Langmuir 1999, 15, 5893-5905
5893
Adsorption-Induced Surface Reconstruction Processes: A Comparison of Different Lattice-Gas Models† F. Nieto,‡,§ A. A. Tarasenko,§,| C. Uebing,§,⊥ and V. Pereyra*,‡ Departamento de Fı´sica and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina, Max-Planck-Institut fu¨ r Eisenforschung, D-40074 Du¨ sseldorf, Germany, Institute of Physics, National Academy of Sciences of Ukraine, and Prospect Nauki 46, 2520288, Kijv, Ukraine, and Lehrstuhl fu¨ r Physikalische Chemie II, Universita¨ t Dortmund, D-44227 Dortmund, Germany Received August 24, 1998. In Final Form: June 3, 1999 In this paper we study three different lattice-gas models for adsorption-induced surface reconstruction in a system like H/W(001). The models are based in the symmetry properties of the local potential for a tungsten surface atom, which present four equilibrium positions where such atom can be located. The two-position, the symmetrical two-position, and the four-position models differ each other in the number and location of the equilibrium positions of both the metal and the adsorption sites. Neglecting the adsorbateadsorbate interactions, an analytical treatment is performed which allows exact solutions for the free energy of the system in each case. We calculate the ground state, and by means of Monte Carlo simulations and finite-size scaling theory, we evaluate the critical temperature as a function of coverage, Tc(θ), when the whole set of the interactions parameters is present. The phase diagrams of the three models are analyzed and compared with the experimental findings.
1. Introduction Surface reconstruction can be a spontaneous or adsorbate-induced phenomenon. The effect of chemisorption can be the extensive restructuring of the substrate forming a completely new surface structure.1-14 Adsorbate-induced reconstruction is usually discussed, in the framework of the lattice-gas model, in terms of phase transition theory. The detailed description of the process depends on whether the phase transition is continuous or first-order type. Systems like H2/Pt(001), CO/Pt(001), or NO/Pt(001)15-17 † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998. * To whom correspondence may be addressed. Phone: 54-2652425109. FAX: 54-2652-430224. Email:
[email protected]. ‡ Universidad Nacional de San Luis. § Max-Planck-Institut fu ¨ r Eisenforchung. | National Academy of Sciences of Ukraine. ⊥ Universita ¨ t Dortmund.
(1) Felter, T. E.; Barker, R. A.; Estrup, P. J. Phys. Rev. Lett. 1977, 38, 1138. (2) Barker, R. A.; Estrup, P. J. J. Chem. Phys. 1981, 74, 1442. (3) Debe, M. K.; King, D. A. J. Phys. C 1977, 10, L303. (4) Barker, R. A.; Estrup, P. J.; Jona, F.; Marcus, P. M. Solid State Commun. 1978, 25, 375. (5) Lau, K. H.; Ying, S. C. Phys. Rev. Lett. 1980, 44, 1222. (6) Roelofs, L. D. Surf. Sci. 1986, 178, 396. (7) Roelofs, L. D.; Wendelken, J. F. Phys. Rev. 1986, B34, 3319. (8) Han, W. K.; Ying, S. C. Phys. Rev. 1990, 41, 9163; 1992, 46, 1849. (9) Singh, D.; Krakauer, H. Phys. Rev. 1988, B37, 3999. (10) Fu, C. L.; Freeman, A. J. Phys. Rev. 1988, B37, 2685. (11) Roelofs, L. D.; Ramseyer, T.; Taylor, L. N.; Singh, D.; Krakauer, H. Phys. Rev. 1989, B40, 9147. (12) Reynolds, A. E.; Kaletta, D.; Erlt, G.; Behm, R. J. Surf. Sci. 1989, 218, 452. (13) Zhdanov, V. P. Elementary Physicochemical Processes on Solid Surfaces. In Fundamental and Applied Catalysis; Twigg, M. V., Spencer, M. S., Eds.; Plenum Press: New York, 1991. (14) Yosimori, A. Prog. Theor. Phys. 1991, 106, 433 and references therein. (15) Estrup, P. J. In Chemistry and Physics on Solid Surfaces V; Vanselow, R., Howe, R., Eds.; Springer: Berlin, 1984; p 126. (16) Ying, S. C. In Chemistry and Physics on Solid Surfaces V; Vanselow, R., Howe, R., Eds.; Springer: Berlin, 1984; p 148.
correspond to the latter case. In fact, adsorption of these species results in (5 × 1) to (1 × 1) reconstruction of the surface. On the other hand, the reconstruction in the H/W(001) system, a very well studied system, is at present assumed to be an order-disorder or continuous phase transition.6-11 The phase diagram of this system has been analyzed by mean of analytical approximations (based on mean-field and transfer matrix ideas)13,18,19 and Monte Carlo simulations.20-24 The models presented in refs 5 and 6 treat the coordinates of a tungsten atoms as continuous variables. The corresponding discrete version is the so-called fourposition (4P) model, which has been used to reproduce part of the phase diagram (at low surface coverages, θ < 1/4).22,24 The simplest version of the 4P model, the twoposition (2P) model, has been succesfully used to analyze the influence of restructuring phenomena on the kinetics of elementary rate process in adsorbed overlayers despite the difference between its symmetry and that of the real system.13,19,21,25-30 Very recently, a new version of the 2P model, the so-called symmetric two position model (2PS), has been also introduced to describe the adsorptioninduced reconstruction process31 in systems such as (17) Behm, R. J.; Thiel, P. A.; Norton, P. R.; Ertl, G. J. Chem. Phys. 1983, 78, 7437. (18) Zhdanov, V. P. Surf. Sci. 1989, 209, 523, L571. (19) Myshlyavtsev, A. V.; Zhdanov, V. P. J. Chem. Phys. 1990, 92, 3909. (20) Bustos, V. PhD Thesis, Universidad Nacional de San Luis, Argentina, 1991. (21) Bustos, V.; Zgrablich, G.; Zhdanov, V. P. Appl. Phys. 1993, A56, 73. (22) Zuppa, C.; Bustos, V.; Zgrablich, G. Phys. Rev. 1995, B51, 2618. (23) Nieto, F.; Pereyra, V. Surf. Sci. 1997, 383, 308. (24) Nieto, F.; Pereyra, V. Surf. Sci. 1998, 399, 96. (25) Zhdanov, V. P. Langmuir 1989, 5, 1044. (26) Zhdanov, V. P. Surf. Sci. Rep. 1991, 12, 183. (27) Zhdanov, V. P. Surf. Sci. 1981, 111, 63. (28) Hellsing, B.; Zhdanov, V. P. Chem. Phys. Lett. 1990, 168. (29) Nieto, F.; Uebing, C.; Pereyra, V. Surf. Sci., in press. (30) Chumak, A. A.; Tarasenko, A. A. Surf. Sci. 1996, 364, 424. (31) Nieto, F. PhD Thesis, Universidad Nacional de San Luis, Argentina, 1998.
10.1021/la981098m CCC: $18.00 © 1999 American Chemical Society Published on Web 08/12/1999
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H/W(001). In this model, as in the classical 2P model, the local potential for a surface atom contains two equilibrium positions, instead of the four proposed by the 4P model. However, the adsorption sites are located in the 2-fold bridge positions between metal atoms as in the 4P models. Therefore, the symmetrical 2P model is a combination of the 2P and 4P models (see Figure 1). Although the 2PS model does not respect the real symmetry of the H/W(001), it is a significant improvement over the well-known 2P model. Despite the well-known limitations shown by the lattice-gas models (they are not able to reproduce the incommensurate region in the experimental phase diagram of H/W(001)), it is interesting to analyze and compare the characteristics of the three versions of the discrete adsorption-induced reconstruction models. The main purpose of the present work is to discuss and to compare the capabilities and scopes of the three lattice-gas models which have been found to be interesting by themselves beyond the origin by which they were developed. In this paper, we present a study of the lattice-gas models for adsorption-induced reconstruction, particularly applicable to the H/W(001) system. Starting from the definition of the 4P model and assuming conditions for both the interaction energies and the symmetry of the lattice, it is possible to recover the 2PS and the 2P models. By means of Monte Carlo simulations and finite-size scaling theory we have obtained the phase diagrams for the three lattice-gas models. The outline of the paper is as follows: in section 2, we present the three lattice- gas models and introduce the order parameters used in the description of the ordered structures present in both the metal and the adsorbate lattices. In section 3, analytical calculations of the phase diagrams for the three models are performed in the particular case when the ad-ad interactions are neglected. The ground state in each model is discussed in section 4. The Monte Carlo procedure and the finite-size scaling techniques are briefly summarized in section 5. The phase diagrams are presented and analyzed in section 6. Finally, we give our conclusions in section 7. 2. The Lattice-Gas Models In the H/W(001) system, the local potential for a surface atom, which can be thought of as arising from underlaying layers in the crystal, contains four equilibrium positions. In Figure 1a) we have plotted the analytical expresion for the surface energy potential, which has been given by Lau and Ying in ref 5. An order-disorder transition can be described in detail based on whether the potential barrier for transition from one equilibrium position to another is higher than or comparable to Tc. Due to the characteristics of the surface energy potential, several properties of the adsorption-induced reconstruction phenomenon can be understood within the framework of the lattice-gas (LG) model. The use of LG models is possible due to the fact that the state of a surface atom can be described by a set of discrete variables. We consider here three simple models of the surface reconstruction, the twoposition (2P), symmetrical two-position (2PS), and fourposition (4P) models of the surface reconstruction, which differ each other in some details. Here we will describe briefly every model. In Figure 1b, we represent a lattice arrangement of particles on the W(001) surface, according to the energetic landscape given in Figure 1a. The crystal surface has square symmetry. Black circles denote tungsten atoms that can be located in one of the four equilibrium positions (empty circles), which correspond to the minimun in the
Nieto et al.
Figure 1. (a) Representation of the local surface energy potential for metal atoms given in ref 5. (b) Arrangement of particles on the metal surface. Each metal atom can be located in four equilibrium positions, corresponding to the local minima in the surface energy potential represented in part a. Such positions are denoted by open circles (black circles denote the metal atoms). Lateral interactions between surface atoms result in the formation of the c(2 × 2) structure for temperatures below the critical one. The diamonds denote the adsorption sites for the adatoms. This representation is the so-called fourposition model. (c) The simplified 2PS model and (d) the twoposition model. Filled diamonds represent occupied adsorption sites. The underlined sites determine one of the two sublattices which are used in the c(2 × 2) order parameter calculation.
surface energy potential given in Figure 1a. The surface potential presents a local minimum between two metal atoms (the so-called “bridge” positions at the middle of the lattice bonds). These are the adsorption sites where a foreign atom can be hosted. The diamonds denote these sites which can be occupied eventually for the hydrogen atoms. A first simplification of the 4P model is the recently presented 2PS model. According to this scheme, every surface atom can be located in any of the two minima of the potential relief placed symmetrically around of the lattice sites. Foreign atoms can be adsorbed in the bridge positions on any bonds of the square lattice as shown in Figure 1c. Note that although the symmetry in the locations of the metal atoms is broken, the corresponding symmetry in the adsorption sites is preserved. A further simplification is decribed in terms of the 2P model, which is schematically outlined in Figure 1d. The only difference between the 2PS and the 2P model is the lack of the symmetry in the adsorption sites. In fact, adatoms can be adsorbed, now, only in the bridge positions on horizontal bonds, as is shown in Figure 1d. Hence, the adsorption sites in the 2P model are reduced by a half compared to the 2PS model. The symmetry in the locations of the equilibrium positions for the metal atoms is exactly as in the 2PS model. The distance between minima is much less than the lattice constant a. At high temperatures, surface atoms are distributed randomly over the minima. As the temperature is lowered, the lateral interactions between surface atoms cause formation of the ordered phase, with a definite symmetry. The symmetry is determined by the signs of the interaction parameters of the surface Hamiltonian. We assume that any metal surface atom interacts only with both its four nearest neighbors (NN) and with the nearest adsorbed atoms on the bonds connecting the surface atom with its NN. The state of any surface atom (for 2P and 2PS models) can be determined by its displacement from the lattice site, normalized to unit si ) 1 if the atom occupies the right minimum and si ) -1
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when it is in the left minimum. According to the definition, the state of the surface is described by a set of the Ising spins {si}, where index i labels the sites of the square lattice. For the 4P model the state of any surface atom is described by the discrete spin variable si ) (six, siy), |si| ) 1, which represents the position vector of a W atom i from the center of the respective square of possible positions. We define the two normalized components of its displacement along the OX direction as six ) (1/21/2 (determined as for the 2P and 2PS models) while in the OY direction, siy ) 1/21/2 if the atom is located on one of the upper minima and siy ) -1/21/2 if the atom occupies one of the lower minima. Therefore the state of the surface is determined completely by two sets of the Ising spins {six} and {siy}, where index i labels the sites as in the previous models. The state of the adparticle subsystem will be described completely by the set of occupation numbers of the bridge sites {nij}. The index ij labels bonds between the ith and the jth surface atoms and
nij )
{
1 if the ijth bond is occupied 0 if the ijth bond is empty
(1)
One can obtain the Hamiltonian of the systems “surface + adatoms” quite easily, expanding the energy of pair interaction between the particles into a series about small displacements of surface atoms. The Hamiltonian H2P for the 2P model has the following form
H2P )
∑[J1sisj + λnij(si - sj) - V0nij] + J2∑sisj +
〈hb〉
〈vb〉
a
∑ nijnkl
(2)
〈ij,kl〉
where V0 is the depth of the adsorption minima, J1,2 is the NN pair interaction energy of the surface atoms in the OX and OY directions, respectively, a is the NN pair interaction energy of the adatoms, λ is the interaction energy between adatoms and surface atoms, and symbols 〈hb〉 and 〈vb〉 mean summation over horizontal and vertical bonds, respectively. Analogously, the Hamiltonian for the 2PS model has the form
H2PS )
∑
〈hb〉
[J1sisj + λnij(si - sj) - V0nij] + (3)
〈ij,kl〉
and the Hamiltonian for the 4P model can be represented as a sum of two similar Hamiltonians for the 2P model
H4P )
∑[J1sixsjx + λnij(six - sjx) - V0nij + J2siysjy] +
〈hb〉
∑[J1siysjy + λnij(siy - sjy) - V0nij + J2sixsjx] +
〈vb〉
a
θσ )
4
∑ni, N i∈σ
θ)
1 4
∑σθσ
∑ nijnkl
(4)
〈ij,kl〉
To analyze the possible ordered states of the system, it is necessary to define the unit cells of both the metal and the adsorbate lattices. The unit cell which allows the identification of different ordered structures in the adsorbed layer is shown in Figure 1b. Let us define four sublattices, al, a2, a3, and a4, to which coverages, θa1, θa2,
(5)
σ ) a1, a2, a3, a4 where N is the number of adsorbate sites. We define order parameters ψ1, ψ2, ψ3, and ψ4 as
ψ1 ) θal + θa2 + θa3 + θa4 ) θ
(6a)
ψ2 ) θal - θa2 + θa3 - θa4
(6b)
ψ3 ) θal + θa2 - θa3 - θa4
(6c)
ψ4 ) -θal + θa2 + θa3 - θa4
(6d)
With the help of these order parameters, all ordered phases occurring in the adsorbed monolayer can be characterized: (i) The phase with 〈ψ2〉 ) 〈ψ3〉 ) 〈ψ4〉 ) 0 has 〈θa1〉 ) 〈θa2〉 ) 〈θa3〉 ) 〈θa4〉 ) θ/4, i.e., there is no preferential occupation of any of the sublattices in Figure 1b. This situation corresponds to the (1 × 1) phase. (ii) In case of (2 × 1) ordered phase we have to consider the following conditions, 〈ψ3〉 ) 〈ψ4〉 ) 0 with 〈ψ2〉 ) (〈ψ1〉 * 0 corresponding to 〈θa1〉 ) 〈θa3〉 * 〈θa2〉 ) 〈θa4〉 and 〈ψ2〉 ) 〈ψ4〉 ) 0 with 〈ψ3〉 ) (〈ψ1〉 * 0 corresponding to 〈θa1〉 ) 〈θa2〉 * 〈θa3〉 ) 〈θa4〉. This results in an alternating diagonal rows of occupation, considering the metal atomic lattice. (iii) In contrast, for the c(2 × 2) phase, the conditions are 〈ψ2〉 ) 〈ψ3〉 ) 0 with 〈ψ4〉 ) (〈ψ1〉 * 0, which means 〈θa3〉 ) 〈θa2〉 * 〈θa1〉 ) 〈θa4〉. For the metal atomic arrangements, the appropriate order parameter related to a given ordered structure is defined according to the position of vector si ) (six, siy). The procedure is the same as that for the adsorbed layer structure discussed above. Let us define, for the metal atoms lattice, four sublattices, b1, b2, b3, and b4, to which the component i (with i ) x, y) of the pseudospin magnetization is defined as
miκ )
4 L
sij, ∑ 2 j∈κ
Mi )
1 4
∑κ miκ
κ ) b1, b2, b3, b4
∑(J2sisj - V0nij) + a ∑ nijnkl
〈vb〉
θa3, and θa4 may be ascribed
(7)
where L2 is the number of surface metal atoms. Let us define in similar way the order parameters, φi1, φi2, φi3, and φi4, as i i i i + mb2 + mb3 + mb4 ) Mi φi1 ) mb1
(8a)
i i i i - mb2 + mb3 - mb4 φi2 ) mb1
(8b)
i i i i φi3 ) mb1 + mb2 - mb3 - mb4
(8c)
i i i i + mb2 + mb3 - mb4 φi4 ) -mb1
(8d)
The definitions of the unit cells and order parameters are rather general and will be used in the analysis of the ordered phase for the three models of reconstruction described above. However, for both the 2P and 2PS models we have to considered only one component of the pseudospin magnetization and the order parameter in the metal lattice, mik and φik with i ) x or y.
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On the other hand, it is important to note that the adsorption sites in the 2PS model are not completely equivalent to the 4P model. The sites located between two metal atoms in the horizontal files are labeled as sites type 1 (sites belonging to sublattices al and a4 in Figure 1b) and the sites located in the vertical rows, as sites type 2 (sites belonging to sublattices a2 and a3 in Figure 1b). As can be expected, the interaction energy between one adsorbed particle and the corresponding nearest metal atoms will be different if the particle is adsorbed in sites 1 or 2.
Q)
∑ exp β(µ∑nij - H),
{ni}
(9)
Q2P ) exp[NA + NFI (I1,I2)]
Λ ) βλ
(17)
k ) (sinh 2I1 sinh 2I2)-1
(18)
1 ln[1 + exp(h + 2Λ)][1 + 4 exp(h - 2Λ)][1 + exp h]-2 (20)
Introducing the free energy of the system F as
F≡
Q4P )
(22)
F2PS ) ln(1 + exp h) + F2P
(23)
for the 2P model
for the 2PS model, and
F4P ) 2F2P
FI(I1,I2) )
1 2π
∫0π ln{2 [cosh 2I1 cosh 2I2 +
{
1 ∂F for 2P and 2PS models N θ ≡ 1 ∂h ∂F for 4P model 2N ∂h
(25)
According to this definition, one can obtain easily the following expressions for the 2P model
θ2P )
(
∂A ∂FI ∂B + sign(βJ1 + B) ∂h ∂I1 ∂h
)
(26)
where
sign(x) )
{
1 if x > 0 -1 if x < 0
(27)
And for the 2PS model
k-1x1 + k2 - 2k cos 2θ]} dθ (13) and other quantities have the following form
h ) β(µ + V0)
(24)
for 4P model. Using the above mentioned expressions one can calculate all thermodynamical quantities. For example, the mean surface density of adatoms, θ, is expressed via the first derivative of the free energy F over the reduced chemical potential, h ) (µ + V0)/kBT
(12)
where FI(I1,I2) is the free energy of the anisotropic Ising model with the pair interaction parameters I1 along the OX direction and I2 along the OY direction
(21)
F2P ) A + FI(I1,I2)
for the 2PS model and for 4P model as
Q22P
kBT ln Q N
one can obtain easily the following expressions for the free energy for the different models of reconstruction:
(10)
(11)
(16)
B)
for the 2P model
Q2PS ) (1 + exp h)NQ2P
I2 ) β|J2|
1 ln[1 + exp(h + 2Λ)][1 + 4 exp(h - 2Λ)][1 + exp h]2 (19)
〈ij〉
where β ≡ 1/kBT, µ is the chemical potential of adatoms, and summation is carried out over all configurations of the system. If one can neglect the pair interaction energy between adatoms (|a| , |J1|, |J2|, |λ|, and/or small or high adatom density), the grand partition function, eq 9, can be reduced to a partition function of an anisotropic Ising model after summation over all adatom occupation numbers nij.30,32,33 The result of the summation for the different models of reconstruction can be written as
(15)
A)
3. Exact Solution for Ea ) 0 The analytical treatment of the three models of reconstruction described in the last section are rather complicated when all the interactions are present. Particularly, the calculation of the critical temperature as a function of coverage, Tc(θ), by using mean-field approximations has been done only for the 2P model.23 However, under certain assumptions about the interaction parameters, it is possible to obtain not only an analytical treatment but the exact solutions for the 2P, 2PS, and 4P models. In this section we will obtain the exact expression for the free energy in the three models by neglecting the adsorbate-adsorbate interactions, a ) 0. The grand partition function Q has the form
I1 ) |βJ1 + B|
(14)
(32) Tarasenko, A. A.; Chumak, A. A. Ukr. Fiz. Zh. 1993, 38, 1741 (in Russian). (33) Tarasenko, A. A.; Chumak, A. A. Ukr. Fiz. Zh. 1995, 40, 995 (in Russian).
θ2PS )
exp h 1 + θ2P 2 1 + exp h
(
)
(28)
The surface density for the 4P model is the same as for the 2PS model and
θ4P ) θ2PS ) 2θ2P
(29)
due to the fact that the number of adsorption sites for the 4P model is two times greater than that for the 2P model.
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Langmuir, Vol. 15, No. 18, 1999 5897
Using obvious expressions:
∂A(h) ∂A(-h) )1∂h ∂h ∂B(h) ∂B(-h) )∂h ∂h
(30)
one can prove easily the validity of the following relation for all three models of surface reconstruction
θ(-h) ) 1 - θ(h)
(31)
3.1. Phase Diagrams. As was already mentioned, at high temperatures surface atoms are randomly distributed over potential minima and the system is in a disordered state. When the temperature is decreased, the interaction between surface atoms causes phase transition into an ordered state. This is the very well known surface reconstruction phenomenon. The critical temperature of the surface reconstruction is determined by the ordinary condition for Ising spin systems34
sinh 2I1 sinh 2I2 ) 1
(32)
The interaction parameter I1 depends on the chemical potential of adatoms, µ. Thus the critical temperature depends on the surface density of adatoms, θ. The dependence is rather complex and differs considerably for different sets of the signs of the interaction parameters I1 and I2. There are four possible combinations of the signs, but as only I1 is dependent on the surface coverage, θ, there are only two different types of dependence of the critical temperature: ferromagnetic (J1 < 0) and antiferromagnetic (J1 > 0). It is possible to solve eq 32 and to obtain the critical values of the chemical potential as a function of the critical temperature and the sign of the reduced interaction parameter βJ1 + B
exp((hc) ) 2 sinh Λ
sinh Λ ( xcosh2 Λ - σ2 -1 σ2 - 1 (33)
where
σ ) exp(-2βJ1) (coth I2)sign(βJ1+B)
(34)
Calculating the chemical potential for a given value of J1, J2, λ, and Tc and substituting them into eq 25, one obtains Tc as a function of adparticle coverage θ: the phase diagram of the system. It should be noted that for the critical value of k ) 1, eq 32, the derivative ∂FI/∂I1 can be calculated easily
∂FI 2 ) cosh 2I2 tan-1(sinh 2I1) ∂I1 π
(35)
As one can see easily from eq 20, the interaction parameter B, B > 0 for all values of the chemical potential h, reaches its maximal value at h ) 0 and decreases to zero as |h| f ∞ and does not depend on the signs of the interaction parameter, λ, and the chemical potential, µ. This can be explained quite easily. For θ < 1/2 - ML, every adatom, adsorbed between surface atoms, tends to increase the antiferromagnetic ordering in the system of surface atoms turning the spins in opposite directions (or occupying the (34) Baxter, R. J. Exactly Solved Models in Statistical Mechanics; Academic Press: London, 1982.
Figure 2. Dependence of the critical temperature vs θ for different values of the ratio λ/J (J ) 1), as indicated, and ) 0. The solid lines represent the exact solution according to section 3 while symbols denote MC calculations: (a) 2P, (b) 2PS, and (c) 4P model.
deeper sites when antiferromagnetic ordering is present). For θ > 1/2 - ML every excess adatom tries to destroy the antiferromagnetic ordering and the effect of the adsorption decreases to zero when surface coverage approaches monolayer density. Therefore for the antiferromagnetic case the picture is rather simple: the critical temperature of the surface reconstruction Tc(θ) is increased with the adatom coverage θ and reaches its maximal value at halfmonolayer coverage. The whole phase diagram is symmetrical around θ ) 1/2 - ML (see Figure 2). A much more complex picture is obtained for the ferromagnetic case. For weak interaction between metal surface atoms and adatom the critical temperature, Tc(θ), decreases with the adatom coverage, θ, and reaches some minimal value at half-monolayer coverage. But if the interaction parameter |λ| exceeds the critical value |λc| ) 2|J1|, the whole dependence of Tc(θ) drastically changes. The most dramatic effect of the adatom-atom interaction, in this case, is a complete destruction of the ordering for all temperatures when adatom density reaches some critical values. In this case, the effective interaction parameter I1 is zero. In order to find the critical density values, let us consider the limit T f 0. Then for the fulfillment of the condition given by eq 32 for the critical temperature, it is necessary that the following relation takes place
lim βJ1 + B ) 0 Tf0
(36)
One can obtain easily from (36) two critical values of the chemical potential
hc(0) ) (hc0 ) (2β(|λ| - 2|J1|)
(37)
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which corresponds to the point of the changing sign of eq 35 due to the condition sign(βJ1 + B). The values of the corresponding derivatives at the critical h can be calculated without any problem
∂A((hc0) 2 ( 1 ) ∂h 4 ∂B((hc0) 1 )∂h 4 ∂FI((hc0) 2 ) ∂I1 π
(38)
As ∂FI((hc0)/∂I1 * 0, every critical value of the chemical potential corresponds to two different values of the critical density. It means that there are four critical points of the surface coverage of adatoms (ending points of the critical lines at T ) 0) and two regions of adparticle density where any ordered structure cannot exist. It is easy to calculate the critical points for the 2P and 4P models
θc2 θc3 θc4
1 2 1 - ≈ 0.091 4 π
( ) 1 2 ) (1 + ) ≈ 0.409 4 π 1 2 ) (3 - ) ≈ 0.591 4 π 1 2 ) (3 + ) ≈ 0.909 4 π
θc1 )
Figure 3. Phase diagrams for the 2PS model and for antiferromagnetic values of the interaction parameter J (J ) -1 in this case). Upon increasing the ad-metal interaction, λ, the system crosses the critical threshold λc defined in section 3 and the corresponding region for the (2 × 1) structure appears (a). (b-f) Upon further increasing of λ it is possible to see how this region grows. The region where the ferromagnetic ordered structure is present has been labeled in the figure as (1 × 1). Further increase of the coupling constant λ leads to sticking of the upper branches and to their splitting in the perpendicular direction. The area without any special label denotes the region where the system is disordered.
(39)
and slightly different values for 2PS model
θc2 θc3 θc4
1 2 1 - ≈ 0.045 8 π
( ) 1 2 ) (1 + ) ≈ 0.205 8 π 1 2 ) (7 - ) ≈ 0.795 8 π 2 1 ) (7 + ) ≈ 0.955 8 π
θc1 )
(40)
The values are exact and do not depend on the interaction parameters J1, J2, and λ (it is necessary, of course, that |λ| > |λc|). In the regions θc1 < θ < θc2 and θc3 < θ < θc4 the surface is disordered even at T ) 0. In the region θc2 < θ < θc3 the ordered structure corresponds to the antiferromagnetic case (if the temperature is less than the critical value). The whole phase diagram is also symmetrical around θ ) 1/2 - ML as in the previous case of antiferromagnetic ordering and has rather complex behavior as the interaction parameter |λ| increases from zero to the direct vicinity of |λc| as shown in Figure 3. For |λ| < |λc| ) 2|J1| there is only one branch of the phase diagram Tc(θ). At T > Tc(θ) surface is disordered. If the temperature is lowered below Tc(θ), the surface became ferromagnetically ordered. If |λ| ) |λc| another branch of the phase diagram appears as two line segments overlapping θ-axis (T ) 0) and extending from θ1 to θ4 and from θ2 to θ3. As adatom-atom interaction is increased, the lines are transformed into convex upward symmetrical branches confining the region of the existence of a
Figure 4. Ground-state map for the 2P model with 1 and λ as variables. This calculations correspond to half coverage (θ ) 0.5), T ) 0 and fixed J > 0. Three regions are clearly delimited, where we can observe the following situation: c(2 × 2) order structure is found in region I; in region II the (2 × 1) order is observed as the most stable phase and in region III the system is disconnected into two independent subsystems (as is explained in the text). In the upper insets we plot the arrangement of the system (metal + adsorbate atoms) in the c(2 × 2) ordered phase. The lower insets correspond to the (2 × 1) ordered structure. The symbols are as in Figure 1.
disordered phase. The region separates ordered phases with different symmetry. The inner curve confines the region of existence of the antiferromagnetic ordered structure. Further increase of the interaction leads to sticking of the upper branches in a common intersection point and to their splitting in the perpendicular direction (see Figure 3). 4. The Ground State In this section we will discuss the effect of different interaction parameters on the ground state of the models. The ground state is presented here as a map of the interaction parameters for fixed values of θ ) 0.5 and T ) 0. 4.1. The 2P Model. In Figure 4, a map of the interaction constants and λ is plotted for fixed J > 0, θ ) 0.5, and T ) 0. In particular, for the 2P and 2PS models we have considered only nearest-neighbor ad-ad interactions, a
Adsorption-Induced Surface Reconstruction Processes
Figure 5. Ground-state map for the 2PS model with and λ as variables. These calculations correspond to half coverage (θ ) 0.5), T ) 0 and fixed J > 0. Three regions are clearly delimited, where we can observe the following situation: (i) in region I both metal atoms and adsorbed particles are forming a c(2 × 2) phase, but, in the adsorbate lattice, the c(2 × 2) structure can be formed only either in sites 1 or in sites 2; (ii) in region II, the metal atoms form a c(2 × 2) phase and half of the adsorbed particles occupy alternatively the adsorption sites of type 1, stabilizing the metal structure (the particles are forming a c(2 × 2) phase on the adsorption sites of type 1). The rest of the adsorption particle can be hosted with equal probability in any adsorption site of type 2 and (iii) region III where the adsorbed particles are forming a (1 × 1) phase and the metal atoms a c(2 × 2) structure.
) 1 ) . The argument about the sign of λ has been discussed in the last sections, so the map is symmetrical around the vertical axis. We can observe that for values of g 0 the system (both metal and adsorbate) is in c(2 × 2) ordered phase and remains in this state for any value of λ (see the upper insets in Figure 4). The energy of the system is given by -2J - |λ|. Any positive increment of the parameters or |λ| leads to a reinforcement of the c(2 × 2) structure. The region I in Figure 4 determines the parameters values corresponding to a c(2 × 2) structure. For < 0 the ground-state presents a very rich behavior due to the competitive interaction effects between the parameters; the coupling constants λ and J force the system to an ordered phase while tries to condense the adsorbate into a (1 × 1)-like ferromagnetic phase. This state corresponds to a disconnection between the metal and adsorbate lattices. In other words, the adsorbate particles do not affect the metal structure (region III in Figure 4). For 0 > > -4J two different configurations appear depending on the values of λ: for λ > - the c(2 × 2) ordered phase is present (region I) and for values smaller than - both lattices, the metal and adsorbate, remain disconnected and the value of the energy is the sum of the energies of both lattices (region III). For < -4J the ordered phase corresponding to a (2 × 1) structure (see the lower insets in Figure 4) appears for λ > -/2 (region II). The corresponding energy for this situation is given by -|λ| + /2. The uncoupled region is determined by the values of λ < -/2. The results that we have shown in the present analysis are obtained for very large size of the lattice. For smaller size, we observe strong finite size effects on the energy of the system. 4.2. The 2PS Model. In Figure 5, we have plotted the map of the interaction constants and λ for fixed J ) 1, θ ) 0.5, and T ) 0. From the ground state map we can distinguish three regions separated by solid lines.
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Figure 6. Ground-state map for the 4P model with 1 and 2 as variables. Three regions are clearly delimited, where we can observe the following situation: (i) region I, characterized by metal atoms presenting c(2 × 2) ordered phase, while adsorbed atoms are in (2 × 1) ordered structure (see upper inset), the region determines where the adsorption-induced reconstruction phenomena will be possible; (ii) region II, where metal and adsorbed atoms are both in a c(2 × 2) phase (see lower inset), the phase diagram of the metal structure does not depend on the adsorption process, but the critical behavior of adsorbed layer is strongly affected by the ordered substrate phase; (iii) region III, where metal atoms present a c(2 × 2) phase and the adsorbed phase is forming (1 × 1) island, the critical behavior of both metal and adsorbed atoms do not affect each others.
(i) Region I, characterized by
> 2|λ|
(41)
where both metal atoms and adsorbed particles are forming a c(2 × 2) phase. However, from the energetic point of view, it is not possible to distinguish between the c(2 × 2) structure formed by the adsorbate in sites 1, where the adsorbate-substrate interactions cancel each other, from the c(2 × 2) phase formed by the adsorbate in sites 2, where the adsorbed particles do not interact with the metal atoms (remember that θ ) 0.5). (ii) Region II, given by the following condition:
2|λ| > > -|λ|
(42)
Here, the metal atoms are forming a c(2 × 2) phase and half of the adsorbed particles occupy alternatively the adsorption sites of type 1, stabilizing the metal structure (the particles are forming a c(2 × 2) phase on the adsorption sites of type 1). The rest of the adsorption particles can be hosted with equal probability in any adsorption site of type 2. From the energy balance, we can see that if < 0 (attractive interaction) or > 0 (repulsive interaction), the sites of type 2 are occupied at random. Therefore the order in the metal structure is determined by the occupation of the sites of type 1. However there is no reason to have a given order in the adsorbed overlayer. (iii) Finally, region III, given by
< -|λ|
(43)
where the adsorbed particles are forming a (1 × 1) phase and the metal atoms a c(2 × 2) phase. The critical temperature of the metal structure is not modified by the presence of the adsorbed molecules. 4.3. The 4P Model. In Figure 6, we have shown the ground-state diagram, at finite coverage θ ) 0.5 and T ) 0, plotted as a function of the interactions constants 1 and 2 (nearest and next nearest neighbors, respectively). In the map we can distinguish three well-defined regions, corresponding to the ordered phases shown in
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the insets of Figure 6. The region I is determined by the following relations
{
λ 1 for > 0 + 1 1 2 2 2 g λ 1 - - 1 for 1 < 0 2 2 -
(44)
The ordered structure in region I is characterized by the fact that metal atom arrangement is a c(2 × 2) phase and adsorbed layer is a (2 × 1) phase. The resulting phase is a strongly anisotropic structure of adsorbed atoms that forms diagonal rows with respect to the metal atoms, as we can see in Figure 6, where both λ > 0 and λ < 0 structures are shown (denoted by empty and filled rows, respectively), presenting a total internal energy Eint ) -2J - 1 - 2λ1 and Eint ) -2J - 1 - 2λ2, respectively. The region II is delimited by the relation
2 e -
λ 1 + 2 2 1
(45)
Here, both metal atoms arrangement and adsorbed layers are in a c(2 × 2) phase. The structure is also an anisotropic array of adsorbed atoms, but forming parallel rows with respect of the metal atoms, as we can see in the inset of Figure 6. The total internal energy is Eint ) -2J - 2 λ1 - λ2. Finally, region III is characterized by the following condition:
λ 1 λ 1 - - 1 g 2 g - + 1 2 2 2 2
used in order to discard any metastable state. The number of MCS for averaging was always taken equal to the one needed for equilibration. Averages have been taken over ≈3 × 102 different initial configurations. These simulations were carried out at the Parix parallel computer (operated by the University of San Luis, Argentina) and Intel Paragon (XPS 10, operated by the Forschungszentrum Ju¨lich, Germany). Since several parameters need to be varied (J, , λ, θ, and T), we could not simulate large systems. However, the results are accurate enough to obtain a good quality finite-size analysis. The first step in the analysis of the critical quantities is the definition of the order parameters related to the possible ordered phase in the surface. Such a phase can be observed either in the metal atoms or, as is usual, in the adsorbed overlayer. Once we define the adequate order parameter φ which represents some ordered structure in the surface, we can define different quantities related to the order parameter such as the susceptibility
χ ) N2[〈φ2〉T - 〈|φ|〉 2T]/T
(47)
and the reduced fourth-order cumulants introduced by Binder35
UL(T) ) 1 -
[〈(φ - 〈φ〉T)4〉T] 3[〈(φ - 〈φ〉T)2〉T]2
(48)
The specific heat is sampled from energy fluctuations
(46)
for 1 < 0. The metal atom arrangement corresponds to a c(2 × 2) phase and the adsorbed layer is in (1 × 1) structure. 5. Numerical Procedure In the next section, (a) the phase diagrams calculated in section 3 will be compared with those obtained by means of Monte Carlo simulations where the critical temperature is determined by finite-size analysis and (b) we will use these techniques in order to determine the phase diagrams for the general case when all the interactions are present. We will summarize the simulational details in the present section. The system represented by the Hamiltonian equation (2-4) is simulated by two (mutually interacting) square sublattices with periodic boundary conditions and L × L ) N metal sites (linear dimensions L ) 16, 20, 24, 32, 40, 48). The numbers of adsorption sites depend on the models (2N for the 2PS and 4P models and N for the 2P model). For a given value of coverage θ and temperature T, a starting configuration is generated which is usually an antiferromagnetic ordered phase for the metal spins and a random one for the adsorbed particles. Successive configurations are then generated by the usual MC thermalization procedure in the canonical ensemble using Metropolis transition probabilities. A MCS (Monte Carlo step per site) is achieved when every site of the two sublattices has been tested once for a transition (for the adsorbate sublattice a transition means the exchange of an occupation state between occupied and empty). The time needed for equilibration varies greatly with the system size, temperature, and coverage. Typically, the smaller lattices, with zero coverage, were equilibrated with runs taking 5 × 104 MCS for each configuration, while ≈106 MCS were used for the larger one, with θ ) 0.5 and low temperatures, T < Tc. The large number of MCS was
Cv ) L2[〈H2〉T - 〈H〉 2T]/T2
(49)
where the thermal average 〈...〉T, in all the quantities, means the time average of the Monte Carlo simulation. Now the standard theory of finite size scaling36-40 implies the following behavior of the above quantities near the critical temperature Tc (t ≡ 1 - T/Tc),
˜ (L1/νt) Cv ) LR/νC
(50)
for L f ∞, t f 0, such that L1/νt ) finite
φ ) L-β/νφ˜ (L1/νt)
(51)
χT ) Lγ/νχ˜ (L1/νt)
(52)
˜ (L1/νt) UL(T) ) U
(53)
Here R, β, γ, and ν are the standard critical exponents of the specific heat (Cv ∝ |t|-R for t f 0, L f ∞), order parameter (φ ∝ tβ for t > 0+, L f ∞), susceptibility (χT ∝ |t|γ for t f 0, L f ∞) and correlation length ξ (ξ ∝ |t|-ν for t f 0, L f ∞), respectively. C ˜ , φ˜ , χ˜ , and U ˜ are scaling functions for the respective quantities. As is well known,37-44 eqs 50, 51, 52, and 53 allow various efficient (35) Binder, K. Z. Phys. Z. Phys. B 1981, 43, 119. (36) Fisher, M. E. In Critical Phenomena; Green, M. S., Eds.; Academic Press: London, 1971; p 1. (37) Barber, M. N. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1983; Vol. 8, p 146. (38) Privman, V. FiniteSize Scaling and Numerical Simulation of Statistical Systems; World Scientific: Singapore, 1990. (39) Binder, K. In Computational Methods in Field Theory; Gausterer, H., Lang, C. B., Eds.; Springer: Berlin, 1992; p 59. (40) Binder, K.; Heermann, D. W. Monte Carlo Simulation in Statistical Physics. An Introduction; Springer: Berlin, 1988. (41) Ferrenberg, A. M.; Landau, D. P. Phys. Rev. 1991, B44, 5081. (42) Deutsch, H.-P.; Binder, K. J. Phys. (Paris) II 1993, 3, 1049.
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routes to extract estimates for both Tc and the critical exponents from Monte Carlo data, one of this route is the extrapolation of the positions Kc(L) of the maxima of the slopes of φ, (dφ/dK)max, where K ) J/T and UL(T), (dUL(T)/dK)max, as well as of the susceptibility maxima, χmax T , respectively. For all these quantities one expects, from eqs 50, 51, 52 and 53, that
Kc(L) ) Kc(∞) + const/L1/ν Lf∞
(54)
where Kc(∞) ) J/Tc and the constant depends on the quantity considered. The other well-known method is the intersection of the reduced fourth-order cumulants. 6. Phase Diagrams at Finite Temperature In the following we will analyze the phase diagrams at finite temperature for the 2P, 2PS, and 4P models in the case when all the interactions are present (including adsorbate-adsorbate interactions a * 0). All the calculations have been done, considering the c(2 × 2) phase as the most stable metal structure at low temperature (antiferromagnetic order, J > 0). 6.1. The 2P Model. The calculations of the critical temperature as a function of the coverage in the 2P model can be done by using mean-field approximations. The results are discussed in ref 23, and here we present the calculations of the phase diagrams using Monte Carlo simulations. In Figure 2a we show the phase diagram obtained for J > 0 and ) 0 and different values of the coupling parameter λ. The ordered phase presents a c(2 × 2) superstructure. This set of values has been used in ref 13 to reproduce the experimental phase diagram of H/W(001) at low coverage, θ < 0.25. The solid lines in Figure 2a represent the exact solution according to section 3 while symbols denote MC calculations. As can be seen, there is excellent agreement between these two very different techniques. The adsorbate-adsorbate interaction increases the critical temperature for positive values (repulsive interaction), with a maximum at θ ) 0.5. The physical reason for this effect is that the substrate at T < Tc spontaneously forms two types of sites for adsorption (inside and outside zigzag rows, see Figure 1d). H atoms may choose the sites which are energetically more favorable. The ordering of H atoms results in an increase of the configurational free energy. However, this increase is completely compensated by the decrease of the potential energy. Thus, adsorption on a bridge site always stabilizes the c(2 × 2) structure. As mentioned above, the sign of the adsorbate-substrate interaction does not affect the phase diagram. The real H-W lateral interaction is attractive and accordingly H atoms prefer to occupy the sites located inside zigzag rows or to be located inside pairs of W atoms.45 We can observe, in Figure 7a, the phase diagram for both negative values of within region I (attractive interaction, > -4J) and zero ad-ad interaction ) 0, respectively. At very low (high) coverage Tc(θ)/Tc(0) > 1, with a small maximum in θ ) 0.1 (0.9). For coverages between 0.2 < θ < 0.8 the c(2 × 2) structure is established for the range of temperatures given by: Tc(θ) < Tc(0), with a minimun in θ ) 0.5. Note that due to the pairwise interaction, the phase diagram is symmetrical around θ (43) Binder, K. In Monte Carlo Methods in Statistical Physics; Binder, K., Ed.; Springer: Berlin, 1979. (44) Binder, K. Phys. Rev. 1984, B29, 5184 and reference therein. (45) Zhdanov, V. P. Langmuir 1996, 12, 101.
Figure 7. (a) Dependence of the critical temperature vs θ for the parameters values corresponding to the region I in Figure 4; for a given value of the relation λ/J ) 4 and for different values of the adsorbate-adsorbate interaction energy as indicated. (b) Phase diagram calculated by MC simulations for the system described by the parameters given in section 6.1. Filled circles (squares) denote the critical temperature corresponding to the (2 × 1) (c(2 × 2)) ordered phase. The solid line separates the disorder from the region where one or both ordered structures are present.
) 0.5. The explanation of this behavior is as follows: at low coverages, the particles do not interact with each other and the critical temperature increases with coverage due to both the metal-adsorbate interactions, λ, and the metal-metal interaction, J; as soon as the coverage increases, the adsorbate-adsorbate interaction forces to the adsorbed particles in a (1 × 1) structure and, as a consequence, weaken the metal c(2 × 2) order. The result is a minimun at θ ) 0.5. Depending on the value of λ, for fixed , the system will be either disconnected (region III) or in region II, according to Figure 4. A more interesting behavior appears for those values of λ and corresponding to the region II, in Figure 4. In Figure 7b we can observe the correponding phase diagram for the following values for the interaction parameters, J ) 1, ) -7, and λ ) 6. As has been noted, at T ) 0, J > 0, and θ ) 0.5, the corresponding structure is the (2 × 1) ordered phase. As the temperature increases the system undergoes a order-disorder phase transitions. Note that at θ ) 0.5, the metal atoms are ordered in a pure (2 × 1) phase. For θ ) 0, the free energy has only the contribution of metal-metal interaction which results in a c(2 × 2) ordered phase. Due to the fact that pairwise interaction results in a symmetrical phase diagram around θ ) 0.5 we conclude that Tc(θ)0) ) Tc(θ)1). The same conclusion can be obtained by the following arguments: the free energy in the c(2 × 2) structure is lower than the corresponding one
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Figure 8. Snapshot of typical equilibrium configuration obtained by MC for the system described in Figure 7b: (a) T ≈ 0, (b) T′ < T < T′′, and (c) T > T′′. The lattice size used here is L ) 16, the coverage θ ) 0.2, and the symbols T′ and T′′ correspond to those given in Figure 7b. For a better comprehension the metal atoms (right hand) and adsorbate molecules (left hand) arrangements are plotted separately. In the adsorbate arrangement, an empty square denotes empty sites and a filled square denotes occupied sites, while in the metal atoms arrangement empty square denotes spin up and filled square denotes spin down.
for the (2 × 1) ordered phase. In fact, at full coverage, the metal-metal interactions cancel each other because of symmetry of the (2 × 1) arrangement. For 0 < θ < 0.5 and T ) 0, the arrangement of the metal atoms presents a coexistence of both structures. The (2 × 1) phase is determined by the adsorbed atoms. In fact, in the presence of adatoms, the system (metal + adsorbate) minimizes the local free energy in such a way that the favorable order is the (2 × 1) structure. The rest of the arrangement (those metal atoms which are not interacting with the adsorbed atoms) presents a c(2 × 2) ordered structure, because only the metal-metal interactions contribute to the internal energy. From Figure 7b, we can observe that the critical temperatures for both structures are different. For the c(2 × 2) phase, Tc(θ) presents a maximum value for θ ) 0 and decreases as θ approaches to half coverage, while for the (2 × 1) structure the behavior of the critical temperature is the opposite. This fact is reflected in the phase diagram shown in Figure 7b, where the filled circles (squares) denote the critical temperature corresponding to the (2 × 1) (c(2 × 2)) ordered phase. The solid line separates the disorder from the region where one or both ordered structures are present. In order to understand such a phase diagram, let us consider the behavior of the system at a given coverage, e.g., θ ) 0.2, Figure 8. At T ) 0, the fraction of the system occupied by the adsorbate is in a (2 × 1) structure, while the rest of the metal arrangement is in a c(2 × 2) phase (see Figure 8a). As the temperature increases, the (2 × 1) phase is
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Figure 9. Phase diagrams corresponding to the 2PS model for a given value of the relation λ/J ) 4 and different ones for the (a) repulsive and (b) attractive ad-ad interaction as indicated.
destroyed by migration of the adsorbed molecules (T > T′, in Figure 8b). The number of metal atoms in the c(2 × 2) structure is increased due to the fact that most of the adsorbate molecules are located in c(2 × 2) sites (see Figure 8b). For higher temperature (T > T′′) the system is completly disordered (see Figure 9c). Finally, the analysis of the phase diagram for 0.5 < θ < 1.0 can be performed by arguments of symmetry. 6.2. The 2PS Model. In Figure 2b, we have shown the phase diagram obtained for J ) 1, ) 0, and different values of the metal-adsorbate interaction parameter, λ, where the solid lines represent the exact solutions according to section 3 while symbols denote MC calculations. As we can observe, the phase diagram is symmetric around θ ) 0.5. In fact, for θ ) 0, 1 the free energy has only the contribution of the metal-metal interactions which results in a c(2 × 2) ordered phase (at full coverage the contribution of the metal-adsorbate interactions cancel). The phase diagrams present a maximum at θ ) 0.5, which is a consequence of the fact that at half coverage the adsorption sites of type 1 are alternatively occupied by adparticles, stabilizing the c(2 × 2) phase in the metal atoms, while the adsorption sites of type 2 are randomly occupied. The value of the maximum increases with λ. To analyze the influence of the adsorbate-adsorbate (ad-ad) interactions on the critical temperature, in Figure 9 we have plotted the phase diagrams, for λ/J ) 4, and different values of repulsive ( > 0) and attractive ( < 0) ad-ad interactions. First, we consider the repulsive case (Figure 9a). At low coverages and for different values of , the curves do not differ significatively from each other, while at half coverage they are strongly dependent on . In fact, for < 2 they have a maximum at θ ) 0.5, while for higher value of the ad-ad interactions they present a minimum that is deeper as increases. Finally, at θ )
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Figure 10. Comparison between MC data obtained in the framework of the 2PS model with the interaction parameters given in section 6 (empty circles) and the experimental phase diagram reported in ref 2 (filled squares).
0.25, 0.75 some curves present a maximum. The explanation is as follow. Let us first consider the behavior at half coverage. For 0 g < 2|λ|, half of the adsorbed particles are forming a c(2 × 2) phase on the adsorption sites of type 1, and the rest are randomly adsorbed on the sites of type 2 (region II in the map of Figure 5). However, as the parameter increases, the energy to destroy the c(2 × 2) phase is lowered and, consequently, the value of Tc(0.5) decreases. However, Tc(0.5) is still greater than Tc(0). For, > 2|λ| (region I in the map of Figure 5) the adsorbed layers are forming c(2 × 2) phases on the sites of type 2 or 1. In both cases, the interactions between adsorbed particles and metal atoms are null on average, so the c(2 × 2) phase is a consequence of the metal-metal interactions and, therefore, Tc(0.5) ) Tc(0). For low coverage, the particles are adsorbed far away from each other (at least on average) and the ad-ad interactions are not relevant. Hence the curves are not sensitive to this parameter. As the coverage increases (near θ ) 0.25), the particles are adsorbed in a c(2 × 2) structure on sites of type 1. The stronger the ad-ad interaction, the higher become the value of the critical temperature. The influence of attractive ad-ad interaction ( < 0) is shown in Figure 9b. The curves are quite similar to the case ) 0. In order to explain the behavior, let us consider, as in the latter analysis, the system at half coverage. For > -|λ| the situation is similar to the repulsive case. Actually, the interaction parameters belong to region II. The critical temperature decreases upon increasing the attractive ad-ad interactions due to the fact that the adsorbed particles try to condense the adsorbate in a (1 × 1) structure, which competes with the c(2 × 2) phase of the metal arrangement. This effect is also present at low coverage, so the curves are sensitive to the variation of the parameter in all the range of coverage. Finally, for < -|λ| both lattices are disconnected and work independently of each other. Finally, in Figure 10 we have plotted the experimental results for the critical temperature in the H/W(001) system (filled squares) and the corresponding values obtained by using the 2PS model with λ/J ) 5.7 and /J ) 7.12 (empty circles). It is very important to emphazise that the experimental phase diagram shows a region where the system presents an incommensurate phase. We are not able to reproduce such incommensurate phase transition in the framework of the lattice-gas scheme (it is necessary to use a continuous model for such a purpose). However, the 2PS model can reproduce the shape of the experimental curve including the maximum at θ ≈ 1/4.
Figure 11. (a) Phase diagram calculated by MC simulations for the 4P model described by the following parameters: λ ) -4, 1 ) 0 and for different values of 2 (the variation is along the 2-axis). The interaction parameters correspond to region I in Figure 6. The inset shows in detail the agreement between experimental data (filled symbols) from ref 2 and the MC simulations (empty symbols) corresponding to: λ ) -4 kcal/ mol, 1 ) 2 kcal/mol, 2 ) 1 kcal/mol, and J ) 0.22 kcal/mol. (b) Phase diagram for the system with interaction parameters corresponding to region II in Figure 6. The surface metal atoms arrangement presents a critical temperature which is not modified by the presence of the adsorbate (horizontal full line). The symbols correspond to phase diagrams for the adsorbed atoms with the following interactions parameters 1 ) 7 and 2 ) 0, λ ) 0 (empty squares), and λ ) -4 (filled squares).
6.3. The 4P Model. In Figure 11a, we have shown the effect of adsorbate-adsorbate interaction on the phase diagram for λ ) -4, 1 ) 0, and different values of 2 (the variation is along the 2-axis), corresponding to region I in Figure 6. For these values of the interaction parameters, the system presents the same critical behavior and undergoes second-order phase transitions from order to disorder for both the metal surface atoms and the adsorbed layer. However, the ordered phases are different: while the surface metal arrangement is in a c(2 × 2) phase, the adsorbed layer is in a (2 × 1) ordered structure. An interesting aspect of the phase diagram is the Gaussian shape of the curves, which is very close to the experimental observation in the phase diagram of H/W(001), at low coverage of H (see the inset in Figure 11). It is important to emphasize that the 4P model predicts adsorptioninduced surface reconstruction only if the adsorbed layer presents a (2 × 1) ordered phase, which corresponds to those parameters belonging to region I in Figure 6. A very different situation is observed for a system characterized by those interaction parameters corresponding to region II. In Figure 11b, we can observe the phase diagrams for both surface metal atoms and the adsorbed layer. The principal feature of the critical regime
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Figure 12. ∆θ (defined in Figure 11) as a function of the interaction parameter λ for the 4P model with interaction parameters corresponding to region II of the map shown in Figure 6.
under this condition is that the surface metal atom arrangement presents a critical temperature which does not depend on adsorbate coverage (Tc(θ) ) Tc(0) ) 2.269J) (full line in Figure 11b), while the critical temperature of the adsorbed layer is strongly dependent on the presence of the surface metal ordered phase. In fact, in the figure we can see the phase diagrams for the adsorbed atoms with interactions parameters given by 1 ) 7 and 2 ) 0. In the case of λ ) λ1 ) λ2 ) 0 we obtain the well-known c(2 × 2) phase diagram, which presents a shape that characterizes the 2D square lattice-gas with nearestneighbor repulsion. This phase diagram is well understood and has been obtained in several numerical and analytical studies.43 For values of λ, such as λ * 0, the situation is completely different. In fact, in the figure we can observe that for λ ) -4 (filled squares) the critical temperature at θ ) 0.5 is lower than for the clean surface. On the other hand, for T < Tc(θ), the metal atom arrangement presents a c(2 × 2) phase and plays the role of an external field which is not randomly but strongly ordered. The effect of the external field is an enhancement of the c(2 × 2) ordered region in the phase diagram as is shown in Figure 11b. At temperature T ) 0, the critical coverages θc for a c(2 × 2) phase (empty squares in Figure 11b) are very well known for a lattice-gas with nearest-neighbor repulsive interactions:43 θ1c ) 0.35 and θ2c ) 0.65. In Figure 12, we have plotted ∆θ ) θ2c - θ1c as a function of the interaction parameter λ. The value of ∆θ increases with |λ|. However, note that the maximum values of ∆θ must be restricted by the percolation threshold of the c(2 × 2) structure. Note that adsorption does not induce surface reconstruction even when the adsorbed layer presents a completely ordered c(2 × 2) phase. This result, which is completely unexpected differs from those obtained for the 2P model,23 which is the simplest version of the 4P model. 7. Summary and Conclusions In this paper, we present a detailed analysis of the lattice-gas models used to study the adsorption-induced reconstruction in a system like H/W(001). The models are based in the symmetry properties of the local potential for a surface atoms, which present four equilibrium positions for a given surface metal atom. The first latticegas approximation is the 2P model, which considers two equilibrium positions for the metal atoms preserving only the adsorption sites that interacts in the horizontal bonds with the metal atoms. The second approximation, is the so-called 2PS model, which has only two equilibrium
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positions for the metal atoms, but the particles can be adsorbed in the bridge of any bonds of the square lattice. Finally, we study the 4P model which considers the whole characteristics of the potential predicted for the metal surface. The influence of the interaction parameters on both the ground state and the phase diagrams was analyzed by analytical and numerical methods. An exact solution for the free energy has been obtained for the three models. This treatment has been possible neglecting the adsorbate-adsorbate interactions, ) 0. These results allow an accurate analysis of the effect of the adatom-atom, λ, and the metal-metal J interactions on the phase diagrams. The behavior of the critical temperature as a function of surface coverage is mainly determined by the coupling between adsorbed and metal atoms. It has been shown that depending on the ordering of the surface atoms there are two different kinds of dependences of Tc(θ) upon adsorption. The presence of adsorbed particles stimulates surface reconstruction when antiferromagnetic interactions, J > 0, are considered, increasing the critical temperature of the reconstruction. On the other hand, incoming adatoms inhibit surface reconstruction for ferromagnetic interactions, J < 0. In the latter case, the ferromagnetic ordering can even be destroyed by adsorption and a new type of ordering can be created if the interaction between surface metal atoms J is sufficiently strong. In this case the phase diagrams are rather complex. The study of the influence of the adsorbate-adsorbate, * 0, on the phase diagrams has been done by using both Monte Carlo simulations and finite-size scaling theory. From the analysis of the critical exponent we conclude that the three models belong to the Ising universality class. The analysis of the ground states for the three models has been done considering that the most stable structure for the metal atoms is the c(2 × 2) phase (J > 0). The three models present different ground state maps depending on the interaction parameters. However, all of them show a common feature: the system remains disconnected for a set of interaction parameters (denoted as region III). In this case, the critical temperature of the metal structure is not affected by the adsorption process. The ground state for the 4P model reveals that the adsorption-induced reconstruction process is only possible when the adsorbed particles are forming a (2 × 1) phase. The c(2 × 2) ordering in the adsorbed overlayer does not affect the critical temperature of the metal atoms. However the phase diagram of the adsorbate is strongly modified by the metal ordered structure. The ground state in the 2PS model is also very interesting. We can distinguish two regions in which the metal atoms form a c(2 × 2) phase. However, the adsorbed overlayer shows also a c(2 × 2) structure only for those values of the interaction parameters which belong to region I on the ground map. On the contrary, in region II the order in the adsorbate structure is determined by the occupation of sites labeled as 1. As a consequence the whole adsorbed layer can be disordered. The map of the 2P model presents also three regions: (a) region I where both metal and adsorbed particles are forming a c(2 × 2) phase, (b) region II where the particles are adsorbed forming a (2 × 1) phase, and finally (c) region III where both subsystems are disconnected. The analysis of the phase diagrams of the 4P model reveals that adsorption-induced reconstruction is verified for a very reduced set of interaction energy parameters. In fact, only for those interaction parameters correspond-
Adsorption-Induced Surface Reconstruction Processes
ing to region I does the critical temperature for metal atoms depend on the adsorbed coverage. In this case, metal atoms present a c(2 × 2) phase and the adsorbed layer shows a (2 × 1) phase with the same critical behavior. For those parameters corresponding to region II, the critical behavior of the metal atoms is not affected by the presence of the adsorbed particles and the surface reconstruction is only induced by temperature. In both lattices, we can observe a c(2 × 2) order at low temperature, T < Tc. One of the peculiarities of the 4P model is the critical behavior of the adsorbed molecules which is strongly affected by the ordered metal phase. The phase diagrams for the 2PS model present, as a main difference with the other models, a very particular dependence on the ad-ad interaction . At half coverage and for a fixed value of λ, the critical temperature exhibits a maximum for ) 0. Attractive or repulsive ad-ad interactions result in a lower critical temperature. The curves of Tc(θ)/Tc(0) vs θ are not very much influenced by repulsive ad-ad interactions for both low and large coverages, θ < 0.15 (θ > 0.85). Upon increasing the values of the repulsive interaction energies the phase diagram presents peaks at θ ) 0.25 and θ ) 0.75. The critical temperature decreases in the whole range of coverage for attractive ad-ad interaction. It is important to emphasize that the 2PS model reproduces nicely the experimental results, at least for the range of coverage where the c(2 × 2) phase is present (0 < θ < 0.35). It should be noted that the improvement in the description of the experimental phase diagram has been achieved due to the increase in the adsorbate degrees of freedom with respect to that of the 2P model at half coverage. One important result observed in the analysis of the 2P model is that the critical temperature at finite coverage can be lower than that corresponding to the metal
Langmuir, Vol. 15, No. 18, 1999 5905
structure at zero coverage, Tc(θ)/Tc(0) < 1, depending on the relation between J and . On the other hand, different ordered phases, such as c(2 × 2) and (2 × 1), can be present for competitive interactions J/ < 0. As a conclusion, in this paper we have analyzed and compared three lattice-gas models for the adsorptioninduced reconstruction transition. The 4P model is the most adequate geometrical representation of the surface potential proposed to represent the metal surface in the experimental system H/W(001). However, the experimental phase diagram can be reproduced only in a small range of coverage, θ < 0.25. The 2P and 2PS models have strong symmetry restrictions. However, one of these two models, the 2PS model, is able to reproduce nicely the experimental results at least for the range of coverage where the c(2 × 2) phase is present (0 < θ < 0.35). Acknowledgment. Two of the authors (F.N. and A.A.T.) are gratefully indebted to the Max-Planck-Institut fu¨r Eisenforschung at Du¨sseldorf, Germany, for financial support and kind hospitality during an extended visit. This work has been partially supported by the Heisenberg program of the Deutsche Forschungsgemeinschaft, by the International Association for the promotion of cooperation with scientists from the New Independent States of the former Soviet Union INTAS-96-0533, by CONICET, Fundacio´n Antorchas, and The National Agency for Promotion of Science and Technology (Argentina). The European Economic Community, Project ITDC-240, is greatly acknowledged for the provision of valuable equipment. F.N. and V.P. acknowledge stimulating discussions with Professor G. Zgrablich. LA981098M
