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Monte Carlo Simulation of Properties of Monolayers and Metal Islands Adsorbed on Metallic (111) Surfaces Mariana I. Rojas, Mario G. Del Po´polo, and Ezequiel P. M. Leiva* Unidad de Matema´ tica y Fı´sica, Facultad de Ciencias Quı´micas, INFIQC, Universidad Nacional de Co´ rdoba, 5000 Co´ rdoba, Argentina Received October 28, 2003. In Final Form: January 30, 2004 To obtain the surface stress changes ∆σ due to the adsorption of metal monolayers onto metallic surfaces, a new model derived from thermodynamic considerations is presented. Such a model is based on continuum Monte Carlo simulations with embedded atom method potentials in the canonical ensemble, and it is extended to consider the behavior on different islands adsorbed onto (111) substrate surfaces. Homoepitaxial and heteroepitaxial systems are studied. Pseudomorphic growth is not observed for small metal islands with considerable positive misfit with the substrate. Instead, the islands become compressed upon increase of their size. A simple model is proposed to interpolate between the misfits of atoms in small islands and the pseudomorphic behavior of the monolayer.
1. Introduction The formation of metal islands on a metal surface represents an important stage in the growth of a phase in the case of homoepitaxy and in the appearance of a new one in the case of heteroepitaxy. In a preliminary study,1 we have developed a Monte Carlo simulation model that allows the properties of small metal islands on metallic surfaces to be studied. Within these studies, we have shown that metal islands adsorbed onto (111) surfaces acquire hexagonal-like equilibrium shapes and may be subject to considerable stress as a result of a balance between the interactions of the atoms in the monolayer with each other and their interaction with the substrate. In the particular case of the heteroatomic Ag/Pt(111) system, we found for all the island sizes considered an outward relaxation of the atoms located at the edge. In addition, we did not find pseudomorphic growth but rather expanded structures which became progressively compressed as the size of the islands increased. On the other hand, for the homoepitaxial systems considered, the islands presented inward relaxation at the edges as expected from bond order conservation analysis. The estimation of the border formation energy was important to understand the thermal stability of the islands because island disintegration is expected to take place when this quantity is of the order of kT. For Ag islands on Pt(111), we calculated a surprisingly low edge energy, in agreement with the experimental finding of Ro¨der et al.2 that Ag islands on Pt(111) disintegrate at relatively low temperatures. On the other hand, Ag on Ag(111) and Cu on Cu(111) presented larger edge formation energies, of the order of 0.2-0.3 eV, which were close to the values found experimentally by Giesen et al.3 A second and more appealing outcome of these simulations was the fact that, for the adsorption of Ag on Pt(111), the nearest neighbor distance, dnn, in the center of the island, was a function of its size. For small islands, dnn presented a value close to the Ag-Ag distance on a Ag(111) surface, approaching the distance between Pt * Corresponding author.
[email protected].
Fax,
54-351-4344972;
e-mail,
(1) Rojas, M. I.; Amilibia, G. E.; Del Po´polo, M. G.; Leiva, E. P. M. Surf. Sci. 2002, 499, L135-L140. (2) Ro¨der, H.; Brune, H.; Kern, K. Phys. Rev. Lett. 1994, 73, 2143. (3) Giesen, M.; Steimer, C.; Ibach, H. Surf. Sci. 2001, 471, 80.
atoms on the Pt(111) surface when the island size increased. This observation was consistent with the fact that a considerably stressed (1 × 1) structure was experimentally observed at full coverage.4 In this respect, the evolution of the surface stress during the deposition of a foreign metal on a metallic surface is a topic of general interest. The surface stress affects the structure of the interface, being the leading force of reconstruction and the source of different kinds of surface defects. Structural changes occurring during the first stage of deposition, which are associated with large changes in the surface stress, affect the shape and the kinetics of growth of subsequent layers in multilayered deposits. As a typical example of the relation between surface stress and structure in heteroepitaxy, the work of Bromann et al.5 can be mentioned. They observed that Ag islands, below a critical size of 200 Å, grow coherently strained on Pt(111), while larger islands relieve strain through the introduction of misfit dislocations. Upon completion of the first monolayer, the dislocations disappear and the Ag film adopts a pseudomorphic structure. These authors’ experimental findings were complemented by effectivemedium theory calculations to analyze the formation of dislocations, found to occur for islands with a pseudomorphic core (diameter above 50 Å). In the present work, we present a model to obtain the change of surface stress by continuum canonical Monte Carlo simulations using the embedded atom method (EAM) potentials.6-7 The model is tested for homoepitaxial and heteroepitaxial monolayers and is extended to consider islands whose sizes vary between 12 and 156 atoms. The homoepitaxial system studied is Ag on Ag(111), and the heteroepitaxial ones are Ag/Pt(111), Au/Pt(111), and Ag/Pd(111). According to the theoretical results of Bromann et al.5 for the first of these systems, our island sizes are expected to be below the limit for the formation of dislocations. The magnitude of the stress is found to increase with the number of atoms in the island, something which is associated with the concomitant structural changes: (4) Ibach, H. Surf. Sci. Rep. 1997, 29, 193-263. (5) Bromann, K.; Brune, H.; Giovannini, M.; Kern, K. Surf. Sci. 1997, 388, L1107-L1114. (6) Daw, M. S.; Baskes, M. I. Phys. Rev. B 1983, 50, 1285. (7) Foiles, S. M.; Baskes, M. I.; Daw, M. S. Phys. Rev. B 1986, 33, 7983.
10.1021/la036021z CCC: $27.50 © 2004 American Chemical Society Published on Web 04/09/2004
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atoms in the islands become progressively locked in registry with the substrate as the islands become larger and larger. In the case of homoatomic systems, this effect takes place earlier (concerning island sizes) and coincides with the disappearance of the change in the surface stress. 2. Thermodynamic Calculation of the Surface Stress Changes Our continuum Monte Carlo simulations were performed in the canonical ensemble. In this ensemble the number of particles N, the temperature T, and the volume V of the system are fixed. The related state function is the Helmholtz free energy A, given by
A(N, T, V) ) -kT ln[Q(N, T, V)]
∫ dp exp(-
)∫
K(p) kT
(
dq exp -
(4)
where Λ is the thermal de Broglie wavelength:
x
h2 2πmkT
dx1...dxN
(
∫0
exp -
dy1...dyN
∫0
Lz
dz1...dzN ×
∫0L
∫0L
z
σ)
x
dx1...dxN
(
dz1...dzN exp -
∫0L
y
dy1...dyN ×
(
)
1 ∂Q ∂A ) -kT ∂S Q(N, T, LzSeq) ∂S
( )∫
∂Q Q 1 ) NBSN-1 N + BSN ∂S kT BS
(
exp -
(9)
1
0
dsx1...dsxN ×
()
∂ν × ∂S ν(sx1...sxN, sy1...syN, z1...zN; Lx, Ly) 1
N
dz1...dzN
)
kT
)
By changing the x and y coordinates by scaled coordinates sx and sy in the previous equation, we obtain (8) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1992. (9) Frenkel, D.; Smit, B. Understanding Molecular Simulations; Academic Press: San Diego, 1996.
(10)
Substituting eq 10 into eq 9, we obtain the equation for the surface stress:
σ ) -kT
(NS) + 〈∂S∂ν 〉
(11)
In the case of monolayer adsorption, the changes of the surface stress can be calculated straightforwardly using eq 11 as follows. A difference of Helmholtz free energy ∆A can be defined as
(12)
where As+a is the free energy of the adsorbate + substrate system and As is the free energy of the naked substrate. From ∆A, a change in the surface stress upon monolayer adsorption can be obtained from
∆σ )
d∆A dS
(13)
where S denotes the surface system. Using eqs 9 and 11 we obtain
(
)
ν(x1...xN, y1...yN, z1... zN) (7) kT
(8)
where the derivative of the partition function ∂Q/∂S is
) 〈
〉
Nad + Nsu dAs+a ∂νad/su + ) -kT dS S ∂S
ν(x1...xN, y1...yN, z1...zN) (6) kT
The canonical partition function for a system of N particles can, therefore, be written as
Q ) QidQex ) B
)
kT
(5)
and the excess part is Ly
dz1...dzN ×
∆A ) As+a - As
VN Q ) N!Λ3N
Λ)
N
ν(sx1...sxN, sy1...syN, z1...zN; Lx, Ly)
(3)
id
∫0
1
In principle, both the surface strain and stress are tensors, but if our model calculations are restricted to epitaxial films on (111) surfaces with face-centered cubid (fcc) symmetry, where the stress tensor is diagonal, we can obtain the surface stress from the derivative of the Helmholtz free energy with respect to the surface area:
)
where h is the Planck’s constant. In the independent particle approach (V ) 0) the ideal part of the partition function is equal to
Q )V
exp -
V(q) ) QidQex (2) kT
1 N!h3N
B)
Lx
(
N
∫01 dsy ...dsy ∫0L
B is a proportionality constant, chosen so that the integral over all the phase space approaches the classical partition function in the limit (hf0). For instance, for a system of N identical atoms
-N
z
1
z
Q(N, V, T) )
ex
∫01 dsx ...dsx ∫01 dsy ...dsy ∫0L
(1)
where Q is the canonical partition function. The total energy can be written as a sum of the kinetic K(p b) and the potential V(q b) contributions, so the partition function is factorized into a product of ideal, Qid, and excess, Qex, parts:8-9
B
Q ) BSN
(14a)
and
( ) 〈 〉
Nsu dAs ∂νsu ) - kT + dSsu Ssu ∂Ssu
(14b)
Substituting eq 14a,b into eq 13 leads to the following expression for the change in the surface stress upon monolayer adsorption:
∆σ ) -kT
( ) 〈
〉 〈 〉
Nad ∂νad/su ∂νsu + S ∂S ∂S
(15)
The terms 〈∂νad/su/∂S〉 and 〈∂νsu/∂S〉 were evaluated from an isotropic stretching and compression of the simulation
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box in the direction parallel to the surface for each configuration of the production run (three points derivative). Consideration of the stress changes upon island adsorption is more involved. In fact, if adsorption of a single island onto a substrate is considered, the stress is a local phenomenon, where the substrate atoms below the island and in its the vicinity have been displaced from their normal positions. At these points, the change of surface stress may be very important. Far from the islands, the surface atoms will be undisturbed; hence, the part of the substrate that is far from the island will not contribute to the change in the surface stress. However, because experiments do not measure a local change of the surface stress but an average answer of the system, such as the bending of a cantilever upon adatom deposition, we must adopt some approach that allows comparison with the experiment. In this respect, it is interesting to ask whether, for the same coverage degree of deposited atoms, the average change of the surface stress is larger or smaller depending on the way in which the atoms are adsorbed (larger or smaller islands, for example). One of the problems within the present approach is that, as a result of computational limitations, only simulations with a single island on a substrate can be performed. A further problem arises with the use of periodic boundary conditions, where the interaction of each island with its images would produce an additional dependence on the substrate size. For this reason, in this work the positions of the atoms have been fixed at the edges of the simulation box so as to be able to predict the answer of a system made of several noninteracting islands. Fixing the atoms at the edge of the simulation box undoubtedly introduces an error in the calculations of the stresses described as follows. However, the smooth transition between the properties of large islands and those of a monolayer described as follows (where none of the surface atoms are fixed) indicates that these effects should not be dramatic for the properties we have calculated here. This approximation should be checked with larger computational power allowing simulations for larger simulation boxes. In the case of simulations with a single island adsorbed, we define a difference of Helmholtz free energy, ∆A, as
∆A ) As+i - As
(16)
where As+i is the free energy of the island + substrate system and As is the free energy of the naked substrate. Thus, if there was no interaction between the island and the substrate, ∆A would correspond to the free energy of the isolated island. From eq 12 for ∆A, an average change in the surface stress can be defined in analogy with eq 13 as
∆σ′ )
d∆A dSsu
limitations and that we could make calculations for a fixed island size and very large sizes of the simulation box, where surface atoms far from the island are undisturbed and make a negligible contribution to d∆A. To get the derivative (17), we would change the area of the system in the quantity dSsu. In this limit and provided we always produced the same surface stretching to get the same final size of the island, we would obtain the same d∆A, independently of the size of the simulation box. However, the condition in italics (same surface stretching...) requires different values of dSsu to get the same d∆A. Thus, islands in the same physical situation (same size, same stretching) deliver the same d∆A for different simulation boxes, but yield different ∆σ′ because different dSsu are required to yield the same physical situation for the island. It is, thus, clear from the definition (17) that for a given island size the change ∆σ′ will depend on the size of the substrate. In other words, if a given (single) island is located on substrates of different sizes and their lattice parameters are stretched to an equivalent amount, the change of the excess of free energy d∆A will be the same, while the changes in the areas of the substrates dSsu will increase with the area of the substrate. Physically speaking, this is tantamount to say that an island of a given size will produce a larger change of the average surface stress on a small substrate than on a large one. For this reason, and as we are interested in comparing the behavior of islands of different sizes, we will define the following quantity:
∆σ′′ )
d∆A dSi
(18)
where Si is the surface occupied by the island. This quantity offers the advantage of not depending on the size of the simulation box, so it allows the comparison between islands of different sizes. It measures the ability of an island to increase (or reduce) the average stress on the area occupied by the island. Because in our simulations the stretching is made on the area of the whole substrate box Ssu, we calculate ∆σ′′ as
∆σ′′ )
dSsu d∆A Ssu d∆A ) dSi dSsu Si dSsu
(19)
where the second equality arises because of the fact that in eq 11 the term ∂ν/∂S implies an isotropic stretching of the island + substrate system (Ssu/Si ) constant ) dSsu/ dSi). Using eqs 9 and 11, we obtain
(
) 〈 〉
Nad + Nsu dAs+i ∂νad/su ) -kT + dSsu Ssu ∂Ssu
(20a)
and
( ) 〈 〉
(17)
where Ssu denotes the surface of the substrate. Equation 17 is formally identical to the definition of (minus) the spreading pressure,10 and it indicates whether the tendency of the surface to contract (or expand) is, on average, larger or smaller in the presence of the island than in its absence. The use of eq 17 to compare the behavior of different island sizes requires some discussion. Let us assume for a while that we had no computational (10) Dash, J. G. Films on Solid Surfaces; Academic Press: New York, 1975.
Nsu dAs ∂νsu ) -kT + dSsu Ssu ∂Ssu
(20b)
where Nad and Nsu denote the number of atoms of the island and of the substrate, respectively. Substituting eq 20a,b into eq 19 leads to the following expression for the change ∆σ′′:
∆σ′′ )
( ( ) 〈 〉 〈 〉)
Ssu Nad ∂νad/su ∂νsu -kT + Si Ssu ∂Ssu ∂Ssu
(21)
The terms 〈∂νad/su/∂Ssu〉 and 〈∂νsu/∂Ssu〉 were evaluated from
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Table 1. Lattice Constants of the Metals Employed in This Work at 0 and 300 K in Angstroms
Table 2. Misfits Emf × 100 for the Different Systems at 0 and 300 K
substrate
0 K (ref 7)
300 K
substrate
0 K (ref 7)
300 K
system
0K
300 K
system
0K
300 K
Ag Pt
4.09 3.92
4.11 3.93
Pd Au
3.89 4.08
3.90 4.10
Ag/Pt(111) Ag/Pd(111)
-4.3 -5.1
-4.5 -5.3
Au/Pt(111)
-4.1
-4.2
an isotropic stretching and compression of the simulation box as in the case of monolayer adsorption. We emphasize that the quantities ∆σ′ and ∆σ′′ are not the diagonal components of a tensor but average quantities obtained in analogy with eq 9, which is rigorously valid for pseudomorphic monolayer adsorption on (111) fcc surfaces. However, the quantity ∆σ′′ is useful to compare the way in which an island enhances or releases the spreading pressure of a surface and can be compared with the excess of stress introduced by a monolayer, which is identical to (minus) the spreading pressure. For the sake of simplicity, we leave out the double prime in the following in the case of the quantity defined in eq 21, but whenever we are comparing island properties with those of monolayers we refer to the definition (21). 3. Model and Simulation Method The substrate was assumed to be a smooth (111) single crystal surface without considering steps or reconstruction. It was represented by a four-layer slab with 480 atoms per layer. The atoms of the first and second plane of the substrate, which belong to a circular region containing about 250 atoms, were allowed to vary their positions during the simulation. The remaining substrate atoms were fixed to their bulk equilibrium configuration to emulate the presence of a semi-infinite crystal. On this substrate, submonolayers of an initially circular shape containing between 12 and 156 atoms were adsorbed to emulate islands of different sizes. In the case of simulations of monolayers, the atoms of the whole monolayer and of the first two surface layers of the substrate were mobile. The potentials employed were given by the EAM which takes into account the many-body interactions characteristic of metallic systems.6-7 Within this model, the total energy is expressed in terms of two contributions: N
Etot )
1
Fi(Fh,i) + ∑∑φij(rij) ∑ 2 i j*i i)1
(22)
where Fh,i is the host electronic density at the position of atom i, Fi(F) is the embedding energy, and φij(rij) is the core-core pair repulsion between the atoms i and j, separated by the distance rij. In all cases, we used the parametrization of Foiles et al.7 The lattice constants at 300 K were calculated and used in the simulations to obtain a stress-free bulk substrate. These values are given in Table 1. All Monte Carlo simulations were performed in the (NVT) ensemble at 300 K. We performed 5000 equilibration steps followed by 10 000 production steps. To let the adatoms overcome high energy barriers, such as those for the displacement on the surface or the detachment from the cluster, we allowed the adatoms to perform long jumps, with displacements ∆r b:
s 1 + n2b s2 ∆r b ) n1b
(23)
s1 and b s2 are the primitive where n1 and n2 are integers and b vectors of a two-dimensional Bravais lattice. These “long” jumps are essential to a proper equilibration of the system. “Short” jumps, as usually employed in Monte Carlo simulations where the positions of the particles are varied
continuously, were also allowed in our studies to describe the vibrational motion of the adsorbate atoms in the neighborhood of their equilibrium positions. This method has been successfully employed to study the growth of Pd submonolayers on Pt(hkl) and Au(hkl),11 yielding results in reasonable agreement with experimental data from literature. 4. Results and Discussion Simulations were performed for the homoatomic system Ag/Ag(111) and for the heteroatomic ones Ag/Pt(111), Au/ Pt(111), and Ag/Pd(111). All the heteroatomic systems considered have an important misfit mf, defined from the bulk lattice parameter a as4
mf )
asubs - aads asubs
(24)
As shown in Table 2, mf slightly increases with temperature. For small-size islands, none of the systems presents a completely pseudomorphic structure, not even the homoatomic case (Ag/Ag(111)). This can be observed in Figure 1a where we show the relaxation pattern of a 129-atom Ag island with respect to the perfectly pseudomorphic configuration on Ag(111). The arrows in the figure denote the displacement of the atoms with respect to their pseudomorphic adsorption sites. In this case, the island presents a small inward relaxation at the edges as expected from the lack of coordination of the atoms at the edges. The heteroepitaxial systems are shown in Figure 1b-d. As in the former case, the selected islands have 129 atoms. In all the cases, the misfit is large and negative and induces a better-defined relaxation pattern. It can be noticed that the heteroatomic islands considered present, in all cases, outward relaxation, being expanded with respect to the (1 × 1) structure. The displacement increases with the distance from the center of the island, indicating that the whole island is expanded. This expansion is found to be a function of the size, and it can be quantified through the average distance between neighboring adatoms, say d h nn(Nad), which is calculated according to
∑
d h nn(Nad) )
dij
i,jneigh
Np
(25)
where the sum runs over all neighboring atomic pairs and Np is the number of pairs. By analogy with eq 24, a misfit of the island can, therefore, be defined as4
is(Nad) )
dsubs - d h nn(Nad) dsubs
(26)
and a similar quantity that will be used in the following is the average relative misfit of the atoms defined as (11) Rojas, M. I.; Del Po´polo, M. G.; Leiva, E. P. M. Langmuir 2000, 16, 9539-9546.
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Figure 1. Atom displacements for the 129-atom island for the different systems studied. The arrows represent the relaxation with respect to the (1 × 1) pseudomorphic configuration in angstroms. (a) Ag/Ag(111); (b) Ag/Pd(111); (c) Ag/Pt(111); (d) Au/Pt(111).
r(Nad) )
h nn(Nad) dsubs - d dsubs - dad
(27)
where dsubs is the nearest neighbor distance between surface substrate atoms and dad is the nearest neighbor distance between adsorbate atoms on a surface of its same nature. r(Nad) equals 0 if the adsorbate is pseudomorphic with the substrate, and equals 1 if it keeps its own structure. According to the simple model presented in Appendix A, the reciprocal of the relative misfit, r(Nad) should follow a linear relation with the number of atoms in the island. For this reason, in Figure 2a, we plotted 1/r(Nad) versus Nad, finding a reasonable agreement with an equation of the type
1 ) A1Nad + B1 r(Nad)
(28)
Values of A1 and B1 for heteroepitaxial systems are reported in Table 3 and can be used together with eq 28 to estimate r(Nad) for islands of arbitrary sizes. Figure 2b shows the misfit defined in eq 26, is(Nad), as a function of the reciprocal of the number of adsorbate atoms. While in the case of Ag/Ag(111) it is clear, from the
simulated points, that is(Nad) tends to 0 for Nad f ∞, this is not so evident for the heteroatomic systems. There, the misfits are still quite large for the largest islands simulated. As stated previously, eq 28 can be heuristically employed in this region to get r(Nad). On the other hand, for relatively small islands (small number of atoms), the misfits tend to be constant, as predicted by eq 28. The energy of the system and, therefore, the surface stress changes are expected to be related to the structure of the adsorbate. It is then illustrative to consider in Figure 2c the average nearest neighbor distance d h nn(Nad) as a function of the number of adsorbed atoms:
d h nn(Nad) )
dsubs(A1Nad + B1 - 1) + dad A1Nad + B1
(29)
From this figure, it can be appreciated that islands in the heteroatomic systems are expanded at the beginning but tend to compress with increasing coverage, reaching the (1 × 1) structure at the monolayer coverage. However, for the simulated systems, the nearest neighbor distances are still considerably different from the values they reach when the monolayer is pseudomorphic with the substrate (arrows). Figure 3a shows the surface stress changes, ∆σ, as a function of the number of atoms in the island, calculated
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Rojas et al. Table 3. Least-Squares Parameters of the Linear Behavior between the Reciprocal Relative Misfit 1/Er(Nad) as a Function of the Number of Adatoms Nad (Eq 28)a system
A1 × 103 [atoms-1]
B1
Ag/Pt(111) Ag/Pd(111) Au/Pt(111)
4.8 ( 0.1 4.3 ( 0.1 3.9 ( 0.8
1.08 ( 0.01 1.29 ( 0.01 2.70 ( 0.08
a A and B are the corresponding slope and intersect, respec1 1 tively.
Figure 2. (a) Reciprocal of the relative misfit r(Nad) defined in eq 27 as a function of the number of atoms constituting the island Nad. These plots were used to fit the parameters A1 and B1 of eq 28, given in Table 3. (b) Misfit is(Nad) defined in eq 26 as a function of the reciprocal of Nad. Note that all the monolayer misfits (1/Nad f 0) are 0 because all the systems are, in this limit, pseudomorphic. (c) Nearest neighbor distances of the atoms in the island versus Nad. The limiting values for Nad f ∞ are d h nn(Pt) ) 2.779 Å and d h nn(Pd) ) 2.758 Å. Triangles, Au on Pt(111); diamonds, Ag on Pd(111); squares, Ag on Pt(111); circles, Ag on Ag(111). The lines denote the fits according to eq 28. Dotted-dashed line, Au on Pt(111); long dashed line, Ag on Pd(111); solid line, Ag on Pt(111).
during the simulation according to eq 21. A clear trend is observed for the heteroatomic systems: islands become
Figure 3. (a) Surface stress change ∆σ in J/m2 evaluated from the simulations according to eq 21 as a function of the number of atoms in the island Nad. (b) ∆σ in J/m2 as a function of the reciprocal of the number of atoms in the island, showing the behavior of the monolayer (1/Nad f 0). Triangles, Au on Pt(111); diamonds, Ag on Pd(111); squares, Ag on Pt(111); circles, Ag on Ag(111). The lines denote the fits according to eq B11. Dotted-dashed line, Au on Pt(111); long dashed line, Ag on Pd(111); solid line, Ag on Pt(111).
more stressed as their size increases. In contrast, the homoatomic Ag/Ag(111) shows the opposite trend: the change in the surface stress disappears as the coverage approaches 1. These results agree with the observation that as the island size increases, the adatoms are locked in registy with the substrate. However, no straightforward prediction about the system behavior for very large islands can be made from this plot. Then some kind of heuristic approach must be used to interpolate between the behavior of small islands and that of the monolayer. This point is discussed in Appendix B. The lines in Figure 3a show the ∆σ obtained from a fitting where the contributions of the
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Table 4. Surface Stress Change for the Adsorption of a Monolayer As Obtained from Monte Carlo Simulations, ∆σMC mon (eq 21), and the Corresponding Prediction from the Fitting According to Eq B11, ∆σadj mon system
2 ∆σMC mon [J/m ]
2 ∆σadj mon [J/m ]
Ag/Pt(111) Ag/Pd(111) Au/Pt(111)
-4.920 -4.884 -3.243
-4.917 -4.877 -3.247
inner and border atoms were considered (see eq B11). Figure 3b shows the usefulness of eq B11 to obtain the stress of large islands. Table 4 shows ∆σ values for the monolayer as obtained by Monte Carlo simulations (using eq 21) and the corresponding values as calculated by the fitting using equation (B11). Both results agree reasonably well. In the case of Ag/Pt(111), our result of -4.9 N/m can be compared with -8.8 N/m, the value measured by Grossmann et al.14 For the systems Ag/Pd(111) and Au/Pt(111), large compressive stresses were obtained, while experiments with these systems also pointed out large and compressive stresses.12,13 At this point, we can relate the results of Figure 3a to those of Figure 2b. In the case of homoepitaxial systems, the fact that larger islands come more and more in registry with the substrate leads to the disappearance of the change in the surface stress. This is quite different in the case of all the heteroepitaxial systems considered. There, atoms in the island also come in registry with the surface, but with a concomitant increase in the compressive stress. Experimental results are usually obtained in a region of coverage where a distribution of islands, f(si), exists. The total number of islands in the system is given by
n)
f(si) ∑ i)1
(30)
where f(si) denotes the number of islands with a size si. We can find now a way to relate the quantities ∆σ as calculated in our simulations to the measured change in the surface stress, say ∆σm. Because the experiment considers the whole surface of the substrate, we have
d(As+i - As) ∆σm ) dSsu
(31)
where As+i labels the Helmholtz free energy of the substrate with the ensemble of islands adsorbed, As is the Helmholtz free energy of the naked substrate, and Ssu denotes the surface of the substrate. If we assume that j , of As+i is just the addition of the contributions, As+i several noninteracting islands, we can write
∆σm ≈
∑j
j d(As+i - As)
dSsu
(32)
or considering the stretching of each individual island, we have (12) Clemens, B. M.; Nix, W. D.; Cho, K. J.; Sabiryanov, R. F.; Larsson, M. I.; Ramaswamy, V.; Hussen, G.; Lee, Y. W.; Chandra, A.; Chrzan, D.; Greaney, P. A. Stress and Nanopattern Formation in Thin Films. Presented at MRS Fall 2002 Meeting, December, 2002. (13) Clemens, B. M.; Ramaswamy, V.; Nix, W. D.; Freitag, J.; Peterson, B.; Phillips, M. A. Stress Evolution During Interface Growth. Presented at MRS Fall 2001 Meeting, November, 2001.
∆σm ≈
j dSj d(As+i - As)
∑j dS
)
dSj
su
Sj
∑j S
∆σj
(33)
su
where ∆σj is the contribution to the stress from island j. Thus, if f(si) were available, the submonolayer stress could be easily calculated from eq B11. According to the present results for heteroepitaxial systems, if a given coverage is obtained with relatively small islands, a smaller change average stress should result than if the same coverage were obtained with larger islands. On the other hand, if island size increases during the deposition process, the initial stress change should be smaller at the beginning, increasing as deposition proceeds. Let us assume for the purpose of a qualitative discussion that we are at the early stages of an instantaneous nucleation process and that the island size distribution is uniform so that eq 33 is reduced to
∆σm ≈ θ∆σ(Nad)
(34)
where ∆σ(Nad) is the stress of an island with Nad atoms and we have defined the coverage degree θ as
θ)
MSN Ssu
(35)
where M is the total number of islands and SN is the island surface. SN is given by
SN ) NadΩ
(36)
where Ω is the area occupied by an atom in the island, which is a slowly varying function of Nad. In fact, Ω is close to the area of an adsorbate atom for small Nad and close to the area of the substrate atom for Nad f ∞ so that it can be considered as a constant in the following argumentation. We can write eq 34 as
( )
∆σm ≈ θ∆σ
θ Ssu Ω M
(37)
In this equation, Ssu/M is the reciprocal of the island density on the surface. If Ssu/M is relatively large (low island densities), a small θ gives already a large value of the argument of ∆σ in eq 37, the islands are large, and their stresses are comparable to that of a monolayer (∆σ ≈ ∆σmonolayer ) constant). Thus, a linear ∆σm versus θ curve should result at all coverages. On the other hand, if Ssu/M in eq 37 is relatively small (high island densities), the dependence of ∆σm on the island size should be apparent and a curvature in the ∆σm versus θ should be evident at low coverage degrees. We have illustrated these features in Figure 4, where we show ∆σm versus θ for different island densities. We show in Figure 4 only the portion corresponding to low coverage degrees. At high coverage degrees, whatever the island size, they should coalesce and present a change in the average surface stress close to that of a monolayer. Thus, depending on the density of nucleation, linear or s-shaped ∆σm versus θ curves may result. An s-shaped curve at low coverage degrees can be appreciated in the inset of Figure 2 of ref 14, and it would be interesting to revisit the study of this experimental system in the light of the present theoretical results. (14) Grossmann, A.; Erley, W.; Hannon, J. B.; Ibach, H. Phys. Rev. Lett. 1996, 77, 127.
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where ks is a constant, r denotes the position of the atoms, and re corresponds to the equilibrium position of the atoms in the absence of other interactions. The interaction of all the atoms in the island with the substrate is given by
Es )
r)R V(r) Fj(2πr) dr ∫r)0
(A2)
Substitution of eq A1 into A2 yields
Es ) πFjks
r)R (r - re)2r dr + NadV0 ∫r)0
(A3)
To determine re, we define as R0 the radius of the islands whose atoms are pseudomorphically adsorbed on the substrate surface. In this case, we have the relation r/re ) R/R0, from which we can get the desired re to be replaced into eq A3 and, thus, obtain Figure 4. Ag/Pt(111) system. Surface stress change ∆σ in J/m2 for different island densities as a function of the coverage degree. The overlap between the islands is neglected. Solid line, 1.5 × 1011 islands/cm2; long dashed line, 1.5 × 1012 islands/cm2; dotteddashed line, 1.5 × 1013 islands/cm2.
5. Outlook The present results can be summarized by stating that the thermodynamic model we have presented allows to obtain the surface stress change for metallic adsorbates on (111) surfaces from Monte Carlo simulations. In the case of islands, a quantity was defined that should be a measure for the average change of the surface stress. In the case of the heteroepitaxial systems with positive lattice misfits considered in this work, the islands are stressed differently depending on their sizes. Larger islands are more stressed than smaller ones because the adatoms are forced to come in registry with the substrate to minimize the total energy. This should lead to a curvature in the plots of the change in the average surface stress versus coverage degree, which should become more evident at high nucleation densities. We have used the EAM to represent the interaction between the particles of the system, but any other type of potential could be used as well. The model can also be used for other, more corrugated, single-crystal surfaces because the higher barriers for adatom motions on the surface are not a hindrance to the simulation scheme proposed. Acknowledgment. This work was supported by PIP and PEI 86/98 CONICET, Program BID 1201/OC-Ar PICT 06-04505, Agencia Co´rdoba Ciencia, SECyT UNC, Argentina. We also wish to thank M. I. Baskes for providing the tables with the potential functions. Language assistance by K. Plasencia is also acknowledged. Appendix A. Simple Model for Predicting the Relationship between Misfit and Island Size In this appendix, we describe a simple model that proved to be useful to analyze and interpret our results on heteroepitaxial systems. Islands made of Nad atoms are considered to have a uniform density Fj and a circular shape of radius R; therefore, Fj ) Nad/(πR2). The atoms in the island are assumed to interact with the substrate via a parabolic potential V(r):
1 V(r) ) ks(r - re)2 + V0 2
(A1)
Es )
Nad k (R - R0)2 + NadV0 4 s
(A4)
We consider now the interaction of the atoms in the island with one another and also in the harmonic approximation. If W(d) denotes the interaction between two neighboring atoms in the island, located at a distance d, we have
1 W(d) ) ka(d - dad)2 + W0 2
(A5)
where ka and W0 are constants and dad denotes the equilibrium distance between the atoms in the isolated island. Because the surface that can be assigned to an adsorbate atom is π(d/2)2, we have the equality
π(d/2)2 )
πR2 Nad
(A6)
so that d ) 2R/(Nad)1/2. If Q0 denotes the radius of the islands at which the adsorbate atoms are in equilibrium in the absence of the substrate, we have
dad )
2Q0
(A7)
xNad
Neglecting border effects, the interaction of the atoms in the island Ei is
Ei )
Nad W(d) 2
(A8)
Thus, the whole interaction energy Etot of the system will be given by
ks Etot ) Nad (R - R0)2 + ka(R - Q0)2 + 4 NadV0 +
Nad 0 W (A9) 2
For a given number of atoms Nad, the energy can be minimized with respect to the island radius R. The minimum condition (dEtot/dR)R)Rmin ) 0 leads to
ka R0 - Rmin ) R0 - Q0 ka + (ksNad/4)
(A10)
where the right-hand side of eq A10 is nothing but the relative misfit r we defined in eq 24, leading to the relationship proposed in eq 25:
Properties of Monolayers and Metal Islands
ksNad 1 )1+ r 4ka
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(A11)
with B ) 1. Appendix B. Model for the Heuristic Fitting of the Change in the Average Surface Stress at Large Island Sizes We shall see that the fitting formulated below requires that the fraction of edge and inner atoms of the islands be expressed in terms of the total number of island atoms Nad. If the shape of the island is assumed to be a circular one, and the atomic density Fj is assumed to be a constant all over the island, the model shown in Figure 5 can be employed. The inner atoms (s) are inscribed in a circle of radius (R - d), leaving the edge atoms (b) in the region [(R - d) < r < R]. The following equalities can be easily shown to be valid:
Fj )
ns nb N ) ) 2 2 πR π(R - d) π(2Rd - d2)
Figure 5. Schematic representation of a circular island of homogeneous atomic density Fj. The circle of radius R - d contains the inner atoms, while the annulus contains the edge atoms between R - d and R.
(B1)
where ns and nb are the number of inner and edge atoms of the island, respectively. The fraction of inner (ns/Nad) and edge (nb/Nad) atoms can be obtained from this equation, taking into account that R ) (Nad)1/2d/2 (see eq A6) to yield
nb 4 4 ) Nad xN Nad ad
(B2a)
nb ns )1Nad Nad
(B2b)
Although the simulated islands are rather hexagonal shaped for the present (111) substrate surfaces, eqs B2a and B2b are a good approximation, as can be seen in Figure 6, where the actual values 〈nb〉/Nad obtained from the simulations are plotted as a function of nb/Nad calculated from eq B2. The straight lines of Figure 6 were fitted according to the equation
(x
〈nb〉 ) A2 Nad
4
Nad
)
4 + B2 Nad
(B3)
The slope A2 and the intercept B2 are shown in Table 5. Although the approach to consider a circular shape to obtain the nb/Nad fraction is good, we used in eq B11 〈nb〉/ Nad values corrected according to the fitting in equation B3. We consider now a heuristic approximation for ∆σ. A first intuitive guess can be obtained from the simple model presented in appendix A. The calculation of stress from this model involves the stretching of the whole system, both the island and the adsorbate. With this purpose, let us consider the derivative of eq A9 at R ) Rmin with respect to the area of the island:
∆σ )
dEtot dEi dR ) dSi dR dSi
where
1 dR ) dSi 2πR
(B4)
Figure 6. Fitting of the fraction of border atoms 〈nb〉/Nad obtained from Monte Carlo simulations as a function of [4/(Nad)1/2] - (4/Na). Table 5. Least-Square Parameters of the Linear Behavior between 〈nb〉/Nad Observed from the Simulation Configurations as a Function of nb/Nad ) [4/(Nad)1/2] (4/Nad) Obtained from Geometric Considerationsa
a
system
A2
B2
Ag/Pt(111) Ag/Pd(111) Au/Pt(111)
0.92 ( 0.02 0.89 ( 0.02 1.01 ( 0.03
0.01 ( 0.01 0.02 ( 0.01 -0.03 ( 0.01
A2 and B2 are the corresponding slope and intersect.
We have to take the derivatives of the four terms on the right-hand side of eq A9. The first term is Nad(ks/4)(R R0)2 and corresponds to the interaction between the adsorbate atoms and the substrate. If we assume that the shape of this parabolic potential is stretched together with the substrate, we can set
d[Nad(ks/4)(R - R0)2] ≈0 dSi
(B5)
In other words, within the present simplified approach, both r and re change in the same way in eq A1, and the interaction Es between the adsorbate atoms and the substrate remains constant.
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The derivative of the second term on the right-hand side of eq A9 is
d[ka(R - Q0)2] ka (R - Q0) ) dSi π R
(B6)
Up to now, the contribution to the stress is given only by the energy change of the interaction of the adsorbate atoms with each other. The derivative of the third term is d(NadV0)/dSi and corresponds to the change of the interaction potential between the adsorbate atoms and the substrate at the equilibrium position when the substrate is expanded. We will approximate to first order the derivative d(NadV0)/dSi by v0′, where v0′ ) dNadV0/dSi|Si)Seq where Seq is the area of the substrate at equilibrium (without deformation). The derivative of the last term is d(NadW0/2)/dSi and is equal to 0. Thus, a simplified approximation for the stress according to the model of appendix A is
∆σ )
ka (Rmin - Q0) + v0′ π Rmin
ka (d - dad) + v0′ π d
system
kas [J/m2]
kab [J/m2]
knh [J/m2]
Ag/Pt(111) Ag/Pd(111) Au/Pt(111)
200.210 ( 0.001 207.934 ( 0.001 142.068 ( 0.001
96.143 ( 0.001 45.817 ( 0.001 10.352 ( 0.001
-2.037 ( 0.001 -1.375 ( 0.001 -1.365 ( 0.001
∆σ )
ns nb ∆σs + ∆σ + knh Nad Nad b
∆σi )
(
)
kai d - dad π d
(B10)
with i ) s, b. In other words, we are allowing atoms located at the edge of the islands to interact with their neighbors with a force constant different from that of the atoms located inside the island. Defining two new variables,
(
)
(
)
x)
ns d - dad πNad d
y)
nb d - dad πNad d
and
(B8)
However, this expression does not take into account that the system contains at least two types of atoms, from the point of view of their coordination, that is, atoms located inside the island (“surface atoms”, labeled s) and atoms located at the edge of the island (“border atoms”, labeled b). According to this, it is reasonable to write the change in the surface stress ∆σ in terms of inner ∆σs and edge ∆σb contributions:
(B9)
where khn is a constant and ∆σi is given by
(B7)
which can be written in terms of the distance between nearest neighbors d using eq A6 to yield
∆σ )
Table 6. Constants Obtained from the Least Squares Fitting of the Surface Stress Change ∆σ as a Function of x and y According to Eq B11
allows eq B9 to be written as
∆σ ) kasx + kaby + knh
(B11)
The parameters kas, kab, and knh were obtained by means of a least-squares fitting and are listed in Table 6. LA036021Z
