0) facets remain trapped there for long times. The diusion from a (1 1 1) to a Marks reentrance (R(110)) can take place either jump or by exchange with the common vertex with the neighbour (1 0 0) facet. The barriers do not depend on the cluster size, and are almost of the same magnitude for both jump and exchange. The reverse processes are quite dicult, because adsorption on the reentrance is more favourable than on (1 1 1) facets. Therefore, once the adatom falls in the Marks reentrance, it very likely remains there, so that diusion along the reentrance is not active for connecting the two caps of the Dh. The only mechanism connecting the caps is the chain process, a multiple exchange process [42] which connects two opposite (1 1 1) facets through the intermediate (1 0 0) facet [28,43±45]. An adatom on a (1 1 1) facet can push a central row (the chain process through a boundary row costs a lot, see Table 4) of the neighbour (1 0 0) facet until the
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
371
Fig. 6. Exchange connecting (1 1 1) and (1 0 0) facets of a 192-Marks Dh. On the left the light adatom moves from a icosahedral site close to the (1 0 0) edge and pushes the dark atom until the last goes onto the (1 0 0) facet.
Table 4 Energy barriers for an adatom on m-Dh (eV) Diusion mechanism
75 Dh
146 Dh
192 Dh
Exchange 1 1 1 ! 1 1 1 Exchange vertex Exchange T ! 1 0 0 Exchange 1 1 1 ! T via (1 0 0) vertex Exchange 1 1 1 ! T via (1 1 1) vertex Exchange 1 1 1 ! 1 0 0 Jump 1 1 1 ! 1 1 1 Jump 1 1 1 ! T Jump 1 1 1 ! 1 0 0
0.56 0.56 0.51 0.41
0.63 0.55 0.52 0.39
0.62 0.56 0.54 0.39
0.25 0.31 0.35 0.38
0.28 0.31 0.35 0.40
0.31 0.31 0.39 0.39
Chain 1 1 1 ! 1 0 0 ! 1 1 1 Via central row Via boundary row
n1 0.07
n2 0.13 0.28
n3 0.19
0.47
atom on the opposite (1 0 0)-edge falls down on the opposite (1 1 1) facet (see Fig. 7). The activation energy depends strongly on the length of the (1 0 0) central row: each additional atom along the central row increases the barrier of 0.06 eV (see Table 4). However, if the Marks integer n is less than 5, the chain process is more frequent than the jump between two adjacent (1 1 1) facets. From our results it turns out that barriers depend on the local arrangement of the surface, but not on internal strain. Processes with the same local geometry have practically the same barriers on both Dh and fcc structures. For example, the
jump between two (1 1 1) facets presents the same barrier on Dh and TO clusters, and this barrier is found also for step descent on a in®nite (1 1 1) crystal surface (see Ref. [46]); the chain mechanism has the same energy barrier on Dh and on TO clusters if the length of the central (1 0 0)-row is the same. Let us now analyse the consequences of our results on the growth of Dh clusters. A given Dh can grow to a larger Dh cluster both by the ®lling of the truncations and reentrances and by the superposition of an umbrella [4] covering either the top or the bottom cap of the cluster on decahedral stacking. This second growth mode gives a Dh with a ®vefold axis longer by one atom, and it is necessary to continue the growth of Dh structures because the number of atoms which can be placed in the truncations and reentrances is obviously limited. Our results show that adatoms can move easily towards truncations and reentrances, where they very likely remain trapped. Moreover, an adatom deposited on a (1 1 1) facet can move very fast between the two caps of the Dh by the chain process and quite easily between neighbouring (1 1 1) facets. This favours the building up of a big island either on the top or on the bottom cap: when the majority of the sites on (1 0 0) facets are ®lled, it is likely to build up a large island covering one of the caps. But this island is more likely grown on hcp sites, and this is like putting an umbrella over the Dh cluster [4], but on the wrong
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
Fig. 7. Chain process connecting two opposite (1 1 1) facets of a 192-Marks Dh: on the left the light adatom on a (1 1 1) pushes the central row (dark atoms) of the neighbour (1 0 0), in the central ®gure the saddle point is represented and ®nally the atom on the opposite edge falls down onto the opposite (1 1 1) facet, on the right.
(icosahedral) stacking. While the umbrella on the decahedral stacking continues the growth of a Dh cluster, the umbrella on icosahedral stacking can initiate the structural transformation of the Dh into an Ih, as seen in MD simulations of growth [21]. Indeed, in this con®guration, further adatoms getting trapped into a Marks reentrance can form a new ®vefold symmetry point: the transformation from a Dh into an Ih is starting. 5.2. Diusion on icosahedral structures The results about single-adatom diusion on Ih clusters are reported in Table 5. We consider Ih of 55, 147 and 309 atoms (see Fig. 4). The surface of the Ih consists of distorted (1 1 1) facets, and in the following we consider both intrafacet and interfacet diusion (see Table 5). Intrafacet diusion takes place by jump, and the barrier is very close to the one on a perfect in®nite (1 1 1) surface, or on the Dh (1 1 1) facets. Again, the internal strain has a little eect on the diusion barriers. Concerning interfacet processes, the exchange with a vertex is much more dicult than the exchange with edge atoms. Because of that, interfacet diusion takes place mainly between adjacent facets sharing a common edge, and not a common vertex only. The diusion between adjacent (1 1 1) facets takes place by jump (see Fig. 8) if the number of atoms of the common (1 1 1) edge is
Table 5 Energy barriers for an adatom on Ih (eV) Diusion mechanism
55 Ih
147 Ih
309 Ih
Exchange 1 1 1 ! 1 1 1 Exchange vertex Jump 1 1 1 ! 1 1 1
0.15 0.44 0.32
0.34 0.47 0.31
0.44 0.50 0.30
larger than 3. Otherwise, the exchange process is more favourable. For example, on the 55 Ih, the common edge has three atoms, and the barrier for exchange is very low (see Table 5), because the facet is so small that the common edge can distort itself to accommodate the adatom (see Fig. 9, where we show the initial, saddle-point and ®nal positions). On larger Ih, this barrier increases a lot, and the jump process (whose barrier does not depend on the length of the common edge) becomes more favourable. Indeed, jumping between (1 1 1) facets has the same barrier for all structures we have considered here and also for fcc clusters [28]. What are the consequences of our ®ndings on the growth of Ih clusters? An Ih can grow shell by shell (SBS, see Ref. [21]) if the adatoms can pass easily among dierent facets in order to have only a single island covering several (1 1 1) facets: the fast diusion between facets guarantees the formation of overlayer structures. From our results it is clear that, at any size, the adatoms can move rather easily from a facet to another allowing to arrange themselves into a big island (this leads to
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
373
Fig. 8. Initial, saddle-point and ®nal con®gurations for the jump process connecting two neighbouring (1 1 1) facets on a 147-Mackay Ih: the dark atom hops on the common edge and then falls down on the opposite facet.
Fig. 9. Initial, saddle-point and ®nal con®gurations for the exchange mechanism between two neighbouring (1 1 1) facets of a 55Mackay Ih: the light adatom on a (1 1 1) pushes the central dark atoms onto a hcp site.
the umbrella growth model see Refs. [4,47]), which more likely grows on hcp stacking at intermediate sizes. This implies that the cluster grows via the SBS mode, provided that temperature is high enough to displace the island on fcc stacking. If not, defected structures are expected to grow. On the other hand, we have found that an Ih can transform itself into a Dh through quasi-melted structures (see Ref. [21]). This transformation can take place if, in conditions (cluster size and temperature) where the Dh structures are favoured, the cluster has sucient kinetic energy to rearrange all atoms around a single ®vefold axis. Because of
that, the Ih ! Dh transformation is more dicult than the Dh ! Ih transformation during growth processes. 6. Discussion and conclusions In this paper, we have studied the energetics of island adsorption and single-adatom diusion on non-crystalline silver clusters by MD simulations. In particular, we have focused our attention on how adatoms can be placed and move, discussing the implications on the cluster growth. Concerning
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F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
island adsorption, we have found that islands can be placed on dierent stackings (fcc and hcp) on both Ih and Dh. The most favourable stacking changes depending on the size of the island. In particular, we have shown that there is always a island size range in which the hcp stacking is by far the best. A preference for hcp stacking was found experimentally on ¯at Ir surfaces (see Ref. [48]) for very small islands, like dimers and trimers, and with energy dierences of the order of a few hundredths of eV. What we ®nd here is completely dierent, and depends crucially on the ®nite extension of the clusters facets. Dimers and trimers stays slightly better on fcc sites, while the hcp stacking is better for islands of the order of 10 or more atoms, with considerable energy dierences (close to 0:5 eV). Here, the cause is not some local relaxation of the structure or some second-third neighbours eect, but is purely geometrical: there are more stable hcp sites than fcc sites on a single facet. Indeed, the fcc stacking becomes again favourable when islands are very large. In the case of Dh, this means almost the complete covering of one cap. This has deep consequences on the growth of Dh. If the island grows on fcc stacking, a larger Dh is obtained [4], with the length of the ®vefold axis increased by one atom. But islands very likely grow on hcp stacking at intermediate sizes, and if temperature is not high enough to let the island displace to fcc stacking, a structural transformation towards an Ih can begin, as seen in the growth simulations in Ref. [21]. Now the question is: how is it possible to build up large islands covering several (1 1 1) facets? The answer is found by inspecting the diusion of single adatoms: if adatoms are able to diuse easily among dierent facets, they can meet and form a single aggregate on the cluster surface; if interfacet diusion is slow, small islands nucleate on dierent facets. This is analogous to what happens in epitaxial crystal growth: fast adatom diusion causes the building up of few large islands. Our results have shown that diusion among dierent facets is rather easy on both Ih and Dh clusters, with the exception of the diusion from the truncations on Marks Dh. On the other hand, in the case of Dh clusters, there is a process (the chain, which is a very interesting example of a multiple exchange
process) which allows a very fast mobility between the two caps. Because of this process, it is likely to have a single island nucleating on only one of the caps. In the case of Ih clusters, interfacet diusion favours the formation of a single large island which grows to the completion of a further shell. Let us compare now adatom diusion on clusters of dierent symmetries: Ih, Dh and TO (for the latter see Ref. [28]). One of the main results of this work is that the diusion on non-crystalline Ag clusters is not very much in¯uenced by the internal strain which characterizes both Ih and Dh. This fact is con®rmed by dierent results about single-facet and interfacet diusion. Intrafacet processes have the same barriers on clusters of dierent structures, and these barriers are very close to those found on in®nite ¯at fcc surfaces. Interfacet jump processes have again the same barriers on dierent clusters, and the same happens for the chain process, whose barrier depends only on the size of the (1 0 0) facet where the chain is passing through, and not on the clusters being a Dh or a truncated octahedron. On the other hand, we have shown that the exchange processes depend strongly on size, local morphology and on the relative inclination between the facets. For any structure, the exchange between two adjacent (1 1 1) is strongly in¯uenced by the length of the (1 1 1) edge and it presents higher and higher barriers when the angle between the facets becomes ¯atter and ¯atter (from the step on a (1 1 1) surface to the exchange on a Dh). In conclusion, our results show that island adsorption and diusion on non-crystalline clusters present very interesting features, due to the peculiar geometry of these structures. The interplay between the energetics of adsorption, which is ruled by geometrical constraints, and the adatom mobility governs the formation of the new shells of atoms and therefore is expected to play a crucial role in the growth of these clusters in their solid state. Acknowledgements We acknowledge ®nancial support from the Italian MURST under the project N. 9902112831.
F. Baletto, R. Ferrando / Surface Science 490 (2001) 361±375
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